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On the Stokes system in cylindrical domains

Joanna Rencławowicz, Wojciech M. Zajączkowski

Abstract

The existence of solutions to some initial-boundary value problem for the Stokes system is proved. The result is shown in Sobolev-Slobodetskii spaces such that the velocity belongs to $W_r^{2+σ,1+σ/2}(Ω^T)$ and gradient of pressure to $W_r^{σ,σ/2}(Ω^T)$, where $r\in(1,\infty)$, $σ\in(0,1)$, $Ω^T=Ω\times(0,T)$. These are special Besov spaces: $B_{r,r}^{2+σ,1+σ/2}(Ω^T)$ and $B_{r,r}^{σ,σ/2}(Ω^T)$, respectively. The existence is proved by the technique of regularizer.

On the Stokes system in cylindrical domains

Abstract

The existence of solutions to some initial-boundary value problem for the Stokes system is proved. The result is shown in Sobolev-Slobodetskii spaces such that the velocity belongs to and gradient of pressure to , where , , . These are special Besov spaces: and , respectively. The existence is proved by the technique of regularizer.
Paper Structure (7 sections, 24 theorems, 313 equations)

This paper contains 7 sections, 24 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The Stokes system in the whole space
  4. The Stokes system in the half-space
  5. The Stokes system in neighborhoods of edges.
  6. The Stokes system in the cylindrical domain $\Omega$
  7. Existence in Sobolev spaces

Key Result

Theorem 1.1

Assume that $f\in W_r^{\sigma,\sigma/2}(\Omega^T)$, $v_0\in W_r^{2+\sigma-2/r}(\Omega)$, $g\in W_r^{1+\sigma,1/2+\sigma/2}(\Omega^T)$, $b_\alpha\in W_r^{1+\sigma-1/r,1/2+\sigma/2-1/2r}(S_i^T)$, $\alpha=1,2$, and $b_3\in W_r^{2+\sigma-1/r,1+\sigma/2-1/2r}(S_i^T)$, where $r\in(1,\infty)$, $\sigma\in(0 where $c$ does not depend on $v$ neither $p$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Besov spaces defined on ${\mathbb R}_+^3\times{\mathbb R}$
  • Definition 2.6
  • Lemma 2.7: see Ch. 4, Sect. 18 BIN
  • Lemma 2.8: see N
  • ...and 41 more