Failure of the Harnack inequality for nonlocal kinetic equations
Published on:
8 May 2024
Primary Category:
Analysis of PDEs
Paper Authors:
Moritz Kassmann,
Marvin Weidner
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Key Details
The Harnack inequality fails for the fractional Kolmogorov equation, a simple nonlocal kinetic equation
This reflects a purely nonlocal phenomenon not present in local kinetic equations
A counterexample is constructed using properties of the fundamental solution
The proof relies on scaling arguments and weak maximum principle estimates
The failure is linked to long-range oscillations caused by the nonlocal operator
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AI generated summary
Failure of the Harnack inequality for nonlocal kinetic equations
The authors prove that the Harnack inequality, an important regularity estimate, fails for nonlocal kinetic equations that arise as models for the Boltzmann equation without cutoff. This is in contrast to local kinetic equations that do satisfy the Harnack inequality. A counterexample is provided for the fractional Kolmogorov equation.
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