Shallow water equations: viscous solutions and inviscid limit

详细信息    查看全文

• 作者：Gui-Qiang Chen (1) (2)
  Mikhail Perepelitsa (3)
  
• 关键词：Primary 35B30 ; 35Q30 ; 35Q31 ; 35L65 ; 35L45 ; 35B35 ; 76N17 ; Secondary 76B15 ; 35L80 ; 35Q35 ; 35B25 ; Shallow water equations ; Inviscid limit ; Viscous ; Inviscid ; Saint ; Venant system ; Friction ; Viscous solutions ; Entropy ; Entropy flux ; Entropy solutions ; Uniform estimates ; Finite energy ; Entropy dissipation measures ; H ? ; compactness ; Measure ; valued solutions
• 刊名：Zeitschrift f眉r angewandte Mathematik und Physik
• 出版年：2012
• 出版时间：December 2012
• 年：2012
• 卷：63
• 期：6
• 页码：1067-1084
• 全文大小：300KB
• 参考文献：1. Bouchut F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhüser Verlag, Basel (2004) CrossRef
  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, 211-23 (2003)
  3. Chen G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 6, 75-20 (1986) (in English)
  4. Chen G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 8, 243-76 (1988) (in Chinese)
  5. 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, 1469-504 (2010) CrossRef
  6. Chen G.-Q., LeFloch Ph.G.: Compressible Euler equations with general pressure law. Arch. Ration. Mech. Anal. 153, 221-59 (2000) CrossRef
  7. Chen G.-Q., LeFloch Ph.G.: Existence theory for the isentropic Euler equations. Arch. Ration. Mech. Anal. 166, 81-8 (2003) CrossRef
  8. Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)
  9. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I)-II). Acta Math. Sci. 5B, 483-00, 501-40 (1985) (in English)
  10. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (I). Acta Math. Sci. 7A, 467-80 (1987) (in Chinese)
  11. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for the isentropic gas dynamics (II). Acta Math. Sci. 8A, 61-4 (1988) (in Chinese)
  12. 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, 63-4 (1989) CrossRef
  13. DiPerna R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1-0 (1983) CrossRef
  14. Gerbeau J.-F., Perthame B.: Derivation of viscous Saint-Venant system for laminar shallow water: numerical validation. Discrete Continuous Dyn. Syst. 1B, 89-02 (2001)
  15. 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, 774-85 (1998) CrossRef
  16. Hugoniot H.: Sur la propagation du movement dans les corps et epécialement dans les gaz parfaits. J. Ecole Polytechnique 58, 1-25 (1889)
  17. LeFloch Ph.G., Westdickenberg M.: Finite energy solutions to the isentropic Euler equations with geometric effects. J. Math. Pures Appl. 88, 386-29 (2007)
  18. Lions P.-L., Perthame B., Tadmor E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163, 415-31 (1994) CrossRef
  19. 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, 599-38 (1996) CrossRef
  20. Mascia C.: A dive into shallow water. Riv. Mat. Univ. Parma. 1, 77-49 (2010)
  21. Mellet A., Vasseur A.: On the barotropic compressible Navier-Stokes equations. Commun. Partial Differ. Equ. 32, 431-52 (2007) CrossRef
  22. Rankine W.J.M.: On the thermodynamic theory of waves of finite longitudinal disturbance. Philos. Trans. R. Soc. Lond. 1960, 277-88 (1870)
  23. Rayleigh, L.(Strutt, J.W.): Aerial plane waves of finite amplitude. Proc. R. Soc. Lond. 84A, 247-84 (1910)
  24. Stokes G.G.: On a difficulty in the theory of sound. Philos. Mag. 33, 349-56 (1848)
  25. Whitham, G.B.: Linear and Nonlinear Waves, Reprint of the 1974 original. Wiley, New York (1999)
  
• 作者单位：Gui-Qiang Chen (1) (2)
  Mikhail Perepelitsa (3)
  
  1. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK
  2. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA
  3. Department of Mathematics, University of Houston, 651 PGH, Houston, TX, 77204-3008, USA
  
• ISSN：1420-9039

文摘

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 ?, 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.