American Institute of Mathematical Sciences

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September  2013, 6(3): 601-623. doi: 10.3934/krm.2013.6.601

Large time behavior of the solution to the Landau Equation with specular reflective boundary condition

Cong He 1, and  Hongjun Yu 2,

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China
2. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631

Received  November 2012 Revised  March 2013 Published  May 2013

In this paper a half space problem for the one-dimensional Landau equation with specular reflective boundary condition is investigated. We show that the solution to the Landau equation converges to a global Maxwellian. Moreover, a time-decay rate is also obtained.
Keywords: Landau equation, time decay rate., global solution.
Mathematics Subject Classification: Primary: 35A01, 82C40; Secondary: 35A0.
Citation: Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601
References:
[1]

R. Alexandre, 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. Ration. Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0.  Google Scholar

[2]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-uniform Gases," Cambridge, 1952. Google Scholar

[3]

P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Ration. Mech. Anal., 138 (1997), 137-167. doi: 10.1007/s002050050038.  Google Scholar

[4]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials I, II, Comm. P.D.E., 25 (2000), 179-298. doi: 10.1080/03605300008821512.  Google Scholar

[5]

R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint, 2012. Google Scholar

[6]

Y. Guo, The Landau equation in periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.  Google Scholar

[7]

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

[8]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Rat. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y.  Google Scholar

[9]

Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812. doi: 10.1090/S0894-0347-2011-00722-4.  Google Scholar

[10]

L. Hsiao and H. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math., 65 (2007), 281-315.  Google Scholar

[11]

F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014.  Google Scholar

[12]

F. Huang and Y. Wang, Large time behavior of the solutions to the Boltzmann equation with specular reflective boundary condition, J. Differential Equations, 240 (2007), 399-429. doi: 10.1016/j.jde.2007.05.032.  Google Scholar

[13]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.  Google Scholar

[14]

P.-L. Lions, On Boltzmann and Landau equations, Phil. Trans. R. Soc. Lond. Ser. A., 346 (1994), 191-204. doi: 10.1098/rsta.1994.0018.  Google Scholar

[15]

T.-P. Liu, and Z. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84.  Google Scholar

[16]

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

[17]

T.-P. Liu, T. Yang, S.-H. Yu and H.-J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181 (2)(2006), 333-371. doi: 10.1007/s00205-005-0414-1.  Google Scholar

[18]

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.  Google Scholar

[19]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13. doi: 10.1007/BF03167088.  Google Scholar

[20]

A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335. doi: 10.1007/BF02101095.  Google Scholar

[21]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.  Google Scholar

[22]

R. M. Strain and K. Y. Zhu, The Vlasov-Poisson-Landau system in $R^3$, preprint, 2012. Google Scholar

[23]

S. Ukai and T. Yang, "Mathematical Theory of Boltzmann Equation," Lecture Notes Series, No. 8, Liu Bie Ju Centre for Math. Sci., City University of Hong Kong, 2006. Google Scholar

[24]

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

[25]

C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Diff. Eq., 1 (1996), 793-816.  Google Scholar

[26]

Y.-J. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $R^3$, SIAM Math. Anal., 44 (2012), 3281-3323. doi: 10.1137/120879129.  Google Scholar

[27]

Y. Wang and Z. Jiang, The specular reflective boundary problem for the Boltzmann equation with soft potentials, Nonlinear Analysis, 75 (2012), 786-805. doi: 10.1016/j.na.2011.09.011.  Google Scholar

[28]

Z. Xin, T. Yang and H. Yu, The Boltzmann equation with soft potentials near the local Maxwellian, Arch. Ration. Mech. Anal., 206 (2012), 239-296. doi: 10.1007/s00205-012-0535-2.  Google Scholar

[29]

Z. Xin, T. Yang and H. Yu, Nonlinear stability of rarefaction waves for the Landau equation, preprint, 2010. Google Scholar

[30]

T. Yang and H.-J. Zhao, A half-space problem for the Boltzmann equation with specular reflection boundary condition, Comm. Math. Phys., 255 (2005), 683-726. doi: 10.1007/s00220-004-1278-1.  Google Scholar

[31]

