June  2019, 12(3): 637-679. doi: 10.3934/krm.2019025

Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation

1. 

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

2. 

CEMS, HCMS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Yi Wang

Received  September 2018 Revised  November 2018 Published  February 2019

Fund Project: The first author is supported by NSFC Grant No. 11601031. The second author is supported by NSFC grants No. 11671385 and 11688101 and CAS Interdisciplinary Innovation Team

We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation, based on the micro-macro decomposition introduced in [24,22] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of planar rarefaction wave for 3D Boltzmann equation, while the corresponding results for the shock and contact discontinuities are still completely open.

Citation: Teng Wang, Yi Wang. Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 637-679. doi: 10.3934/krm.2019025
References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003. Google Scholar

[2]

J. Brezina, E. Chiodaroli and O. Kreml, On contact discontinuities in multi-dimensional isentropic Euler equations, Electronic Journal of Differential Equations, (2018), Paper No. 94, 11 pp. Google Scholar

[3]

R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. doi: 10.1007/BF01206009. Google Scholar

[4]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edition, Cambridge University Press, 1990. Google Scholar

[5]

G. Q. Chen and J. Chen, Stability of rarefaction waves and vacuum states for the multidimensional Euler equations, J. Hyperbolic Differ. Equ., 4 (2007), 105-122. doi: 10.1142/S0219891607001070. Google Scholar

[6]

E. ChiodaroliC. DeLellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math., 68 (2015), 1157-1190. doi: 10.1002/cpa.21537. Google Scholar

[7]

E. Chiodaroli and O. Kreml, Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations, Nonlinearity, 31 (2018), 1441-1460. doi: 10.1088/1361-6544/aaa10d. Google Scholar

[8]

C. DeLellis and L. Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math.(2), 170 (2009), 1417–1436. doi: 10.4007/annals.2009.170.1417. Google Scholar

[9]

E. Feireisl and O. Kreml, Uniqueness of rarefaction waves in multidimensional compressible Euler system, J. Hyperbolic Differ. Equ., 12 (2015), 489-499. doi: 10.1142/S0219891615500149. Google Scholar

[10]

E. FeireislO. Kreml and A. Vasseur, Stability of the isentropic Riemann solutions of the full multidimensional Euler system, SIAM J. Math. Anal., 47 (2015), 2416-2425. doi: 10.1137/140999827. Google Scholar

[11]

H. Grad, Asymptotic theory of the boltzmann equation Ⅱ, in Rarefied Gas Dynamics (J. A. Laurmann, ed.), Academic Press, New York, 1 (1963), 26–59. Google Scholar

[12]

F. M. HuangY. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phy., 295 (2010), 293-326. doi: 10.1007/s00220-009-0966-2. Google Scholar

[13]

F. M. HuangY. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: Ⅰ. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728. doi: 10.3934/krm.2010.3.685. Google Scholar

[14]

F. M. HuangY. Wang and T. Yang, Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to Riemann problem, Arch. Rational Mech. Anal., 203 (2012), 379-413. doi: 10.1007/s00205-011-0450-y. Google Scholar

[15]

F. M. HuangY. WangY. Wang and T. Yang, The limit of the Boltzmann equation to the Euler equations for Riemann problems, SIAM J. Math. Anal., 45 (2013), 1741-1811. doi: 10.1137/120898541. Google Scholar

[16]

F. M. HuangZ. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014. Google Scholar

[17]

F. M. Huang and T. Yang, Stability of contact discontinuity for the Boltzmann equation, J. Differ. Equations, 229 (2006), 698-742. doi: 10.1016/j.jde.2005.12.004. Google Scholar

[18]

C. Klingenberg and S. Markfelder, The Riemann problem for the multi- dimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch Rational Mech Anal., 227 (2018), 967-994. doi: 10.1007/s00205-017-1179-z. Google Scholar

[19]

P. D. Lax, Hyperbolic systems of conservation laws, Ⅱ., Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. Google Scholar

[20]

L. A. LiT. Wang and Y. Wang, Stability of planar rarefaction wave to 3D full compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 230 (2018), 911-937. doi: 10.1007/s00205-018-1260-2. Google Scholar

[21]

L. A. Li and Y. Wang, Stability of the planar rarefaction wave to the two-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 4937-4963. doi: 10.1137/18M1171059. Google Scholar

[22]

T. P. LiuT. 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

[23]

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

[24]

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

[25]

T. P. Liu and S. H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997. doi: 10.1007/s00205-013-0640-x. Google Scholar

[26]

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

[27]

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

[28]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[29]

T. Wang and Y. Wang, Stability of superposition of two viscous shock waves for the Boltzmann equation, SIAM J. Math. Anal., 47 (2015), 1070-1120. doi: 10.1137/140963005. Google Scholar

[30]

Z. P. Xin, Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions, Trans. Amer. Math. Soc., 319 (1990), 805-820. doi: 10.1090/S0002-9947-1990-0970270-8. Google Scholar

[31]

Z. P. Xin and H. H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations, J. Differential Equations, 249 (2010), 827-871. doi: 10.1016/j.jde.2010.03.011. Google Scholar

[32]

S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations, Commun. Pure Appl. Math, 58 (2005), 409-443. doi: 10.1002/cpa.20027. Google Scholar

[33]

S. H. Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1041-1118. doi: 10.1090/S0894-0347-2010-00671-6. Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003. Google Scholar

[2]

J. Brezina, E. Chiodaroli and O. Kreml, On contact discontinuities in multi-dimensional isentropic Euler equations, Electronic Journal of Differential Equations, (2018), Paper No. 94, 11 pp. Google Scholar

[3]

R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. doi: 10.1007/BF01206009. Google Scholar

