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arXiv · 2407.07001

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

Kyungkeun Kang·Baishun Lai·Chen-Chih Lai·Tai-Peng Tsai
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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.

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Kyungkeun Kang, Baishun Lai, Chen-Chih Lai, Tai-Peng Tsai. 2024-07-09. Applications of the Green tensor estimates of the nonstationary Stokes system in the half space. https://arxiv.org/abs/2407.07001

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