Maximal $L^1$ regularity for solutions to inhomogeneous incompressible Navier-Stokes equations

TL;DR

This paper establishes new maximal $L^1$ regularity results for solutions to inhomogeneous incompressible Navier-Stokes equations, providing global estimates without smallness constraints on initial density fluctuations.

Contribution

It introduces a novel global $L^1$-in-time estimate for the velocity's Lipschitz norm, independent of initial density fluctuation size, using advanced semigroup and elliptic estimate techniques.

Findings

Derived a new $L^1$ regularity estimate for velocity

Achieved estimates without small initial density assumptions

Developed tools applicable to other fluid dynamics problems

Abstract

This paper is devoted to the maximal $L^1$ regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new global $L^1$-in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the initial density fluctuation. In the derivation of this estimate, we study the maximal $L^1$ regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method might be used to study some other issues arising from incompressible or compressible viscous fluids.