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Local solutions for nonhomogeneous Navier-Stokes equations with large flux

Joanna Rencławowicz, Wojciech M. Zajączkowski

Abstract

The local existence of solutions to nonhomogeneous Navier-Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev-Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the $x_3$-axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that $x_3$-coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to $W_σ ^{2+{s},1+{s}/2}(Ω^t)$, the gradient of pressure to $W_σ ^{{s},{s}/2}(Ω^t)$ and the density to $W_{r,\infty}^{1,1}(Ω^t)$, where ${s}\in(0,1)$, $σ >3/{s}$, $r>5/{s}$, $r>σ $.

Local solutions for nonhomogeneous Navier-Stokes equations with large flux

Abstract

The local existence of solutions to nonhomogeneous Navier-Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev-Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the -axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that -coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to , the gradient of pressure to and the density to , where , , , .
Paper Structure (11 sections, 16 theorems, 375 equations)

This paper contains 11 sections, 16 theorems, 375 equations.

Table of Contents

  1. Introduction
  2. Notation and auxiliary results
  3. Method of successive approximations
  4. Estimates for first derivatives of $\varrho_n$
  5. Estimates for solutions to (\ref{['3.3']})
  6. Analysis of inequality (\ref{['5.9']}): term $\|\varrho_nv_{n+1}\|_{W_{\sigma}^{{s},{s}/2}(\Omega^t)}$
  7. Analysis of inequality (\ref{['5.9']}): term $\|\varrho_nv_n\cdot\nabla v_{n+1}\|_{W_{\sigma}^{{s},{s}/2}(\Omega^t)}$
  8. Analysis of inequality (\ref{['5.9']}): term $\|\varrho_nf\|_{W_{\sigma}^{{s},{s}/2}(\Omega^t)}$
  9. Estimates for $v_{n+1}$ and $p_{n+1}$
  10. A lower bound for $v_{3;n+1}$
  11. Existence of local solution to problem (\ref{['1.1']})

Key Result

Theorem 1.1

Assume Then there exists a local solution $(v,p,\varrho)$ to the nonhomogeneous Navier-Stokes problem (1.1) such that with and Moreover, the density remains bounded and the velocity and the pressure satisfy where data are described by assumptions 1, 2, 3. Finally, the $x_3$-coordinate of velocity is positive

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2: Besov spaces (see BIN, N, G, Tr1)
  • Remark 2.3: Sobolev-Slobodetskii spaces
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6: The Korn inequality, see SS
  • Remark 2.7
  • ...and 30 more