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Weak-Strong Uniqueness for a Rigid Body Immersed in an Inviscid Compressible Fluid

Qianfeng Li, Emil Wiedemann

Abstract

We consider the coupled motion of a free rigid body immersed in an inviscid compressible isentropic fluid. By means of a vanishing viscosity limit, we obtain the local-in-time existence of a dissipative measure-valued solution to the model. Moreover, we establish the weak-strong uniqueness property of the obtained measure-valued solution. To our knowledge, this is the first mathematical result on compressible inviscid fluid-structure interaction. The key novel technique is the construction of a suitable approximation of the test function in the weak formulation of the inviscid system, as the space of test functions depends on the viscosity parameter.

Weak-Strong Uniqueness for a Rigid Body Immersed in an Inviscid Compressible Fluid

Abstract

We consider the coupled motion of a free rigid body immersed in an inviscid compressible isentropic fluid. By means of a vanishing viscosity limit, we obtain the local-in-time existence of a dissipative measure-valued solution to the model. Moreover, we establish the weak-strong uniqueness property of the obtained measure-valued solution. To our knowledge, this is the first mathematical result on compressible inviscid fluid-structure interaction. The key novel technique is the construction of a suitable approximation of the test function in the weak formulation of the inviscid system, as the space of test functions depends on the viscosity parameter.
Paper Structure (12 sections, 13 theorems, 186 equations)

This paper contains 12 sections, 13 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Mathematical description
  3. Literature review and main results
  4. Existence of Young Measure-Valued Solutions
  5. Fundamental Results for Generalized Young Measures
  6. Definition of Dissipative Young Measure-Valued Solutions
  7. Existence of Young Measure-Valued Solutions
  8. Weak-Strong Uniqueness
  9. Change of Coordinates
  10. Proof of Theorem \ref{['thm:Weak-strongUniqueness']} (Relative energy analysis)
  11. Reformulation of the solid's momentum equation
  12. Proof of Lemma \ref{['lem:ConvergeBou1']}

Key Result

Theorem 1.1

Assume that the adiabatic exponent satisfies $\gamma > \tfrac{3}{2}$, and that the domains $\Omega$ and $\mathcal{B}_0$ are regular satisfying for some positive constant $\sigma$. Suppose that the initial data fulfill the following conditions: Define the initial Young measure by Then the compressible inviscid fluid-structure interaction system eq:model–eq:inertia admits a local-in-time dissipat

Theorems & Definitions (24)

  • Remark 1.1
  • Theorem 1.1: Existence of Young Measure-Valued Solutions
  • Remark 1.2
  • Theorem 1.2: Weak-Strong Uniqueness
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 14 more