Time analyticity for inhomogeneous parabolic equations and the Navier-Stokes equations in the half space

TL;DR

This paper establishes the time analyticity of weak solutions to inhomogeneous parabolic equations and bounded mild solutions of the Navier-Stokes equations in the half space, extending previous results from the whole space.

Contribution

It extends the time analyticity results to inhomogeneous equations and Navier-Stokes solutions in the half space with boundary conditions, under exponential growth assumptions.

Findings

Proves time analyticity for weak solutions of inhomogeneous parabolic equations in half space.

Establishes time analyticity for bounded mild solutions of Navier-Stokes equations in the half space.

Extends previous whole space results to half space with boundary conditions.

Abstract

We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (2020)] and Dong-Zhang [J. Funct. Anal. (2020)], where the authors proved the time analyticity of solutions to the homogeneous heat equation and the Navier-Stokes equations in the whole space.