December  2019, 14(4): 677-707. doi: 10.3934/nhm.2019027

A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method

Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin, 53706, USA

* Corresponding author: Yuhua Zhu

Received  August 2018 Revised  April 2019 Published  October 2019

Fund Project: The author was supported in part by Prof. Shi Jin's NSF grants DMS-1522184 and DMS1107291: RNMS KI-Net.

We study the Vlasov-Poisson-Fokker-Planck (VPFP) system with uncertainty and multiple scales. Here the uncertainty, modeled by multi-dimensional random variables, enters the system through the initial data, while the multiple scales lead the system to its high-field or parabolic regimes. We obtain a sharp decay rate of the solution to the global Maxwellian, which reveals that the VPFP system is decreasingly sensitive to the initial perturbation as the Knudsen number goes to zero. The sharp regularity estimates in terms of the Knudsen number lead to the stability of the generalized Polynomial Chaos stochastic Galerkin (gPC-SG) method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC-SG method has spectral accuracy uniform in the Knudsen number.

Citation: Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks and Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
References:
[1]

A. ArnoldJ. A. CarrilloI. Gamba and C.-W. Shu, Low and high field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck systems, Transport Theory and Statistical Physics, 30 (2001), 121-153.  doi: 10.1081/TT-100105365.

[2]

I. BabuskaR. Tempone and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM Journal on Numerical Analysis, 42 (2004), 800-825.  doi: 10.1137/S0036142902418680.

[3]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics (Paris), 4. Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000.

[4]

S. Chandrasekhar, Stochastic probems in physics and astronomy, Reviews of Modern Physics, 15 (1943), 1-89. 

[5]

A. CohenR. DeVore and C. Schwab, Convergence rates of best $N$-term galerkin approximations for a class of elliptic sPDEs, Foundations of Computational Mathematics, 10 (2010), 615-646.  doi: 10.1007/s10208-010-9072-2.

[6]

A. CohenR. DeVore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE's, Analysis and Applications, 9 (2011), 11-47.  doi: 10.1142/S0219530511001728.

[7]

R. J. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Communications in Mathematical Physics, 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[8]

T. GoudonJ. NietoF. Poupaud and J. Soler, Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system, Journal of Differential Equations, 213 (2005), 418-442.  doi: 10.1016/j.jde.2004.09.008.

[9]

J. W. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, Journal of Computational Physics, 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.

[10]

J. W. Hu and S. Jin, Uncertainty quantification for kinetic equations, Uncertainty Quantification for Kinetic and Hyperbolic Equations, SEMA SIMAI Springer Ser., Springer, Cham, 14 (2017), 193-229.  doi: 10.1007/978-3-319-67110-9_6.

[11]

H. Ju Hwang and J. H. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete & Continuous Dynamical Systems-Series B, 18 (2013), 681-691.  doi: 10.3934/dcdsb.2013.18.681.

[12]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Math. Univ. Parma (N.S.), 2 (2012), 177-216. 

[13]

S. Jin, J.-G. Liu and Z. Ma, 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, Research in the Mathematical Sciences, 4 (2017), 25 pp. doi: 10.1186/s40687-017-0105-1.

[14]

S. JinD. B. Xiu and X. Y. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, Journal of Computational Physics, 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.

[15]

S. Jin and Y. H. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales, SIAM Journal on Mathematical Analysis, 50 (2018), 1790-1816.  doi: 10.1137/17M1123845.

[16]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.

[17]

L. Liu and S. Jin, Hypocoercivity based sensitivity analysis and spectral convergence of the stochastic Galerkin approximation to collisional kinetic equations with multiple scales and random inputs, SIAM Multiscale Modeling and Simulation, 16 (2018), 1085-1114.  doi: 10.1137/17M1151730.

[18]

J. NietoF. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Archive for Rational Mechanics and Analysis, 158 (2001), 29-59.  doi: 10.1007/s002050100139.

[19]

R. W. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Mathematical Modelling and Numerical Analysis, 52 (2018), 1651-1678.  doi: 10.1051/m2an/2018024.

[20]

J. Soler, Asymptotic behaviour for the Vlasov-Poisson-Fokker-Planck system, Nonlinear Analysis: Theory, Methods & Applications, 30 (1997), 5217-5228.  doi: 10.1016/S0362-546X(97)00239-3.

[21]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.

