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Shock waves in conservation laws with physical viscosity
About this Title
Tai-Ping Liu, Institute of Mathematics, Academia Sinica, Taiwan – and – Department of Mathematics, Stanford University and Yanni Zeng, Department of Mathematics, University of Alabama at Birmingham
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 234, Number 1105
ISBNs: 978-1-4704-1016-2 (print); 978-1-4704-2032-1 (online)
DOI: https://doi.org/10.1090/memo/1105
Published electronically: August 25, 2014
Keywords: Conservation laws,
physical viscosity,
shock waves,
nonlinear stability,
large time behavior,
wave interactions,
pointwise estimates,
Green’s function,
compressible Navier-Stokes equations,
magneto-hydrodynamics,
quasilinear hyperbolic-parabolic systems
MSC: Primary 35K59, 35L67; Secondary 35L65, 35Q35, 35A08, 35B40, 35B35, 76W05, 76N15
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Green’s functions for Systems with Constant Coefficients
- 4. Green’s Function for Systems Linearized Along Shock Profiles
- 5. Estimates on Green’s Function
- 6. Estimates on Crossing of Initial Layer
- 7. Estimates on Truncation Error
- 8. Energy Type Estimates
- 9. Wave Interaction
- 10. Stability Analysis
- 11. Application to Magnetohydrodynamics
Abstract
We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In particular, we show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small, but independent. Our assumptions on the viscosity matrix are general so that our results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. Our analysis depends on accurate construction of an approximate Green’s function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that we can close the nonlinear term through the Duhamel’s principle.- I-Liang Chern, Multiple-mode diffusion waves for viscous non-strictly hyperbolic conservation laws, Comm. Math. Phys. 138 (1991), no. 1, 51–61. MR 1108036
- Julian D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225–236. MR 42889, DOI 10.1090/S0033-569X-1951-42889-X
- Charles C. Conley and Joel A. Smoller, On the structure of magnetohydrodynamic shock waves, Comm. Pure Appl. Math. 27 (1974), 367–375. MR 368586, DOI 10.1002/cpa.3160270306
- Heinrich Freistühler, Christian Fries, and Christian Rohde, Existence, bifurcation, and stability of profiles for classical and non-classical shock waves, Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp. 287–309, 814. MR 1850311
- Heinrich Freistühler and Tai-Ping Liu, Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws, Comm. Math. Phys. 153 (1993), no. 1, 147–158. MR 1213739
- K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1686–1688. MR 285799, DOI 10.1073/pnas.68.8.1686
- David Gilbarg, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73 (1951), 256–274. MR 44315, DOI 10.2307/2372177
- 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
- David Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), no. 3-4, 301–315. MR 866843, DOI 10.1017/S0308210500018953
- David Hoff and Tai-Ping Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J. 38 (1989), no. 4, 861–915. MR 1029681, DOI 10.1512/iumj.1989.38.38041
- Eberhard Hopf, The partial differential equation $u_t+uu_x=\mu u_{xx}$, Comm. Pure Appl. Math. 3 (1950), 201–230. MR 47234, DOI 10.1002/cpa.3160030302
- Tatsuo Iguchi and Shuichi Kawashima, On space-time decay properties of solutions to hyperbolic-elliptic coupled systems, Hiroshima Math. J. 32 (2002), no. 2, 229–308. MR 1925900
- Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
- S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral thesis, Kyoto University, (1983)
- Shuichi Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 169–194. MR 899951, DOI 10.1017/S0308210500018308
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- 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, Interactions of nonlinear hyperbolic waves, Nonlinear analysis (Taipei, 1989) World Sci. Publ., Teaneck, NJ, 1991, pp. 171–183. MR 1103571
- 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
- T.-P. Liu and S.-H Yu, Green’s function for Boltzmann equation, 3-D waves, Bulletin, Inst. Math. Academia Sinica 1 (2006), 1–78
- Tai-Ping Liu and Shih-Hsien Yu, Viscous rarefaction waves, Bull. Inst. Math. Acad. Sin. (N.S.) 5 (2010), no. 2, 123–179. MR 2681405
- Tai-Ping Liu and Shih-Hsien Yu, Solving Boltzmann equation, Part I: Green’s function, Bull. Inst. Math. Acad. Sin. (N.S.) 6 (2011), no. 2, 115–243. MR 2850554
- Tai-Ping Liu, Shih-Hsien Yu, and Yanni Zeng, Viscous conservation laws, Part I: scalar laws, Bull. Inst. Math. Acad. Sin. (N.S.) 5 (2010), no. 3, 233–310. MR 2766341
- 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, Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations 153 (1999), no. 2, 225–291. MR 1683623, DOI 10.1006/jdeq.1998.3554
- 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 Yanni Zeng, On Green’s function for hyperbolic-parabolic systems, Acta Math. Sci. Ser. B (Engl. Ed.) 29 (2009), no. 6, 1556–1572. MR 2589092, DOI 10.1016/S0252-9602(10)60003-3
- 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
- F. Rellich, Perturbation theory of eigenvalue problems, Lecture notes, New York University, (1953)
- Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. MR 798756, DOI 10.14492/hokmj/1381757663
- Chi-Wang Shu and Yanni Zeng, High-order essentially non-oscillatory scheme for viscoelasticity with fading memory, Quart. Appl. Math. 55 (1997), no. 3, 459–484. MR 1466143, DOI 10.1090/qam/1466143
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
- Anders Szepessy and Zhou Ping Xin, Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993), no. 1, 53–103. MR 1207241, DOI 10.1007/BF01816555
- 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
- Yanni Zeng, $L^p$ asymptotic behavior of solutions to hyperbolic-parabolic systems of conservation laws, Arch. Math. (Basel) 66 (1996), no. 4, 310–319. MR 1380573, DOI 10.1007/BF01207832
- Yanni Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150 (1999), no. 3, 225–279. MR 1738119, DOI 10.1007/s002050050188
- Yanni Zeng, Gas flows with several thermal nonequilibrium modes, Arch. Ration. Mech. Anal. 196 (2010), no. 1, 191–225. MR 2601073, DOI 10.1007/s00205-009-0247-4
- Yanni Zeng, Large time behavior of solutions to nonlinear viscoelastic model with fading memory, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 1, 219–236. MR 2921874, DOI 10.1016/S0252-9602(12)60014-9