Dissipative Measure-valued Solutions Research Articles

Existence and Stability of Dissipative Measure-Valued Solutions to the Full Compressible Magnetohydrodynamic Flows

Existence and Stability of Dissipative Measure-Valued Solutions to the Full Compressible Magnetohydrodynamic Flows

Dissipative Measure-Valued Solutions to the Euler–Poisson Equation

Dissipative Measure-Valued Solutions to the Euler–Poisson Equation

A semi-implicit finite volume scheme for dissipative measure-valued solutions to the barotropic Euler system

A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [Feireisl et al., Calc. Var. Part. Differ. Equ. 55 (2016) 141] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax’s equivalence theorem under some physically reasonable boundedness assumptions. The concept of Ƙ-convergence [Feireisl et al., IMA J. Numer. Anal. 40 (2020) 2227–2255] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak–strong uniqueness principle and relative entropy are presented.

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.

Asymptotic limits of the Euler–Poisson equations for dissipative measure-valued solutions

We prove that the dissipative measure-valued solutions of the 3D compressible Euler–Poisson equations converge to the solution of the incompressible Euler equations in an asymptotic regime, where the Mach number and Debye length tend to zero simultaneously. Furthermore, the convergence rates are also obtained.

Convergence of a spectral method for the stochastic incompressible Euler equations

We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by amultiplicativenoise. We show that the SVM solution converges to adissipative measure-valued martingalesolution of the underlying problem. These solutions are weak in the probabilistic sensei.e.the probability space and the driving Wiener process are an integral part of the solution. We also exhibit a weak (measure-valued)-strong uniqueness principle. Moreover, we establishstrongconvergence of approximate solutions to the regular solution of the limit system at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system.

Measure-valued solutions to the stochastic compressible Euler equations and incompressible limits

We introduce a new concept of dissipative measure-valued martingale solutions to the stochastic compressible Euler equations. These solutions are weak in the probabilistic sense i.e., the probability space and the driving Wiener process are an integral part of the solution. We derive the relative energy inequality for the stochastic compressible Euler equations and, as a corollary, we exhibit pathwise weak-strong uniqueness principle. Moreover, making use of the relative energy inequality, we investigate the low Mach limit (incompressible limit) of underlying system of equations. As a main novelty with respect to the related literature, our results apply to general nonlinear multiplicative stochastic perturbations of Nemytskij type.

On Weak-Strong Uniqueness for Stochastic Equations of Incompressible Fluid Flow

We introduce a concept of dissipative measure-valued martingale solution to the stochastic Euler equations describing the motion of an inviscid incompressible fluid. These solutions are characterized by a parametrized Young measure and a concentration defect measure in the total energy balance. Moreover, they are weak in the probabilistic sense i.e., the underlying probability space and the driving Wiener process are intrinsic parts of the solution. We first exhibit the relative energy inequality for the incompressible Euler equations driven by a multiplicative noise and then demonstrate the pathwise weak-strong uniqueness principle. Finally, we also provide a sufficient condition, á la Prodi (Ann Mat Pura Appl 48:173–182, 1959) and Serrin (in: Nonlinear problems, University of Wisconsin Press, Madison, Wisconsin, pp 69–98, 1963), for the uniqueness of weak martingale solutions to the stochastic Navier–Stokes system in the class of finite energy weak martingale solutions.

Weak-strong uniqueness for measure-valued solutions to the equations of quasiconvex adiabatic thermoelasticity

This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measure-valued solutions.

On the quasineutral limit for the compressible Euler-Poisson equations

<p style='text-indent:20px;'>In this paper, we consider the quasineutral limit of compressible Euler-Poisson equations based on the concept of dissipative measure-valued solutions. In the case of well-prepared initial data under periodic boundary condictions, we prove that dissipative measure-valued solutions of the compressible Euler-Poisson equations converge to the smooth solution of the incompressible Euler system when the Debye length tends to zero.</p>

On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law

<p style='text-indent:20px;'>In this paper, we define a renormalized dissipative measure-valued (rDMV) solution of the compressible magnetohydrodynamics (MHD) equations with non-monotone pressure law. We prove the existence of the rDMV solutions and establish a suitable relative energy inequality. And we obtain the weak (measure-valued)-strong uniqueness property of this rDMV solution with the help of the relative energy inequality.</p>

Correction to: On the Existence of Dissipative Measure-Valued Solutions to the Compressible Micropolar System

The article “On the Existence of Dissipative Measure-Valued Solutions to the Compressible Micropolar System”, written by Bingkang Huang, was originally published electronically on the publisher’s internet portal (currently SpringerLink) on 14 September 2020 with open access

Low Mach and thin domain limit for the compressible Euler system

We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer $$\Omega _{\delta }=(0,\delta )\times {\mathbb {R}}^2, \delta >0$$ . In the framework of dissipative measure-valued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and $$\delta \rightarrow 0$$ .

On the Existence of Dissipative Measure-Valued Solutions to the Compressible Micropolar System

Compressible micropolar equations model a class of fluids with microstructure. In this paper we establish the dissipative measure-valued solution to the micropolar fluids. We also give the weak-strong uniqueness principle to this system which means its dissipative measure-valued solution is the same as the classical solution, provided they emanate from the same initial data.

On a Singular Limit for the Compressible Rotating Euler System

The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in the case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated.

Low stratification of the complete Euler system

The present paper takes advantage of the concept of dissipative measure-valued solutions to show the rigorous derivation of the Euler–Boussinesq (EB) system that has been successfully used in various meteorological models. In particular, we show that EB system can be obtained as a singular limit of the complete Euler system. We provide two types of result—firstly, we treat the case of well-prepared initial data for any sufficiently regular bounded domain. Secondly, we use the dispersive estimates for acoustic equation to tackle the case of the ill-prepared initial data on an unbounded exterior domain.

On Weak (Measure-Valued)\u2013Strong Uniqueness for Compressible Navier\u2013Stokes System with Non-monotone Pressure Law

In this paper our goal is to define a renormalized dissipative measure-valued (rDMV) solution of compressible Navier–Stokes system for fluids with non-monotone pressure–density relation. We prove existence of rDMV solutions and establish a suitable relative energy inequality. Moreover we obtain the weak (measure-valued)–strong uniqueness property of this rDMV solution with the help of relative energy inequality.

Stability of Strong Solutions to the Navier--Stokes--Fourier System

We identify a large class of objects---dissipative measure-valued (DMV) solutions to the Navier--Stokes--Fourier system---in which the strong solutions are stable. More precisely, a DMV solution co...

Dissipative measure-valued solutions for general conservation laws

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued – strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system et al. and there are also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems, and most importantly, we give examples of equations, for which the aspect of measure-valued – strong uniqueness has not been considered before, like incompressible magnetohydrodynamics and shallow water magnetohydrodynamics.

Convergence of a finite volume scheme for the compressible Navier–Stokes system

We study convergence of a finite volume scheme for the compressible (barotropic) Navier–Stokes system. First we prove the energy stability and consistency of the scheme and show that the numerical solutions generate a dissipative measure-valued solution of the system. Then by the dissipative measure-valued-strong uniqueness principle, we conclude the convergence of the numerical solution to the strong solution as long as the latter exists. Numerical experiments for standard benchmark tests support our theoretical results.