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A regularity criterion for the angular component of velocity in the norm $L_\infty(0,T;L_p(Ω)),\;\frac 3 p <1$ in axisymmetric Navier Stokes equations in a cylinder

Wiesław J. Grygierzec, Wojciech M. Zajączkowski

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $Ω\subset\R^3$. We assume that $v_r$, $v_\varphi$, $ω_\varphi$ vanish on the lateral part of boundary $\partialΩ$ of the cylinder, and that $v_z$, $ω_\varphi$, $\partial_zv_\varphi$ vanish on the top and bottom parts of the boundary $\partialΩ$, where we used standard cylindrical coordinates, and we denoted by $ω=\curl v$ the vorticity field. We use $H^3$ Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Using the estimate \[ \|v_\varphi\|_{L_\infty(0,T;L_p)Ω)}\les A, \] where A is a given number and $p>3$ we prove the existence of global regular axially-symmetric solutions.

A regularity criterion for the angular component of velocity in the norm $L_\infty(0,T;L_p(Ω)),\;\frac 3 p <1$ in axisymmetric Navier Stokes equations in a cylinder

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder . We assume that , , vanish on the lateral part of boundary of the cylinder, and that , , vanish on the top and bottom parts of the boundary , where we used standard cylindrical coordinates, and we denoted by the vorticity field. We use Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Using the estimate where A is a given number and we prove the existence of global regular axially-symmetric solutions.

Paper Structure

This paper contains 14 sections, 24 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation 2
  4. Inequalities
  5. Basic estimates
  6. Estimates for the modified stream function $\psi_1$
  7. Weighted Sobolev estimates for $\psi_1$
  8. Elliptic estimates for the modified stream function $\psi_1$
  9. Energy estimates for $\Phi$ and $\Gamma$
  10. Estimates for swirl $u=rv_\varphi$
  11. Order reduction estimates
  12. Global estimate for regular solutions
  13. Preliminaries
  14. Proof of (\ref{['7.1']})

Key Result

Theorem 1.2

Let $v$ be a smooth solution to (1.1)--(1.3) on time interval $(0,T)$. Let the expansions (1.21) near the axis of symmetry hold. Let the quantities from Notation n1.1 be finite for $t\in(0,T)$. Let (1.23') hold with ${\sigma}>3$. Then there exists a function $\phi=\phi(A,D_1,\dots,D_7)$ such that where $\phi$ denotes an increasing positive function.

Theorems & Definitions (39)

  • Theorem 1.2
  • proof
  • Theorem 1.3
  • Lemma 2.1: Hardy inequality, see Lemma 2.16 in BIN
  • Lemma 2.2: Sobolev interpolation, see Sect. 15 in BIN
  • Lemma 2.3: Hardy interpolation, see Lemma 2.4 in CFZ
  • Lemma 2.4: see Lemma 2.2 in Z1Z2
  • proof
  • Lemma 2.5: Maximum principle for the swirl
  • proof
  • ...and 29 more