On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E.S. Daus, Arch. Ration. Mech. Anal. 3 (2016) 1367–1443] for the deterministic problem in the perturbative regime, and in [E.S. Daus, S. Jin and L. Liu, Kinet. Relat. Models 12 (2019) 909–922] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far neither in the single species case nor in the multi-species case.

[1]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[2]  V. Giovangigli.  Multicomponent flow modeling , 1999 .

[3]  C. Cercignani.  Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations , 2000 .

[4]  Francesco Salvarani,et al.  A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions , 2005 .

[5]  Cl'ement Mouhot,et al.  Explicit Coercivity Estimates for the Linearized Boltzmann and Landau Operators , 2006, math/0607538.

[6]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[7]  Maria Pia Gualdani,et al.  Factorization for non-symmetric operators and exponential H-theorem , 2010, 1006.5523.

[8]  Francesco Salvarani,et al.  Diffusion asymptotics of a kinetic model for gaseous mixtures , 2012 .

[9]  Ralph C. Smith,et al.  Uncertainty Quantification: Theory, Implementation, and Applications , 2013 .

[10]  Guannan Zhang,et al.  Stochastic finite element methods for partial differential equations with random input data* , 2014, Acta Numerica.

[11]  F. Salvarani,et al.  Compactness of Linearized Kinetic Operators , 2016 .

[12]  Ansgar Jüngel,et al.  Hypocoercivity for a Linearized Multispecies Boltzmann System , 2015, SIAM J. Math. Anal..

[13]  M. Briant.  Ju l 2 01 6 STABILITY OF GLOBAL EQUILIBRIUM FOR THE MULTI-SPECIES BOLTZMANN EQUATION IN L ∞ SETTINGS , 2016 .

[14]  Esther S. Daus,et al.  The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium , 2016, Archive for Rational Mechanics and Analysis.

[15]  M. Briant.  Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions , 2015, 1507.03153.

[16]  Bruno Després,et al.  Uncertainty Propagation; Intrusive Kinetic Formulations of Scalar Conservation Laws , 2016, SIAM/ASA J. Uncertain. Quantification.

[17]  M. Briant.  Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings , 2016, 1603.01497.

[18]  F. Salvarani,et al.  The Maxwell-Stefan Diffusion Limit for a Kinetic Model of Mixtures , 2014, Acta Applicandae Mathematicae.

[19]  Li Wang,et al.  Uniform Regularity for Linear Kinetic Equations with Random Input Based on Hypocoercivity , 2016, SIAM/ASA J. Uncertain. Quantification.

[20]  Liu Liu,et al.  DG-IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings , 2017, J. Sci. Comput..

[21]  Shi Jin,et al.  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 , 2017, Research in the Mathematical Sciences.

[22]  Giacomo Dimarco,et al.  Uncertainty Quantification for Kinetic Models in Socio–Economic and Life Sciences , 2017, 1706.07500.

[23]  Jingwei Hu,et al.  Uncertainty Quantification for Kinetic Equations , 2017 .

[24]  Abubakr Gafar Abdalla,et al.  Probability Theory , 2017, Encyclopedia of GIS.

[25]  Liu Liu,et al.  An Asymptotic-Preserving Stochastic Galerkin Method for the Semiconductor Boltzmann Equation with Random Inputs and Diffusive Scalings , 2017, Multiscale Model. Simul..

[26]  C. Baranger,et al.  On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases , 2018, 31ST INTERNATIONAL SYMPOSIUM ON RAREFIED GAS DYNAMICS: RGD31.

[27]  LIU LIU,et al.  Hypocoercivity Based Sensitivity Analysis and Spectral Convergence of the Stochastic Galerkin Approximation to Collisional Kinetic Equations with Multiple Scales and Random Inputs , 2017, Multiscale Model. Simul..

[28]  I. Gamba,et al.  On Existence and Uniqueness to Homogeneous Boltzmann Flows of Monatomic Gas Mixtures , 2018, Archive for Rational Mechanics and Analysis.

[29]  Liu Liu,et al.  A stochastic asymptotic-preserving scheme for the bipolar semiconductor Boltzmann-Poisson system with random inputs and diffusive scalings , 2018, J. Comput. Phys..

[30]  Esther S. Daus,et al.  Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel , 2018, Kinetic & Related Models.

[31]  Liu Liu,et al.  Hypocoercivity for a BGK model for gas mixtures , 2018, Journal of Differential Equations.

[32]  M. Briant,et al.  Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions , 2018, Communications on Pure & Applied Analysis.