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arXiv · 2406.19805

The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space

Daniele Barbera·Miho Murata
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Abstract

In this paper, we consider the model describing viscous incompressible liquid crystal flows, called the Beris-Edwards model, in the half-space.This model is a coupled system by the Navier-Stokes equations with the evolution equation of the director fields $Q$. The purpose of this paper is to prove that the linearized problem has a unique solution satisfying the maximal $L^p$ -$L^q$ regularity estimates, which is essential for the study of quasi-linear parabolic or parabolic-hyperbolic equations. Our method relies on the $\mathcal R$-boundedness of the solution operator families to the resolvent problem in order to apply operator-valued Fourier multiplier theorems. Consequently, we also have the local well-posedness for the Beris-Edwards model with small initial data.

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Daniele Barbera, Miho Murata. 2024-06-28. The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space. https://arxiv.org/abs/2406.19805

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