March  2011, 4(1): 17-40. doi: 10.3934/krm.2011.4.17

Bounded solutions of the Boltzmann equation in the whole space

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

2. 

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501

3. 

17-26 Iwasaki, Hodogaya, Yokohama 240-0015

4. 

School of Mathematics, Wuhan University, 430072, Wuhan

5. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  October 2010 Revised  October 2010 Published  January 2011

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.
Citation: Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17
References:
[1]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff,, J. Stat. Physics, 104 (2001), 327.  doi: 10.1023/A:1010317913642.  Google Scholar

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327.  doi: 10.1007/s002050000083.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation,, Arch. Rational Mech. Anal., 198 (2010), 39.  doi: 10.1007/s00205-010-0290-1.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff,, to appear in Comm. Math. Phys., ().   Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff,, C. R. Math. Acad. Sci. Paris, ().  doi: 10.1016/j.crma.2010.07.008.  Google Scholar

[6]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Boltzmann equation without angular cutoff in the whole space: I. Global existence for soft potential,, Preprint HAL, ().   Google Scholar

[7]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, The Boltzmann equation without angular cutoff in the whole space: II. Global existence for hard potential,, to appear in Analysis and Applications, ().   Google Scholar

[8]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions,, Preprint HAL, ().   Google Scholar

[9]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interaction,, Communications on Pure and Applied Mathematics, 55 (2002), 30.  doi: 10.1002/cpa.10012.  Google Scholar

[10]

C. Cercignani, "The Boltzmann Equation and its Applications,", Applied mathematical sciences, 67 (1988).   Google Scholar

[11]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Applied mathematical sciences, 106 (1994).   Google Scholar

[12]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability,, Ann. Math., 130 (1989), 321.  doi: 10.2307/1971423.  Google Scholar

[13]

H. Grad, Asymptotic Theory of the Boltzmann Equation II,, in: Laurmann J. A. (ed.) Rarefied Gas Dynamics, 1 (1963), 26.   Google Scholar

[14]

P.-T. Gressman and R.-M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions,, Proc. Nat. Acad. Sci., 107 (2010), 5744.  doi: 10.1073/pnas.1001185107.  Google Scholar

[15]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Maths. J., 53 (2004), 1081.  doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[16]

Y. Guo, Bounded solutions for the Boltzmann equation,, Quaterly of Applied Mathematics, LXVIII (2010), 143.   Google Scholar

[17]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181.  doi: 10.1007/BF00280740.  Google Scholar

[18]

P. L. Lions, Regularity and compactness for Boltzmann collision kernels without angular cut-off,, C. R. Acad. Sci. Paris Series I, 326 (1998), 37.   Google Scholar

[19]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation,, Phys. D, 188 (2004), 178.  doi: 10.1016/j.physd.2003.07.011.  Google Scholar

[20]

Y. P. Pao, Boltzmann collision operator with inverse power intermolecular potential,, I, 27 (1974).   Google Scholar

[21]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179.  doi: 10.3792/pja/1195519027.  Google Scholar

[22]

S. Ukai, Les solutions globales de l'equation de Boltzmann dans l'espace tout entier et dans le demi-espace,, C. R. Acad. Sci. Paris Ser. A-B, 282 (1976).   Google Scholar

[23]

S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,, Japan J. Appl. Math., 1 (1984), 141.   Google Scholar

[24]

S. Ukai, Solutions of the Boltzmann equation,, in, 18 (1986), 37.   Google Scholar

[25]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002).   Google Scholar

show all references

References:
[1]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff,, J. Stat. Physics, 104 (2001), 327.  doi: 10.1023/A:1010317913642.  Google Scholar

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327.  doi: 10.1007/s002050000083.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation,, Arch. Rational Mech. Anal., 198 (2010), 39.  doi: 10.1007/s00205-010-0290-1.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff,, to appear in Comm. Math. Phys., ().   Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff,, C. R. Math. Acad. Sci. Paris, ().  doi: 10.1016/j.crma.2010.07.008.  Google Scholar

[6]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Boltzmann equation without angular cutoff in the whole space: I. Global existence for soft potential,, Preprint HAL, ().   Google Scholar

[7]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, The Boltzmann equation without angular cutoff in the whole space: II. Global existence for hard potential,, to appear in Analysis and Applications, ().   Google Scholar

[8]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions,, Preprint HAL, ().   Google Scholar

[9]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interaction,, Communications on Pure and Applied Mathematics, 55 (2002), 30.  doi: 10.1002/cpa.10012.  Google Scholar

[10]

C. Cercignani, "The Boltzmann Equation and its Applications,", Applied mathematical sciences, 67 (1988).   Google Scholar

[11]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Applied mathematical sciences, 106 (1994).   Google Scholar

[12]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability,, Ann. Math., 130 (1989), 321.  doi: 10.2307/1971423.  Google Scholar

[13]

H. Grad, Asymptotic Theory of the Boltzmann Equation II,, in: Laurmann J. A. (ed.) Rarefied Gas Dynamics, 1 (1963), 26.   Google Scholar

[14]

P.-T. Gressman and R.-M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions,, Proc. Nat. Acad. Sci., 107 (2010), 5744.  doi: 10.1073/pnas.1001185107.  Google Scholar

[15]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Maths. J., 53 (2004), 1081.  doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[16]

Y. Guo, Bounded solutions for the Boltzmann equation,, Quaterly of Applied Mathematics, LXVIII (2010), 143.   Google Scholar

[17]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Rational Mech. Anal., 58 (1975), 181.  doi: 10.1007/BF00280740.  Google Scholar

[18]

P. L. Lions, Regularity and compactness for Boltzmann collision kernels without angular cut-off,, C. R. Acad. Sci. Paris Series I, 326 (1998), 37.   Google Scholar

[19]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation,, Phys. D, 188 (2004), 178.  doi: 10.1016/j.physd.2003.07.011.  Google Scholar

[20]

Y. P. Pao, Boltzmann collision operator with inverse power intermolecular potential,, I, 27 (1974).   Google Scholar

[21]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation,, Proc. Japan Acad., 50 (1974), 179.  doi: 10.3792/pja/1195519027.  Google Scholar

[22]

S. Ukai, Les solutions globales de l'equation de Boltzmann dans l'espace tout entier et dans le demi-espace,, C. R. Acad. Sci. Paris Ser. A-B, 282 (1976).   Google Scholar

[23]

S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,, Japan J. Appl. Math., 1 (1984), 141.   Google Scholar

[24]

S. Ukai, Solutions of the Boltzmann equation,, in, 18 (1986), 37.   Google Scholar

[25]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002).   Google Scholar

[1]

Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291

[2]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[3]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[4]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[5]

Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326

[6]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[7]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[8]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048

[9]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[11]

Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088

[12]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005

[13]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400

[14]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001

[15]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

[16]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021005

[17]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003

[18]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021002

[19]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[20]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (7)

[Back to Top]