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Existence of martingale solutions to a nonlinearly coupled stochastic fluid-structure interaction problem

Krutika Tawri, Suncica Canic

Abstract

In this paper we study a nonlinear stochastic fluid-structure interaction problem with a multiplicative, white-in-time noise. The problem consists of the Navier-Stokes equations describing the flow of an incompressible, viscous fluid in a 2D cylinder interacting with an elastic lateral wall whose elastodynamics is described by a membrane/shell equation. The flow is driven by the inlet and outlet data, and by the stochastic forcing. The stochastic noise is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic conditions assumed at the moving interface, which is a random variable not known a priori. The geometric nonlinearity due to the nonlinear coupling requires the development of new techniques to capture martingale solutions for this class of stochastic fluid-structure interaction problems. We introduce a constructive approach based on a Lie splitting scheme and prove the existence of martingale solutions to the system. To the best of our knowledge, this is the first result in the field of stochastic PDEs that addresses existence of solutions on moving fluid domains involving incompressible viscous fluids, where the displacement of the boundary and the fluid domain are random variables that are not known a priori and are parts of the solution itself.

Existence of martingale solutions to a nonlinearly coupled stochastic fluid-structure interaction problem

Abstract

In this paper we study a nonlinear stochastic fluid-structure interaction problem with a multiplicative, white-in-time noise. The problem consists of the Navier-Stokes equations describing the flow of an incompressible, viscous fluid in a 2D cylinder interacting with an elastic lateral wall whose elastodynamics is described by a membrane/shell equation. The flow is driven by the inlet and outlet data, and by the stochastic forcing. The stochastic noise is applied both to the fluid equations as a volumetric body force, and to the structure as an external forcing to the deformable fluid boundary. The fluid and the structure are nonlinearly coupled via the kinematic and dynamic conditions assumed at the moving interface, which is a random variable not known a priori. The geometric nonlinearity due to the nonlinear coupling requires the development of new techniques to capture martingale solutions for this class of stochastic fluid-structure interaction problems. We introduce a constructive approach based on a Lie splitting scheme and prove the existence of martingale solutions to the system. To the best of our knowledge, this is the first result in the field of stochastic PDEs that addresses existence of solutions on moving fluid domains involving incompressible viscous fluids, where the displacement of the boundary and the fluid domain are random variables that are not known a priori and are parts of the solution itself.
Paper Structure (13 sections, 16 theorems, 185 equations, 3 figures)

This paper contains 13 sections, 16 theorems, 185 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. The deterministic model and a weak formulation
  4. The stochastic framework and definition of martingale solutions
  5. ALE mappings
  6. Definition of martingale solutions
  7. Operator splitting scheme
  8. Definition of the splitting scheme
  9. Approximate solutions
  10. Passing to the limit as $N\rightarrow\infty$
  11. Tightness results
  12. Almost sure convergence
  13. Passing to the limit as $\varepsilon \rightarrow 0$

Key Result

Lemma 3.1

Assume that $\eta^n$ and $v^{n}$ are $H^2_0(0,L)$ and $L^2(0,L)$-valued $\mathcal{F}_{t^n}$-measurable random variables, respectively. Then there exist $H^2_0(0,L)$- valued $\mathcal{F}_{t^n}$-measurable random variables $\eta^{n+\frac{1}{2}},v^{n+\frac{1}{2}}$ that solve first, and the following se where corresponds to numerical dissipation.

Figures (3)

  • Figure 1: A sketch of the fluid domain $\mathcal{O}_{\eta}(t)$ with the elastic lateral boundary $\Gamma_\eta(t)$, the inlet and outlet boundaries $\Gamma_{in}$ and $\Gamma_{out}$, and the bottom (symmetry) boundary $\Gamma_b$. The lightly shaded region represents a confidence interval of where the structure is likely to be.
  • Figure 2: A realization of the vertical squeezing and the mollification operator.
  • Figure 3: Distance between the curves $R+\eta^n_*$ and $\frac{R+{\eta^k_*}}{\sigma}$.

Theorems & Definitions (28)

  • Definition 1: Martingale solution
  • Remark 1
  • Lemma 3.1: Existence for the structure subproblem
  • proof
  • Lemma 3.2: Existence for the fluid subproblem
  • proof
  • Theorem 3.3: Uniform Estimates
  • proof
  • Lemma 3.4
  • proof
  • ...and 18 more