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On the local existence of solutions to the Navier-Stokes-wave system with a free interface

Igor Kukavica, Linfeng Li, Amjad Tuffaha

Abstract

We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space $H^{2+ε}$ and the initial structure velocity is in $H^{1.5+ε}$ , where $ε\in (0, 1/2)$.

On the local existence of solutions to the Navier-Stokes-wave system with a free interface

Abstract

We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space and the initial structure velocity is in , where .
Paper Structure (11 sections, 16 theorems, 299 equations)

This paper contains 11 sections, 16 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. The model and main results
  3. Space-time trace, interpolation, and hidden regularity inequalities
  4. The nonhomogeneous parabolic problem
  5. Solution to a parabolic-wave system
  6. Uniform boundedness of the iterative sequence
  7. Contracting property
  8. Solution to the Navier-Stokes-wave system
  9. The Lagrangian flow map, Jacobian matrix, and density estimates
  10. Uniform boundedness of the iterative sequence
  11. Contracting property

Key Result

Theorem 2.1

Let $s\in (2, 2+ \epsilon_0]$ for $\epsilon_0 \in (0,1/2)$. Assume that $R_0 \in H^s (\Omega_{\text{f}})$, $R_0^{-1} \in H^s (\Omega_{\text{f}})$, $w_1 \in H^{s-1/2} (\Omega_{\text{e}})$, $v_0 \in H^s (\Omega_{\text{f}})$, $v_0 |_{\Gamma_{\text{c}}} \in H^{s+1/2} (\Gamma_{\text{c}})$, $\partial_3 v_ for $j=1,2,3$. Then the system EQ260--EQ23 with the coupling conditions EQ262--EQ263, boundary cond

Theorems & Definitions (29)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • Corollary 3.2
  • proof : Proof of Lemma \ref{['L06']}
  • Lemma 3.3
  • Corollary 3.4
  • proof : Proof of Lemma \ref{['L12']}
  • Lemma 3.5: LLT
  • Lemma 3.6: RV
  • ...and 19 more