Existence and uniqueness of solutions to the Fermi-Dirac Boltzmann equation for soft potentials
Author:
Zongguang Li
Journal:
Quart. Appl. Math. 82 (2024), 735-787
MSC (2020):
Primary 35Q20, 35Q40; Secondary 35B20, 35B45
DOI:
https://doi.org/10.1090/qam/1681
Published electronically:
October 27, 2023
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Abstract: In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter $\delta$ that can decrease from $\delta =1$ for the Fermi-Dirac particles to $\delta =0$ for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space $L^{\infty }_{T}L^{\infty }_{v,x}\cap L^{\infty }_{T}L^{\infty }_{x}L^1_v$ with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper bound $F(t,x,v)\leq \frac {1}{\delta }$. The key point is that the obtained estimates are uniform in the quantum parameter $0< \delta \leq 1$.
References
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- Renjun Duan and Yong Wang, The Boltzmann equation with large-amplitude initial data in bounded domains, Adv. Math. 343 (2019), 36–109. MR 3880824, DOI 10.1016/j.aim.2018.11.007
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- Xuguang Lu, On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles, J. Statist. Phys. 105 (2001), no. 1-2, 353–388. MR 1861208, DOI 10.1023/A:1012282516668
- Xuguang Lu, On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: global existence of weak solutions, J. Differential Equations 245 (2008), no. 7, 1705–1761. MR 2433484, DOI 10.1016/j.jde.2008.06.028
- Xuguang Lu, On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: averaging compactness of weak solutions, J. Stat. Phys. 124 (2006), no. 2-4, 517–547. MR 2264618, DOI 10.1007/s10955-006-9039-5
- Xuguang Lu and Bernt Wennberg, On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles, Arch. Ration. Mech. Anal. 168 (2003), no. 1, 1–34. MR 2029003, DOI 10.1007/s00205-003-0247-8
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- Zhimeng Ouyang and Lei Wu, On the quantum Boltzmann equation near Maxwellian and vacuum, J. Differential Equations 316 (2022), 471–551. MR 4377165, DOI 10.1016/j.jde.2022.01.056
- Yasushi Shizuta, On the classical solutions of the Boltzmann equation, Comm. Pure Appl. Math. 36 (1983), no. 6, 705–754. MR 720591, DOI 10.1002/cpa.3160360602
- Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (2006), no. 1-3, 417–429. MR 2209761, DOI 10.1080/03605300500361545
- Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287–339. MR 2366140, DOI 10.1007/s00205-007-0067-3
- Seiji Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad. 50 (1974), 179–184. MR 363332
- Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci. 18 (1982), no. 2, 477–519 (57–99). MR 677262, DOI 10.2977/prims/1195183569
- Seiji Ukai and Tong Yang, The Boltzmann equation in the space $L^2\cap L^\infty _\beta$: global and time-periodic solutions, Anal. Appl. (Singap.) 4 (2006), no. 3, 263–310. MR 2239407, DOI 10.1142/S0219530506000784
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- E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I, Phys. Rev. 43 (1933), 522.
- Yingkui Zhang and Xuguang Lu, Boltzmann equations with quantum effects. I. Long time behavior of spatial decay solutions, Tsinghua Sci. Technol. 7 (2002), no. 3, 215–218. MR 1908521
- Yingkui Zhang and Xuguang Lu, Boltzmann equations with quantum effects. II. Entropy identity, existence and uniqueness of spatial decay solutions, Tsinghua Sci. Technol. 7 (2002), no. 3, 219–222. MR 1908522
References
- R. Alonso, V. Bagland, L. Desvillettes, and B. Lods, About the use of entropy production for the Landau-Fermi-Dirac equation, J. Stat. Phys. 183 (2021), no. 1, Paper No. 10, 27. MR 4242165, DOI 10.1007/s10955-021-02751-z
- Céline Baranger and Clément Mouhot, Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials, Rev. Mat. Iberoamericana 21 (2005), no. 3, 819–841. MR 2231011, DOI 10.4171/RMI/436
- Nicola Bellomo, Andrzej Palczewski, and Giuseppe Toscani, Mathematical topics in nonlinear kinetic theory, World Scientific Publishing Co., Singapore, 1988. MR 996631
- Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys. 74 (1980), no. 1, 71–95. MR 575897
- Russel E. Caflisch, The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic, Comm. Math. Phys. 74 (1980), no. 2, 97–109. MR 576265
- Dingqun Deng and Renjun Duan, Spectral gap formation to kinetic equations with soft potentials in bounded domain, Comm. Math. Phys. 397 (2023), no. 3, 1441–1489. MR 4541924, DOI 10.1007/s00220-022-04519-2
- 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
- J. Dolbeault, Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles, Arch. Rational Mech. Anal. 127 (1994), no. 2, 101–131. MR 1288807, DOI 10.1007/BF00377657
- Renjun Duan, Feimin Huang, Yong Wang, and Tong Yang, Global well-posedness of the Boltzmann equation with large amplitude initial data, Arch. Ration. Mech. Anal. 225 (2017), no. 1, 375–424. MR 3634029, DOI 10.1007/s00205-017-1107-2
- Renjun Duan and Yong Wang, The Boltzmann equation with large-amplitude initial data in bounded domains, Adv. Math. 343 (2019), 36–109. MR 3880824, DOI 10.1016/j.aim.2018.11.007
- Richard S. Ellis and Mark A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl. (9) 54 (1975), 125–156. MR 609540
