Bounded Solutions in Incompressible Hydrodynamics

TL;DR

This paper investigates the uniqueness of bounded, non-integrable solutions to Euler-type equations, revealing the necessity of boundary conditions at infinity and establishing sharp criteria for solution uniqueness in various fluid dynamics contexts.

Contribution

It identifies the boundary condition at infinity required for uniqueness of bounded solutions and applies this to prove full uniqueness results for Besov-Lipschitz and Serfati solutions, extending to ideal MHD.

Findings

Bounded solutions lack uniqueness without boundary conditions at infinity.

A sharp boundary condition at infinity ensures solution uniqueness.

Techniques apply to both Euler equations and ideal MHD variables.

Abstract

In this article, we study bounded solutions of Euler-type equations on $\mathbb{R}^d$ which have no integrability at $|x| \rightarrow +\infty$. As has been previously noted, such solutions fail to achieve uniqueness in an initial value problem, even under strong smoothness conditions. This contrasts with well-posedness results that have been obtained by using the Leray projection operator in these equations. This apparent paradox is solved by noting that using the Leray projector requires an extra condition the solutions must fulfill at $|x| \rightarrow + \infty$. Our goal is to find one such condition which is sharp. We then apply the methods we develop to prove a full uniqueness result for Besov-Lipschitz solutions, as to the theory of Serfati solutions. In the last Section, we see how these techniques also apply to the Els\"asser variables used in ideal MHD.