Dissipative measure-valued solutions to the Euler-Poisson equation

Abstract

We consider several pressureless variants of the compressible Euler equation driven by nonlocal repulsionattraction and alignment forces with Poisson interaction. Under an energy admissibility criterion, we prove existence of global measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We then investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a (partial) weak-strong uniqueness principle.

Analysis of the generalised Aw-Rascle model

“…The weak-strong property of measure-valued solutions to fluid equations was first observed by Brennier et al [5] for the incompressible Euler system. After that, analogous results were proven for the whole variety of fluid models [12,15,29] including ultimately the compressible Navier-Stokes equations [19] and general conservation laws [28].…”

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

“…The approximating system. To construct a measure-valued solution we use a method as outlined in [62, Section 5.5], see also [18,53]. This is a fairly standard procedure based on regularizing density by a positive parameter…”

“…isentropic Euler system [48], polyconvex elastodynamics [30], Euler-Poisson system [18], general hyperbolic conservation laws [54]. Moreover, for many equations describing compressible fluids, the measure-valued formulation has been significantly simplified [1,8,42]: it boils down to the usual distributional identity modulo the so-called Reynolds stress tensor.…”

The role of Riesz potentials in the weak-strong uniqueness for the Euler-Poisson system

“…The weak-strong uniqueness property lives behind the stability estimates on relaxation; see [7] for the weak-strong uniqueness of a Euler-Poisson system, for periodic solutions, as a consequence of its relaxation limit towards a Keller-Segel system. The weak-strong uniqueness principle has also been studied for the Euler-Poisson system with linear damping and confinement in a bounded smooth domain, for classes of measure-valued solutions; in this case being called measure-valued-strong uniqueness [3]. Furthermore, it has as well been studied for a similar set of equations, the Navier-Stokes-Poisson system, for classes of weak solutions that are continuous in time with respect to the weak topology [2].…”