-
Previous Article
Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators
- KRM Home
- This Issue
-
Next Article
On the uniqueness for coagulation and multiple fragmentation equation
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[15] |
T.-P. Liu, and Z. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws,, Asian J. Math., 1 (1997), 34.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence,, Adv. Diff. Eq., 1 (1996), 793.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[15] |
T.-P. Liu, and Z. Xin, Pointwise decay to contact discontinuities for systems of viscous conservation laws,, Asian J. Math., 1 (1997), 34.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence,, Adv. Diff. Eq., 1 (1996), 793.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[32] |
H. Yu, Cauchy problem of the Vlasov-Poisson-Landau system,, preprint, (2012). Google Scholar |
[1] |
Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267 |
[2] |
Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019230 |
[3] |
Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic & Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043 |
[4] |
Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks & Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002 |
[5] |
Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 |
[6] |
Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307 |
[7] |
Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089 |
[8] |
Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825 |
[9] |
Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133 |
[10] |
Ellen Baake, Michael Baake, Majid Salamat. The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63 |
[11] |
Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic & Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57 |
[12] |
Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Irena Lasiecka, Flávio A. Falcão Nascimento. Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1987-2011. doi: 10.3934/dcdsb.2014.19.1987 |
[13] |
Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21 |
[14] |
Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169 |
[15] |
Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100 |
[16] |
Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 |
[17] |
J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 |
[18] |
Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885 |
[19] |
Nan Chen, Cheng Wang, Steven Wise. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1689-1711. doi: 10.3934/dcdsb.2016018 |
[20] |
Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 |
2018 Impact Factor: 1.38
Tools
Metrics
Other articles
by authors
[Back to Top]