Stationary solutions to the Boltzmann equation in the plane
Authors:
Leif Arkeryd and Anne Nouri
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 76P05
DOI:
https://doi.org/10.1090/qam/1692
Published electronically:
March 25, 2024
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Abstract: The paper proves existence of stationary solutions to the Boltzmann equation in a bounded set of $\mathbb {R}^2$ for given indata, hard forces and truncation in the collision kernel for small velocities and close to parallel colliding velocities. It does not use any averaging in velocity lemma. Instead, it is based on stability techniques employing the Kolmogorov-Riesz-Fréchet theorem, from the discrete velocity stationary case, where the averaging in velocity lemmas are not valid.
References
- Leif Arkeryd, On the stationary Boltzmann equation in $\textbf {R}^n$, Internat. Math. Res. Notices 12 (2000), 625–641. MR 1772079, DOI 10.1155/S1073792800000349
- Leif Arkeryd and Anne Nouri, The stationary Boltzmann equation in $\Bbb R^n$ with given indata, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 2, 359–385. MR 1991144
- Leif Arkeryd and Anne Nouri, $L^1$ solutions to the stationary Boltzmann equation in a slab, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 3, 375–413 (English, with English and French summaries). MR 1842024
- Leif Arkeryd and Anne Nouri, The stationary Boltzmann equation in the slab with given weighted mass for hard and soft forces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 3-4, 533–556 (1999). MR 1677990
- L. Arkeryd and A. Nouri, A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory, Indiana Univ. Math. J. 44 (1995), no. 3, 815–839. MR 1375351, DOI 10.1512/iumj.1995.44.2010
- L. Arkeryd and A. Nouri, On the stationary Povzner equation in $\textbf {R}^n$, J. Math. Kyoto Univ. 39 (1999), no. 1, 115–153. MR 1684160, DOI 10.1215/kjm/1250517956
- Leif Arkeryd and Anne Nouri, Stationary solutions to the two-dimensional Broadwell model, Doc. Math. 25 (2020), 2023–2048. MR 4187717
- Leif Arkeryd and Anne Nouri, On stationary solutions to normal, coplanar discrete Boltzmann equation models, Commun. Math. Sci. 18 (2020), no. 8, 2215–2234. MR 4195571, DOI 10.4310/CMS.2020.v18.n8.a6
- Leif Arkeryd and Anne Nouri, Discrete velocity Boltzmann equations in the plane: stationary solutions, Anal. PDE 16 (2023), no. 8, 1869–1884. MR 4657147, DOI 10.2140/apde.2023.16.1869
- R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321–366. MR 1014927, DOI 10.2307/1971423
- R. Esposito, Y. Guo, C. Kim, and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm. Math. Phys. 323 (2013), no. 1, 177–239. MR 3085665, DOI 10.1007/s00220-013-1766-2
- Raffaele Esposito, Yan Guo, Chanwoo Kim, and Rossana Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE 4 (2018), no. 1, Paper No. 1, 119. MR 3740632, DOI 10.1007/s40818-017-0037-5
- J. P. Guiraud, Interior boundary problem for the linear Boltzmann equation (Problème aux limites intérieur pour l’équation de Boltzmann linéaire), Jour. de Mécanique 9 (1970), 183–231.
- J. P. Guiraud, Interior boundary problem for the stationary weakly nonlinear Boltzmann equation (Problème aux limites intérieur pour l’équation de Boltzmann en régime stationnaire, faiblement non linéaire), Jour. de Mécanique 11 (1972), 443–490.
- François Golse and Laure Saint-Raymond, Velocity averaging in $L^1$ for the transport equation, C. R. Math. Acad. Sci. Paris 334 (2002), no. 7, 557–562 (English, with English and French summaries). MR 1903763, DOI 10.1016/S1631-073X(02)02302-6
- A. N. Kolmogorov, Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel, Nachr. Ges. Wiss. Göttingen 9 (1931), 60–63.
- M. Riesz, Sur les ensembles compacts de fonctions sommables, Acta Univ. Szeged Sect. Sci. Math. 6 (1933), 136–142.
