Hölder Continuity for a Family of Nonlocal Hypoelliptic Kinetic Equations

Source abstract

In this work, Hölder continuity is obtained for solutions to the nonlocal kinetic Fokker--Planck equation and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker--Planck equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonlocal operator than in the work of Imbert and Silvestre [ The Weak Harnack Inequality for the Boltzmann Equation Without Cut-Off, arXiv:1608.07571, 2016] but allows unbounded source terms.