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This article is concerned with the rigorous justification of the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main difference with respect to earlier works is that we neglect viscosity effects but rather include diffusivity effects induced by an advection component proposed by Gent and McWilliams in the 90's to model effective eddy correlations for non-eddy-resolving systems. We also crucially make use of the assumption that the fluid is stably stratified, work in functional spaces of high (but finite) regularity and keep track of the various parameters at stake. A key ingredient of our analysis is the reformulation of our problem using isopycnal coordinates, on which careful energy estimates are performed.

We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously unexplored case of more general shear flows close to Couette, the inviscid damping results follow by a weighted energy estimate. Each outcome concerning the stably stratified regime applies to the Boussinesq equations as well. Remarkably, our results hold under the celebrated Miles-Howard criterion for stratified fluids., 28 pages, some typos corrected

We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-Kawashima condition, i.e. such that some eigendirections do not exhibit dissipation at all. In the space-time resonances framework introduced by Germain, Masmoudi and Shatah, we prove that, when the source term has a Nonresonant Bilinear Form, as proposed by Pusateri and Shatah CPAM 2013, the formation of singular-ities is prevented, despite the lack of dissipation. This allows us to show that smooth solutions to this preliminary case-study model exist globally in time.

This article deals with the asymptotic behavior of the two-dimensional inviscid Boussinesq equations with a damping term in the velocity equation. Precisely, we provide the time-decay rates of the smooth solutions to that system. The key ingredient is a careful analysis of the Green kernel of the linearized problem in Fourier space, combined with bilinear estimates and interpolation inequalities for handling the nonlinearity.

International audience; We obtain sharp decay estimates in time in the context of Sobolev spaces, for smooth 4 solutions to the one dimensional Jin-Xin model under the diffusion scaling, which are uniform with 5 respect to the singular parameter of the scaling. This provides convergence to the limit nonlinear 6 parabolic equation both for large time, and for the vanishing singular parameter. The analysis is 7 performed by means of two main ingredients. First, a crucial change of variables highlights the 8 dissipative property of the Jin-Xin system, and allows to observe a faster decay of the dissipative 9 variable with respect to the conservative one, which is essential in order to close the estimates. Next, 10 the analysis relies on a deep investigation on the Green function of the linearized Jin-Xin model, 11 depending on the singular parameter, combined with the Duhamel formula in order to handle the 12 nonlinear terms. 13