Applications of the Green tensor estimates of the nonstationary Stokes system in the half space

TL;DR

This paper applies Green tensor estimates to analyze nonstationary Stokes and Navier-Stokes systems in half space, deriving new estimates, constructing solutions with decay properties, and extending to coupled fluid systems.

Contribution

It introduces new pointwise Green tensor estimates for the nonstationary Stokes system and applies them to solve various fluid models in the half space with decay and boundary conditions.

Findings

Established $L^1$-$L^q$ estimates for non-solenoidal initial data.

Constructed mild solutions with decay properties for Navier-Stokes and MHD systems.

Extended Green tensor applications to coupled fluid systems with boundary effects.

Abstract

In this paper, we present a series of applications of the pointwise estimates of the (unrestricted) Green tensor of the nonstationary Stokes system in the half space, established in our previous work [CMP 2023]. First, we show the $L^1$-$L^q$ estimates for the Stokes flow with possibly non-solenoidal $L^1$ initial data, generalizing the results of Giga-Matsui-Shimizu [Math. Z. 1999] and Desch-Hieber-Pr\"uss [J. Evol. Equ. 2001]. Second, we construct mild solutions of the Navier-Stokes equations in the half space with mixed-type pointwise decay or with pointwise decay alongside boundary vanishing. Finally, we explore various coupled fluid systems in the half space including viscous resistive magnetohydrodynamics equations, a coupled system for the flow and the magnetic field of MHD type, and the nematic liquid crystal flow. For each of these systems, we construct mild solutions in $L^q$, pointwise decay, and uniformly local $L^q$ spaces.