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38 results on '"Takéo Takahashi"'

1. Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation*

Author: Arnab Roy, Takéo Takahashi, and Debayan Maity
Subjects: Cuboid, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Physics and Astronomy, Statistical and Nonlinear Physics, Wave equation, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Nonlinear system, Barotropic fluid, Fluid–structure interaction, Compressibility, 0101 mathematics, Convection–diffusion equation, Displacement (fluid), Mathematical Physics, Mathematics
Abstract: In this article, we consider a fluid-structure interaction system where the fluid is viscous and compressible and where the structure is a part of the boundary of the fluid domain and is deformable. The fluid is governed by the barotropic compressible Navier-Stokes system whereas the structure displacement is described by a wave equation. We show that the corresponding coupled system admits a unique strong solution for an initial fluid density and an initial fluid velocity in $H^3$ and for an initial deformation and an initial deformation velocity in $H^4$ and $H^3$ respectively. The reference configuration for the fluid domain is a rectangular cuboid with the elastic structure being the top face. We use a modified Lagrangian change of variables to transform the moving fluid domain into the rectangular cuboid and then analyze the corresponding linear system coupling a transport equation (for the density), a heat-type equation, and a wave equation. The corresponding results for this linear system and estimations of the coefficients coming from the change of variables allow us to perform a fixed point argument and to prove the existence and uniqueness of strong solutions for the nonlinear system, locally in time.
Published: 2021
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2. Stabilization of a rigid body moving in a compressible viscous fluid

Author: Takéo Takahashi, Arnab Roy, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Takahashi, Takéo
Subjects: Viscous liquid, 01 natural sciences, Damper, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Position (vector), Fluid-structure interaction, Fluid–structure interaction, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ball (mathematics), [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, 010102 general mathematics, Mathematical analysis, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Rigid body, MSC : 35Q35, 35D30, 35D35, 35R37, 76N10, 93D15, 93D20, compressible Navier-Stokes system, stabilization, 010101 applied mathematics, Spring (device), Compressibility, global solutions, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Analysis of PDEs (math.AP)
Abstract: We consider the stabilizability of a fluid-structure interaction system where the fluid is viscous and compressible and the structure is a rigid ball. The feedback control of the system acts on the ball and corresponds to a force that would be produced by a spring and a damper connecting the center of the ball to a fixed point $$h_1$$ . We prove the global-in-time existence of strong solutions for the corresponding system under a smallness condition on the initial velocities and on the distance between the initial position of the center of the ball and $$h_1$$ . Then, we show with our feedback law, that the fluid and the structure velocities go to 0 and that the center of the ball goes to $$h_1$$ as $$t\rightarrow \infty $$ .
Published: 2020
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3. Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation

Author: Takéo Takahashi, Debayan Maity, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute for Fundamental Research (TIFR), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Debayan Maity was partially supported by INSPIRE faculty fellowship (IFA18-MA128) and by Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520. Takéo Takahashi was partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010)., and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Change of variables, fluid-structure interaction, R-sectorial operators, Fixed point, 01 natural sciences, Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Physics, Small data, Applied Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, Linear system, maximal regularity, General Engineering, General Medicine, compressible NavierStokesFourier system, 010101 applied mathematics, Computational Mathematics, Compressibility, strong solutions, General Economics, Econometrics and Finance, Displacement (fluid), Analysis, Analysis of PDEs (math.AP)
Abstract: The article is devoted to the mathematical analysis of a fluid–structure interaction system where the fluid is compressible and heat conducting and where the structure is deformable and located on a part of the boundary of the fluid domain. The fluid motion is modeled by the compressible Navier–Stokes–Fourier system and the structure displacement is described by a structurally damped plate equation. Our main results are the existence of strong solutions in an L p − L q setting for small time or for small data. Through a change of variables and a fixed point argument, the proof of the main results is mainly based on the maximal regularity property of the corresponding linear systems. For small time existence, this property is obtained by decoupling the linear system into several standard linear systems whereas for global existence and for small data, the maximal regularity property is proved by showing that the corresponding linear coupled fluid–structure operator is R -sectorial.
Published: 2021
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4. $$L^{p}$$ Theory for the Interaction Between the Incompressible Navier–Stokes System and a Damped Plate

Author: Takéo Takahashi, Debayan Maity, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute for Fundamental Research (TIFR), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Incompressible Navier-Stokes System, Physics, Change of variables, AMS subject classifications. 35Q35, 76D03, 76D05, 74F1, Applied Mathematics, Operator (physics), 010102 general mathematics, Linear system, Mathematical analysis, Boundary (topology), Fixed point, Condensed Matter Physics, 01 natural sciences, Action (physics), Physics::Fluid Dynamics, 010101 applied mathematics, Computational Mathematics, Fluid-structure interaction, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Maximal Lp regularity, Uniqueness, 0101 mathematics, Strong solutions, Mathematical Physics
Abstract: International audience; We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary is moving as a damped beam under the action of the fluid. We prove the existence and uniqueness of global strong solutions for the corresponding fluid-structure interaction system in an Lp-Lq setting. The main point in the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped beam. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.
Published: 2021
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5. Well-posedness for the coupling between a viscous incompressible fluid and an elastic structure

