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

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.
Citation: Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic and 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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

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

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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

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

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