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Proceedings of 47th Sapporo Symposium on Partial Differential Equations

Ei, S.-I. Giga, Y. Hamamuki, N. Jimbo, S. Kubo, H. Kuroda, H. Liu, Y. Ozawa, T. Sakajo, T. Tsutaya, K. 北海道大学

2022.07.28

概要

In this talk, I will concern about forward and inverse analyses for anisotropic elastic systems and anisotropic viscoelastic systems, especially for the dynamical case. In most of inverse problems, basic tools for them are the unique continuation property (UCP) of solutions and the exact boundary controllability (EBC). The first one is well known in PDE. The second one is to find a boundary control which connect the given initial state to the given final state. Moving from a scalar equation to a system and the system becomes anisotropic, it becomes hard to have these tools. Nevertheless, in order to have these tools, it is necessary to consider a good setting that does not lose generality as much as possible.

The settings which I propose here are as follows. For having the Holmgren-John UCP, the setting would be piecewise homogeneous coefficients or piecewise analytic coefficients. This UCP should be the optimal one which can give the optimal time for continuing the zero set of solutions in terms of the travel time of the slowest wave. I do believe that this setting is general enough because it includes the first order finite element model, and it can also describe the medium discontinuity. Combining the Holmgren-John UCP with some geometric tools and algebraic tools, it is possible to obtain some global uniqueness result for the coefficient identification problem (referred as inverse coefficient problem which is abbreviated by ICP) by using the so-called localized Neumann to Dirichlet map (ND-map) as measured data, and also some local recovery of coefficients if we know the medium discontinuity. Here the geometric tools are the theory of analytic sets ([3]) and continuity of symmetric axis of tensors ([12]), and the algebraic tools are the Stroh formalism ([20]) and the theory of matrix polynomials ([10]). This ICP result gives a mathematical foundation for the vibroseis reflection exploration in geophysics.

The EBC can imply the observability inequality which estimates the initial data by some lateral boundary integral of the solution to the initial boundary value problem for a system. Further, this inequality is useful to analyze the stability estimate for the inverse source problem for the system, which is the first step toward analyzing the stability estimate for the ICP ([1]). The biggest advantage of the EBC is that it is not necessary to assume the piecewise homogeneous or analytic condition for the coefficients of the system.

For an elastic system coupled with a conductivity equation, i.e. piezoelectric system, it is possible to obtain the EBC by the Russell principle ([11]). This principle says whenever there is a uniform decay estimate of solutions for the initial boundary value problem as the time goes to infinity for a hyperbolic equation or system, it is possible to have the EBC. Then, the result given in ([11]) is very interesting, because the piezoelectric system is not hyperbolic and it has to be anisotropic due to the anisotropy of materials which have piezoelectric property. Further, the conductivity equation is giving physically very natural dissipation which yields the uniform decay of solutions. As for the viscoelastic system, the viscosity gives the dissipation but it is an obstruction for having the time reversibility in the complete sense. Hence, it is interesting to analyze the possibility of having the EBC for the viscoelastic system.

The rest of this paper is organized as follows. In Section 2, anisotropic elastic/ viscoelastic systems are introduces. Then in Section 3, some results and possible results will be presented for the ICP for elastic/viscoelastic systems with some ex- planation of the tools for their proofs. In Section 4, I give a very short description of current situation of the ongoing study on the EBC for viscoelastic systems.

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