Abstract
In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim [11] and under the assumption that the velocity of a given reference solution stays far from $0$ on a part of the boundary where Robin conditions are prescribed.
Highlights
Let us consider an open Lipschitz bounded connected domain Ω of Rd, d ≥ 2
We prove an identifiability result and a logarithmic stability estimate for the Stokes equations with Robin boundary conditions 1.1 under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed
We have proved, under some regularity assumptions on the open set Ω and on the solution (u, p) of system 1.1, logarithmic stability estimates for the Stokes system with mixed Neumann and Robin boundary conditions
Summary
Let us consider an open Lipschitz bounded connected domain Ω of Rd, d ≥ 2. In [5], the authors use a result proved by K.D. Phung in [21] to obtain a logarithmic stability estimate which is valid in any dimension for an open set Ω of class C∞. Phung in [21] to obtain a logarithmic stability estimate which is valid in any dimension for an open set Ω of class C∞ This result has been generalized in [8] and [9] to C1,1 and Lipschitz domains. Under some regularity assumptions and using the theorem of unique continuation for the Stokes equations proved in [18], we prove that if two measurements of the velocity are equal on (0, T ) × Γ, where Γ ⊆ Γe is a non-empty open subset of the boundary, the two corresponding Robin coefficients are equal on Γ0. We are going to start with some preliminary results which will be useful in the subsequent sections
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