American Institute of Mathematical Sciences

Advanced
March  2009, 2(1): 205-214. doi: 10.3934/krm.2009.2.205

Local Hilbert expansion for the Boltzmann equation

Yan Guo 1, , Juhi Jang 2, and  Ning Jiang 2,

1. 

Brown University, Providence, RI 02912, United States
2. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York City, NY 10012, United States, United States

Received  September 2008 Revised  October 2008 Published  January 2009

We revisit the classical work of Caflisch [1] for compressible Euler limit of the Boltzmann equation. By using a new $L^{2}\mbox{-}L^{\infty }$ method, we prove the validity of the Hilbert expansion before shock formations in the Euler system with moderate temperature variation.
Keywords: Boltzmann equations., fluid dynamic limit.
Mathematics Subject Classification: Primary: 76P05, 35L67; Secondary: 82A4.
Citation: Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205
[1]

Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic & Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
[2]

Vincent Giovangigli. Persistence of Boltzmann entropy in fluid models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 95-114. doi: 10.3934/dcds.2009.24.95
[3]

Ciro D'Apice, Rosanna Manzo. A fluid dynamic model for supply chains. Networks & Heterogeneous Media, 2006, 1 (3) : 379-398. doi: 10.3934/nhm.2006.1.379
[4]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046
[5]

Jérôme Renault. General limit value in dynamic programming. Journal of Dynamics & Games, 2014, 1 (3) : 471-484. doi: 10.3934/jdg.2014.1.471
[6]

Alberto Bressan, Marco Mazzola, Hongxu Wei. A dynamic model of the limit order book. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1015-1041. doi: 10.3934/dcdsb.2019206
[7]

Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581
[8]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869
[9]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[10]

Jean Ginibre, Giorgio Velo. Modified wave operators without loss of regularity for some long range Hartree equations. II. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1357-1376. doi: 10.3934/cpaa.2015.14.1357
[11]

Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933
[12]

Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001
[13]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[14]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159
[15]

Marzia Bisi, Giampiero Spiga. A Boltzmann-type model for market economy and its continuous trading limit. Kinetic & Related Models, 2010, 3 (2) : 223-239. doi: 10.3934/krm.2010.3.223
[16]

Angelo Morro. Nonlinear diffusion equations in fluid mixtures. Evolution Equations & Control Theory, 2016, 5 (3) : 431-448. doi: 10.3934/eect.2016012
[17]

Jin Ma, Xinyang Wang, Jianfeng Zhang. Dynamic equilibrium limit order book model and optimal execution problem. Mathematical Control & Related Fields, 2015, 5 (3) : 557-583. doi: 10.3934/mcrf.2015.5.557
[18]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737
[19]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020349
[20]

Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic & Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences