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Viscous evolution of a point vortex in a half-plane

Anne-Laure Dalibard, Thierry Gallay

Abstract

As a model for vortex-wall interactions, we consider the two-dimensional incompressible Navier--Stokes equations in the half-plane $R^2_+$ with no-slip boundary condition and point vortices as initial data. We focus on the paradigmatic example of a single vortex in an otherwise stagnant fluid, which is already quite challenging from a mathematical point of view. We prove that this system has a unique global solution for all values of the Reynolds number $|Γ|/ν$, where $Γ$ is the circulation of the vortex and $ν$ the kinematic viscosity of the fluid. The solution we construct has finite energy for positive times and converges to zero in energy norm as $t \to +\infty$. Uniqueness holds under the assumption that the solution is close to a Lamb--Oseen vortex for small times. To our knowledge, all previous results in domains with boundaries assume that the initial vorticity has small or zero atomic part. In our particular situation, we remove the smallness condition by decomposing the solution into a vortex and a boundary layer term, so that we can apply the techniques developed in the whole plane $R^2$ to avoid the difficulties related to the large circulation of the vortex.

Viscous evolution of a point vortex in a half-plane

Abstract

As a model for vortex-wall interactions, we consider the two-dimensional incompressible Navier--Stokes equations in the half-plane with no-slip boundary condition and point vortices as initial data. We focus on the paradigmatic example of a single vortex in an otherwise stagnant fluid, which is already quite challenging from a mathematical point of view. We prove that this system has a unique global solution for all values of the Reynolds number , where is the circulation of the vortex and the kinematic viscosity of the fluid. The solution we construct has finite energy for positive times and converges to zero in energy norm as . Uniqueness holds under the assumption that the solution is close to a Lamb--Oseen vortex for small times. To our knowledge, all previous results in domains with boundaries assume that the initial vorticity has small or zero atomic part. In our particular situation, we remove the smallness condition by decomposing the solution into a vortex and a boundary layer term, so that we can apply the techniques developed in the whole plane to avoid the difficulties related to the large circulation of the vortex.
Paper Structure (24 sections, 31 theorems, 255 equations)

This paper contains 24 sections, 31 theorems, 255 equations.

Table of Contents

  1. Introduction and main results
  2. Formulation of the problem
  3. The Stokes semigroup in vorticity formulation
  4. Statement of the main results
  5. Sketch of the proof of the main result
  6. Perspectives
  7. Outline of the paper
  8. The Stokes semigroup and the Biot--Savart law
  9. Estimates of the boundary term
  10. Smoothing properties
  11. Continuity properties
  12. Estimates on the Biot--Savart law
  13. The linearized evolution at the Lamb--Oseen vortex
  14. Existence and uniqueness theory
  15. Integral equation and decomposition of the solution
  16. ...and 9 more sections

Key Result

Proposition 1.1

(Properties of the Stokes semigroup) 1) The family $\bigl(S(t)\bigr)_{t \ge 0}$ defined by Stokes is a $C_0$-semigroup in the space $L^1_\perp(\mathbb{R}^2_+)$. 2) If $\omega_0 \in L^1(\mathbb{R}^2)$ the maps $t \mapsto S(t)\omega_0$ and $t \mapsto S(t)\nabla\omega_0$ belong to $C^0((0,\infty),L^p(\

Theorems & Definitions (69)

  • Proposition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Lemma 2.1
  • Remark 2.2
  • proof
  • ...and 59 more