Author: Takéo Takahashi, Muriel Boulakia, Sergio Guerrero, Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), COmputational Mathematics for bio-MEDIcal Applications (COMMEDIA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Coupling, Elastic structure, Applied Mathematics, 010102 general mathematics, Linear elasticity, Mathematical analysis, 2010 Mathematics Subject Classification. 76D03, 76D05, 35Q74, 76D27, Structure (category theory), General Physics and Astronomy, Statistical and Nonlinear Physics, Viscous incompressible fluid, 01 natural sciences, Domain (mathematical analysis), Navier-Stokes system, Physics::Fluid Dynamics, 010101 applied mathematics, Bounded function, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Fluid-structure, Displacement (fluid), Mathematical Physics, Mathematics
Abstract: International audience; In this paper, we consider a system modeling the interaction between a viscous incompressible fluid and an elastic structure. The fluid motion is represented by the classical Navier-Stokes equations while the elastic displacement is described by the linearized elasticity equation. The elastic structure is immersed in the fluid and the whole system is confined into a general bounded smooth domain of R3. Our main result is the local in time existence and uniqueness of a strong solution of the corresponding system.
Published: 2019
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6. Existence of contacts for the motion of a rigid body into a viscous incompressible fluid with the Tresca boundary conditions

Author: Matthieu Hillairet, Takéo Takahashi, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Physics, Gravity (chemistry), Cauchy stress tensor, General Mathematics, 2010 Mathematics Subject Classification : 74F10, 35R35, 35Q30, 76D05, 010102 general mathematics, Boundary (topology), Newton's laws of motion, Mechanics, Slip (materials science), Rigid body, 01 natural sciences, Domain (mathematical analysis), fluid-structure, 010101 applied mathematics, Physics::Fluid Dynamics, Navier-Stokes system, Mathematics - Analysis of PDEs, Tresca's boundary conditions, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Analysis of PDEs (math.AP)
Abstract: International audience; We consider a fluid-structure interaction system composed by a rigid ball immersed into a viscous in-compressible fluid. The motion of the structure satisfies the Newton laws and the fluid equations are the standard Navier-Stokes system. At the boundary of the fluid domain, we use the Tresca boundary conditions, that permit the fluid to slip tangentially on the boundary under some conditions on the stress tensor. More precisely, there is a threshold determining if the fluid can slip or not and there is a friction force acting on the part where the fluid can slip. Our main result is the existence of contact in finite time between the ball and the exterior boundary of the fluid for this system in the bidimensional case and in presence of gravity.
Published: 2021
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7. On the Existence of Strong Solutions to a Fluid Structure Interaction Problem with Navier Boundary Conditions

Author: Takéo Takahashi, Imene Aicha Djebour, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Physics, Small data, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Condensed Matter Physics, 01 natural sciences, Navier-Stokes system, Physics::Fluid Dynamics, 010101 applied mathematics, Strong solutions, Computational Mathematics, damped beam equation, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Fluid motion, strong solutions, Uniqueness, Boundary value problem, 0101 mathematics, fluid-structure interaction systems, Plate equation, Mathematical Physics
Abstract: International audience; We consider a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas we add a damping in the plate equation. We use here Navier-slip boundary conditions instead of the standard no-slip boundary conditions. The main results are the local in time existence and uniqueness of strong solutions of the corresponding system and the global in time existence and uniqueness of strong solutions for small data and if we assume the presence of frictions in the boundary conditions.
Published: 2019
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8. Analysis of a system modelling the motion of a piston in a viscous gas

Author: Marius Tucsnak, Debayan Maity, Takéo Takahashi, Center for Applicable Mathematics [Bangalore] (TIFR-CAM), Tata Institute for Fundamental Research (TIFR), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), and Tata Institute of Fundamental Research [Bombay] (TIFR)
Subjects: Physics, fluid-particle interaction, Compressible Navier-Stokes System, Applied Mathematics, 010102 general mathematics, Motion (geometry), Context (language use), Mechanics, Condensed Matter Physics, 01 natural sciences, Cylinder (engine), law.invention, 010101 applied mathematics, Computational Mathematics, Piston, law, Free boundary problem, Compressibility, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Uniqueness, 0101 mathematics, Mathematical Physics
Abstract: International audience; We study a free boundary problem modelling the motion of a piston in a viscous gas. The gas-piston system fills a cylinder with fixed extremities, which possibly allow gas from the exterior to penetrate inside the cylinder. The gas is modeled by the 1D compressible Navier-Stokes system and the piston motion is described by the second Newton's law. We prove the existence and uniqueness of global in time strong solutions. The main novelty brought in by our results is that they include the case on nonhomogeneous boundary conditions which, as far as we know, have not been studied in this context. Moreover, even for homogeneous boundary conditions, our results require less regularity of the initial data than those obtained in previous works.
Published: 2016
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9. An optimal control approach to ciliary locomotion

