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Inverse problems for one-dimensional fluid-solid interaction models

J. Apraiz, A. Doubova, E. Fernández-Cara, M. Yamamoto

Abstract

We consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss on the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one end point of the spatial interval. In particular, we establish uniqueness results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities.

Inverse problems for one-dimensional fluid-solid interaction models

Abstract

We consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss on the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one end point of the spatial interval. In particular, we establish uniqueness results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities.
Paper Structure (4 sections, 7 theorems, 82 equations, 4 figures)

This paper contains 4 sections, 7 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lateral estimates and uniqueness
  4. Global estimates and uniqueness

Key Result

Lemma 2.1

Let us assume that with $a, b\in L^{\infty}(Q_{\ell}(p))$, $u\in H^2(Q_{\ell}(p))$ and there exist constants $M>0$ and $\delta \in (0,1)$ such that Then:

Figures (4)

  • Figure 1: The set $Q(\eta)$ when $\bar{t} \leq T/2$.
  • Figure 2: The curves $\psi = \eta$, $\psi = 2\eta$ and $\psi = 3\eta$.
  • Figure 3: The sets $Q(2\eta)$, $Q(3\eta)$ and $Q(4\eta)$.
  • Figure 4: The set $Q(\eta)$ when $\bar{t} > T/2$.

Theorems & Definitions (11)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1: Local stability for the lateral inverse problem
  • Remark 3.2
  • Corollary 3.3
  • Corollary 3.4: Lateral uniqueness
  • Corollary 3.5
  • Remark 3.6
  • Theorem 4.1: Conditional stability
  • ...and 1 more