T. Yang and H.-J. Zhao, A new energy method for the Boltzmann equation, J. Math. Phys., 47 (2006), 053301, 19 pp. doi: 10.1063/1.2195528.  Google Scholar

[32]

H. Yu, Cauchy problem of the Vlasov-Poisson-Landau system, preprint, 2012. Google Scholar

show all references

References:
[1]

R. Alexandre, 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. Ration. Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0.  Google Scholar

[2]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-uniform Gases," Cambridge, 1952. Google Scholar

[3]

P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Ration. Mech. Anal., 138 (1997), 137-167. doi: 10.1007/s002050050038.  Google Scholar

[4]

L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials I, II, Comm. P.D.E., 25 (2000), 179-298. doi: 10.1080/03605300008821512.  Google Scholar

[5]

R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint, 2012. Google Scholar

[6]

Y. Guo, The Landau equation in periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9.  Google Scholar

[7]

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

[8]

Y. Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Rat. Mech. Anal., 197 (2010), 713-809. doi: 10.1007/s00205-009-0285-y.  Google Scholar

[9]

Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812. doi: 10.1090/S0894-0347-2011-00722-4.  Google Scholar

[10]

L. Hsiao and H. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math., 65 (2007), 281-315.  Google Scholar

[11]

F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014.  Google Scholar

[12]

F. Huang and Y. Wang, Large time behavior of the solutions to the Boltzmann equation with specular reflective boundary condition, J. Differential Equations, 240 (2007), 399-429. doi: 10.1016/j.jde.2007.05.032.  Google Scholar

[13]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.  Google Scholar

[14]

P.-L. Lions, On Boltzmann and Landau equations, Phil. Trans. R. Soc. Lond. Ser. A., 346 (1994), 191-204. doi: 10.1098/rsta.1994.0018.  Google Scholar

[15]

T.-P. Liu, and Z. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws, Asian J. Math., 1 (1997), 34-84.  Google Scholar

[16]

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

[17]

T.-P. Liu, T. Yang, S.-H. Yu and H.-J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181 (2)(2006), 333-371. doi: 10.1007/s00205-005-0414-1.  Google Scholar

[18]

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.  Google Scholar

[19]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13. doi: 10.1007/BF03167088.  Google Scholar

[20]

A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335. doi: 10.1007/BF02101095.  Google Scholar

[21]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3.  Google Scholar

[22]

R. M. Strain and K. Y. Zhu, The Vlasov-Poisson-Landau system in $R^3$, preprint, 2012. Google Scholar

[23]

S. Ukai and T. Yang, "Mathematical Theory of Boltzmann Equation," Lecture Notes Series, No. 8, Liu Bie Ju Centre for Math. Sci., City University of Hong Kong, 2006. Google Scholar

[24]

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

[25]

C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Diff. Eq., 1 (1996), 793-816.  Google Scholar

[26]

Y.-J. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $R^3$, SIAM Math. Anal., 44 (2012), 3281-3323. doi: 10.1137/120879129.  Google Scholar

[27]

Y. Wang and Z. Jiang, The specular reflective boundary problem for the Boltzmann equation with soft potentials, Nonlinear Analysis, 75 (2012), 786-805. doi: 10.1016/j.na.2011.09.011.  Google Scholar

[28]

Z. Xin, T. Yang and H. Yu, The Boltzmann equation with soft potentials near the local Maxwellian, Arch. Ration. Mech. Anal., 206 (2012), 239-296. doi: 10.1007/s00205-012-0535-2.  Google Scholar

[29]

Z. Xin, T. Yang and H. Yu, Nonlinear stability of rarefaction waves for the Landau equation, preprint, 2010. Google Scholar

[30]

T. Yang and H.-J. Zhao, A half-space problem for the Boltzmann equation with specular reflection boundary condition, Comm. Math. Phys., 255 (2005), 683-726. doi: 10.1007/s00220-004-1278-1.  Google Scholar

[31]

T. Yang and H.-J. Zhao, A new energy method for the Boltzmann equation, J. Math. Phys., 47 (2006), 053301, 19 pp. doi: 10.1063/1.2195528.  Google Scholar

[32]

H. Yu, Cauchy problem of the Vlasov-Poisson-Landau system, preprint, 2012. Google Scholar

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