Inverse problems for advection diffusion equations in admissible geometries

Abstract

We study inverse boundary problems for the advection diffusion equation on an admissible manifold, i.e. a compact Riemannian manifold with boundary of dimension ≥ 3, which is conformally embedded in a product of the Euclidean real line and a simple manifold. We prove the unique identifiability of the advection term of class H 1 ∩ L ∞ and of class H 2/3 ∩ C 0,1/3 from the knowledge of the associated Dirichlet-to-Neumann map on the boundary of the manifold. This seems to be the first global identifiability result for possibly discontinuous advection terms.Here and in what follows H s (M 0 ), s ∈ R, is the standard Sobolev space on M 0 , see [28, Chapter 4], and M 0 = M \ ∂M stands for the interior of M. We also let ν be the unit outer normal to the boundary of M. We shall define the trace of the normal derivative ∂ ν u ∈ H −1/2 (∂M) as follows. Let ϕ ∈ H 1/2 (∂M). Then letting v ∈ H 1 (M 0 ) be a continuous extension of ϕ, we set ∂ ν u, ϕ H −1/2 (∂M )×H 1/2 (∂M ) = M ∇u, ∇v dV + M X(u)vdV, (1.2) 1

Global Identifiability of Low Regularity Fluid Parameters in Acoustic Tomography of Moving Fluid

“…Remark. The assumption A ∈ (H 1 ∩ L ∞ )(Ω; C n ) in Theorem 1.1 corresponds to the optimal space on the scale of spaces H s ∩ L ∞ , s ≥ 0, for which the inverse boundary problem for the operator L A,q can be solved by means of the techniques of L 2 Carleman estimates.This could be seen in particular from the estimate (3.31) which is of purely qualitative nature, see also the discussion in [23].…”

“…and therefore, C V,W, q 2 (B) = C A 1 ,q 1 (B). Similarly to [23] this equality of the sets of the Cauchy data implies that the following integral identity holds,…”

“…The purpose of this paper is to weaken the regularity assumptions (1.5) on the fluid parameters c, v, ρ, and α in the results of [1], [3], and in particular, to allow some parameters to be discontinuous. In doing so, motivated by the recent work [23], we shall impose regularity assumptions on the fluid parameters, weaker than those in (1.5), implying that A(ω) ∈ (H 1 ∩ L ∞ )(Ω; C n ) and q(ω) ∈ L ∞ (Ω; C). In this case the reduction to the magnetic Schrödinger operator (1.6) is not useful since q(ω) given by (1.7) is no longer in L ∞ (Ω) and we only have q(ω) ∈ L 2 (Ω).…”

“…The paper is organized as follows. Section 2 is devoted to the construction of complex geometric optics solutions for our equations, relying on the techniques developed in [19], [21], [23], and [35]. Theorem 1.1 together with Corollary 1.2 are established in Section 3, and Section 4 is concerned with the proofs of Corollary 1.4 and Corollary 1.5, following the approach of [3] and noticing that it still works in our low regularity setting.…”

The Attenuated Geodesic Ray Transform on Tensors: Generic Injectivity and Stability

“…In [11], unique determination of the conductivity from the boundary measurements was reduced to injectivity of I a . In a similar way the latter is related to inverse problems for other elliptic equations and systems [4,19,20,21] including nonlinear ones [5]. The transform I a arises in several problems as well.…”