The Landau equation and Fisher information

In recent work with Luis Silvestre we rule out blow ups for space homogeneous Landau equation. This result follows from the  monotonicity along the flow of an important functional in statistics — the Fisher information. This realization was made possible by a new lifting procedure which (somewhat surprisingly) relates kinetic equations to linear elliptic equations on families of two-dimensional spheres foliating six dimensional Euclidean space. The convexity of the Fisher information functional as well as the symmetries of the equation play an important role in the proof. Time permitting I will also discuss parts of work of Imbert, Silvestre, and Villani, which improves on the Landau result by obtaining new nonlocal functional inequalities that allow them to rule out blow ups for the space homogeneous Boltzmann equation.

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