- Miguel Escobedo, Stéphane Mischler, and Manuel A. Valle, Homogeneous Boltzmann equation in quantum relativistic kinetic theory, Electronic Journal of Differential Equations. Monograph, vol. 4, Southwest Texas State University, San Marcos, TX, 2003. MR 1958975
- Robert T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR 1379589, DOI 10.1137/1.9781611971477
- Harold Grad, Asymptotic theory of the Boltzmann equation. II, Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962) Adv. Appl. Mech., Supplement 2, Academic Press, New York-London, 1963, pp. 26–59. MR 156656
- Yan Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal. 169 (2003), no. 4, 305–353. MR 2013332, DOI 10.1007/s00205-003-0262-9
- Yan Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J. 53 (2004), no. 4, 1081–1094. MR 2095473, DOI 10.1512/iumj.2004.53.2574
- Yan Guo, Bounded solutions for the Boltzmann equation, Quart. Appl. Math. 68 (2010), no. 1, 143–148. MR 2598886, DOI 10.1090/S0033-569X-09-01180-4
- Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal. 197 (2010), no. 3, 713–809. MR 2679358, DOI 10.1007/s00205-009-0285-y
- Ling-Bing He, Xuguang Lu, and Mario Pulvirenti, On semi-classical limit of spatially homogeneous quantum Boltzmann equation: weak convergence, Comm. Math. Phys. 386 (2021), no. 1, 143–223. MR 4287184, DOI 10.1007/s00220-021-04029-7
- Ning Jiang, Linjie Xiong, and Kai Zhou, The incompressible Navier-Stokes-Fourier limit from Boltzmann-Fermi-Dirac equation, J. Differential Equations 308 (2022), 77–129. MR 4339598, DOI 10.1016/j.jde.2021.10.061
- N. Jiang and K. Zhou, The compressible Euler and acoustic limits from quantum Boltzmann equation with Fermi-Dirac statistics, arXiv:2111.07784.
- Zongguang Li, Large amplitude solutions in $L^p_vL^\infty _TL^\infty _x$ to the Boltzmann equation for soft potentials, SIAM J. Math. Anal. 54 (2022), no. 4, 4163–4197. MR 4451308, DOI 10.1137/21M1443790
- Tai-Ping Liu, Tong Yang, and Shih-Hsien Yu, Energy method for Boltzmann equation, Phys. D 188 (2004), no. 3-4, 178–192. MR 2043729, DOI 10.1016/j.physd.2003.07.011
- Xuguang Lu, On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles, J. Statist. Phys. 105 (2001), no. 1-2, 353–388. MR 1861208, DOI 10.1023/A:1012282516668
- Xuguang Lu, On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: global existence of weak solutions, J. Differential Equations 245 (2008), no. 7, 1705–1761. MR 2433484, DOI 10.1016/j.jde.2008.06.028
- Xuguang Lu, On the Boltzmann equation for Fermi-Dirac particles with very soft potentials: averaging compactness of weak solutions, J. Stat. Phys. 124 (2006), no. 2-4, 517–547. MR 2264618, DOI 10.1007/s10955-006-9039-5
- Xuguang Lu and Bernt Wennberg, On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles, Arch. Ration. Mech. Anal. 168 (2003), no. 1, 1–34. MR 2029003, DOI 10.1007/s00205-003-0247-8
- Takaaki Nishida and Kazuo Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 1, 229–239. MR 432105, DOI 10.2977/prims/1195190965
- Zhimeng Ouyang and Lei Wu, On the quantum Boltzmann equation near Maxwellian and vacuum, J. Differential Equations 316 (2022), 471–551. MR 4377165, DOI 10.1016/j.jde.2022.01.056
- Yasushi Shizuta, On the classical solutions of the Boltzmann equation, Comm. Pure Appl. Math. 36 (1983), no. 6, 705–754. MR 720591, DOI 10.1002/cpa.3160360602
- Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations 31 (2006), no. 1-3, 417–429. MR 2209761, DOI 10.1080/03605300500361545
- Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal. 187 (2008), no. 2, 287–339. MR 2366140, DOI 10.1007/s00205-007-0067-3
- Seiji Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad. 50 (1974), 179–184. MR 363332
- Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci. 18 (1982), no. 2, 477–519 (57–99). MR 677262, DOI 10.2977/prims/1195183569
- Seiji Ukai and Tong Yang, The Boltzmann equation in the space $L^2\cap L^\infty _\beta$: global and time-periodic solutions, Anal. Appl. (Singap.) 4 (2006), no. 3, 263–310. MR 2239407, DOI 10.1142/S0219530506000784
- E. A. Uehling, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. II, Phys. Rev. 46 (1934), 917.
- E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I, Phys. Rev. 43 (1933), 522.
- Yingkui Zhang and Xuguang Lu, Boltzmann equations with quantum effects. I. Long time behavior of spatial decay solutions, Tsinghua Sci. Technol. 7 (2002), no. 3, 215–218. MR 1908521
- Yingkui Zhang and Xuguang Lu, Boltzmann equations with quantum effects. II. Entropy identity, existence and uniqueness of spatial decay solutions, Tsinghua Sci. Technol. 7 (2002), no. 3, 219–222. MR 1908522
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Additional Information
Zongguang Li
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People’s Republic of China
MR Author ID:
1257574
Email:
zgli@math.cuhk.edu.hk
Keywords:
Quantum Boltzmann equation,
large amplitude solutions,
existence,
uniform estimates
Received by editor(s):
March 6, 2023
Received by editor(s) in revised form:
September 21, 2023
Published electronically:
October 27, 2023
Additional Notes:
This work was supported by the Hong Kong PhD Fellowship Scheme
Article copyright:
© Copyright 2023
Brown University
