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

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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.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Carrillo, J., Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147, 1999, 269–361.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Michel, A. and 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., 41, 2003, 2262–2293.

    Article  MathSciNet  MATH  Google Scholar 

  18. Mascia, C., Porretta, A. and Terracina, A., Nonhomogeneous Dirichlet problems for degenerate parabolichyperbolic equations, Arch. Ration. Mech. Anal., 163, 2002, 87–124.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Zhigang Wang & Yachun Li

  2. Department of Mathematics, Fuyang Teachers College, Fuyang, 236029, Anhui, China

    Zhigang Wang

Authors
  1. Zhigang Wang
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  2. Yachun Li
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Correspondence to Yachun Li.

Additional information

Project supported by the National Natural Science Foundation of China (No. 10971135), the Program for New Century Excellent Talents of the Ministry of Education of China (No. NCET-07-0546), the University Young Teacher Sciences Foundation of Anhui Province (No. 2010SQRL145), Shanghai Jiao Tong University Innovation Fund for Postgraduates (No. AE071202) and the Quality Project Fund of Fuyang Teachers College (No. 2010JPKC07).

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Wang, Z., Li, Y. Applications of the kinetic formulation for scalar conservation laws with a zero-flux type boundary condition. Chin. Ann. Math. Ser. B 33, 351–366 (2012). https://doi.org/10.1007/s11401-012-0714-3

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  • DOI: https://doi.org/10.1007/s11401-012-0714-3

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