Well-posedness of the Stokes–Coriolis system in the half-space over a rough surface

Abstract

This paper is devoted to the well-posedness of the stationary 3d Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of Gérard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks to a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes-Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes-Coriolis system with data in the Kato space H

Nonlinear boundary layers for rotating fluids

“…But contrary to the Stokes case, there is no easy integral representation. Still, such linear problem was tackled in the recent paper [14], by the first author and C. Prange. To solve the Dirichlet problem, they use a Fourier transform in variables y 1 , y 2 , leading to accurate formulas.…”

“…The linear study [14] is a starting point for our study of the nonlinear system (1.4), but we will need many refined estimates, combined with a fixed point argument. More precisely, the outline of the paper is the following.…”

“…For simplicity, we take M = 0. The case without source term (F = 0) was partially analyzed in [14], but we will establish new estimates, notably related to low frequencies. Let us emphasize that the difficulty induced by low frequencies already appeared in [14, Proposition 2.1, page 6], even in the case of classical Sobolev data: in such case, some cancellation of the Fourier transformv 0,3 at frequency ξ = 0 was assumed.…”

“…Our results extend the older paper [18], restricted to periodic variations of the bottom. It ponders on the recent linear analysis carried in [14].…”

Cauchy theory for the gravity water waves system with non-localized initial data

“…We think that this section could be useful for further studies in this framework. In Section 3 we prove that the Dirichlet-Neumann operator is well defined in this framework (notice that this fact is not straightforward, see [18,15] for related works), and we give a precise description (including sharp elliptic estimates in very rough domains) on these spaces. In Section 4, we symmetrize the system and prove a priori estimates.…”