[22]

Y. H. Zhu, Sensitivity analysis and uniform regularity for the Boltzmann equation with uncertainty and its stochastic Galerkin approximation, Preprint.

show all references

References:
[1]

A. ArnoldJ. A. CarrilloI. Gamba and C.-W. Shu, Low and high field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck systems, Transport Theory and Statistical Physics, 30 (2001), 121-153.  doi: 10.1081/TT-100105365.

[2]

I. BabuskaR. Tempone and G. E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM Journal on Numerical Analysis, 42 (2004), 800-825.  doi: 10.1137/S0036142902418680.

[3]

F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Series in Applied Mathematics (Paris), 4. Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000.

[4]

S. Chandrasekhar, Stochastic probems in physics and astronomy, Reviews of Modern Physics, 15 (1943), 1-89. 

[5]

A. CohenR. DeVore and C. Schwab, Convergence rates of best $N$-term galerkin approximations for a class of elliptic sPDEs, Foundations of Computational Mathematics, 10 (2010), 615-646.  doi: 10.1007/s10208-010-9072-2.

[6]

A. CohenR. DeVore and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE's, Analysis and Applications, 9 (2011), 11-47.  doi: 10.1142/S0219530511001728.

[7]

R. J. DuanM. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Communications in Mathematical Physics, 300 (2010), 95-145.  doi: 10.1007/s00220-010-1110-z.

[8]

T. GoudonJ. NietoF. Poupaud and J. Soler, Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system, Journal of Differential Equations, 213 (2005), 418-442.  doi: 10.1016/j.jde.2004.09.008.

[9]

J. W. Hu and S. Jin, A stochastic Galerkin method for the Boltzmann equation with uncertainty, Journal of Computational Physics, 315 (2016), 150-168.  doi: 10.1016/j.jcp.2016.03.047.

[10]

J. W. Hu and S. Jin, Uncertainty quantification for kinetic equations, Uncertainty Quantification for Kinetic and Hyperbolic Equations, SEMA SIMAI Springer Ser., Springer, Cham, 14 (2017), 193-229.  doi: 10.1007/978-3-319-67110-9_6.

[11]

H. Ju Hwang and J. H. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete & Continuous Dynamical Systems-Series B, 18 (2013), 681-691.  doi: 10.3934/dcdsb.2013.18.681.

[12]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Math. Univ. Parma (N.S.), 2 (2012), 177-216. 

[13]

S. Jin, J.-G. Liu and Z. Ma, 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, Research in the Mathematical Sciences, 4 (2017), 25 pp. doi: 10.1186/s40687-017-0105-1.

[14]

S. JinD. B. Xiu and X. Y. Zhu, Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings, Journal of Computational Physics, 289 (2015), 35-52.  doi: 10.1016/j.jcp.2015.02.023.

[15]

S. Jin and Y. H. Zhu, Hypocoercivity and uniform regularity for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales, SIAM Journal on Mathematical Analysis, 50 (2018), 1790-1816.  doi: 10.1137/17M1123845.

[16]

Q. Li and L. Wang, Uniform regularity for linear kinetic equations with random input based on hypocoercivity, SIAM/ASA J. Uncertainty Quantification, 5 (2017), 1193-1219.  doi: 10.1137/16M1106675.

[17]

L. Liu and S. Jin, Hypocoercivity based sensitivity analysis and spectral convergence of the stochastic Galerkin approximation to collisional kinetic equations with multiple scales and random inputs, SIAM Multiscale Modeling and Simulation, 16 (2018), 1085-1114.  doi: 10.1137/17M1151730.

[18]

J. NietoF. Poupaud and J. Soler, High-field limit for the Vlasov-Poisson-Fokker-Planck system, Archive for Rational Mechanics and Analysis, 158 (2001), 29-59.  doi: 10.1007/s002050100139.

[19]

R. W. Shu and S. Jin, Uniform regularity in the random space and spectral accuracy of the stochastic galerkin method for a kinetic-fluid two-phase flow model with random initial inputs in the light particle regime, Mathematical Modelling and Numerical Analysis, 52 (2018), 1651-1678.  doi: 10.1051/m2an/2018024.

[20]

J. Soler, Asymptotic behaviour for the Vlasov-Poisson-Fokker-Planck system, Nonlinear Analysis: Theory, Methods & Applications, 30 (1997), 5217-5228.  doi: 10.1016/S0362-546X(97)00239-3.

[21]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970.

[22]

Y. H. Zhu, Sensitivity analysis and uniform regularity for the Boltzmann equation with uncertainty and its stochastic Galerkin approximation, Preprint.

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