Alexandre Dutrifoy

On 3-D Vortex Patches in Bounded Domains

Abstract This article concerns the equations of motion of perfect incompressible fluids in a smooth, bounded, simply connected domain of . So we study the Euler system where v is the velocity field and p is the pressure, along with an initial datum, and a condition at the boundary of the domain Ω, meaning that the fluid particles cannot cross the boundary (n denotes the unit outward normal). We suppose that the curl of v 0 is a vortex patch, which involves some conormal smoothness implying but not , and examine the classical problems of the existence of a solution, either locally or globally in time, and of the persistence of the initial regularity.

Fast Wave Averaging for the Equatorial Shallow Water Equations

The equatorial shallow water equations in a suitable limit are shown to reduce to zonal jets as the Froude number tends to zero. This is a theorem of a singular limit with a fast variable coefficient due to the vanishing of the Coriolis force at the equator. Although it is not possible to get uniform estimates in classical Sobolev spaces (other than L2) by differentiating the system, a new method exploiting the particular structure of the fast coefficient leads to uniform estimates in slightly different functional spaces. The computation of resonances shows that fast waves may interact with a strong external forcing, introduced to mimic the effects of moisture, to create zonal jets.

Precise regularity results for the Euler equations

Abstract It has already been proved, under various assumptions, that no singularity can appear in an initially regular perfect fluid flow, if the L∞ norm of the velocitys curl does not blow up. Here that result is proved for flows in smooth bounded domains of R d (d⩾2) when the regularity is expressed in terms of Besov (or Triebel–Lizorkin) spaces.