Carleman estimates and some inverse problems for the coupled quantitative thermoacoustic equations by boundary data. Part I: Carleman estimates

Abstract

In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function φ ⁢ ( x , t ) = e λ ⁢ ψ ⁢ ( x , t ) , ψ ⁢ ( x , t ) = | x - x 0 | 2 - β ⁢ | t - t 0 | 2 + β ⁢ t 0 2 \varphi(x,t)={\mathrm{e}}^{\lambda\psi(x,t)},\quad\psi(x,t)=\lvert x-x_{0}% \rvert^{2}-\beta\lvert t-t_{0}\rvert^{2}+\beta t_{0}^{2} for x in a bounded domain in ℝ n {\mathbb{R}^{n}} with C 3 {C^{3}} -boundary and t ∈ ( 0 , T ) {t\in(0,T)} , where t 0 = T 2 {t_{0}=\frac{T}{2}} . We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding part II paper [M. Cristofol, S. Li and Y. Shang, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. Part II: Inverse problems, preprint 2020, https://hal.archives-ouvertes.fr/hal-02863385].

Carleman estimates and some inverse problems for the coupled quantitative thermoacoustic equations by partial boundary layer data. Part II: Some inverse problems

“…All the technical conditions listed below are required in the proof of the Carleman estimate established in Theorem 1.1 of the part I ([5]). We introduce the set $$$ \mathcal{U} $$$ which is concerned with the coefficients $$$ {a}_j(x),j=1 $$$, 2, 3, 4: …”

“…Now we recall the following Carleman estimate established in [5] by Li and Shang with the notation $$$ {Q}_{\omega }=\omega \times \left(0,T\right) $$$.…”

“…Let T > 0 be given and denote t 0 = T 2 . All the technical conditions listed below are required in the proof of the Carleman estimate established in Theorem 1.1 of the part I ( [5]). We introduce the set  which is concerned with the coefficients a 𝑗 (x), 𝑗 = 1, 2, 3, 4:…”

“…The first one concerns the new Carleman estimates that we need to prove for such systems. We aim to solve our different inverse problems using the observation of only one component, the pressure $$$ P $$$, or the temperature $$$ \theta $$$ by adapting the Carleman estimate proved in part I, Theorem 1.1 in [5]. The second series of difficulties appear when we address the reconstruction of the coefficients and particularly the case of the thermal conductivity which is in the principal part of the system.…”