References
- Leif Arkeryd, On the stationary Boltzmann equation in $\mathbf {R}^n$, Internat. Math. Res. Notices 12 (2000), 625–641. MR 1772079, DOI 10.1155/S1073792800000349
- Leif Arkeryd and Anne Nouri, The stationary Boltzmann equation in $\mathbb {R}^n$ with given indata, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 2, 359–385. MR 1991144
- Leif Arkeryd and Anne Nouri, $L^1$ solutions to the stationary Boltzmann equation in a slab, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 3, 375–413 (English, with English and French summaries). MR 1842024
- Leif Arkeryd and Anne Nouri, The stationary Boltzmann equation in the slab with given weighted mass for hard and soft forces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 3-4, 533–556 (1999). MR 1677990
- L. Arkeryd and A. Nouri, A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory, Indiana Univ. Math. J. 44 (1995), no. 3, 815–839. MR 1375351, DOI 10.1512/iumj.1995.44.2010
- L. Arkeryd and A. Nouri, On the stationary Povzner equation in $\mathbf {R}^n$, J. Math. Kyoto Univ. 39 (1999), no. 1, 115–153. MR 1684160, DOI 10.1215/kjm/1250517956
- Leif Arkeryd and Anne Nouri, Stationary solutions to the two-dimensional Broadwell model, Doc. Math. 25 (2020), 2023–2048. MR 4187717
- Leif Arkeryd and Anne Nouri, On stationary solutions to normal, coplanar discrete Boltzmann equation models, Commun. Math. Sci. 18 (2020), no. 8, 2215–2234. MR 4195571, DOI 10.4310/CMS.2020.v18.n8.a6
- Leif Arkeryd and Anne Nouri, Discrete velocity Boltzmann equations in the plane: stationary solutions, Anal. PDE 16 (2023), no. 8, 1869–1884. MR 4657147, DOI 10.2140/apde.2023.16.1869
- R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. 130 (1989), no. 2, 321–366. MR 1014927, DOI 10.2307/1971423
- R. Esposito, Y. Guo, C. Kim, and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm. Math. Phys. 323 (2013), no. 1, 177–239. MR 3085665, DOI 10.1007/s00220-013-1766-2
- Raffaele Esposito, Yan Guo, Chanwoo Kim, and Rossana Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE 4 (2018), no. 1, Paper No. 1, 119. MR 3740632, DOI 10.1007/s40818-017-0037-5
- J. P. Guiraud, Interior boundary problem for the linear Boltzmann equation (Problème aux limites intérieur pour l’équation de Boltzmann linéaire), Jour. de Mécanique 9 (1970), 183–231.
- J. P. Guiraud, Interior boundary problem for the stationary weakly nonlinear Boltzmann equation (Problème aux limites intérieur pour l’équation de Boltzmann en régime stationnaire, faiblement non linéaire), Jour. de Mécanique 11 (1972), 443–490.
- François Golse and Laure Saint-Raymond, Velocity averaging in $L^1$ for the transport equation, C. R. Math. Acad. Sci. Paris 334 (2002), no. 7, 557–562 (English, with English and French summaries). MR 1903763, DOI 10.1016/S1631-073X(02)02302-6
- A. N. Kolmogorov, Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel, Nachr. Ges. Wiss. Göttingen 9 (1931), 60–63.
- M. Riesz, Sur les ensembles compacts de fonctions sommables, Acta Univ. Szeged Sect. Sci. Math. 6 (1933), 136–142.
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Additional Information
Leif Arkeryd
Affiliation:
Department of Mathematical Sciences, University of Göteborg, , 41296 Göteborg, Sweden
MR Author ID:
27070
Email:
arkeryd@chalmers.se
Anne Nouri
Affiliation:
Département de Mathématiques, Aix-Marseille University, CNRS, I2M UMR 7373, 13453 Marseille, France
MR Author ID:
333335
Email:
anne.nouri@univ-amu.fr
Received by editor(s):
May 17, 2023
Received by editor(s) in revised form:
February 4, 2024
Published electronically:
March 25, 2024
Article copyright:
© Copyright 2024
Brown University
