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Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow. (English) Zbl 1205.35188

The vanishing viscosity limit of Navier-Stokes equations to the Euler isentropic equations in 1D is established. Some new uniform estimation, in terms of the physical viscosity coefficient, is given for the solutions of Navier-Stokes equations with finite-energy initial data. The isentropic Euler equations are studied related with nonlinear systems of conservation laws. Let \(n\) be the exponent of the density in the pressure state equation. The existence of a global finite-energy entropy solution to Euler equations with general initial data is obtained for \(n > 3\), extending some previous results of P. G. LeFloch and M. Westdickenberg [J. Math. Pures Appl. (9) 88, No. 5, 389–429 (2007; Zbl 1188.35150)]. In section 4, the \(H^{-1}\) compactness of entropy dissipation measures for solutions of Navier-Stokes equations, under some particular conditions, is given. In section 5, it is proved that the measure-valued solutions are confined by the Murat-Tartar commutator relation for any two pairs of weak entropy-entropy flux kernels by using compensated compactness.

MSC:

35Q30 Navier-Stokes equations
35Q31 Euler equations
76N15 Gas dynamics (general theory)
35B45 A priori estimates in context of PDEs

Citations:

Zbl 1188.35150
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References:

[1] Alberti, A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 (7) pp 761– (2001) · Zbl 1021.49012
[2] Ball, PDEs and continuum models of phase transitions (Nice, 1988) pp 207– (1989)
[3] Bianchini, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (1) pp 223– (2005) · Zbl 1082.35095
[4] Chen, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics. III, Acta Math. Sci. 6 (1) pp 75– (1986) · Zbl 0643.76086
[5] Chinese translation: Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics. III, Acta Math. Sci. (Chinese) 8 (3) pp 243– (1988)
[6] Chen, Remarks on DiPerna’s paper: ”Convergence of the viscosity method for isentropic gas dynamics” [Comm. Math. Phys. 91 (1983), 1-30], Proc. Amer. Math. Soc. 125 pp 2981– (1997)
[7] Chen , G.-Q. 1990
[8] Chen, Compressible Euler equations with general pressure law, Arch. Ration. Mech. Anal. 153 pp 221– (2000) · Zbl 0970.76082
[9] Existence theory for the isentropic Euler equations, Arch. Ration. Mech. Anal. 166 pp 81– (2003) · Zbl 1027.76043
[10] Dafermos, Hyperbolic conservation laws in continuum physics (2010) · Zbl 1196.35001
[11] Ding, On a lemma of DiPerna and Chen, Acta Math. Sci. Ser. B Engl. Ed. 26 (1) pp 188– (2006) · Zbl 1152.35431 · doi:10.1016/S0252-9602(06)60040-4
[12] Ding, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics I, II, Acta Math. Sci. 5 (4) pp 415– (1985) · Zbl 0643.76085
[13] Chinese translations: Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics. I, Acta Math. Sci. (Chinese) 7 (4) pp 467– (1987)
[14] Convergence of the Lax-Friedrichs scheme for the system of equations of isentropic gas dynamics. II, Acta Math. Sci. (Chinese) 8 (1) pp 61– (1988)
[15] Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys. 121 (1) pp 63– (1989) · Zbl 0689.76022
[16] DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1) pp 27– (1983) · Zbl 0519.35054
[17] DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1) pp 1– (1983) · Zbl 0533.76071
[18] Gilbarg, The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73 pp 256– (1951) · Zbl 0044.21504
[19] Guès, Navier-Stokes regularization of multidimensional Euler shocks, Ann. Sci. École Norm. Sup. (4) 39 (1) pp 75– (2006) · Zbl 1173.35082
[20] 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 (5) pp 774– (1998) · Zbl 0913.35031
[21] Hoff, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J. 38 (4) pp 861– (1989) · Zbl 0674.76047
[22] Hugoniot, Mémoire sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits. 2e Partie, J. École Polytechnique Paris 58 pp 1– (1889)
[23] Kanel’, On a model system of equations for one-dimensional gas motion, Differ. Uravn. 4 (4) pp 721– (1968)
[24] Lax, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) pp 603– (1971) · doi:10.1016/B978-0-12-775850-3.50018-2
[25] LeFloch, Finite energy solutions to the isentropic Euler equations with geometric effects, J. Math. Pures Appl. (9) 88 (5) pp 386– (2007) · Zbl 1188.35150 · doi:10.1016/j.matpur.2007.07.004
[26] Lions, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (6) pp 599– (1996) · Zbl 0853.76077
[27] Lions, Kinetic formulation of the isentorpic gas dynamics and p-systems, Comm. Math. Phys. 163 (2) pp 415– (1994)
[28] Morawetz, An alternative proof of DiPerna’s theorem, Comm. Pure Appl.Math. 44 (8-9) pp 1081– (1991) · Zbl 0763.35056
[29] Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (3) pp 489– (1978)
[30] Perthame, Kinetic formulation for systems of two conservation laws and elastodynamics, Arch. Ration. Mech. Anal. 155 (1) pp 1– (2000) · Zbl 0980.35092
[31] Rankine, Classic papers in shock compression science pp 133– (1998) · doi:10.1007/978-1-4612-2218-7_5
[32] Rayleigh, Classic papers in shock compression science pp 361– (1998) · doi:10.1007/978-1-4612-2218-7_9
[33] Serre, La compacité par compensation pour les systèmes hyperboliques non linéaires de deux èquations à une dimension d’espace, J. Math. Pures Appl. (9) 65 (4) pp 423– (1986)
[34] Serre , D. Shearer , J. 1993
[35] Stokes, Classic papers in shock compression science pp 71– (1998) · doi:10.1007/978-1-4612-2218-7_2
[36] Tartar , L. Compensated compactness and applications to partial differential equations Nonlinear analysis and mechanics: Heriot-Watt Symposium IV 136 212 Pitman Boston-London 1979
[37] Taylor, The conditions necessary for discontinuous motions in gases, Proc. Roy. Soc. London Ser. A 84 pp 371– (1910) · JFM 41.0846.01
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