Wellposedness for density-dependent incompressible viscous fluids on the torus $\T^3$

TL;DR

This paper proves local and global well-posedness of density-dependent incompressible viscous fluids on the 3D torus, using critical Besov space initial data, with small velocity and arbitrary density variations.

Contribution

It establishes well-posedness results for inhomogeneous Navier-Stokes equations on the torus with initial data in critical Besov spaces, including global existence under small velocity assumptions.

Findings

Global-in-time existence of solutions under small velocity conditions.

Initial density can be large, not requiring smallness.

Solutions exist in critical Besov spaces.

Abstract

We investigate the local wellposedness of incompressible inhomogeneous Navier-Stokes equations on the Torus $\T^3$, with initial data in the critical Besov spaces. Under some smallness assumption on the velocity in the critical space $B^{\frac{1}{2}}\_{2,1}(\T^3)$, the global-in-time existence of the solution is proved. The initial density is required to belong to~$B^{\frac{3}{2}}\_{2,1}(\T^3)$ but not supposed to be small.