Author: Marius Tucsnak, Jorge San Martín, Takéo Takahashi, Departamento de Ingeniería Matemática, Facultad de Ciencias Fisicas y Matemáticas, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), and ANR-11-BS03-0002,HAMECMOPSYS,Approche Hamiltonienne pour l'analyse et la commande des systèmes multiphysiques à paramètres distribués(2011)
Subjects: 0209 industrial biotechnology, Control and Optimization, media_common.quotation_subject, Boundary (topology), 02 engineering and technology, controllability, 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], optimal control, symbols.namesake, 020901 industrial engineering & automation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Sensitivity (control systems), 0101 mathematics, Eccentricity (behavior), Astrophysics::Galaxy Astrophysics, Mathematics, media_common, Applied Mathematics, Mathematical analysis, Ode, Scalar (physics), Stokes equations, Reynolds number, Optimal control, 010101 applied mathematics, Controllability, Gegenbauer functions, symbols, ciliates, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]
Abstract: We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The ``best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.
Published: 2016
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10. Mathematical analysis of the motion of a rigid body in a compressible Navier-Stokes-Fourier fluid

Author: Debayan Maity, Marius Tucsnak, Takéo Takahashi, Bernhard H. Haak, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
Subjects: General Mathematics, Motion (geometry), R-sectorial operators, 76N10, strong, 01 natural sciences, AMS subject classifications 35Q30, symbols.namesake, Mathematics - Analysis of PDEs, Thermal insulation, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, solutions, Mathematics, fluid-particle interaction, Small data, business.industry, 010102 general mathematics, Mathematical analysis, Linear system, maximal regularity, Rigid body, 76D05, 010101 applied mathematics, Fourier transform, Compressible Navier-Stokes-Fourier System, symbols, Compressibility, business, Analysis of PDEs (math.AP)
Abstract: International audience; We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier-Stokes-Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an L p-L q setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the R-sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.
Published: 2019
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11. Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation

Author: Takéo Takahashi, Mehdi Badra, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
Subjects: Coupling, Gevrey class semigroups, Semigroup, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Fixed point, 01 natural sciences, Domain (mathematical analysis), fluid-structure, 010101 applied mathematics, Physics::Fluid Dynamics, Navier-Stokes system, Computational Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Gevrey class, 2010 Mathematics Subject Classification : 76D03, 76D05, 35Q74, 76D27, Analysis, Beam (structure), Mathematics
Abstract: International audience; We analyse a bi-dimensional fluid-structure interaction system composed by a viscous incompressible fluid and a beam located at the boundary of the fluid domain. Our main result is the existence and uniqueness of strong solutions for the corresponding coupled system. The proof is based on a the study of the linearized system and a fixed point procedure. In particular, we show that the linearized system can be written with a Gevrey class semigroup. The main novelty with respect to previous results is that we do not consider any approximation in the beam equation.
Published: 2019
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12. Analysis of a simplified model of rigid structure floating in a viscous fluid

Author: Takéo Takahashi, Jorge San Martín, Marius Tucsnak, Debayan Maity, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre de modélisation mathématique (CMM), Universitad de Chile-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-16-CE92-0028,INFIDHEM,Systèmes interconnectés de dimension infinie pour les milieux hétérogènes(2016), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Function space, Differential equation, Applied Mathematics, Mathematical analysis, return to equilibrium, General Engineering, Viscous shallow water equations, fluid-structure interaction, floating structure, Viscous liquid, 01 natural sciences, 010305 fluids & plasmas, 010101 applied mathematics, Viscosity, AMS subject classifications 35Q35 74F10, Flow (mathematics), Linearization, Modeling and Simulation, 0103 physical sciences, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], strong solutions, 0101 mathematics, Shallow water equations, Mathematics
Abstract: International audience; We study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. The first novelty we bring in is that we propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Our well-posedness results are local in time for every initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, we show that the linearization of our system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space. Finally, we describe some numerical tests, performed on the original nonlinear system, which illustrate the return to equilibrium and the influence of the viscosity coefficient.
Published: 2019
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13. Existence of weak solutions for a Bingham fluid-rigid body system

Author: Benjamin Obando, Takéo Takahashi, Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE)
Subjects: Physics, Viscoplasticity, Applied Mathematics, Weak solution, 010102 general mathematics, Mathematical analysis, Newton's laws of motion, Weak formulation, Rigid body, 01 natural sciences, Displacement (vector), 010101 applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Penalty method, 0101 mathematics, Bingham plastic, Mathematical Physics, Analysis
Abstract: International audience; We consider the motion of a rigid body in a viscoplastic material. This material is modeled by the 3D Bingham equations, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is an inequality (due to the plasticity of the fluid), and it involves a free boundary (due to the motion of the rigid body). We approximate it by regularizing the convex terms in the Bingham fluid and by using a penalty method to take into account the presence of the rigid body.
Published: 2019
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14. Elastic energy of a convex body

Author: Antoine Henrot, Chiara Bianchini, and Takéo Takahashi
Subjects: 010101 applied mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Elastic energy, Convex body, 0101 mathematics, 01 natural sciences, Mathematics
Published: 2015
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15. Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid

