Vanishing viscosity limit of the compressible Navier-Stokes equations with finite energy and total mass. (English) Zbl 1481.35330

Summary: Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., \( \alpha = 1\), \(\gamma = 2\)).


35Q35 PDEs in connection with fluid mechanics
35Q31 Euler equations
35B25 Singular perturbations in context of PDEs
35B44 Blow-up in context of PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35R09 Integro-partial differential equations
35R35 Free boundary problems for PDEs
35D30 Weak solutions to PDEs
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76N17 Viscous-inviscid interaction for compressible fluids and gas dynamics
Full Text: DOI


[1] Bianchini, S.; Bressan, A., Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. Math. (2), 161, 223-342 (2005) · Zbl 1082.35095
[2] Bresch, D.; Desjardins, B., Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Commun. Math. Phys., 238, 1-2, 211-223 (2003) · Zbl 1037.76012
[3] Bressan, A.; Yang, T., On the convergence rate of vanishing viscosity approximations, Commun. Pure Appl. Math., 57, 1075-1109 (2004) · Zbl 1060.35109
[4] Bressan, A.; Huang, F.; Wang, Y.; Yang, T., On the convergence rate of vanishing viscosity approximations for nonlinear hyperbolic systems, SIAM J. Math. Anal., 44, 3537-3563 (2012) · Zbl 1264.35017
[5] Chapman, S.; Cowling, T., The Mathematical Theory of Non-uniform Gases (1970), Cambridge University Press: Cambridge University Press London
[6] G.-Q. Chen, The compensated compactness method and the system of isentropic gas dynamics, Lecture notes, Preprint MSRI-00527-91, Berkeley, October 1990.
[7] Chen, G.-Q.; Perepelista, M., Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Commun. Pure Appl. Math., 1469-1504 (2010) · Zbl 1205.35188
[8] Chen, G.-Q.; He, L.; Wang, Y.; Yuan, D.-F., Global solutions of the compressible Euler-Poisson equations with large initial data of spherical symmetry (2021)
[9] Ding, X.; Chen, G.-Q.; Luo, P., Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I)-(II), Acta Math. Sci., 8A, 61-94 (1989), (in Chinese); Ding, X.; Chen, G.-Q.; Luo, P., Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Commun. Math. Phys., 121, 63-84 (1989) · Zbl 0689.76022
[10] DiPerna, R. J., Convergence of the viscosity method for isentropic gas dynamics, Commun. Math. Phys., 91, 1-30 (1983) · Zbl 0533.76071
[11] Fang, D.; Zhang, T., Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data, J. Differ. Equ., 222, 63-94 (2006) · Zbl 1357.35245
[12] Fang, D.; Zhang, T., Compressible Navier-Stokes equations with vacuum state in the case of general pressure law, Math. Methods Appl. Sci., 9, 1081-1106 (2006) · Zbl 1184.35259
[13] Hoff, D., 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, 774-785 (1998) · Zbl 0913.35031
[14] Hoff, D.; Liu, T.-P., The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J., 38, 861-915 (1989) · Zbl 0674.76047
[15] Huang, F.; Pan, R.; Wang, T.; Wang, Y.; Zhai, X., Vanishing viscosity limit for isentropic Navier-Stokes equations with density-dependent viscosity
[16] Huang, F.; Wang, Z., Convergence of viscosity solutions for isothermal gas dynamics, SIAM J. Math. Anal., 34, 3, 595-610 (2002) · Zbl 1036.35129
[17] Guo, Z. H.; Jiang, S.; Xie, F., Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier-Stokes equations with degenerate viscosity coefficient, Asymptot. Anal., 60, 1-2, 101-123 (2008) · Zbl 1166.35357
[18] Jiang, S., Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity, Math. Nachr., 190, 169-183 (1998) · Zbl 0927.35014
[19] Jiang, S.; Xin, Z.; Zhang, P., Global weak solution to 1D compressible isentropic Navier-Stokes with density-dependent viscosity, Methods Appl. Anal., 12, 239-252 (2005) · Zbl 1110.35058
[20] Jiu, Q.; Xin, Z., The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, Kinet. Relat. Models, 1, 313-330 (2008) · Zbl 1144.35440
[21] Kanel, Y., On a model system of equations of one-dimensional gas motion, Differ. Uravn., 4, 721-734 (1968)
[22] LeFloch, P.; Westdickenberg, M., Finite energy solutions to the isentropic Euler equations with geometric effects, J. Math. Pures Appl., 88, 386-429 (2007) · Zbl 1188.35150
[23] Lions, P. L.; Perthame, B.; Tadmor, E., Kinetic formulation of the isentropic gas dynamics and p-systems, Commun. Math. Phys., 163, 415-431 (1994) · Zbl 0799.35151
[24] Lions, P. L.; Perthame, B.; Souganidis, P., Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Commun. Pure Appl. Math., 49, 599-638 (1996) · Zbl 0853.76077
[25] Liu, T.; Xin, Z.; Yang, T., Vacuum states for compressible flow, Discrete Contin. Dyn. Syst., 4, 1, 1-32 (1998) · Zbl 0970.76084
[26] Mellet, A.; Vasseur, A., Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39, 1344-1365 (2008) · Zbl 1141.76054
[27] Qin, X. L.; Yao, Z. A.; Zhao, H. X., One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries, Commun. Pure Appl. Anal., 7, 373-381 (2008) · Zbl 1141.35060
[28] Xin, Z., Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Commun. Pure Appl. Math., XLVI, 621-C665 (1993) · Zbl 0804.35108
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