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Existence and regularity of steady-state solutions of the Navier-Stokes equations arising from irregular data

Gael Y. Diebou

Abstract

We analyze the forced incompressible stationary Navier-Stokes flow in $\mathbb{R}^n_+$, $n>2$. Existence of a unique solution satisfying a global integrabilty property measured in a scale of tent spaces is established for small data in homogenous Sobolev space with $s=-\frac{1}{2}$ degree of smoothness. Moreover, the velocity field is shown to be locally Hölder continuous while the pressure belongs to $L^p_{loc}$ for any $p\in (1,\infty)$. Our approach is based on the analysis of the inhomogeneous Stokes system for which we derive a new solvability result involving Dirichlet data in Triebel-Lizorkin classes with negative amount of smoothness and is of independent interest.

Existence and regularity of steady-state solutions of the Navier-Stokes equations arising from irregular data

Abstract

We analyze the forced incompressible stationary Navier-Stokes flow in , . Existence of a unique solution satisfying a global integrabilty property measured in a scale of tent spaces is established for small data in homogenous Sobolev space with degree of smoothness. Moreover, the velocity field is shown to be locally Hölder continuous while the pressure belongs to for any . Our approach is based on the analysis of the inhomogeneous Stokes system for which we derive a new solvability result involving Dirichlet data in Triebel-Lizorkin classes with negative amount of smoothness and is of independent interest.
Paper Structure (8 sections, 15 theorems, 163 equations)

This paper contains 8 sections, 15 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Tent spaces and functional settings
  3. Main results
  4. Auxiliary results
  5. Homogeneous Stokes system and linear estimates
  6. Inhomogeneous Stokes system
  7. Proofs of main results
  8. Appendix

Key Result

Lemma 1.1

Let $K$ be a compact set in $\mathbb{R}^{n}_+$ and assume that $F\in T^{p,q}_{s_q}$ for $p,q\in [1,\infty)$. Then where the constant $C_1,C_2$ only depend on $p,q,n$ and $K$.

Theorems & Definitions (34)

  • Lemma 1.1
  • Remark 1.1
  • Lemma 1.2
  • Remark 1.2
  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.3
  • ...and 24 more