Uniform Local Existence for Inhomogeneous Rotating Fluid Equations

TL;DR

This paper proves the global and uniform local existence of strong solutions for inhomogeneous rotating fluid equations in anisotropic Sobolev spaces, under small initial data and specific inhomogeneity conditions.

Contribution

It establishes uniform local existence results with respect to the Rossby number for anisotropic inhomogeneous rotating fluids, introducing a new refined product law for Sobolev regularity.

Findings

Global existence of strong solutions for small initial data.

Uniform local existence with respect to Rossby number under specific conditions.

Propagation of isotropic Sobolev regularity using a new product law.

Abstract

We investigate the equations of anisotropic incompressible viscous fluids in $\R^3$, rotating around an inhomogeneous vector $B(t, x_1, x_2)$. We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniformlocal existence result with respect to the Rossby number in the same functional spaces under the additional assumption that $B=B(t,x_1)$ or $B=B(t,x_2)$. We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law.