Carleman estimate for Biot consolidation system in poro‐elasticity and application to inverse problems

Abstract

ABSTRACT. In this paper, we consider a coupled system of mixed hyperbolic-parabolic type which describes the Biot consolidation model in poro-elasticity. We establish a local Carleman estimate for Biot consilidation system. Using this estimate, we prove the uniqueness and a Hölder stability in determining on the one hand a physical parameter arising in connection with secondary consolidation effects λ * and on the other hand the two spatially varying density by a single measurement of solution over ω × (0, T ), where T > 0 is a sufficiently large time and a suitable subbdomain ω satisfying ∂ω ⊃ ∂Ω.

Logarithmic stability of an inverse problem for Biot’s consolidation system in poro-elasticity

“…Here T 0 > is a sufficiently large time. We note that the stability in determining * l and , 1 2 r r in [22] is established for each parameter separately. In this paper, we extend the result further.…”

“…In [21], Bellassoued and Yamamoto established a Hölder stability estimate for the inverse source problem based on a new Carleman estimate for Biot's system. In [22], Bellassoued and Riahi established a Carleman estimate for Biot's system and proved the Lipschitz stability and the uniqueness in determining secondary consolidation effects * l and two spatially varying densities , 1 2 r r by a single measurement of solution over T T , w´-( ) and a suitable subdomain ω satisfying w ¶W Ì ¶ . Here T 0 > is a sufficiently large time.…”

Lipschitz stability in determination of coefficients in a two-dimensional Boussinesq system by arbitrary boundary observation

“…They not only give quantitative results of single extension but they are also used for the study of inverse problems as well as in control theory for PDEs. Moreover the Carleman estimates and their applications to inverse problems are considered by many authors and there are many works that are relevant to this topic [1,2,3]. We list some works for the well-known equations in mathematical physics.…”

“…Then, in a thermoelastic model, Wu and Liu [32] study the inverse problem of determining two spatially varying coefficients by proving the Lipschitz stability and the uniqueness for this inverse problem and based on Carleman estimates. In addition Bellassoued and Riahi in [3] proved the uniqueness and a Hölder stability in determining spatially varying coefficients for coupled system of mixed hyperbolicparabolic type, which describes the Biot consolidation model in poro-elasticity. For elasticity, we refer to Isakov and Kim in [22] and Imanuvilov, Isakov and Yamamoto in [20].…”

Iterative methods for solving a poroelastic shell model of Naghdi's type

“…All these models are two-dimensional models (defined on a two-dimensional domain) and describe behavior of three-dimensional elastic bodies that are thin in one direction.In geomechanics, there are many examples of elastic bodies which are saturated by a fluid, see e.g., references [14,15]. Their overall behavior can be modeled by the quasi-static Biot's system of partial differential equations, see reference [1] or [16]. It couples the equations of linearized elasticity, containing the pressure gradient of the flow, with the mass conservation equation involving the fluid…”