American Institute of Mathematical Sciences

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April  2016, 36(4): 2229-2256. doi: 10.3934/dcds.2016.36.2229

On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces

Hao Tang 1, and  Zhengrong Liu 2,

1. 

Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
2. 

Department of Mathematics, South China University of Technology, Guangzhou 510640

Received  March 2015 Revised  May 2015 Published  September 2015

In this paper, motivated by [13], we use the Littlewood-Paley theory to investigate the Cauchy problem of the Boltzmann equation. When the initial data is a small perturbation of an equilibrium state, under the Grad's angular cutoff assumption, we obtain the unique global strong solution to the Boltzmann equation for the hard potential case in the Chemin-Lerner type spaces $C([0,\infty);\widetilde{L}^{2}_{\xi}(B_{2,r}^{s}))$ with $1\leq r\leq2$ and $s>3/2$ or $s=3/2$ and $r=1$. Besides, we also prove the Lipschitz continuity of the solution map. Our results extend some previous works on the Boltzmann equation in Sobolev spaces.
Keywords: Boltzmann equation, Littlewood-Paley theory., angular cutoff assumption, Cauchy problem, hard potential.
Mathematics Subject Classification: Primary: 35Q20, 35A01; Secondary: 35B3.
Citation: Hao Tang, Zhengrong Liu. On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2229-2256. doi: 10.3934/dcds.2016.36.2229
References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.

[2]

R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Rational Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0.

[3]

R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010. doi: 10.1016/j.jfa.2011.10.007.

[4]

R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinet. Relat. Models., 6 (2013), 1011-1041. doi: 10.3934/krm.2013.6.1011.

[5]

D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces, J. Math.Pures Appl., 101 (2014), 495-551. doi: 10.1016/j.matpur.2013.06.012.

[6]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[8]

J. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998.

[9]

J. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Diff. Equs., 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.

[10]

R. Duan, The Boltzmann equation near equilibrium states in $R^n$, Methods Appl. Anal., 14 (2007), 227-249. doi: 10.4310/MAA.2007.v14.n3.a2.

[11]

R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189. doi: 10.1088/0951-7715/24/8/003.

[12]

R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in $L^2(H_x^N)$, J. Diff. Equs., 244 (2008), 3204-3234. doi: 10.1016/j.jde.2007.11.006.

[13]

R. Duan, S. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation, arXiv:1310.2727.

[14]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6.

[15]

R. Duan and T. Yang, Stability of the one species Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal., 41 (2010), 2353-2387. doi: 10.1137/090745775.

[16]

N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab., 25 (2015), 860-897. doi: 10.1214/14-AAP1012.

[17]

R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[18]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.

[19]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040.

[20]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353. doi: 10.1007/s00205-003-0262-9.

[21]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.

[22]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z.

[23]

T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, in: Spectral Theory and Differential Equations, in: Lecture Notes in Math., Springer-Verlag, Berlin, 448 (1975), 25-70.

[24]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D., 188 (2004), 178-192. doi: 10.1016/j.physd.2003.07.011.

[25]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.

[26]

H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley in Chichester, New York, 1987.

[27]

V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$, Adv. in Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012.

[28]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567. doi: 10.1007/s00220-006-0109-y.

[29]

T. Tao, Global well-posedness of the Benjamin-ono equation in $H^1(\mathbbR)$, J. Hyperbolic Diff. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.

[30]

H. Tang and Z. Liu, Continuous properties of the solution map for the Euler equations, J. Math. Phys., 55 (2014), 031504, 10pp. doi: 10.1063/1.4867622.

[31]

H. Tang and Z. Liu, Well-posedness of the modfied Camassa-Holm equation in Besov spaces, Z. Angew. Math. Phys., 66 (2015), 1559-1580. doi: 10.1007/s00033-014-0483-9.

[32]

H. Tang, Y. Zhao and Z. Liu, A note on the solution map for the periodic Camassa-Holm equation, Appl. Anal., 93 (2014), 1745-1760. doi: 10.1080/00036811.2013.847923.

[33]

H. Tang, S. Shi and Z. Liu, The dependences on initial data for the b-family equation in critical Besov space, Monatsh. Math., 177 (2015), 471-492. doi: 10.1007/s00605-014-0673-8.

[34]

H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[35]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.

[36]

S. Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 37-96. doi: 10.1016/S0168-2024(08)70128-0.

[37]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310. doi: 10.1142/S0219530506000784.

[38]

S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces, Anal. Appl., 3 (2005), 157-193. doi: 10.1142/S0219530505000522.

[39]

S. Ukai, T. Yang and H. Zhao, Convergence rate to stationary solutions for Boltzmann equation with external force, Chinese Ann. Math. Ser. B., 27 (2006), 363-378. doi: 10.1007/s11401-005-0199-4.