Author: Toshiaki Hishida, Takéo Takahashi, Ana L. Silvestre, Nagoya University, Center for Computacional and Stochastic Mathematics (CEMAT), Instituto Superior Técnico, Universidade Técnica de Lisboa (IST), Departamento de Matemática [Lisbonne] (DM/IST), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), T. Hishida is partially supported by Grant-in-Aid for Scientific Research, 18K03363, from JSPS., A. L. Silvestre acknowledges the financial support of the Portuguese FCT - Fundação para a Ciência e a Tecnologia, through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID., T. Takahashi is partially supported by the project IFSMACS ANR-15-CE40-0010, financed by the French Agence Nationale de la Recherche, and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Control and Optimization, Gâteaux derivative, 2010 Mathematics Subject Classification : 76D05, 49K21, 76D55, 49J21, Motion (geometry), Boundary (topology), 01 natural sciences, Domain (mathematical analysis), Rotating body, Mathematics - Analysis of PDEs, Drag reduction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 3-D Navier-Stokes equations, Self-propelled motion, 0101 mathematics, Mathematics - Optimization and Control, Mathematics, Boundary control, 010102 general mathematics, Mathematical analysis, Rigid body, 010101 applied mathematics, Computational Mathematics, Control and Systems Engineering, Drag, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Unit (ring theory), Exterior domain, Reference frame
Abstract: Consider a rigid body 𝒮 ⊂ ℝ3 immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to 𝒮. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain Ω = ℝ3 \ 𝒮. This paper deals with the problem of using boundary controls v*, acting on the whole ∂Ω or just on a portion Γ of ∂Ω, to generate a self-propelled motion of 𝒮 with a target velocity V (x) := ξ + ω × x and to minimize the drag about 𝒮. Firstly, an appropriate drag functional is derived from the energy equation of the fluid and the problem is formulated as an optimal boundary control problem. Then the minimization problem is solved for localized controls, such that supp v*⊂ Γ, and for tangential controls, i.e, v*⋅ n|∂Ω = 0, where n is the outward unit normal to ∂Ω. We prove the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map, establish the well-posedness of the corresponding adjoint equations and, finally, derive the first order optimality conditions. The results are obtained under smallness restrictions on the objectives |ξ| and |ω| and on the boundary controls.
Published: 2020
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16. On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Author: Erica L. Schwindt, Takéo Takahashi, Jorge San Martín, Departamento de Ingeniería Matemática [Santiago] (DIM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE), Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), and Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Convex hull, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Boundary (topology), Inverse problem, Rigid body, 01 natural sciences, 010101 applied mathematics, Physics::Fluid Dynamics, Nonlinear system, Classical mechanics, complex geometrical solutions, Geometrical inverse problems, fluid-structure interaction, Navier–Stokes system, enclosure method, Obstacle, Fluid–structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Point (geometry), 0101 mathematics, Mathematics
Abstract: We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.
Published: 2017
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17. A boundary control problem for the steady self-propelled motion of a rigid body in a Navier-Stokes fluid

Author: Ana L. Silvestre, Toshiaki Hishida, Takéo Takahashi, Nagoya University, Center for Computacional and Stochastic Mathematics (CEMAT), Instituto Superior Técnico, Universidade Técnica de Lisboa (IST), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Boundary (topology), Motion (geometry), 01 natural sciences, Rotating body, Physics::Fluid Dynamics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Self-propelled motion, Navier stokes, 0101 mathematics, Mathematical Physics, Mathematics, 3-D Navier–Stokes equations, Steady state, Boundary control, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Rigid body, Asymptotic behavior, 010101 applied mathematics, Distribution (mathematics), Classical mechanics, Flow velocity, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], Exterior domain, Analysis
Abstract: International audience; We study the self-propelled motions of a rigid body immersed in a viscous incompressible fluid which fills the exterior domain of the rigid body.The mechanism used by the body to reach the desired motion is modeled through a distribution of velocities at its boundary. The fluid motion is modeled by the stationary Navier-Stokes system. These equations are coupled with two relations for the balance of forces and torques.We prove that there exists a control allowing the rigid body to move with a prescribed rigid velocity provided the velocity is small enough. We also show that since the net force exerted by the fluid to the rigid body vanishes, we have a better summability of the fluid velocity than the classical summability result for the solutions of the stationary Navier-Stokes system in exterior domains.
Published: 2017
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18. Feedback boundary stabilization of 2d fluid-structure interaction systems

Author: Mehdi Badra, Takéo Takahashi, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015)
Subjects: Applied Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), fluid-structure interaction, feedback stabilization, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Physics::Fluid Dynamics, Nonlinear system, Classical mechanics, Position (vector), Fluid–structure interaction, No-slip condition, beam equation, Discrete Mathematics and Combinatorics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, Navier-Stokes equations, Navier–Stokes equations, Analysis, Stationary state, Mathematics
Abstract: International audience; We study the feedback stabilization of a system composed by an incompressible viscous fluid and a deformable structure located at the boundary of the fluid domain. We stabilize the position and the velocity of the structure and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our result concerns weak solutions for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. We prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the deformable structure and the velocity of the fluid.
Published: 2017
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19. Fluid-rigid structure interaction system with Coulomb's law

