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In Jayanti and Trivisa (2022 J. Math. Fluid Mech. 24 46), the authors proved the existence of local-in-time weak solutions to a model of superfluidity. The system of governing equations was derived in Pitaevskii (1959 Sov. Phys. JETP 8 282â€"287) and couples the nonlinear Schrödinger equation and the Navierâ€"Stokes equations. In this article, we prove a weakâ€"strong type uniqueness theorem for these weak solutions. Only some of their regularity properties are used, allowing room for improved existence theorems in the future, with compatible uniqueness results. [ABSTRACT FROM AUTHOR]

The Hall–Vinen–Bekharevich–Khalatnikov equations are a macroscopic model of superfluidity at nonzero temperatures. For smooth, compactly supported data, we prove the global well-posedness of strong solutions to these equations in R 2 , in the incompressible and isothermal case. The proof utilises a contraction mapping argument to establish local well-posedness for high-regularity data, following which we demonstrate global regularity using an analogue of the Beale–Kato–Majda criterion in this context. In the Appendix, we address the sufficient conditions on a 2D vorticity field, in order to have a finite kinetic energy. [ABSTRACT FROM AUTHOR]

This article establishes the global existence of weak solutions to a model proposed by Rosensweig [Ferrohydrodynamics, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1985] for the dynamics of ferrofluids. The system is expressed by the conservation of linear momentum, the incompressibility condition, the conserv ation of angular momentum, and the evolution of the magnetization. The existence proof is inspired by the DiPerna--Lions theory of renormalized solutions. In addition, the rigorous relaxation limit of the equations of ferrohydrodynamics towards the quasi-equilibrium is investigated. The proof relies on the relative entropy method, which involves constructing a suitable functional, analyzing its time evolution, and obtaining convergence results for the sequence of approximating solutions. [ABSTRACT FROM AUTHOR]

We investigate the growth of tumors using a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the effect of nutrient and drug applications on the progression of the tumor. We construct a convergent finite difference scheme to approximate solutions of the nonlinear system of partial differential equations. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The same numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug both shrinkage and growth of tumors can occur. The role of inhomogeneous boundary conditions, vascularization, and anisotropies in the development of tumor shape irregularities are discussed. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the 'tumor' is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the 'waste' and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum $$\Omega $$ with boundary $$\partial \Omega $$ both of which evolve in time. In particular, the evolution of the boundary $$\partial \Omega $$ is prescibed by a given velocity $${{{\varvec{V}}}.}$$ The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79-91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787-803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)]. [ABSTRACT FROM AUTHOR]

Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on image analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumor cells, whereas the velocity field is given by a Brinkman regularization of the classical Darcy’s law. An efficient finite difference scheme is proposed and shown to converge to a weak solution of the system. Our approach relies on convergence and compactness arguments in the spirit of Lions [P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Vol. 10 of Oxford Lecture Series Math. Appl. The Clarendon Press, Oxford University Press, New York (1998)]. [ABSTRACT FROM AUTHOR]

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505-527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357-1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235-277] for systems of hyperbolic conservation laws without source terms. [ABSTRACT FROM AUTHOR]

Abstract: This article deals with a fluid–particle interaction model for the evolution of particles dispersed in a fluid. The fluid flow is governed by the Navier–Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually. The existence of weakly dissipative solutions is established under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, a weak–strong uniqueness result is established via the relative entropy method yielding that a weakly dissipative solution agrees with a classical solution with the same initial data when such a classical solution exists. [Copyright &y& Elsevier]

Polymeric fluids arise in many practical applications in biotechnology, medicine, chemistry, industrial processes, and atmospheric sciences. In this paper, the Doi model for the suspensions of rod-like molecules in a compressible fluid is investigated. The model under consideration describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained. Prescribing arbitrarily the initial density of the fluid, the initial velocity, and the initial orientation distribution in suitable spaces, we establish the global-in-time existence of a weak solution to our model defined on a bounded domain in the three-dimensional space. The proof relies on the construction of an approximate sequence of solutions by introducing appropriate regularization and the establishment of compactness. [ABSTRACT FROM AUTHOR]

