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Martingale solutions in stochastic fluid-structure interaction

Dominic Breit, Prince Romeo Mensah, Thamsanqa Castern Moyo

Abstract

We consider a viscous incompressible fluid interacting with a linearly elastic shell of Koiter type which is located at some part of the boundary. Recently models with stochastic perturbation in the shell equation have been proposed in the literature but only analysed in simplified cases. We investigate the full model with transport noise, where (a part of) the boundary of the fluid domain is randomly moving in time. We prove the existence of a weak martingale solution to the underlying system.

Martingale solutions in stochastic fluid-structure interaction

Abstract

We consider a viscous incompressible fluid interacting with a linearly elastic shell of Koiter type which is located at some part of the boundary. Recently models with stochastic perturbation in the shell equation have been proposed in the literature but only analysed in simplified cases. We investigate the full model with transport noise, where (a part of) the boundary of the fluid domain is randomly moving in time. We prove the existence of a weak martingale solution to the underlying system.
Paper Structure (22 sections, 15 theorems, 162 equations)

This paper contains 22 sections, 15 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Deterministic models
  3. Stochastic models
  4. The weak formulation
  5. Plan
  6. Mathematical framework and the main result
  7. Stratonovich integrals
  8. Geometric setup
  9. Solenoidal extension
  10. Weak martingale solutions
  11. The linearised problem
  12. The linearised Galerkin problem
  13. Tightness of $\partial_t\eta^N$
  14. Stochastic compactness
  15. Pathwise uniqueness
  16. ...and 7 more sections

Key Result

Theorem 1

Let $X,Z$ be two Banach spaces, such that $X'\subset Z'$. Assume that $f_n:I\to X$ and $g_n: I\to X'$, such that $g_n\in L^\infty(I;Z')$ uniformly. Moreover assume the following: Then there is a subsequence, such that

Theorems & Definitions (21)

  • Remark 1: The 2D case
  • Definition 1
  • Theorem 1
  • Proposition 2
  • Corollary 3
  • Lemma 4
  • Definition 2: Weak martingale solution
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 11 more