Abstract
In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.
Highlights
Introduction and main resultsFor a nonempty bounded open subset Ω of RN (N = 2 or N = 3), we consider a pair velocity-pressure (v, p) ∈ H2(Ω) × H1(Ω) solution of the following linearized Navier-Stokes equations:−ν∆v + (z1 · ∇) v + (v · ∇) z2 + ∇p = f in Ω, div v = d in Ω. (1.1)Above and in the following, ν > 0 is a constant which represents the kinematic viscosity of the fluid, f ∈ L2(Ω), d ∈ H1(Ω) and z1 ∈ L∞(Ω) and z2 ∈ W1,r(Ω) with r > 2 if N = 2, r = 3 if N = 3. (1.2)In the following, z1 and z2 will be two solutions of the Navier-Stokes equations in Ω
The main aim of this section is to prove a Carleman inequality for the non homogeneous Oseen equations
We recall that Ω is a nonempty bounded open subset of RN (N = 2 or N = 3) with a boundary ∂Ω of class C2, that Γobs is a nonempty open subset of ∂Ω and that ω is a nonempty open subset of Ω
Summary
We stress the fact that the previous estimate (1.17) depends on the solutions z1 and z2 through the choice of the compact set K and the constant m To complete this result, it would be interesting to obtain a quantitative estimate of the vanishing rate of z, like what is done in [4] in the case of the Laplace equation.
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