Global well-posedness for 2D fractional inhomogeneous Navier-Stokes equations with rough density

TL;DR

This paper proves the global existence and uniqueness of solutions for 2D fractional inhomogeneous Navier-Stokes equations with rough density, establishing maximal regularity estimates and analyzing density patch regularity.

Contribution

It introduces new maximal regularity estimates for fractional Stokes systems and demonstrates global well-posedness and regularity persistence for rough density in 2D INS equations.

Findings

Global existence of solutions with large velocity fields.

Uniqueness of solutions under small density perturbations.

Persistence of density patch regularity over time.

Abstract

The paper concerns with the global well-posedness issue of the 2D incompressible inhomogeneous Navier-Stokes (INS) equations with fractional dissipation and rough density. We first establish the $L^q_t(L^p)$-maximal regularity estimate for the generalized Stokes system with fractional dissipation, and then we employ it to obtain the global existence of solution for the 2D fractional INS equations with large velocity field, provided that the $L^2\cap L^\infty$-norm of density minus constant 1 is small enough. Moreover, by additionally assuming that the density minus 1 is sufficiently small in the norm of some multiplier spaces, we prove the uniqueness of the constructed solution by using the Lagrangian coordinates approach. We also consider the density patch problem for the 2D fractional INS equations, and show the global persistence of $C^{1,\gamma}$-regularity of the density patch boundary when the piecewise jump of density is small enough.