## Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation

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- by Ran Duan, Hongxia Liu and Huijiang Zhao PDF
- Trans. Amer. Math. Soc.
**361**(2009), 453-493 Request permission

## Abstract:

The expansion waves for the compressible Navier-Stokes equations have recently been shown to be nonlinear stable. The nonlinear stability results are called local stability or global stability depending on whether the $H^1-$norm of the initial perturbation is small or not. Up to now, local stability results have been well established. However, for global stability, only partial results have been obtained. The main purpose of this paper is to study the global stability of rarefaction waves for the compressible Navier-Stokes equations. For this purpose, we introduce a positive parameter $t_0$ in the construction of smooth approximations of the rarefaction wave solutions for the compressible Euler equations so that the quantity $\ell =\frac {t_0}{\delta }$ ($\delta$ denotes the strength of the rarefaction waves) is sufficiently large to control the growth induced by the nonlinearity of the system and the interaction of waves from different families. Then by using the energy method together with the continuation argument, we obtain some nonlinear stability results provided that the initial perturbation satisfies certain growth conditions as $\ell \to +\infty$. Notice that the assumption that the quantity $\ell$ can be chosen to be sufficiently large implies that either the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially.## References

- K. N. Chueh, C. C. Conley, and J. A. Smoller,
*Positively invariant regions for systems of nonlinear diffusion equations*, Indiana Univ. Math. J.**26**(1977), no. 2, 373–392. MR**430536**, DOI 10.1512/iumj.1977.26.26029 - Ran Duan, Xuan Ma, and Huijiang Zhao,
*A case study of global stability of strong rarefaction waves for $2\times 2$ hyperbolic conservation laws with artificial viscosity*, J. Differential Equations**228**(2006), no. 1, 259–284. MR**2254431**, DOI 10.1016/j.jde.2006.02.005 - David Hoff,
*Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states*, Z. Angew. Math. Phys.**49**(1998), no. 5, 774–785. MR**1652200**, DOI 10.1007/s000330050120 - David Hoff and Joel Smoller,
*Solutions in the large for certain nonlinear parabolic systems*, Ann. Inst. H. Poincaré Anal. Non Linéaire**2**(1985), no. 3, 213–235 (English, with French summary). MR**797271** - Ling Hsiao and Song Jiang,
*Nonlinear hyperbolic-parabolic coupled systems*, Evolutionary equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, pp. 287–384. MR**2103699** - Ling Hsiao and Ronghua Pan,
*Zero relaxation limit to centered rarefaction waves for a rate-type viscoelastic system*, J. Differential Equations**157**(1999), no. 1, 20–40. MR**1710012**, DOI 10.1006/jdeq.1998.3615 - Feimin Huang, Akitaka Matsumura, and Zhouping Xin,
*Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations*, Arch. Ration. Mech. Anal.**179**(2006), no. 1, 55–77. MR**2208289**, DOI 10.1007/s00205-005-0380-7 - F. M. Huang, Z. P. Xin, and T. Yang, Contact discontinuity with general perturbations for gas motions. Preprint 2004.
- A. M. Il$’$in and O. A. Oleĭnik, Behavior of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time.
*Amer. Math. Soc. Transl. Ser. 2***42**(1964), 19-23. - Alan Jeffrey and Huijiang Zhao,
*Global existence and optimal temporal decay estimates for systems of parabolic conservation laws. I. The one-dimensional case*, Appl. Anal.**70**(1998), no. 1-2, 175–193. MR**1671563**, DOI 10.1080/00036819808840683 - Song Jiang, Guoxi Ni, and Wenjun Sun,
*Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids*, SIAM J. Math. Anal.**38**(2006), no. 2, 368–384. MR**2237152**, DOI 10.1137/050626478 - Song Jiang and Reinhard Racke,
*Evolution equations in thermoelasticity*, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 112, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR**1774100** - Song Jiang and Ping Zhang,
*Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas*, Quart. Appl. Math.**61**(2003), no. 3, 435–449. MR**1999830**, DOI 10.1090/qam/1999830 - Yoshiyuki Kagei and Shuichi Kawashima,
*Local solvability of an initial boundary value problem for a quasilinear hyperbolic-parabolic system*, J. Hyperbolic Differ. Equ.**3**(2006), no. 2, 195–232. MR**2229854**, DOI 10.1142/S0219891606000768 - Ja. I. Kanel′,