Author: Takéo Takahashi, Loredana Bălilescu, Jorge San Martín, Department of Mathematics [Pitesti], Faculty of Mathematics, Universitatea din Pitesti [Roumanie] (UPIT)-Universitatea din Pitesti [Roumanie] (UPIT), Departamento de Matemática = Mathematics Department [Florianópolis] (MTM), Centro de Ciências Físicas e Matemáticas [Florianópolis] (CFM), Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC)-Universidade Federal de Santa Catarina = Federal University of Santa Catarina [Florianópolis] (UFSC), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-15-CE40-0010,IFSMACS,Interaction Fluide-Structure : Modélisation, analyse, contrôle et simulation(2015), and Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE)
Subjects: Coulomb's law 2010 Mathematics Subject Classification: 35Q30, 01 natural sciences, Coulomb's law, Physics::Fluid Dynamics, Navier-Stokes system, symbols.namesake, 76D03, Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Fluid-structure system, Boundary value problem, 0101 mathematics, Physics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mixed boundary condition, 16. Peace & justice, Robin boundary condition, 76D27, 010101 applied mathematics, Computational Mathematics, Classical mechanics, Dirichlet boundary condition, No-slip condition, symbols, Cauchy boundary condition, Cauchy theory, Analysis
Abstract: International audience; We propose a new model in a fluid-structure system composed by a rigid body and a viscous incompress-ible fluid using a boundary condition based on Coulomb's law. This boundary condition allows the fluid to slip on the boundary if the tangential component of the stress is too large. In the opposite case, we recover the standard Dirichlet boundary condition. The governing equations are the Navier-Stokes system for the fluid and the Newton laws for the body. The corresponding coupled system can be written as a variational inequality. We prove that there exists a weak solution of this system.
Published: 2017
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20. Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid

Author: Olivier Glass, Takéo Takahashi, Franck Sueur, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), ANR-09-BLAN-0213,CISIFS,Controle et Identification pour les Systemes d'Interaction Fluide-Structure(2009), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est
Subjects: Smoothness (probability theory), General Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Motion (geometry), Perfect fluid, Rigid body, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Mathematics - Analysis of PDEs, Bounded function, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, Analysis of PDEs (math.AP), Mathematics
Abstract: International audience; We consider the motion of a rigid body immersed in an incompressible perfect fluid which occupies a threedimensional bounded domain. For such a system the Cauchy problem is well-posed locally in time if the initial velocity of the fluid is in the H¨older space C1,r. In this paper we prove that the smoothness of the motion of the rigid body may be only limited by the smoothness of the boundaries (of the body and of the domain). In particular for analytic boundaries the motion of the rigid body is analytic (till the classical solution exists and till the solid does not hit the boundary). Moreover in this case this motion depends smoothly on the initial data.
Published: 2012
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21. Collisions in Three-Dimensional Fluid Structure Interaction Problems

Author: Matthieu Hillairet and Takéo Takahashi
Subjects: Partial differential equation, Applied Mathematics, Weak solution, 010102 general mathematics, Mathematical analysis, Equations of motion, Horizontal plane, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Computational Mathematics, Fluid–structure interaction, Ball (mathematics), 0101 mathematics, Reynolds-averaged Navier–Stokes equations, Analysis, CFD-DEM, Mathematics
Abstract: This paper deals with a system composed of a rigid ball moving into a viscous incompressible fluid over a fixed horizontal plane. The equations of motion for the fluid are the Navier–Stokes equations, and the equations for the motion of the rigid ball are obtained by applying Newton's laws. We show that for any weak solution of the corresponding system satisfying the energy inequality, the rigid ball never touches the plane. This result is the extension of that obtained in [M. Hillairet, Comm. Partial Differential Equations, 32 (2007), pp. 1345–1371] in the two-dimensional setting.
Published: 2009
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22. Existence of global weak solutions for a phase-field model of a vesicle moving into a viscous incompressible fluid

Author: Takéo Takahashi, Yuning Liu, Universität Regensburg (UR), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), CORIDA, ANR-09-BLAN-0213,CISIFS,Controle et Identification pour les Systemes d'Interaction Fluide-Structure(2009), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM)
Subjects: Field (physics), General Mathematics, Navier-Stokes, Monotonic function, 01 natural sciences, Quantitative Biology::Subcellular Processes, Physics::Fluid Dynamics, Phase (matter), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Membrane vesicle, 0101 mathematics, Physics, Physics::Biological Physics, phase-field, Vesicle, 010102 general mathematics, General Engineering, Mechanics, Viscous incompressible fluid, vesicle membrane, 010101 applied mathematics, Compact space, Classical mechanics, local incompressibility, fluid vesicle interaction, Convection–diffusion equation
Abstract: International audience; We consider in this article a model of vesicle moving into a viscous incompressible fluid. The vesicle is described through a phase-field equation and through a transport equation modeling the local incompressibility of its membrane. The equations for the fluid are the classical Navier--Stokes equations with a force resulting from the presence of the vesicle. The model for the system vesicle--fluid is an approximation of a model obtained by Jamet and Misbah. Our main result states the existence of weak solutions for the corresponding system. The proof is based on compactness/monotonicity arguments
Published: 2014
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23. On the identifiability of a rigid body moving in a stationary viscous fluid

