On Mixed Pressure-Velocity Regularity Criteria to the Navier-stokes Equations in Lorentz Spaces, Part II: The Non-slip Boundary Value Problem

Abstract

This paper is a continuation of the authors recent work [Beirão da Veiga, H. and Yang, J., On mixed pressure-velocity regularity criteria to the Navier-Stokes equations in Lorentz spaces, Chin. Ann. Math., 42(1), 2021, 1–16], in which mixed pressure-velocity criteria in Lorentz spaces for Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations, in the whole space ℝ³ and in the periodic torus ${\mathbb{T}^3}$, are established. The purpose of the present work is to extend the result of mentioned above to smooth, bounded domains, under the non-slip boundary condition. Let π denote the fluid pressure and v the fluid velocity. It is shown that if ${\pi \over {{{\left( {1 + \left| v \right|} \right)}^\theta }}} \in {L^p}\left( {0,T;{L^{q,\infty }}\left( \Omega \right)} \right)$;, where 0 ≤ θ ≤ 1, and ${2 \over p} + {3 \over q} = 2 - \theta$ with p ≥ 2, then v is regular on Ω × (0, T].