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

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