# American Institute of Mathematical Sciences

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 & 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.  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.  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.  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.  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.  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.  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.  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.   Google Scholar [11] F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions,, Adv. Math., 219 (2008), 1246.  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.  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.  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.  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.   Google Scholar [16] T.-P. Liu, T. Yang and S.-H. Yu, Energy method for the Boltzmann equation,, Physica D, 188 (2004), 178.  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 (2006), 333.  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.  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.  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.  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.  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, (2006).   Google Scholar [24] C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71.  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.   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.  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.  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.  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.  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).  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.  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.  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.  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.  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.  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.  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.  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.   Google Scholar [11] F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions,, Adv. Math., 219 (2008), 1246.  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.  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.  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.  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.   Google Scholar [16] T.-P. Liu, T. Yang and S.-H. Yu, Energy method for the Boltzmann equation,, Physica D, 188 (2004), 178.  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 (2006), 333.  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.  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.  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.  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.  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, (2006).   Google Scholar [24] C. Villani, A review of mathematical topics in collisional kinetic theory,, in, (2002), 71.  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.   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.  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.  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.  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.  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).  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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