Dissipative measure-valued solutions for general conservation laws

Abstract

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 magentohydrodynamics and shallow water magnetohydrodynamics.O.K. acknowledges the support of the Neuron Impuls Junior project "Mathematical analysis of hyperbolic conservation laws" in the general framework of RVO: 67985840.

Relative Entropy Method for Measure Solutions of the Growth-Fragmentation Equation

“…Let us remark that the method of analysis employed in the current paper is inspired by the classical relative entropy method introduced by Dafermos in [10]. In recent years this method was extended to yield results on measure-valued-strong uniqueness for equations of fluid dynamics [6,17,22] and general conservation laws [9,12,19]. See also [11] and refereces therein.…”

Dissipative measure-valued solutions to the Euler-Poisson equation

“…Indeed, this was the case for most of the recent studies on weak-strong uniqueness for measure-valued solutions in hydrodynamics or more general conservation laws. Starting from the works of Brenier et al [4] on the incompressible Euler and Demoulini et al [9] on polyconvex elastodynamics, and their somewhat surprising observation that measure-valued solutions can enjoy the weak-strong uniqueness property (under an admissibility condition), this property has been proved for a variety of other equations, see [8,11,16,17]. Notably the weak-strong uniqueness property for dissipative measure-valued solutions has been recently put to practical use in proving convergence of finite volume numerical schemes for the Euler and Navier-Stokes equations [12,13].…”

Weak-strong uniqueness for energy-reaction-diffusion systems

“…We refer to the survey by Wiedemann [45] for more details and further references. For recent advances on conditional uniqueness results for dissipative measure-valued solutions to conservation laws, see [27] and references therein. Quite interesting in the thermodynamics context is moreover the relative entropy technique employed by DiPerna [18] and Dafermos [16] for hyperbolic conservation laws.…”