[4]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edition, Cambridge University Press, 1990. Google Scholar

[5]

G. Q. Chen and J. Chen, Stability of rarefaction waves and vacuum states for the multidimensional Euler equations, J. Hyperbolic Differ. Equ., 4 (2007), 105-122. doi: 10.1142/S0219891607001070. Google Scholar

[6]

E. ChiodaroliC. DeLellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math., 68 (2015), 1157-1190. doi: 10.1002/cpa.21537. Google Scholar

[7]

E. Chiodaroli and O. Kreml, Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations, Nonlinearity, 31 (2018), 1441-1460. doi: 10.1088/1361-6544/aaa10d. Google Scholar

[8]

C. DeLellis and L. Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math.(2), 170 (2009), 1417–1436. doi: 10.4007/annals.2009.170.1417. Google Scholar

[9]

E. Feireisl and O. Kreml, Uniqueness of rarefaction waves in multidimensional compressible Euler system, J. Hyperbolic Differ. Equ., 12 (2015), 489-499. doi: 10.1142/S0219891615500149. Google Scholar

[10]

E. FeireislO. Kreml and A. Vasseur, Stability of the isentropic Riemann solutions of the full multidimensional Euler system, SIAM J. Math. Anal., 47 (2015), 2416-2425. doi: 10.1137/140999827. Google Scholar

[11]

H. Grad, Asymptotic theory of the boltzmann equation Ⅱ, in Rarefied Gas Dynamics (J. A. Laurmann, ed.), Academic Press, New York, 1 (1963), 26–59. Google Scholar

[12]

F. M. HuangY. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phy., 295 (2010), 293-326. doi: 10.1007/s00220-009-0966-2. Google Scholar

[13]

F. M. HuangY. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: Ⅰ. Superposition of rarefaction waves and contact discontinuity, Kinet. Relat. Models, 3 (2010), 685-728. doi: 10.3934/krm.2010.3.685. Google Scholar

[14]

F. M. HuangY. Wang and T. Yang, Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to Riemann problem, Arch. Rational Mech. Anal., 203 (2012), 379-413. doi: 10.1007/s00205-011-0450-y. Google Scholar

[15]

F. M. HuangY. WangY. Wang and T. Yang, The limit of the Boltzmann equation to the Euler equations for Riemann problems, SIAM J. Math. Anal., 45 (2013), 1741-1811. doi: 10.1137/120898541. Google Scholar

[16]

F. M. HuangZ. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014. Google Scholar

[17]

F. M. Huang and T. Yang, Stability of contact discontinuity for the Boltzmann equation, J. Differ. Equations, 229 (2006), 698-742. doi: 10.1016/j.jde.2005.12.004. Google Scholar

[18]

C. Klingenberg and S. Markfelder, The Riemann problem for the multi- dimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch Rational Mech Anal., 227 (2018), 967-994. doi: 10.1007/s00205-017-1179-z. Google Scholar

[19]

P. D. Lax, Hyperbolic systems of conservation laws, Ⅱ., Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. Google Scholar

[20]

L. A. LiT. Wang and Y. Wang, Stability of planar rarefaction wave to 3D full compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 230 (2018), 911-937. doi: 10.1007/s00205-018-1260-2. Google Scholar

[21]

L. A. Li and Y. Wang, Stability of the planar rarefaction wave to the two-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 50 (2018), 4937-4963. doi: 10.1137/18M1171059. Google Scholar

[22]

T. P. LiuT. 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

[23]

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

[24]

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

[25]

T. P. Liu and S. H. Yu, Invariant manifolds for steady Boltzmann flows and applications, Arch. Rational Mech. Anal., 209 (2013), 869-997. doi: 10.1007/s00205-013-0640-x. Google Scholar

[26]

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

[27]

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

[28]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[29]

T. Wang and Y. Wang, Stability of superposition of two viscous shock waves for the Boltzmann equation, SIAM J. Math. Anal., 47 (2015), 1070-1120. doi: 10.1137/140963005. Google Scholar

[30]

Z. P. Xin, Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions, Trans. Amer. Math. Soc., 319 (1990), 805-820. doi: 10.1090/S0002-9947-1990-0970270-8. Google Scholar

[31]

Z. P. Xin and H. H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations, J. Differential Equations, 249 (2010), 827-871. doi: 10.1016/j.jde.2010.03.011. Google Scholar

[32]

S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations, Commun. Pure Appl. Math, 58 (2005), 409-443. doi: 10.1002/cpa.20027. Google Scholar

[33]

S. H. Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1041-1118. doi: 10.1090/S0894-0347-2010-00671-6. Google Scholar

[1]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[2]

Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035

[3]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401

[4]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693

[5]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985

[6]

Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075

[7]

Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673

[8]

Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579

[9]

Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control & Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010

[10]

Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028

[11]

Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007

[12]

Zhigang Wu, Wenjun Wang. Uniform stability of the Boltzmann equation with an external force near vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 811-823. doi: 10.3934/cpaa.2015.14.811

[13]

Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141

[14]

Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271

[15]

Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163-173. doi: 10.3934/proc.2011.2011.163

[16]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[17]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063

[18]

Denis Matignon, Christophe Prieur. Asymptotic stability of Webster-Lokshin equation. Mathematical Control & Related Fields, 2014, 4 (4) : 481-500. doi: 10.3934/mcrf.2014.4.481

[19]

Yoshihiro Ueda, Tohru Nakamura, Shuichi Kawashima. Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space. Kinetic & Related Models, 2008, 1 (1) : 49-64. doi: 10.3934/krm.2008.1.49

[20]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (38)
  • HTML views (33)
  • Cited by (0)

Other articles
by authors

[Back to Top]