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Your search keyword '"ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019)"' showing total 9 results
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9 results on '"ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019)"'

1. Regularization estimates and hydrodynamical limit for the Landau equation

Author

Carrapatoso, Kleber, Rachid, Mohamad, Tristani, Isabelle, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, Applied Mathematics, General Mathematics, FOS: Mathematics, Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and optimal, indeed, we obtain the instantaneous expected anisotropic gain of regularity (see [53] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system.

Published
2022
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2. From Boltzmann equation for granular gases to a modified Navier-Stokes-Fourier system

Author

Ricardo J. Alonso, Bertrand Lods, Isabelle Tristani, Texas A&M University at Qatar, Science Department, Doha, Qatar., Departamento de Matemática (PUC-RIO), Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Università degli studi di Torino (UNITO), Collegio Carlo Alberto, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), O Conselho Nacional de Desenvolvimento Cient ́ıfico e Tec- nolo ́gico, Bolsa de Produtividade em Pesquisa - CNPq (303325/2019-4), Italian Ministry of Education, University and Research (MIUR), 'Dipartimenti di Eccellenza' grant 2018-2022, ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Tristani, Isabelle, Entropie, flots, inégalités - - EFI2017 - ANR-17-CE40-0030 - AAPG2017 - VALID, and Singularités dans des limites asymptotiques d'équations de Vlasov - - SALVE2019 - ANR-19-CE40-0004 - AAPG2019 - VALID

Subjects
Mathematics - Analysis of PDEs, Nearly elastic regime, Knudsen number, FOS: Mathematics, Inelastic Boltzmann equation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical and Nonlinear Physics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Granular flows, Incompressible Navier-Stokes hydrodynamical limit, Mathematical Physics, Long-time asymptotic, Analysis of PDEs (math.AP)
Abstract

In this paper, we give an overview of the results established in [3] which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier-Stokes-Fourier system., arXiv admin note: text overlap with arXiv:2008.05173

Published
2021

3. On the linearized Vlasov-Poisson system on the whole space around stable homogeneous equilibria

Author

Toan T. Nguyen, Daniel Han-Kwan, Frédéric Rousset, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), and ANR-18-CE40-0020,ODA,Ondes déterministes et aléatoires(2018)

Subjects
Physics, 010102 general mathematics, Complex system, Statistical and Nonlinear Physics, Space (mathematics), 01 natural sciences, 010101 applied mathematics, Mathematics - Analysis of PDEs, Homogeneous, Physical space, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Poisson system, Mathematical Physics, Analysis of PDEs (math.AP), Mathematical physics
Abstract

We study the linearized Vlasov–Poisson system around suitably stable homogeneous equilibria on $${\mathbb {R}}^d\times {\mathbb {R}}^d$$ (for any $$d \ge 1$$ ) and establish dispersive $$L^\infty $$ decay estimates in the physical space.

Published
2021
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4. From Newton's second law to Euler's equations of perfect fluids

Author

Mikaela Iacobelli, Daniel Han-Kwan, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), and ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019)

Subjects
General Mathematics, FOS: Physical sciences, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Fluid dynamics, Coulomb, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Incompressible euler equations, Limit (mathematics), 0101 mathematics, Mathematical Physics, Physics, Heuristic, Applied Mathematics, 010102 general mathematics, Dynamics (mechanics), Mathematical Physics (math-ph), 010101 applied mathematics, Classical mechanics, Energy method, Euler's formula, symbols, Analysis of PDEs (math.AP)
Abstract

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty., Minor typos corrected

Published
2021
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5. The non-relativistic limit of the Vlasov-Maxwell system with uniform macroscopic bounds

Author

Brigouleix, Nicolas, Han-Kwan, Daniel, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), and ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

We study in this paper the non-relativistic limit from Vlasov-Maxwell to Vlasov-Poisson, which corresponds to the regime where the speed of light is large compared to the typical velocities of particles. In contrast with \cite{Asano-Ukai-86-SMA}, \cite{Degond-86-MMAS}, \cite{Schaeffer-86-CMP} which handle the case of classical solutions, we consider measure-valued solutions, whose moments and electromagnetic fields are assumed to satisfy some uniform bounds. To this end, we use a functional inspired by the one introduced by Loeper in his proof of uniqueness for the Vlasov-Poisson system \cite{Loeper-2006}. We also build a special class of measure-valued solutions, that enjoy no higher regularity with respect to the momentum variable, but whose moments and electromagnetic fields satisfy all required conditions to enter our framework.

Published
2020

6. Large time behavior of small data solutions to the Vlasov-Navier-Stokes system on the whole space

Author

Han-Kwan, Daniel, Centre de Mathématiques Laurent Schwartz (CMLS), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019), Han-Kwan, Daniel, and Singularités dans des limites asymptotiques d'équations de Vlasov - - SALVE2019 - ANR-19-CE40-0004 - AAPG2019 - VALID

Subjects
Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Analysis of PDEs (math.AP)
Abstract

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system on $\R^3 \times \R^3$. We prove that the kinetic distribution function concentrates in velocity to a Dirac mass supported at $0$, while the fluid velocity homogenizes to $0$, both at a polynomial rate. The proof is based on two steps, following the general strategy laid out in \cite{HKMM}: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, (2) a bootstrap argument allows to obtain such a control. This last step requires a fine understanding of the structure of the so-called Brinkman force, which follows from a family of new identities for the dissipation (and higher versions of it) associated to the Vlasov-Navier-Stokes system.

