Neumann–Poincaré operator - Wikipedia
The system is.
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Here the function f is the control function which acts over a fixed small nonempty open subset w with characteristic function 1 w. See Figure 5.
The existence of solutions and regularity for this system has been recently studied in several papers see ,  and the references therein. The controllability result is the following, saying that it is possible to drive the structure and the fluid at rest and the immersed solid up to its reference position in arbitrarily small time with a localized control f , provided the initial conditions are sufficiently small. Theorem 3. Suppose that: i the initial body solid shape satisfies.
The last condition in 12 is a symmetry restriction over the shape of the solid needed in the proof of the Carleman inequality satisfied by the adjoint problem of the linearized system to get estimates on the structure motion from estimates on the fluid velocity on the interface. The result only holds for small initial data because we want to keep the non-collision condition on the whole interval 0, T.
The first result of this kind using global Carleman estimates was obtained in  for a one-dimensional Burgers-particle system studied in . Also, similar results to the one presented here has been simultaneously and independently obtained in the preprint  see the preprint version of . The idea of the proof is the following and follows ideas for the controllability of the Navier-Stokes equations recently used in ,  and the ideas of . We first consider a linearized problem.
We define the angle associated to the rotation velocity defined up to a constant. We also consider a given velocity satisfying regularity properties and compatibility conditions with ,. We prove a null controllability result for the linearized problem with the help of a Carleman inequality shown on the adjoint system associated to a linearized system. Finally, Theorem 3. We will give the Carleman inequality here for two reasons.
First, once Navier-Stokes equations are involved, the corresponding global Carleman inequality is different from 6 because of the pressure or because of incompressibility. Indeed, the exponential weights appearing at the left and right hand sides of the inequality are no more the same. Secondly, since we are working with variable domains, this is in fact a Carleman inequality in moving domains. Let us first introduce the corresponding adjoint operator on this case given by the solution v, q, b , g of the linear system. The following extremal weights appear when eliminating the explicit dependence on the pressure q of the Carleman inequality.
For a given field v x, t we take the notation:. The Carleman inequality is expressed on the moving domains S t and F t and the transport theorem is used on its deduction. More precisely, the weight F used here is of the form compare with 3. The time dependent weight function y x, t is chosen as the standard weight for the heat equation but it follows the shape of S t. This concept come up from multipliers technique and controllability , .
In order to solve the Dirichlet to Neumann one measurement inverse problem, it suffices to measure on a rotated exit part of the boundary G r. More precisely. There are of course an infinite number of intermediate cases depending on the localization of x 0 , the direction of a and the angle q. Remark also that the rotated exit condition is a particular case of the geometrical optics BLR condition , .
Theorem 4. The proof of this Theorem is based on a global Carleman estimate using the variant of the Carleman weight for the wave equation compare with 4. The main steps in the deduction of such inequality are taken from [35, 23] following a well known technique due to Bukhgeim and Klibanov , . Roughly speaking, the technique in this case consists in reducing the problem to a source inverse problem for the perturbed equation around u and then take its time derivative in order to obtain a quasi-observability inequality after use of the Carleman inequality.
Quasi, since the initial condition is bounded not only by the observation but also by the unknown source term, but a sufficiently large time and the properties of the Carleman weight allow to get rid of this source term. Other completely different problem is the case when you have Dirichlet to Newmann map measurements. But recently, it has been shown that you can solve the one measurement problem for the wave equation with an arbitrarily boundary measurement region in the case of Neumann boundary conditions and Dirichlet measurements , but the corresponding inequality analogous to 22 is logarithmic .
Notice that recently, Global Carleman estimates and application to one measurement inverse problems for the wave equation were obtained in the case of variable but still regular coefficients , .
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The inverse problem of retrieving coefficients from a wave equation with discontinuous coefficients from boundary measurements arise naturally in geophysics and more precisely, in seismic prospection of earth inner layers . To fix ideas we assume that W 1 is simply connected.
We set:. The following inverse stability result holds see the preprint  :. Theorem 5. This Theorem is proved by combining the Carleman inequality for the wave equation with discontinuous coefficients proved in  and the method of Bukhgeim-Klibanov explained in section 4. To this end, system 27 is viewed as two wave equations with constant coefficients coupled with transmission conditions see . Then, a global Carleman inequality is found out for this transmission problem by working with variants of Carleman weights of the form compare with 4.
We also combine the Carleman inequalities obtained from two different interior points as we did in section 2, see also Figure 1 , left. The convexity hypothesis on W 1 comes from the fact that the positiveness of the Hessian of the weight F is related with the curvature of G 1 with respect to x 0. There are a lot of important works concerning this inverse problem in the case that a wide class of measurements are available.
In these cases, microlocal analysis has been used and it gives positive answer to the problem of retrieving coefficients and discontinuity interfaces without restrictive hypothesis of convexity of the interfaces or monotonicity of the speed of waves. These kinds of results are fundamental for seismic prospection. For an overview on this subject see  and the references therein, in particular related to thie study are the works of , , , , .
Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Baudouin, A. Mercado and A.
Osses, Global Carleman estimates in a transmission problem for the wave equation. Application to a one-measurement inverse problem , Inverse Problems, 23 , Bauduoin and J. Inverse Problems, 18 6 , Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation.
Pures Appl. Yamamoto, Inverse source problem for a transmission problem for a parabolic equation.
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