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International Journal for Uncertainty Quantification
IF: 3.259 5-Year IF: 2.547 SJR: 0.417 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2019026936
pages 123-142

RANDOM REGULARITY OF A NONLINEAR LANDAU DAMPING SOLUTION FOR THE VLASOV-POISSON EQUATIONS WITH RANDOM INPUTS

Zhiyan Ding
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA
Shi Jin
School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China

ABSTRACT

In this paper, we study the nonlinear Landau damping solution of the Vlasov-Poisson equations with random inputs from the initial data or equilibrium, for the solution studied by Hwang and Velázquez smoothly on the random input, if the long-time limit distribution function has the same smoothness, under some smallness assumptions. We also establish the decay of the higher-order derivatives of the solution in the random variable, with the same decay rate as its deterministic counterpart.

REFERENCES

  1. Landau, L.D., On the Vibration of the Electronic Plasma, J. Phys. USSR, 10(25):25–34, 1946.

  2. Degond, P., Spectral Theory of the Linearized Vlasov-Poisson Equation, Trans. Am. Math. Sci., 294(2):435–453, 1986.

  3. Glassey, R. and Schaeff J., Time Decay for Solutions to the Linearized Vlasov Equation, Transp. Theory Stat. Phys., 23(4):411–453, 1994.

  4. Caglioti, E. and Maffei, C., Time Asymptotics for Solutions of Vlasov-Poisson Equation in a Circle, J. Stat. Phys., 92(1):301– 323, 1998.

  5. Hwang, H.J. and Velazquez, J., On the Existence of Exponentially Decreasing Solutions of the Nonlinear Landau Damping Problem, Indiana Univ. Math. J., 58(6):2623–2660, 2009.

  6. Mouhot, C. and Villani, C., On Landau Damping, Acta Math., 207(1):29–201, 2011.

  7. Bedrossian, J., Masmoudi, N., and Mouhot, C., Landau Damping: Paraproducts and Gevrey Regularity, Ann. PDE, 2:4, 2016.

  8. Smith, R., Uncertainty Quantification: Theory, Implementation, and Applications, Philadelphia: SIAM, 2013.

  9. Babuska, I., Tempone, R., and Zouraris, G., Galerkin Finite Element Approximations of Stochastic Elliptic Diffitial Equations, SIAM J. Numer. Anal., 42(2):800–825, 2004.

  10. Nobile, F. and Tempone, R., Analysis and Implementation Issues for the Numerical Approximation of Parabolic Equations with Random Coefficients, Int. J. Numer. Methods. Eng., 80(6):979–1006, 2009.

  11. Motamed, M., Nobile, F., and Tempone, R., Stochastic Collocation Method for the Second Wave Equation with a Discontinuous Random Speed, Numer. Math., 123(3):493–536, 2013.

  12. Tang, T. and Zhou, T., Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed, Commun. Comput. Phys., 8(1):226–248, 2010.

  13. Jin, S., Liu, J., and Ma, Z., Uniform Spectral Convergence of the Stochastic Galerkin Method for the Linear Transport Equations with Random Inputs in Diffusive Regime and a Micro-Macro Decomposition based Asymptotic Preserving Method, Res. Math. Sci., 4:15,2017.

  14. Li, Q. and Wang, L., Uniform Regularity for Linear Kinetic Equations with Random Input based on Hypocoercivity, SIAM/ASA J. Uncertainty Quantif., 5(1):1193–1219, 2018.

  15. Jin, S. and Zhu, Y., Hypocoercivity and Uniform Regularity for the Vlasov-Poisson-Fokker-Planck System with Uncertainty and Multiple Scales, SIAM J. Math. Anal., 50(2):1790–1816, 2018.

  16. Liu, L. and Jin, S., Hypocoercivity based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs, Multiscale Model. Simul., 16(3):1085–1114, 2018.

  17. Shu, R. and Jin, S., Uniform Regularity in the Random Space and Spectral Accuracy of Thestochastic Galerkin Method for a Kinetic-Fluid Two-Phase Flow Model with Random Initial Inputs in the Light Particle Regime, ESAIM: M2AN, 52(5):1651– 1678, 2018.

  18. Shu, R. and Jin, S., A Study of Landau Damping with Random Initial Inputs, J. Differ. Eqs., 266(4):1922–1945, 2019.

  19. Shu, R. and Jin, S., Random Regularity of Landau Damping Solution of Mouhot and Villani with Random Inputs, in preparation.

  20. Xiu, D., Fast Numerical Methods for Stochastic Computations: A Review, Commun. Comput. Phys, 5(2):242–272, 2009.

  21. Xiu, D. and Karniadakis, G.E., The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations, SIAM J. Sci. Comput., 24(2):619–644, 2002.


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