Author: Takéo Takahashi, Carlos Conca, Erica L. Schwindt, Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Universidad de Chile = University of Chile [Santiago] (UCHILE)
Subjects: Physics, Applied Mathematics, MSC: 49K40, 65J22, 76D05, 010102 general mathematics, Boundary (topology), Fluid mechanics, Herschel–Bulkley fluid, Mechanics, Viscous liquid, Rigid body, 01 natural sciences, Computer Science Applications, Theoretical Computer Science, 010101 applied mathematics, Physics::Fluid Dynamics, [INFO.INFO-AU]Computer Science [cs]/Automatic Control Engineering, Signal Processing, Compressibility, Identifiability, Convex body, 0101 mathematics, Mathematical Physics
Abstract: International audience; This paper is devoted to a geometrical inverse problem associated with a fluid-structure system. More precisely, we consider the interaction between a moving rigid body and a viscous and incompressible fluid. Assuming a low Reynolds regime, the inertial forces can be neglected and, therefore, the fluid motion is modelled by the Stokes system. We first prove the well posedness of the corresponding system. Then we show an identifiability result: with one measure of the Cauchy forces of the fluid on one given part of the boundary and at some positive time, the shape of a convex body and its initial position are identified.
Published: 2012
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24. Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid

Author: Takéo Takahashi, Muriel Boulakia, Erica L. Schwindt, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical simulation of biological flows (REO), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centro de Modelamiento Matemático [Santiago] (CMM), Centre National de la Recherche Scientifique (CNRS)-Universidad de Santiago de Chile [Santiago] (USACH), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Universidad de Santiago de Chile [Santiago] (USACH)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: existence and uniqueness of strong solutions, Applied Mathematics, deformable structure, 010102 general mathematics, Mathematical analysis, Linear elasticity, MSC: 74F10, 76D03, 35Q30, 35Q35, 37N15, 76D05, Structure (category theory), Motion (geometry), Fluid mechanics, 01 natural sciences, Domain (mathematical analysis), 010101 applied mathematics, Physics::Fluid Dynamics, Classical mechanics, incompressible Navier-Stokes equations, Fluid–structure interaction, Fluid-structure interaction, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Boundary value problem, 0101 mathematics, Mathematics
Abstract: International audience; In this paper we study a three-dimensional fluid-structure interaction problem. The motion of the fluid is modeled by the Navier-Stokes equations and we consider for the elastic structure a finite dimensional approximation of the equation of linear elasticity. The time variation of the fluid domain is not known a priori, so we deal with a free boundary value problem. Our main result yields the local in time existence and uniqueness of strong solutions for this system.
Published: 2012
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25. Weak solutions for the motion of a self-propelled deformable structure in a viscous Incompressible fluid

Author: Takéo Takahashi, Marius Tucsnak, Šárka Nečasová, Mathematical Institute [Praha] - Academy of Sciences, Czech Academy of Sciences [Prague] (CAS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: Physics, Partial differential equation, Deformation (mechanics), Applied Mathematics, 010102 general mathematics, Motion (geometry), Equations of motion, Mechanics, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Classical mechanics, Position (vector), Fluid–structure interaction, Free boundary problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Navier–Stokes equations, ComputingMilieux_MISCELLANEOUS
Abstract: We consider the three-dimensional motion of a self-propelled deformable structure into a viscous incompressible fluid. The deformation of the solid is given whereas its position is unknown. Such a system could model the propulsion of fish-like swimmers. The equations of motion of the fluid are the Navier-Stokes equations and the equations for the structure are deduced from Newton's laws. The corresponding system is a free boundary problem and the main result of the paper is the existence of weak solutions for this problem.
Published: 2011
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26. Inverse problem and null-controllability for parabolic systems

Author: Takéo Takahashi, Galina C. García, Departamento de Matemática y Ciencia de la Computación, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), and CORIDA
Subjects: Generalized inverse, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Null (mathematics), Boundary (topology), Inverse problem, controllability, 01 natural sciences, Parabolic partial differential equation, 010101 applied mathematics, Controllability, Inverse scattering problem, inverse problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Initial value problem, 0101 mathematics, 93C25, 93B07, 93B05, 93C20, 35R30, Mathematics
Abstract: In this paper, we present some abstract results giving a general connection between null-controllability and several inverse problems for a class of parabolic equations. We obtain some conditional stability estimates for the inverse problems consisting of determining the initial condition and the source term, from interior or boundary measurements. We apply this framework for Stokes system with interior and boundary observations, for a coupling of two Stokes system and a linear fluid-structure system.
Published: 2011
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27. Homogenization and singular limits for the complete Navier-Stokes-Fourier system

Author: Eduard Feireisl, Takéo Takahashi, Antonin Novotny, Mathematical Institute [Praha] - Academy of Sciences, Czech Academy of Sciences [Prague] (CAS), Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), Robust control of infinite dimensional systems and applications (CORIDA), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), CORIDA, Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Czech Academy of Sciences [Prague] ( ASCR ), Institut de Mathématiques de Toulon - EA 2134 ( IMATH ), Université de Toulon ( UTLN ), Institut Élie Cartan de Nancy ( IECN ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ), Robust control of infinite dimensional systems and applications ( CORIDA ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Université Henri Poincaré - Nancy 1 ( UHP ) -Université Nancy 2-Institut National Polytechnique de Lorraine ( INPL ) -Centre National de la Recherche Scientifique ( CNRS ) -Laboratoire de Mathématiques et Applications de Metz ( LMAM ), Université Paul Verlaine - Metz ( UPVM ) -Centre National de la Recherche Scientifique ( CNRS ) -Université Paul Verlaine - Metz ( UPVM ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique ( Inria )
Subjects: Mathematics(all), Compressible Navier-Stokes-Fourier equations, General Mathematics, Viscous liquid, 01 natural sciences, Homogenization (chemistry), Compressible flow, Physics::Fluid Dynamics, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, Homogenization, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Porous medium equation, 010101 applied mathematics, Nonlinear system, Fourier transform, Fourier analysis, symbols, Compressibility, Porous medium, Compressible Navier–Stokes–Fourier equations
Abstract: International audience; In this paper we apply the methods of homogenization to the full Navier-Stokes-Fourier system describing the motion of a general viscous, compressible, and heat conducting fluid. We study the asymptotic behavior of solutions in perforated domains with tiny holes, where the diameter of the holes is proportional to their mutual distance. As a limit system, we identify a porous medium type equation with a nonlinear Darcy's law.
Published: 2010
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28. Blow up and grazing collision in viscous fluid solid interaction systems

