Nonlinear wave propagations over a Boltzmann shock profile

Yu, Shih-Hsien

doi:10.1090/S0894-0347-2010-00671-6

Nonlinear wave propagations over a Boltzmann shock profile
HTML articles powered by AMS MathViewer

by Shih-Hsien Yu

J. Amer. Math. Soc. 23 (2010), 1041-1118

DOI: https://doi.org/10.1090/S0894-0347-2010-00671-6

Published electronically: May 24, 2010

PDF | Request permission

Abstract:

In this paper we study the wave propagation over a Boltzmann shock profile and obtain pointwise time-asymptotic stability of Boltzmann shocks. We design a ${\mathbb T}$-${\mathbb C}$ scheme to study the coupling of the transverse and compression waves. The pointwise information of the Green’s functions of the Boltzmann equation linearized around the end Maxwellian states of the shock wave provides the basic estimates for the transient waves. The compression of the Boltzmann shock profile together with a low order damping allows for an accurate energy estimate by a localized scalar equation. These two methods are combined to construct an exponentially sharp pointwise linear wave propagation structure around a Boltzmann shock profile. The pointwise estimates thus obtained are strong enough to study the pointwise nonlinear wave coupling and to conclude the convergence with an optimal convergent rate $O(1)[(1+t)(1+\varepsilon t)]^{-1/2}$ around the Boltzmann shock front, where $\varepsilon$ is the strength of a shock wave.

References

Russel E. Caflisch and Basil Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys. 86 (1982), no. 2, 161–194. MR 676183, DOI 10.1007/BF01206009
Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI 10.1007/BF00276840
Jonathan Goodman, Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 683–695. MR 978372, DOI 10.1090/S0002-9947-1989-0978372-9
Harold Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids 6 (1963), 147–181. MR 155541, DOI 10.1063/1.1706716
Harold Grad, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Sympos. Appl. Math., Vol. XVII, Amer. Math. Soc., Providence, R.I., 1965, pp. 154–183. MR 0184507
Harold Grad, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Sympos. Appl. Math., Vol. XVII, Amer. Math. Soc., Providence, R.I., 1965, pp. 154–183. MR 0184507
Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108. MR 791863, DOI 10.1090/memo/0328
Tai-Ping Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (1997), no. 11, 1113–1182. MR 1470318, DOI 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.3.CO;2-8
Tai-Ping Liu and Yanni Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc. 125 (1997), no. 599, viii+120. MR 1357824, DOI 10.1090/memo/0599
Tai-Ping Liu and Yanni Zeng, Time-asymptotic behavior of wave propagation around a viscous shock profile, Comm. Math. Phys. 290 (2009), no. 1, 23–82. MR 2520507, DOI 10.1007/s00220-009-0820-6
Tai-Ping Liu and Shih-Hsien Yu, Propagation of a stationary shock layer in the presence of a boundary, Arch. Rational Mech. Anal. 139 (1997), no. 1, 57–82. MR 1475778, DOI 10.1007/s002050050047
Tai-Ping Liu and Shih-Hsien Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004), no. 1, 133–179. MR 2044894, DOI 10.1007/s00220-003-1030-2
Tai-Ping Liu and Shih-Hsien Yu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608. MR 2082240, DOI 10.1002/cpa.20011
Tai-Ping Liu and Shih-Hsien Yu, Green’s function of Boltzmann equation, 3-D waves, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 1, 1–78. MR 2230121
Tai-Ping Liu and Shih-Hsien Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math. 60 (2007), no. 3, 295–356. MR 2284213, DOI 10.1002/cpa.20172
Tai-Ping Liu and Shih-Hsien Yu, Viscous shock wave tracing, local conservation laws, and pointwise estimates, Bull. Inst. Math. Acad. Sin. (N.S.) 4 (2009), no. 3, 235–297. MR 2571697
Liu, T.-P.; Yu, S.-H. Invariant manifolds for steady Boltzmann flows and its applications, preprint.
Akitaka Matsumura and Kenji Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 2 (1985), no. 1, 17–25. MR 839317, DOI 10.1007/BF03167036
Sone, Y. Kinetic theory of evaporation and condensation linear and nonlinear problems, J. Phys. Soc. Jpn. 45 (1978), 315-320.
Sone, Y.; Aoki, K; Yamashita, K. A study of unsteady strong condensation on a plane condensed phase with special interest in formation of steady profile, in: V. Bofli and C. Cercignani, eds., Rarefied Gas Dynamics (Teubner, Stuttgart), Vol. II (1986), 323-333.
Yoshio Sone, François Golse, Taku Ohwada, and Toshiyuki Doi, Analytical study of transonic flows of a gas condensing onto its plane condensed phase on the basis of kinetic theory, Eur. J. Mech. B Fluids 17 (1998), no. 3, 277–306. MR 1632303, DOI 10.1016/S0997-7546(98)80260-4
Yoshio Sone, Molecular gas dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. Theory, techniques, and applications. MR 2274674, DOI 10.1007/978-0-8176-4573-1
Yanni Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous $1$-D flow, Comm. Pure Appl. Math. 47 (1994), no. 8, 1053–1082. MR 1288632, DOI 10.1002/cpa.3160470804