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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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