*A model system of equations for the one-dimensional motion of a gas*, Differencial′nye Uravnenija**4**(1968), 721–734 (Russian). MR**0227619** - S. Kawashima,
*Systems of a Hyperbolic-Parabolic Composite, with Applications to the Equations of Magnetohydrodynamics.*Thesis, Kyoto University, 1985. - Shuichi Kawashima and Akitaka Matsumura,
*Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion*, Comm. Math. Phys.**101**(1985), no. 1, 97–127. MR**814544** - Shuichi Kawashima, Akitaka Matsumura, and Kenji Nishihara,
*Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas*, Proc. Japan Acad. Ser. A Math. Sci.**62**(1986), no. 7, 249–252. MR**868811** - Shuichi Kawashima and Takaaki Nishida,
*Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases*, J. Math. Kyoto Univ.**21**(1981), no. 4, 825–837. MR**637519**, DOI 10.1215/kjm/1250521915 - Tai-Ping Liu,
*Shock waves for compressible Navier-Stokes equations are stable*, Comm. Pure Appl. Math.**39**(1986), no. 5, 565–594. MR**849424**, DOI 10.1002/cpa.3160390502 - Tai-Ping Liu and Zhou Ping Xin,
*Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations*, Comm. Math. Phys.**118**(1988), no. 3, 451–465. MR**958806** - Tai-Ping Liu and Zhouping Xin,
*Pointwise decay to contact discontinuities for systems of viscous conservation laws*, Asian J. Math.**1**(1997), no. 1, 34–84. MR**1480990**, DOI 10.4310/AJM.1997.v1.n1.a3 - 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 - Akitaka Matsumura and Takaaki Nishida,
*The initial value problem for the equations of motion of viscous and heat-conductive gases*, J. Math. Kyoto Univ.**20**(1980), no. 1, 67–104. MR**564670**, DOI 10.1215/kjm/1250522322 - 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 - Akitaka Matsumura and Kenji 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), no. 1, 1–13. MR**899210**, DOI 10.1007/BF03167088 - Akitaka Matsumura and Kenji Nishihara,
*Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas*, Comm. Math. Phys.**144**(1992), no. 2, 325–335. MR**1152375** - Akitaka Matsumura and Kenji Nishihara,
*Global asymptotics toward the rarefaction wave for solutions of viscous $p$-system with boundary effect*, Quart. Appl. Math.**58**(2000), no. 1, 69–83. MR**1738558**, DOI 10.1090/qam/1738558 - Kenji Nishihara, Tong Yang, and Huijiang Zhao,
*Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations*, SIAM J. Math. Anal.**35**(2004), no. 6, 1561–1597. MR**2083790**, DOI 10.1137/S003614100342735X - Kenji Nishihara, Huijiang Zhao, and Yinchuan Zhao,
*Global stability of strong rarefaction waves of the Jin-Xin relaxation model for the $p$-system*, Comm. Partial Differential Equations**29**(2004), no. 9-10, 1607–1634. MR**2103847**, DOI 10.1081/PDE-200037772 - Mari Okada and Shuichi Kawashima,
*On the equations of one-dimensional motion of compressible viscous fluids*, J. Math. Kyoto Univ.**23**(1983), no. 1, 55–71. MR**692729**, DOI 10.1215/kjm/1250521610 - Denis Serre,
*Relaxations semi-linéaire et cinétique des systèmes de lois de conservation*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**17**(2000), no. 2, 169–192 (French, with English and French summaries). MR**1753092**, DOI 10.1016/S0294-1449(99)00105-5 - 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**, DOI 10.1007/978-1-4612-0873-0 - G. B. Whitham,
*Linear and nonlinear waves*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR**0483954** - Zhou Ping Xin,
*Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases*, Comm. Pure Appl. Math.**46**(1993), no. 5, 621–665. MR**1213990**, DOI 10.1002/cpa.3160460502

## Additional Information

**Ran Duan**- Affiliation: Laboratory of Nonlinear Analysis, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People’s Republic of China
**Hongxia Liu**- Affiliation: Department of Mathematics, Jinan University, Guangzhou 510632, People’s Republic of China
**Huijiang Zhao**- Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
- Email: hhjjzhao@hotmail.com
- Received by editor(s): April 9, 2007
- Published electronically: August 15, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**361**(2009), 453-493 - MSC (2000): Primary 35L65, 35L60
- DOI: https://doi.org/10.1090/S0002-9947-08-04637-0
- MathSciNet review: 2439413