Well/ill posedness for the Euler-Korteweg-Poisson system and related problems

TL;DR

This paper investigates the well-posedness of the Euler-Korteweg-Poisson system, revealing the existence of infinitely many weak solutions and establishing weak-strong uniqueness in certain cases, highlighting the system's complex solution structure.

Contribution

It demonstrates the existence of infinitely many global weak solutions for the Euler-Korteweg-Poisson system, including cases with vacuum, and proves weak-strong uniqueness without vacuum.

Findings

Existence of infinitely many global weak solutions.

Presence of infinitely many dissipative weak solutions.

Weak-strong uniqueness in non-vacuum cases.

Abstract

We consider a general Euler-Korteweg-Poisson system in $R^3$, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum.