Abstract
We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C 2 test-functions, are confined in a compact set in H −1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.
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References
Bouchut F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhüser Verlag, Basel (2004)
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, 211–223 (2003)
Chen G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 6, 75–120 (1986) (in English)
Chen G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 8, 243–276 (1988) (in Chinese)
Chen G.-Q., Perepelitsa M.: Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow. Commun. Pure Appl. Math. 63, 1469–1504 (2010)
Chen G.-Q., LeFloch Ph.G.: Compressible Euler equations with general pressure law. Arch. Ration. Mech. Anal. 153, 221–259 (2000)
Chen G.-Q., LeFloch Ph.G.: Existence theory for the isentropic Euler equations. Arch. Ration. Mech. Anal. 166, 81–98 (2003)
Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)
Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I)–(II). Acta Math. Sci. 5B, 483–500, 501–540 (1985) (in English)
Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I). Acta Math. Sci. 7A, 467–480 (1987) (in Chinese)
Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (II). Acta Math. Sci. 8A, 61–94 (1988) (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)
DiPerna R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)
Gerbeau J.-F., Perthame B.: Derivation of viscous Saint-Venant system for laminar shallow water: numerical validation. Discrete Continuous Dyn. Syst. 1B, 89–102 (2001)
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)
Hugoniot H.: Sur la propagation du movement dans les corps et epécialement dans les gaz parfaits. J. Ecole Polytechnique 58, 1–125 (1889)
LeFloch Ph.G., Westdickenberg M.: Finite energy solutions to the isentropic Euler equations with geometric effects. J. Math. Pures Appl. 88, 386–429 (2007)
Lions P.-L., Perthame B., Tadmor E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163, 415–431 (1994)
Lions P.-L., Perthame B., Souganidis P.E.: 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)
Mascia C.: A dive into shallow water. Riv. Mat. Univ. Parma. 1, 77–149 (2010)
Mellet A., Vasseur A.: On the barotropic compressible Navier-Stokes equations. Commun. Partial Differ. Equ. 32, 431–452 (2007)
Rankine W.J.M.: On the thermodynamic theory of waves of finite longitudinal disturbance. Philos. Trans. R. Soc. Lond. 1960, 277–288 (1870)
Rayleigh, L.(Strutt, J.W.): Aerial plane waves of finite amplitude. Proc. R. Soc. Lond. 84A, 247–284 (1910)
Stokes G.G.: On a difficulty in the theory of sound. Philos. Mag. 33, 349–356 (1848)
Whitham, G.B.: Linear and Nonlinear Waves, Reprint of the 1974 original. Wiley, New York (1999)
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Chen, GQ., Perepelitsa, M. Shallow water equations: viscous solutions and inviscid limit. Z. Angew. Math. Phys. 63, 1067–1084 (2012). https://doi.org/10.1007/s00033-012-0209-9
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DOI: https://doi.org/10.1007/s00033-012-0209-9