Published
2020

7. Fluid dynamic limit of Boltzmann equation for granular hard--spheres in a nearly elastic regime

Author

Alonso, Ricardo J., Lods, Bertrand, Tristani, Isabelle, Texas A&M University at Qatar, Department of Statistics and Economics, Università degli studi di Torino (UNITO), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Nearly elastic regime, MSC: 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05], 76T25 Granular flows [See also 74C99, 74E20], 47H20 Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07], 35Q35 PDEs in connection with fluid mechanics, 35Q30 Navier-Stokes equations [See also 76D05, 76D07, 76N10], FOS: Physical sciences, Inelastic Boltzmann equation, Mathematical Physics (math-ph), Incompressible Navier-Stokes hydrodynamical limit, Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, Knudsen number, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Granular flows, Mathematical Physics, Long-time asymptotic, Analysis of PDEs (math.AP)
Abstract

In this paper, we provide the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and, to our knowledge, it is thus the first hydrodynamic system that properly describes rapid granular flows consistent with the kinetic formulation. To this end, we write our Boltzmann equation in a non dimensional form using the dimensionless Knudsen number which is intended to be sent to $0$. There are several difficulties in such derivation, the first one coming from the fact that the original Boltzmann equation is free-cooling and, thus, requires a self-similar change of variables to introduce an homogeneous steady state. Such a homogeneous state is not explicit and is heavy-tailed, which is a major obstacle to adapting energy estimates and spectral analysis. Additionally, a central challenge is to understand the relation between the restitution coefficient, which quantifies the energy loss at the microscopic level, and the Knudsen number. This is achieved by identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. We are, then, able to prove exponential stability uniformly with respect to the Knudsen number for solutions of the rescaled Boltzmann equation in a close to equilibrium regime. Finally, we prove that solutions to the Boltzmann equation converge in a specific weak sense towards a hydrodynamic limit which depends on time and space variables only through macroscopic quantities. Such macroscopic quantities are solutions to a suitable modification of the incompressible Navier-Stokes-Fourier system which appears to be new in this context.

Published
2020

8. A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium

Author

Ayi, Nathalie, Herda, Maxime, Hivert, Hélène, Tristani, Isabelle, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Maxime Herda thanks the LabEx CEMPI (ANR-11-LABX-0007-01). Hélène Hivert thanks the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC starting grant MESOPROBIO no 639638). Isabelle Tristani thanks the ANR EFI: ANR-17-CE40-0030 and the ANR SALVE: ANR-19-CE40-0004 for their support. This work is part of a collaborative research project that was initiated for the Junior Trimester Program in Kinetic Theory at the Hausdorff Research Institute for Mathematics in Bonn. Part of the work wascarried out during the time spent at the institute. The authors are grateful for this opportunity and warmly acknowledge the HIM for the financial support and the hospitality they benefited during their stay., ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure - Paris (ENS Paris), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
Fokker-Planck operator, Hypocoercivity, fractional diffusion, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], heavy-tailed distribution, linear kinetic equations
Abstract

International audience; In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.; Dans cet article, on s’intéresse au comportement en temps long d’équations cinétiques linéaires dont les équilibres locaux sont à queue lourde. Notre contribution principale concerne l’équation de Lévy–Fokker–Planck cinétique, pour laquelle nous adaptons des techniques d’hypocoercivité afin de démontrer la convergence exponentielle des solutions vers un équilibre global. En comparant au cas de l’équation de Fokker–Planck cinétique classique, les enjeux ici sont liés au manque de symétrie de l’opérateur non-local de Lévy–Fokker–Planck et à la compréhension de ses propriétés de régularisation. En complément de notre analyse, nous traitons également le cas de l’équation de BGK à queue lourde.

Published
2020
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9. Large time behavior of the Vlasov-Navier-Stokes system on the torus

Author

Iván Moyano, Ayman Moussa, Daniel Han-Kwan, Centre de Mathématiques Laurent Schwartz (CMLS), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), 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), Statistical Laboratory [Cambridge], Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)-Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and ANR-19-CE40-0004,SALVE,Singularités dans des limites asymptotiques d'équations de Vlasov(2019)

Subjects
Physics, Mechanical Engineering, Dirac (video compression format), 010102 general mathematics, Mathematical analysis, Complex system, Structure (category theory), Mathematics::Analysis of PDEs, Torus, Type (model theory), 01 natural sciences, Exponential function, 010101 applied mathematics, Mathematics (miscellaneous), Distribution function, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Analysis, Energy (signal processing), Analysis of PDEs (math.AP)
Abstract

International audience; We study the large time behavior of Fujita–Kato type solutions to the Vlasov–Navier–Stokes system set on $\T^3 \times \R^3$. Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing us to get global bounds on moments.

Published
2019
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