972 research outputs found
Well-posedness for mean-field evolutions arising in superconductivity
We establish the existence of a global solution for a new family of
fluid-like equations, which are obtained in a joint work with Serfaty in
certain regimes as the mean-field evolution of the supercurrent density in a
(2D section of a) type-II superconductor with pinning and with imposed electric
current. We also consider general vortex-sheet initial data, and investigate
the uniqueness and regularity properties of the solution. For some choice of
parameters, the equation under investigation coincides with the so-called lake
equation from 2D shallow water fluid dynamics, and our analysis then leads to a
new existence result for rough initial data.Comment: 53 pages; revised version, including an appendix jointly written with
Julian Fischer about global existence for the degenerate parabolic cas
Inverse coefficient problem for Grushin-type parabolic operators
The approach to Lipschitz stability for uniformly parabolic equations
introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates,
seems hard to apply to the case of Grushin-type operators studied in this
paper. Indeed, such estimates are still missing for parabolic operators
degenerating in the interior of the space domain. Nevertheless, we are able to
prove Lipschitz stability results for inverse coefficient problems for such
operators, with locally distributed measurements in arbitrary space dimension.
For this purpose, we follow a strategy that combines Fourier decomposition and
Carleman inequalities for certain heat equations with nonsmooth coefficients
(solved by the Fourier modes)
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