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In this paper we study families of solutions to heat equation with a small parameter and a propagation term consisting in a discontinuous vector field b through a smooth compact hypersurface S. Our purpose is to describe the evolution of the energy density as h goes to 0 of a family of solutions for a bounded square integrable family of inital data. Outside of S it is classical to calculate this limit by using semi-classical measures associated with the family of solutions. The discontinuity of b through S induces a difficulty that we overcome provided a second microlocalization. We introduce two-microlocal items describing the concentration of a square integrable bounded sequence on a hypersurface. By using these items we calculate for convenient times semi-classical measures of the family of solutions in the whole cotangent space and the limit of the energy density as small parameter goes to 0.
Asymptotic Analysis – IOS Press
Published: Jan 1, 2000
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