June  2010, 3(2): 195-222. doi: 10.3934/krm.2010.3.195

On half-space problems for the weakly non-linear discrete Boltzmann equation

1. 

Department of Mathematics, Karlstad University, 651 88 Karlstad, Sweden

Received  September 2009 Revised  December 2009 Published  May 2010

Existence of solutions of weakly non-linear half-space problems for the general discrete velocity (with arbitrarily finite number of velocities) model of the Boltzmann equation are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. Applications to axially symmetric models are studied in more detail.
Citation: Niclas Bernhoff. On half-space problems for the weakly non-linear discrete Boltzmann equation. Kinetic and Related Models, 2010, 3 (2) : 195-222. doi: 10.3934/krm.2010.3.195
[1]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure and Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957

[2]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic and Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35

[3]

Niclas Bernhoff. Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures. Kinetic and Related Models, 2012, 5 (1) : 1-19. doi: 10.3934/krm.2012.5.1

[4]

Nicola Abatangelo, Serena Dipierro, Mouhamed Moustapha Fall, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1205-1235. doi: 10.3934/dcds.2019052

[5]

Davide Bellandi. On the initial value problem for a class of discrete velocity models. Mathematical Biosciences & Engineering, 2017, 14 (1) : 31-43. doi: 10.3934/mbe.2017003

[6]

Vasily Denisov and Andrey Muravnik. On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations. Electronic Research Announcements, 2003, 9: 88-93.

[7]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems and Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[8]

Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure and Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663

[9]

Gael Diebou Yomgne. On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space. Communications on Pure and Applied Analysis, 2022, 21 (2) : 517-539. doi: 10.3934/cpaa.2021186

[10]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[11]

Chérif Amrouche, Huy Hoang Nguyen. Elliptic problems with $L^1$-data in the half-space. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 369-397. doi: 10.3934/dcdss.2012.5.369

[12]

Gershon Kresin, Vladimir Maz’ya. Optimal estimates for the gradient of harmonic functions in the multidimensional half-space. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 425-440. doi: 10.3934/dcds.2010.28.425

[13]

Diego D. Felix, Marcelo F. Furtado, Everaldo S. Medeiros. Semilinear elliptic problems involving exponential critical growth in the half-space. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4937-4953. doi: 10.3934/cpaa.2020219

[14]

Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic and Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037

[15]

Tsukasa Iwabuchi. On analyticity up to the boundary for critical quasi-geostrophic equation in the half space. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1209-1224. doi: 10.3934/cpaa.2022016

[16]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939

[17]

Ziwei Zhou, Jiguang Bao, Bo Wang. A Liouville theorem of parabolic Monge-AmpÈre equations in half-space. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1561-1578. doi: 10.3934/dcds.2020331

[18]

Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353

[19]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145

[20]

Laurent Gosse. Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic and Related Models, 2012, 5 (2) : 283-323. doi: 10.3934/krm.2012.5.283

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]