Lorenzo Marino

We study the long time behaviour of the solutions to a linearized Boltzmann equation in one spatial dimension, subject to a random mechanism of transmission, reflection or absorption/generation at an interface. Assuming a fast enough degeneracy of the scattering kernel and a slow decay for the probability of absorption at low frequencies, we show that the solutions to the interface model exhibit a super-diffusive behaviour in the long time limit, with the scaling parameter depending only on the interplay between the decay velocities of the scattering kernel and the drift. We also characterise such a limit as the unique weak solution to a non-local in space evolution equation, subject to a reflection-transmission condition at the interface. Our proof relies on a merger between probabilistic techniques, exploiting that the kinetic dynamics away from the interface is indeed the Kolmogorov equation for a classic jump process, and analytical results on Dirichlet forms. This talk is based on a work in collaboration with Tomasz Komorowski (Polish Academy of Sciences) and K Bogdan (Wroclaw University of Science and Technology). The activity was carried out within the project: NoisyFluid ”Noise in Fluids”, Grant Agreement 101053472, CUP E53C22001720006.

Super-diffusive limit for a kinetic interface model

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