Abstract: In this article the incompressible limits of weak solutions to the governing equations for magnetohydrodynamics flows on both bounded and unbounded domains are established. The governing equations for magnetohydrodynamic flows are expressed by the full Navier–Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and with an additional equation which describes the evolution of the magnetic field. The scaled analogues of the governing equations for magnetohydrodynamic flows involve the Mach number, Froude number and Alfven number. In the case of bounded domains the establishment of the singular limit relies on a detail analysis of the eigenvalues of the acoustic operator, whereas the case of unbounded domains is being treated by their suitable approximation by a family of bounded domains and the derivation of uniform bounds. [Copyright &y& Elsevier]

The rate of convergence for vanishing viscosity approximations to hyperbolic balance laws is established. The result applies to systems that are strictly hyperbolic and genuinely nonlinear with a source term satisfying a special mechanism that induces dissipation. The proof relies on error estimates that measure the interaction of waves. Shock waves are treated by monitoring the evolution of suitable Lyapunov functionals, whereas interactions involving rarefaction waves are accommodated by employing a sharp decay estimate. [ABSTRACT FROM AUTHOR]

Abstract: We consider strictly hyperbolic and genuinely nonlinear systems of hyperbolic balance laws in one-space dimension. Sharp decay estimates are derived for the positive waves in an entropy weak solution. The result is obtained by introducing a partial ordering within the family of positive Radon measures, using symmetric rearrangements and a comparison with a solution of Burgers''s equation with impulsive sources as well as lower semicontinuity properties of continuous Glimm-type functionals. [Copyright &y& Elsevier]

Abstract: A model for the dynamics of gaseous stars is introduced and formulated by the Navier–Stokes–Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus is investigated. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data. [Copyright &y& Elsevier]

We consider a multidimensional model for the dynamics of liquid-vapor phase transitions. In the present context, liquid and vapor are treated as different species with different volume fractions and different molecular weights. The model presented here is a prototype of a "binary fluid mixture" and is formulated by a system that generalizes the Navier--Stokes(--Fourier) equations in Eulerian coordinates. This system takes now a new form due to the choice of rather complex constitutive relations that can accommodate appropriately the physical context. The setting of the problem presented in this work is motivated by physical considerations. The transport fluxes satisfy rather general constitutive laws, the viscosity and heat conductivity depend on the temperature, and the pressure law is a nonlinear function of the temperature depending on the mass density fraction of the vapor (liquid) in the fluid as well as the molecular weights of the individual species. The existence of globally defined weak solutions of the relevant system of partial differential equations that generalizes the Navier--Stokes(--Fourier) equations for compressible fluids is established by using weak convergence methods, and compactness and interpolation arguments in the spirit of Feireisl [Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004] and Lions [Mathematical Topics in Fluid Mechanics, Vol. 2, The Clarendon Press, Oxford University Press, New York, 1998]. [ABSTRACT FROM AUTHOR]

We consider a multidimensional model for the combustion of compressible reacting fluids. The flow is governed by the Navier–Stokes in Eulerian coordinates and the chemical reaction is irreversible and is governed by the Arrhenius kinetics. The existence of globally defined weak solutions is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl [16] and P.L. Lions [24]. [ABSTRACT FROM AUTHOR]

A multidimensional model is introduced for the dynamic combustion of compressible, radiative and reactive gases. In the macroscopic description adopted here, the radiation is treated as a continuous field, taking into account both the wave (classical) and photonic (quantum) aspects associated with the gas [20, 36]. The model is formulated by the Navier-Stokes equations in Euler coordinates, which is now expressed by the conservation of mass, the balance of momentum and energy and the two species chemical kinetics equation. In this context, we are dealing with a one way irreversible chemical reaction governed by a very general Arrhenius-type kinetics law. The analysis in the present article extends the earlier work of the authors [17], since it now covers the general situation where, both the heat conductivity and the viscosity depend on the temperature, the pressure now depends not only on the density and temperature but also on the mass fraction of the reactant, while the two species chemical kinetics equation is of higher order. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible reacting fluids is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl [26] and P.L. Lions [35]. [ABSTRACT FROM AUTHOR]

