#### Vol. 13, No. 4, 2020

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Estimates for the Navier–Stokes equations in the half-space for nonlocalized data

### Yasunori Maekawa, Hideyuki Miura and Christophe Prange

Vol. 13 (2020), No. 4, 945–1010
##### Abstract

This paper is devoted to the study of the Stokes and Navier–Stokes equations, in a half-space, for initial data in a class of locally uniform Lebesgue integrable functions, namely ${L}_{uloc,\sigma }^{q}\left({ℝ}_{+}^{d}\right)$. We prove the analyticity of the Stokes semigroup ${e}^{-tA}$ in ${L}_{uloc,\sigma }^{q}\left({ℝ}_{+}^{d}\right)$ for $1. This follows from the analysis of the Stokes resolvent problem for data in ${L}_{uloc,\sigma }^{q}\left({ℝ}_{+}^{d}\right)$, $1. We then prove bilinear estimates for the Oseen kernel, which enables us to prove the existence of mild solutions. The three main original aspects of our contribution are: the proof of Liouville theorems for the resolvent problem and the time-dependent Stokes system under weak integrability conditions, the proof of pressure estimates in the half-space, and the proof of a concentration result for blow-up solutions of the Navier–Stokes equations. This concentration result improves a recent result by Li, Ozawa and Wang and provides a new proof.

##### Keywords
Navier–Stokes equations, resolvent estimates, analyticity, Stokes semigroup, local uniform Lebesgue spaces, mild solutions, concentration, Liouville theorems, pressure, half-space
##### Mathematical Subject Classification 2010
Primary: 35A01, 35A02, 35B44, 35B53, 35Q30
Secondary: 35C99, 76D03, 76D05