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Carleman estimate with piecewise weight and applications to inverse problems for first-order transport equations
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DOI：10.1088/1361-6420/ae5083.png)
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Abstract
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We consider a first-order transport equation partial derivative(t)u(x,t)+(H(x)& sdot;del u(x,t))+p(x)u(x,t)=F(x,t) for x is an element of Omega subset of R-d, where d >= 2, Omega is a bounded domain and 0 < t < T. We prove a Carleman estimate for more general condition on the principal coefficients H(x) than in the existing works. The key is the construction of a piecewise smooth weight function in x according to a suitable decomposition of Omega. Our assumptions on H generalize the conditions in the existing articles, and require that a directed graph created by the corresponding stream field has no closed loops. Then, we apply our Carleman estimate to two inverse problems of determination of an initial value and one of a spatial factor of a source term, so that we establish Lipschitz stability estimates for the inverse problems.
Keywords:
first-order transport equation
Carleman estimate
inverse source problem
stability
piecewise weight function
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