Propagation and absorption of concentration effects near shock hypersurfaces for the heat equation

Abstract

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.