Maximum Norm Estimates of the Solution of the Navier-Stokes Equations in the Halfspace with Bounded Initial Data

Santosh Pathak

doi:10.1155/2021/6686526

Abstract

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in ℝ + n for n ≥ 3 with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in T n and ℝ n respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

Highlights

We consider the Cauchy problem of the incompressible Navier-Stokes equations in Rn+, n ≥ 3: ut + u · ∇u+∇p ∇·u ujt=0 ujxn=0 = = = =Δu for x 0 for x ∈ f for x ∈ 0 for t >∈ Rn+, Rn+, t Rn+, 0, t > > 0, 0,9 >>>>>= >>>>>; ð1Þ where u = uðx, tÞ = ðu1ðx, tÞ,⋯,unðx, tÞÞ and p = pðx, tÞ stand for the unknown velocity vector field of the fluid and its pressure, while f = f ðxÞ = ð f1ðxÞ,⋯f nðxÞÞ is the given initial velocity vector field, with ∇·f = 0 and f jxn=0 = 0
Auxiliary Results,” we introduce some auxiliary results which will be labelled as propositions
We note that each gij is quadratic in components of u

Summary

Introduction

We consider the Cauchy problem of the incompressible Navier-Stokes equations in Rn+, n ≥ 3: ut + u · ∇u+∇p ∇·u ujt=0 ujxn=0. There is a large literature on the existence and uniqueness of solution of the Navier-Stokes equations in Rn. For the given initial data, solutions of (1) have been constructed in various function spaces. Bae and Jin, the local-in-time existence of mild (strong) solution of the halfspace problem was provided in [9] by Solonikov for continuous bounded initial data in Rn+. We will generalize the techniques of obtaining the uniform estimates on ∇e−AtP∇div of the paper [8] by Bae and Jin to obtain our desired uniform estimates on Dje−AtP∇div In their paper, they require the uniform estimates to prove the existence of the local solution of the Navier-Stokes equations in halfspace for bounded initial data. Appendices A, B, and C contain proofs of the propositions which are introduced in “Some Auxiliary Results.”

Some Auxiliary Results

Estimates for the Navier-Stokes Equations

Proof of Theorem 1

Proof of Proposition 3

Proof of Proposition 5