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Holder Continuity Smoothing for Hypoelliptic Kinetic Equations

By Alexis Vasseur, The University of Texas at Austin

We consider generalized Fokker-Planck equations with rough elliptic coefficients. Typically, the parabolic regularization takes effect in the velocity space only. We show that the Liouville transport operator extends the regularity to the space variable as well. Precisely, we show that any solution with integrable and bounded initial data, becomes Holder continuous for positive times. The proof is based on the De Giorgi method for elliptic operators. The method is extended to the hypoelliptic case, using the averaging lemmas. This is a joint work with Francois Golse.