The Harnack inequality fails for nonlocal kinetic equations
Moritz Kaßmann, Marvin Weidner
Abstract
We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest equation in this theory, the fractional Kolmogorov equation. Our result reflects a purely nonlocal phenomenon since the Harnack inequality holds true for local kinetic equations like the Kolmogorov equation.
Topics & Concepts
Harnack's inequalityHarnack's principleMathematicsInequalityKinetic energyMathematical analysisApplied mathematicsCalculus (dental)Pure mathematicsClassical mechanicsPhysicsDentistryMedicineGas Dynamics and Kinetic TheoryNumerical methods in inverse problemsNonlinear Partial Differential Equations