Global existence for non-homogeneous incompressible inviscid fluids in presence of Ekman pumping
Marco Bravin, Francesco Fanelli
TL;DR
The paper analyzes global solvability for the density-dependent incompressible Euler equations with a damping term $\mathfrak{D}_{\alpha}^{\gamma}(\rho,u)=\alpha\rho^{\gamma}u$ (with $\alpha>0$ and $\gamma\in\{0,1\}$), interpreting the damping as Ekman pumping. Employing Besov spaces and harmonic-analytic tools, it proves global well-posedness under nonlinear smallness conditions that tie the initial velocity, density variations, and damping strength, with additional results in planar flows for $\gamma=1$ allowing arbitrarily large $\|u_0\|$ when $\rho_0-1$ is small. The authors develop a three-step strategy—local well-posedness, a Beale–Kato–Majda-type continuation criterion, and control of low-regularity norms—to obtain global existence and exponential decay in $B^{s}_{p,r}$ norms for velocity and pressure gradient, plus decay of higher-order norms. Distinctions between $\gamma=1$ and $\gamma=0$ are treated carefully, with the latter requiring refined commutator and interpolation techniques due to a linear pressure term, and additional planar-case refinements. The results contribute a robust global well-posedness theory for non-homogeneous, damped incompressible Euler equations in a critical Besov setting, with implications for geophysical flow modeling under 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}_α^γ(ρ, u) = αρ^γ u $, where $α>0$ and $ γ\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 $d\geq 2$, 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 $ρ_0-1$ and of the damping coefficient $α$. On the other hand, in the specific situation of planar motions and damping term with $γ=1$, 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 $ρ_0-1$ are small enough. The formulated smallness conditions rely only on the endpoint Besov norm $B^1_{\infty,1}$ 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 $B^s_{p,r}$.