Global existence for non-homogeneous incompressible inviscid fluids in presence of Ekman pumping

Abstract

In this paper, we study the global solvability of the density-dependent incompressible Euler equations, supplemented with a damping term of the form ${\mathfrak{D}}_{\alpha }^{\gamma }(\rho ,u)=\alpha {\rho }^{\gamma }u$, where $\alpha >0$ and $\gamma \in \{0,1\}$. To some extent, this system can be seen as a simplified model describing the mean dynamics in the ocean; from this perspective, the damping term can be interpreted as a term encoding the effects of the celebrated Ekman pumping in the system. On the one hand, in the general case of space dimension $d\ge 2$, we establish global well-posedness in the Besov spaces framework, under a non-linear smallness condition involving the size of the initial velocity field $u_0$, of the initial non-homogeneity ${\rho }_0-1$ and of the damping coefficient $\alpha$. On the other hand, in the specific situation of planar motions and damping term with $\gamma =1$, we exhibit a second smallness condition implying global existence, which in particular yields global well-posedness for arbitrarily large initial velocity fields, provided the initial density variations ${\rho }_0-1$ are small enough. The formulated smallness conditions rely only on the endpoint Besov norm $B_{\infty ,1}^1$ of the initial datum, whereas, as a byproduct of our analysis, we derive exponential decay of the velocity field and of the pressure gradient in the high regularity norms $B_{p,r}^s$.

Modifications in Lemma 4.2: (i) corrected a gross mistake in the proof; (ii) added, when $p=+\infty$, an integrability assumption on $\nabla u_0$; (iii) proof improved to avoid integrability assumptions on $\nabla {\rho }_0$