[40]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.

[41]

L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the fokker-planck-boltzmann equation, Kinet. Relat. Models., 7 (2014), 169-194. doi: 10.3934/krm.2014.7.169.

show all references

References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.

[2]

R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Rational Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0.

[3]

R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010. doi: 10.1016/j.jfa.2011.10.007.

[4]

R. Alexandre R. Y. Morimoto, S. Ukai, C. J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinet. Relat. Models., 6 (2013), 1011-1041. doi: 10.3934/krm.2013.6.1011.

[5]

D. Arsénio and N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces, J. Math.Pures Appl., 101 (2014), 495-551. doi: 10.1016/j.matpur.2013.06.012.

[6]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[7]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[8]

J. Chemin, Perfect Incompressible Fluids, Clarendon Press, Oxford, 1998.

[9]

J. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Diff. Equs., 121 (1995), 314-328. doi: 10.1006/jdeq.1995.1131.

[10]

R. Duan, The Boltzmann equation near equilibrium states in $R^n$, Methods Appl. Anal., 14 (2007), 227-249. doi: 10.4310/MAA.2007.v14.n3.a2.

[11]

R. Duan, Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189. doi: 10.1088/0951-7715/24/8/003.

[12]

R. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: global existence and uniform stability in $L^2(H_x^N)$, J. Diff. Equs., 244 (2008), 3204-3234. doi: 10.1016/j.jde.2007.11.006.

[13]

R. Duan, S. Liu and J. Xu, Global well-posedness in spatially critical Besov space for the Boltzmann equation, arXiv:1310.2727.

[14]

R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6.

[15]

R. Duan and T. Yang, Stability of the one species Vlasov-Poisson-Boltzmann system. SIAM J. Math. Anal., 41 (2010), 2353-2387. doi: 10.1137/090745775.

[16]

N. Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab., 25 (2015), 860-897. doi: 10.1214/14-AAP1012.

[17]

R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[18]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.

[19]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040.

[20]

Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353. doi: 10.1007/s00205-003-0262-9.

[21]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.

[22]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: 10.1007/s00222-003-0301-z.

[23]

T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, in: Spectral Theory and Differential Equations, in: Lecture Notes in Math., Springer-Verlag, Berlin, 448 (1975), 25-70.

[24]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D., 188 (2004), 178-192. doi: 10.1016/j.physd.2003.07.011.

[25]

T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.

[26]

H. J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley in Chichester, New York, 1987.

[27]

V. Sohinger and R. M. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $\mathbbR_x^n$, Adv. in Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012.

[28]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567. doi: 10.1007/s00220-006-0109-y.

[29]

T. Tao, Global well-posedness of the Benjamin-ono equation in $H^1(\mathbbR)$, J. Hyperbolic Diff. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.

[30]

H. Tang and Z. Liu, Continuous properties of the solution map for the Euler equations, J. Math. Phys., 55 (2014), 031504, 10pp. doi: 10.1063/1.4867622.

[31]

H. Tang and Z. Liu, Well-posedness of the modfied Camassa-Holm equation in Besov spaces, Z. Angew. Math. Phys., 66 (2015), 1559-1580. doi: 10.1007/s00033-014-0483-9.

[32]

H. Tang, Y. Zhao and Z. Liu, A note on the solution map for the periodic Camassa-Holm equation, Appl. Anal., 93 (2014), 1745-1760. doi: 10.1080/00036811.2013.847923.

[33]

H. Tang, S. Shi and Z. Liu, The dependences on initial data for the b-family equation in critical Besov space, Monatsh. Math., 177 (2015), 471-492. doi: 10.1007/s00605-014-0673-8.

[34]

H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[35]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: 10.3792/pja/1195519027.

[36]

S. Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 37-96. doi: 10.1016/S0168-2024(08)70128-0.

[37]

S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310. doi: 10.1142/S0219530506000784.

[38]

S. Ukai, T. Yang and H. Zhao, Global solutions to the Boltzmann equation with external forces, Anal. Appl., 3 (2005), 157-193. doi: 10.1142/S0219530505000522.

[39]

S. Ukai, T. Yang and H. Zhao, Convergence rate to stationary solutions for Boltzmann equation with external force, Chinese Ann. Math. Ser. B., 27 (2006), 363-378. doi: 10.1007/s11401-005-0199-4.

[40]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, North-Holland, Amsterdam, I (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.

[41]

L. Xiong, T. Wang and L. Wang, Global existence and decay of solutions to the fokker-planck-boltzmann equation, Kinet. Relat. Models., 7 (2014), 169-194. doi: 10.3934/krm.2014.7.169.

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