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Prandtl-Batchelor flows with a point vortex on a disk

Zhi Chen, Mingwen Fei, Zhiwu Lin, Jianfeng Zhao

Abstract

In this paper we aim to construct a very weak solution to the steady two-dimensional Navier-Stokes equations which is affected by an external force induced by a point vortex on the unit disk. Such a solution is also the form of Prandtl-Batchelor type, i.e. the vorticity in the limit of vanishing viscosity is constant in an inner region separated from the boundary layer. Multi-scale asymptotic analysis which will capture well the singular behavior of the solution is used to construct higher order approximate solutions of the Navier-Stokes equations firstly, and then stability analysis is performed for the error system, finally the existence of Prandtl-Batchelor flows with a point vortex on a disk with the wall velocity slightly different from the rigid-rotation is proved. To overcome the singularity at the origin from the point vortex and strong singular behaviors near the boundary from viscous flow, we introduce a coordinate transformation which converts a bounded singularity problem to an unbounded one, and use a new decomposition for the error and perform basic energy estimate and positivity estimate for each part of the decomposition separately. The equation in the vorticity form shall be used to deal with $L^\infty$ estimate of the vorticity error.

Prandtl-Batchelor flows with a point vortex on a disk

Abstract

In this paper we aim to construct a very weak solution to the steady two-dimensional Navier-Stokes equations which is affected by an external force induced by a point vortex on the unit disk. Such a solution is also the form of Prandtl-Batchelor type, i.e. the vorticity in the limit of vanishing viscosity is constant in an inner region separated from the boundary layer. Multi-scale asymptotic analysis which will capture well the singular behavior of the solution is used to construct higher order approximate solutions of the Navier-Stokes equations firstly, and then stability analysis is performed for the error system, finally the existence of Prandtl-Batchelor flows with a point vortex on a disk with the wall velocity slightly different from the rigid-rotation is proved. To overcome the singularity at the origin from the point vortex and strong singular behaviors near the boundary from viscous flow, we introduce a coordinate transformation which converts a bounded singularity problem to an unbounded one, and use a new decomposition for the error and perform basic energy estimate and positivity estimate for each part of the decomposition separately. The equation in the vorticity form shall be used to deal with estimate of the vorticity error.

Paper Structure

This paper contains 24 sections, 23 theorems, 361 equations.

Table of Contents

  1. Introduction
  2. Construction of approximate solutions
  3. Euler expansions for leading order and 1-order
  4. Equations for $(u_e^{(0)},v_e^{(0)},p_e^{(0)})$
  5. Equations for $(u_e^{(1)},v_e^{(1)},p_e^{(1)})$
  6. Prandtl expansions near $r=1$ for $\varepsilon^{-1}$-order, leading order and 1-order
  7. Equations for $(v_p^{(0)}, p_p^{(0)})$
  8. Equations for $(u_p^{(0)},v_p^{(1)},p_p^{(1)})$
  9. Equations for $(u_p^{(1)},v_p^{(2)})$
  10. Solvabilities of leading order, 1-order Prandtl equations and 1-order Euler equations
  11. Linearized Euler equations for $(u_e^{(2)}, v_e^{(2)}, p_e^{(2)})$ and their solvabilities
  12. Linearized Prandtl equations for $(u_p^{(2)},v_p^{(3)})$ and their solvabilities
  13. Approximate solutions
  14. Linear stability estimates of error equations
  15. Error equations
  16. ...and 9 more sections

Key Result

Theorem 1.1

Assume that $\varpi(\theta)$ is a $2\pi$-periodic smooth function which satisfies $\int_0^{2\pi}\varpi(\theta)d\theta=0$, then there exist $\varepsilon_0>0, \eta_0>0$ such that for any $\varepsilon\in (0,\varepsilon_0), \eta\in (0,\eta_0)$, the Navier-Stokes equations (NS-curvilnear) with $\alpha>0$ where $(u_e(r),0)$ is the Couette flow (Couette flow) with constant choose, and $u_p^{(0)}$ is the

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 34 more