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

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 scalinginvariant smallness assumption on the initial velocity. We obtain a new global L 1 -in-time estimate for the Lipschitz seminorm of the velocity field. In the derivation of this estimate, we study the maximal 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 can be used to study maximal L 1 regularity for other incompressible or compressible viscous fluids.

Gaussian bounds of fundamental matrix and maximal $L^1$ regularity for Lamé system with rough coefficients

“…What makes the maximal L 1 regularity for (1.5) possible is the observation that the composite operator ρ −1 A behaves similarly to some operator with constant coefficients, in the sense that certain Besov-type norms defined via the semigroups generated by both operators are equivalent. This was one of the key observations made in [25] in which the author of the present paper proved the first maximal L 1 regularity result concerning viscous incompressible fluids with truly variable densities.…”

“…In the incompressible case, the operator A in (1.4) is different from the one in (1.2). Indeed, we need to introduce a socalled Stokes operator to unify the internal force (viscosity and pressure) in (1.2) (see [25]). But A is just what it used to be in the compressible pressureless case.…”

“…In [25], we were only able to work in the L 2 (in space) framework due to the presence of pressure. In this paper, we mainly focus on the pressureless case, and we will work in the general L p (in space) framework.…”

“…A practical benefit of doing so is that one can lower the regularity of the density (see [7]). For the analysis in [25] to adapt to the L p framework, we need to make the extra effort to obtain pointwise bounds for the kernel of the semigroup generated by ρ −1 A. Let us consider two concrete examples.…”

“…For us, working in spaces with negative regularity can weaken the regularity of the density. Our method is motivated by our prior work [25] in which we obtained maximal L 1 regularity for a generalized Stokes operator with variable coefficients in homogeneous Besov spaces. It turns out that the strategy of the proof of the concrete result in [25] works equally well for the abstract problem.…”