Bounded Solutions in Incompressible Hydrodynamics

Dimitri Cobb

Published 2021-05-07Version 1

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 $|x| \rightarrow + \infty$. We find this condition to be necessary and sufficient. We deduce a full well-posedness result for the Euler equations in the space $C^0_T(B^1_{\infty, 1})$. We also apply the methods developed for the Euler system to other hydrodynamic models: the Navier-Stokes equations and magnetohydrodynamics.