\(C^\infty \) Smoothing for Weak Solutions of the Inhomogeneous Landau Equation

  • Christopher Henderson
  • Stanley SnelsonEmail author


We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth, and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.


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Conflict of interest

The authors declare that they have no conflict of interest.


Both authors were partially supported by National Science Foundation Grant DMS-1246999. CH was partially supported by NSF grant DMS-1907853. SS was partially supported by a Ralph E. Powe Award from ORAU.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA
  3. 3.Department of Mathematical SciencesFlorida Insitute of TechnologyMelbourneUSA

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