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Global motions for nonhomogeneous Navier-Stokes equations with large flux

Joanna Rencławowicz, Wojciech M. Zajączkowski

Abstract

The nonhomogeneous Navier-Stokes equations are considered in a cylindrical domain in ${\mathbb R}^3$, parallel to the $x_3$-axis with large inflow and outflow on the top and the bottom. Moreover, on the lateral part of the cylinder the slip boundary conditions are assumed. The global existence of regular solutions is proved under assumptions that inflow and outflow are close to homogeneous and norms of derivative with respect to $x_3$ of the external force and initial velocity are sufficiently small. The key point of this paper is to verify that $x_3$-coordinate of velocity remains positive.

Global motions for nonhomogeneous Navier-Stokes equations with large flux

Abstract

The nonhomogeneous Navier-Stokes equations are considered in a cylindrical domain in , parallel to the -axis with large inflow and outflow on the top and the bottom. Moreover, on the lateral part of the cylinder the slip boundary conditions are assumed. The global existence of regular solutions is proved under assumptions that inflow and outflow are close to homogeneous and norms of derivative with respect to of the external force and initial velocity are sufficiently small. The key point of this paper is to verify that -coordinate of velocity remains positive.
Paper Structure (10 sections, 30 theorems, 437 equations)

This paper contains 10 sections, 30 theorems, 437 equations.

Table of Contents

  1. Introduction
  2. Notation and auxiliary results
  3. Energy estimate for solutions to (\ref{['1.1']})
  4. Energy estimate for solutions to problem (\ref{['2.6']})
  5. A priori estimates for first derivatives of $\varrho$
  6. A lower bound for $v_3$
  7. Estimates for $\chi$ - the third component of vorticity
  8. Auxiliary results
  9. Increasing regularity for velocity and pressure
  10. Global estimate

Key Result

Theorem 1.1

(local existence, see RZ3) Assume Then there exists a local solution $(v,p,\varrho)$ to the nonhomogeneous Navier-Stokes problem (1.1) such that Moreover, the density remains bounded and the velocity and the pressure satisfy where data are described by assumptions 1,2 and 3, $\phi$ is an increasing positive function, $t\le T$ and $T$ is sufficiently small. Finally, the $x_3$-coordinate of velo

Theorems & Definitions (65)

  • Theorem 1.1
  • Remark 1.2
  • proof
  • proof
  • proof
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • ...and 55 more