Dissipative measure-valued solutions to the Euler-Poisson equation

TL;DR

This paper establishes the existence of global measure-valued solutions for pressureless Euler-Poisson equations with nonlocal forces and explores a weak-strong uniqueness principle using a relative energy functional.

Contribution

It introduces the concept of dissipative measure-valued solutions for Euler-Poisson equations with nonlocal interactions and proves a weak-strong uniqueness result.

Findings

Existence of global measure-valued solutions under energy admissibility.

Development of a relative energy functional for solution comparison.

Partial weak-strong uniqueness principle established.

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.