We investigate the global existence of weak solutions to a free boundary problem governing the evolution of finitely extensible bead-spring chains in dilute polymers. We construct weak solutions of the two-phase model by performing the asymptotic limit as the adiabatic exponent γ goes to ∞ for a macroscopic model which arises from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids. In this context the polymeric molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. This class of models involves the unsteady, compressible, isentropic, isothermal Navier-Stokes system in a bounded domain Ω in R d , d = 2 , 3 coupled with a Fokker-Planck-Smoluchowski-type diffusion equation (cf. Barrett and Süli [3] , [4] , [7]). The convergence of these solutions, up to a subsequence, to the free-boundary problem is established using weak convergence methods, compactness arguments which rely on the monotonicity properties of certain quantities in the spirit of [12]. [ABSTRACT FROM AUTHOR]

We investigate the long-time behavior of solutions to a stochastically forced one-dimensional Navier–Stokes system, describing the motion of a compressible viscous fluid, in the case of linear pressure law. We prove existence of an invariant measure for the Markov process generated by strong solutions. We overcome the difficulties of working with non-Feller Markov semigroups on non-complete metric spaces by generalizing the classical Krylov–Bogoliubov method, and by providing suitable polynomial and exponential moment bounds on the solution, together with pathwise estimates. [ABSTRACT FROM AUTHOR]

The hydrodynamic limit of a kinetic Cucker-Smale flocking model is investigated. The starting point is the model considered in [Existence of weak solutions to kinetic flocking models, SIAM Math. Anal. 45 (2013) 215-243], which in addition to free transport of individuals and a standard Cucker-Smale alignment operator, includes Brownian noise and strong local alignment. The latter was derived in [On strong local alignment in the kinetic Cucker-Smale equation, in Hyperbolic Conservation Laws and Related Analysis with Applications (Springer, 2013), pp. 227-242] as the singular limit of an alignment operator first introduced by Motsch and Tadmor in [A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 141 (2011) 923-947]. The objective of this work is the rigorous investigation of the singular limit corresponding to strong noise and strong local alignment. The proof relies on a relative entropy method. The asymptotic dynamics is described by an Euler-type flocking system. [ABSTRACT FROM AUTHOR]

We prove a low Mach number limit for a dispersive fluid system [3] which contains third-order corrections to the compressible Navier-Stokes. We show that the classical solutions to this system in the whole space ℝn converge to classical solutions to ghost-effect systems [7]. Our analysis follows the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and to simplify the analysis in [4]. [ABSTRACT FROM AUTHOR]

We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under consideration include the kinetic Cucker--Smale equation [Cucker and Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862, Cucker and Smale, Japan. J. Math., 2 (2007), pp. 197-227] with possibly nonsymmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor [J. Statist. Phys., 141 (2011), pp. 923-947]. The main tools employed in the analysis are the velocity-averaging lemma and the Schauder fixed-point theorem along with various integral bounds. [ABSTRACT FROM AUTHOR]

We establish a low Mach number limit for classical solutions over the whole space of a compressible fluid dynamic system that includes dispersive corrections to the Navier-Stokes equations. The limiting system is similar to a ghost effect system [Y. Sone, Kinetic Theory and Fluid Dynamics, Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, 2002]. Our analysis builds upon the framework developed by Métivier and Schochet [Arch. Ration. Mech. Anal., 158 (2001), pp. 61-90] and Alazard [Arch. Ration. Mech. Anal., 180 (2006), pp. 1-73] for nondispersive systems. The strategy involves establishing a priori estimates for the slow motion as well as a priori estimates for the fast motion. The desired convergence is obtained by establishing the local decay of the energy of the fast motion. [ABSTRACT FROM AUTHOR]

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log T, log n, log 1=ε)=ε, where T is the evolution time, ε is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differe pntial equations, showing that the problem is intractable for R<2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R. [ABSTRACT FROM AUTHOR]