Author: Matthieu Hillairet, Takéo Takahashi, Mathématiques pour l'Industrie et la Physique (MIP), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM)
Subjects: Applied Mathematics, 010102 general mathematics, Mathematics::Analysis of PDEs, Interaction systems, Viscous liquid, Viscous incompressible fluid, Collision, 01 natural sciences, 010101 applied mathematics, Strong solutions, Physics::Fluid Dynamics, Classical mechanics, Bounded function, Compressibility, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ball (mathematics), 0101 mathematics, Mathematical Physics, Analysis, Mathematics
Abstract: International audience; In this paper we investigate finite time blow up of strong solutions to the system describing the motion of a rigid ball inside a bounded cavity filled with a viscous incompressible fluid. The equations of motion for the fluid are of Navier--Stokes type and the equations for the motion of the rigid ball are obtained by applying Newton's laws. The whole system evolves under the action of gravity. First, we prove contact between the ball and the boundary of the cavity implies the blow up of the strong solution. Then we prove for some configurations such a contact has to occur in finite time.
Published: 2010
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29. Small solids in an inviscid fluid

Author: Takéo Takahashi, Nicolas Seguin, Frédéric Lagoutière, Boris Andreianov, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Laboratoire de Recherche en Informatique (LRI), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), CORIDA, Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM)
Subjects: Statistics and Probability, Solid-fluid interaction, 01 natural sciences, symbols.namesake, Quadratic equation, 35F25, 35L80, 65M99, Inviscid flow, singular source term, well-balanced scheme, random-choice method, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, adapted entropy, Mathematics, Partial differential equation, Finite volume method, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, Riemann solver, Computer Science Applications, Burgers' equation, Burgers equation, 010101 applied mathematics, Drag, Ordinary differential equation, symbols
Abstract: We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.
Published: 2010
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30. Topological sensitivity analysis for time-dependent problems

Author: Samuel Amstutz, Boris Vexler, Takéo Takahashi, Laboratoire d'Analyse non linéaire et Géométrie (LANLG), Avignon Université (AU), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences (OeAW), CORIDA, and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: Control and Optimization, 010102 general mathematics, Topology optimization, parabolic equations, Topological space, Topology, 01 natural sciences, Domain (mathematical analysis), hyperbolic equations, 010101 applied mathematics, Computational Mathematics, 49Q10, 49Q12, 35K05, 35L05, Control and Systems Engineering, Topological sensitivity, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], General topology, Sensitivity (control systems), 0101 mathematics, Particular point topology, Topological quantum number, Topology (chemistry), ComputingMilieux_MISCELLANEOUS, topology optimization, Mathematics
Abstract: The topological sensitivity analysis consists in studying the behavior of a given shape functional when the topology of the domain is perturbed, typically by the nucleation of a small hole. This notion forms the basic ingredient of different topology optimization/reconstruction algorithms. From the theoretical viewpoint, the expression of the topological sensitivity is well-established in many situations where the governing p.d.e. system is of elliptic type. This paper focuses on the derivation of such formulas for parabolic and hyperbolic problems. Different kinds of cost functionals are considered.
Published: 2008
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31. Convergence of a Lagrange--Galerkin method for a fluid-rigid body system in ALE formulation

Author: Guillaume Legendre, Takéo Takahashi, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), The work of the first author was supported by FONDECYT under grant n° 3060018 and by the Centro de Modelamiento Matemático., Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), and Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM)
Subjects: Mathematical optimization, 010103 numerical & computational mathematics, 01 natural sciences, Physics::Fluid Dynamics, Method of characteristics, Convergence (routing), Fluid–structure interaction, Fluid-structure interaction, Arbitrary Lagrangian Eulerian, Applied mathematics, Lagrange–Galerkin method, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Galerkin method, Mathematics, Numerical Analysis, Partial differential equation, Applied Mathematics, Numerical analysis, Rigid body, Finite element method, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Incompressible Navier–Stokes equations, Analysis
Abstract: International audience; In this paper, we propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid, modeled by the Navier-Stokes equations. Our method combines a finite element approximations and the use of the method of characteristics to solve an Arbitrary Lagrangian Eulerian formulation of the problem. We derive error estimates for this scheme which imply its convergence.
Published: 2008
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32. On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid

