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

  • Christopher Henderson
  • Stanley SnelsonEmail author
Article
  • 37 Downloads

Abstract

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.

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Funding

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.

References

  1. 1.
    Alexandre, R., Villani, C.: On the Landau approximation in plasma physics. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 21(1), 61–95, 2004ADSMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bramanti , M., Brandolini , L.: Schauder estimates for parabolic nondivergence operators of Hörmander type. J. Differ. Equ. 234(1), 177–245, 2007ADSCrossRefGoogle Scholar
  3. 3.
    Cameron , S., Silvestre , L., Snelson , S.: Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéare35(3), 625–642, 2018ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, 3rd edn. Cambridge University Press, Cambridge 1970zbMATHGoogle Scholar
  5. 5.
    Chen , Y., Desvillettes , L., He , L.: Smoothing effects for classical solutions of the full Landau equation. Arch. Ration. Mech. Anal. 193(1), 21–55, 2009MathSciNetCrossRefGoogle Scholar
  6. 6.
    Desvillettes , L., Villani , C.: On the spatially homogeneous Landau equation for hard potentials part I: existence, uniqueness and smoothness. Commun. Partial Differ. Equ. 25(1–2), 179–259, 2000CrossRefGoogle Scholar
  7. 7.
    Di Francesco , M., Polidoro , S.: Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Differ. Equ. 11(11), 1261–1320, 2006MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin 2001zbMATHGoogle Scholar
  9. 9.
    Golse , F., Imbert , C., Mouhot , C., Vasseur , A.: Harnack inequality for kinetic Fokker–Planck equations with rough coefficients and application to the Landau equation. Annali della Scuola Normale Superiore di PisaXIX(1), 253–295, 2019MathSciNetzbMATHGoogle Scholar
  10. 10.
    Guo , Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231(3), 391–434, 2002ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Han , Q., Lin , F.-H.: Elliptic Partial Differential Equations. Courant Lecture Notes, 2nd edn. Courant Institute of Mathematical Sciences, New York University, New York 2011Google Scholar
  12. 12.
    He , L., Yang , X.: Well-posedness and asymptotics of grazing collisions limit of Boltzmann equation with Coulomb interaction. SIAM J. Math. Anal. 46(6), 4104–4165, 2014MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hörmander , L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171, 1967MathSciNetCrossRefGoogle Scholar
  14. 14.
    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. American Mathematical Society, Providence 1996Google Scholar
  15. 15.
    Lanconelli , E., Polidoro , S.: On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino52(1), 29–63, 1994. (Partial differential equations, II (Turin, 1993))MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lifshitz, E.M., Pitaevskii, L.P.: Course of Theoretical Physics: Physical Kinetics, vol. 10, 1st edn. Butterworth-Heinemann, Oxford 1981Google Scholar
  17. 17.
    Liu , S., Ma , X.: Regularizing effects for the classical solutions to the Landau equation in the whole space. J. Math. Anal. Appl. 417(1), 123–143, 2014MathSciNetCrossRefGoogle Scholar
  18. 18.
    Manfredini , M.: The Dirichlet problem for a class of ultraparabolic equations. Adv. Differ. Equ. 2(5), 831–866, 1997MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mouhot , C., Neumann , L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity19(4), 969, 2006ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Silvestre , L.: Upper bounds for parabolic equations and the Landau equation. J. Differ. Equ. 262(3), 3034–3055, 2017ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Villani , C.: On the Cauchy problem for Landau equation: sequential stability, global existence. Adv. Differ. Equ. 1(5), 793–816, 1996MathSciNetzbMATHGoogle Scholar
  22. 22.
    Villani , C.: On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 08(06), 957–983, 1998MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang , W., Zhang , L.: The \(C^\alpha \) regularity of weak solutions of ultraparabolic equations. Discrete Contin. Dyn. Syst. 29(3), 1261–1275, 2011MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

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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