Applications of the kinetic formulation for scalar conservation laws with a zero-flux type boundary condition

Abstract

The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws. Firstly, a kinetic formulation of entropy solutions is established. Secondly, by using the kinetic formulation and kinetic techniques, the uniqueness of entropy solutions is obtained. Finally, the parabolic approximation is studied and an error estimate of order $\eta ^{\tfrac{1}{3}}$ between the entropy solution and the viscous approximate solutions is established by using kinetic techniques, where η is the size of artificial viscosity.

Keywords

Scalar conservation laws / Entropy solutions / Kinetic formulation / Uniqueness / Error estimate

References

[1] Anzellotti G.. Pairings between measures and functions and compensated compactness. Ann. Mat. Pura Appl., 1983, 135: 293-318

[2] Bardos C., Le Roux A. Y., Nedelec J. C.. First order quasilinear equations with boundary conditions. Comm. Partial Differential Equatians, 1979, 4: 1017-1034

[3] Bürger R., Evje S., Karlsen K. H.. On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl., 2000, 247: 517-556

[4] Bürger R., Frid H., Karlsen K. H.. On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition. J. Math. Anal. Appl., 2007, 326: 108-120

[5] Bürger R., Karlsen K. H.. On some upwind schemes for the phenomenological sedimentationconsolidation model. J. Engrg. Math., 2001, 41: 145-166

[6] Bürger R., Kunik M.. A critical look at the kinematic-wave theory for sedimentation-consolidation processes in closed vessels. Math. Methods Appl. Sci., 2001, 24: 1257-1273

[7] Carrillo J.. Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal., 1999, 147: 269-361

[8] Chen G. Q., Frid H.. Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal., 1999, 147: 89-118

[9] Chen G. Q., Perthame B.. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2003, 20: 645-668

[10] Chen G. Q., Karlsen K.. L ¹-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations. Trans. Amer. Math. Soc., 2006, 358: 937-963

[11] Droniou J., Imbert C., Vovelle J.. An error estimate for the parabolic approximation of mutidimensional scalar conservation laws with boundary conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2004, 21: 689-714

[12] Imbert C., Vovelle J.. A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications. SIAM J. Math. Anal., 2004, 36(1): 214-232

[13] Kobayasi K.. A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions. J. Diff. Eqs., 2006, 230: 682-701

[14] Kružkov S. N.. First order quasilinear equation in several independent variables. Math. USSR-Sb., 1970, 10: 217-243

[15] Kwon Y. S.. Well-posedness for entropy solutions to multidimensional scalar conservation laws with a strong boundary condition. J. Math. Anal. Appl., 2008, 340: 543-549

[16] Lions P. L., Perthame B., Tadmor E.. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc., 1994, 7: 169-191

[17] Michel A., Vovelle J.. Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods. SIAM J. Numer. Anal., 2003, 41: 2262-2293

[18] Mascia C., Porretta A., Terracina A.. Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic equations. Arch. Ration. Mech. Anal., 2002, 163: 87-124

[19] Murray J. D.. Mathematical Biology II: Spatial Models and Biomedical Applications, 2003 3rd ed. New York: Springer-Verlag

[20] Otto F.. Initial boundary value problem for a scalar conservation law. C. R. Acad. Sci. Pairs Ser. I Math., 1996, 322: 729-734

[21] Perthame B.. Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure. J. Math. Pures Appl., 1998, 77: 1055-1064

[22] Perthame B., Souganidis P. E.. Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws. Arch. Ration. Mech. Anal., 2003, 170(4): 359-370

[23] Vasseur A.. Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal., 2001, 160: 181-193