Hidden asymptotics for the weak solutions of the strongly stratified Boussinesq system without rotation

Abstract

The asymptotics of the strongly stratified Boussinesq system when the Froude number goes to zero have been previously investigated, but the resulting limit system surprisingly did not depend on the thermal diffusivity ${\nu }^{\prime }$. In this article we obtain richer asymptotics (depending on ${\nu }^{\prime }$) for more general ill-prepared initial data.

As for the rotating fluids system, the only way to reach this limit consists in finding suitable non-conventional initial data: here, to a function classically depending on the full space variable, we add a second one only depending on the vertical coordinate.

Thanks to a refined study of the structure of the limit system and to new adapted Strichartz estimates, we obtain convergence in the context of weak Leray-type solutions providing explicit convergence rates when possible. In the usually simpler case $\nu ={\nu }^{\prime }$ we are able to improve the Strichartz estimates and the convergence rates. The last part of the appendix is devoted to the proof of a new and crucial dispersion estimate, as classical methods fail.

Finally, our theorems can also be rewritten as a global existence result and asymptotic expansion for the classical Boussinesq system near an explicit stationary solution and for large non-conventional vertically stratified initial data.