Author: Jaime H. Ortega, Lionel Rosier, Takéo Takahashi, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Centre de modélisation mathématique (CMM), and Universitad de Chile-Centre National de la Recherche Scientifique (CNRS)
Subjects: Conservation law, Angular momentum, Applied Mathematics, 010102 general mathematics, Perfect fluid, Vorticity, Rigid body, 01 natural sciences, Euler equations, Physics::Fluid Dynamics, 010101 applied mathematics, Euler's laws of motion, symbols.namesake, Classical mechanics, Fluid–structure interaction, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematical Physics, Analysis, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: We consider the motion of a rigid body immersed in a bidimensional incompressible perfect fluid. The motion of the fluid is governed by the Euler equations and the conservation laws of linear and angular momentum rule the dynamics of the rigid body. We prove the existence and uniqueness of a global classical solution for this fluid–structure interaction problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid–structure interaction problem obtained by incorporating some viscosity.
Published: 2007
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33. Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid

Author: Patricio Cumsille, Takéo Takahashi, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Centro de Modelamiento Matemático [Santiago] (CMM), and Universidad de Santiago de Chile [Santiago] (USACH)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Conservation law, General Mathematics, 010102 general mathematics, Mathematical analysis, Geometry, Fluid mechanics, Reynolds stress, Viscous liquid, Rigid body, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Navier–Stokes equations, Poinsot's ellipsoid, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract: In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space ℝd, d = 2 or 3. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem.
Published: 2007

34. A Control Theoretic Approach to the Swimming of Microscopic Organisms

Author: Marius Tucsnak, Takéo Takahashi, Jorge San Martín, Takahashi, Takéo, Centre de Modélisation Mathématique / Centro de Modelamiento Matemático (CMM), Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Centre de modélisation mathématique (CMM), and Universitad de Chile-Centre National de la Recherche Scientifique (CNRS)
Subjects: Applied Mathematics, 010102 general mathematics, Relative velocity, Boundary (topology), Reynolds number, Thrust, Fluid mechanics, [MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC], Rigid body, 01 natural sciences, 010101 applied mathematics, symbols.namesake, Classical mechanics, Position (vector), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Vector field, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics
Abstract: Papier accepté pour la publication dans Quarterly of Applied Mathematics; International audience; In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that, there exists a large class of finite dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body in any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers. Our aim is to give a control theoretic interpretation of the swimming mechanism of micro-organisms (like ciliata) which is one of the fascinating problems in fluid mechanics.
Published: 2006

35. Liquid jet generation and break-up

Author: Céline Baranger, G. Baudin, L. Boudin, Bruno Després, Frédéric Lagoutière, E. Lapébie, and Takéo Takahashi
Subjects: 010101 applied mathematics, 010102 general mathematics, 0101 mathematics, 01 natural sciences
Published: 2005
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36. Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid

Author: Takéo Takahashi, Marius Tucsnak, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Tucsnak, Marius, and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
Subjects: Physics, Conservation law, Angular momentum, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Dynamics (mechanics), Viscous liquid, Condensed Matter Physics, Rigid body, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Computational Mathematics, Classical mechanics, Cylinder, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Poinsot's ellipsoid, Mathematical Physics
Abstract: In this paper, we consider a two-dimensional fluid-rigid body problem. The motion of the fluid is modelled by the Navier-Stokes equations, whereas the dynamics of the rigid body is governed by the conservation laws of linear and angular momentum. The rigid body is supposed to be an infinite cylinder of circular cross-section. Our main result is the existence and uniqueness of global strong solutions.
Published: 2005

37. Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid

Author: Lionel Rosier, Takéo Takahashi, Jaime H. Ortega, Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Centre de modélisation mathématique (CMM), and Universitad de Chile-Centre National de la Recherche Scientifique (CNRS)
Subjects: Numerical Analysis, Partial differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Perfect fluid, Vorticity, 01 natural sciences, Euler equations, Physics::Fluid Dynamics, 010101 applied mathematics, Computational Mathematics, symbols.namesake, Modeling and Simulation, symbols, Fluid dynamics, Compressibility, Uniqueness, Ball (mathematics), 0101 mathematics, [MATH]Mathematics [math], Analysis, Mathematics
Abstract: In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying R 2 . We prove the global in time existence and the uniqueness of the classical solution for this fluid-structure problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid-structure problem obtained by incorporating some dissipation.
Published: 2005
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38. A simple 1D model of inviscid fluid–solid interaction

Author: Frédéric Lagoutière, Takéo Takahashi, Nicolas Seguin, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), CORIDA, and Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est
Subjects: Shock wave, 01 natural sciences, Riemann problem, symbols.namesake, Inviscid flow, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Fluid-particle interaction, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Fluid–particle interaction, Interaction model, 16. Peace & justice, Non-conservative product, Burgers' equation, Burgers equation, 010101 applied mathematics, Classical mechanics, Flow velocity, Ordinary differential equation, symbols, Analysis, 76N20, 35B40, 35Q53
Abstract: International audience; We analyze a one-dimensional fluid-particle interaction model, composed by the Burgers equation for the fluid velocity and an ordinary differential equation which governs the particle movement. The coupling is achieved through a friction term. One of the novelties is to consider entropy weak solutions involving shock waves. The difficulty is the interaction between these shock waves and the particle. We prove that the Riemann problem with arbitrary data always admits a solution, which is explicitly constructed. Besides, two asymptotic behaviors are described: the long-time behavior and the behavior for large friction coefficients.
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