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Global existence and convergence near equilibrium for the moving interface problem between Navier-Stokes and the linear wave equation

Daniel Coutand

Abstract

We first establish existence for all positive time near equilibrium for the moving interface problem between the Navier-Stokes equations for the evolving fluid phase (moved by the fluid velocity) and an elastic body modelled by the linear wave equation. This problem has an infinite number of simple solutions with a flat interface (with zero velocity in the fluid, and zero horizontal velocity in the solid), that we call flat interface solutions. We then show that if the initial data is close enough to the canonical equilibrium, the solution converges towards a flat interface solution in large time, showing that these flat interface solutions capture the long time behaviour of this fluid-structure problem near the canonical equilibrium. This result is established with gravity (which can be set to zero or not). It is established for the case where the solid has initial volume close to the volume of its reference configuration (where the linear wave equation is naturally written).

Global existence and convergence near equilibrium for the moving interface problem between Navier-Stokes and the linear wave equation

Abstract

We first establish existence for all positive time near equilibrium for the moving interface problem between the Navier-Stokes equations for the evolving fluid phase (moved by the fluid velocity) and an elastic body modelled by the linear wave equation. This problem has an infinite number of simple solutions with a flat interface (with zero velocity in the fluid, and zero horizontal velocity in the solid), that we call flat interface solutions. We then show that if the initial data is close enough to the canonical equilibrium, the solution converges towards a flat interface solution in large time, showing that these flat interface solutions capture the long time behaviour of this fluid-structure problem near the canonical equilibrium. This result is established with gravity (which can be set to zero or not). It is established for the case where the solid has initial volume close to the volume of its reference configuration (where the linear wave equation is naturally written).
Paper Structure (39 sections, 289 equations, 1 figure)

This paper contains 39 sections, 289 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Notation for the gradient vector
  4. Notation for partial differentiation and the Einstein summation convention
  5. Tangential (or horizontal) derivatives
  6. Sobolev spaces
  7. Sobolev spaces on $\Gamma$, or on any horizontal plane
  8. The unit normal vectors
  9. Conventions on constants and polynomials
  10. Cofactor matrix and Piola's identity
  11. Description of the Navier-Stokes-linear wave interaction problem, special solutions with a flat interface
  12. description of the problem
  13. The canonical equilibrium of height $h_s$
  14. A general class of special solutions with flat interface of height $h_e$
  15. Arbitrary Lagrangian formulation of the problem and statement of our results
  16. ...and 24 more sections

Figures (1)

  • Figure :