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The Brio system with initial conditions involving Dirac masses: A result afforded by a distributional product

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Abstract

The Brio system is a 2 × 2 fully nonlinear system of conservation laws which arises as a simplified model in the study of plasmas. The present paper offers explicit solutions to this system subjected to initial conditions containing Dirac masses. The concept of a solution emerges within the framework of a distributional product and represents a consistent extension of the concept of a classical solution. Among other features, the result shows that the space of measures is not sufficient to contain all solutions of this problem.

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References

  1. Bouchut, F. and James, F., Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. PDE, 24, 1999, 2173–2190.

    Article  MathSciNet  MATH  Google Scholar 

  2. Brenier, Y. and Grenier, E., Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35, 1998, 2317–2328.

    Article  MathSciNet  MATH  Google Scholar 

  3. Bressan, A. and Rampazzo, F., On differential systems with vector-valued impulsive controls, Bull. Un. Mat. Ital. Ser. B, 2(7), 1988, 641–656.

    MathSciNet  MATH  Google Scholar 

  4. Brio, M., Admissibility conditions for weak solutions of nonstrictly hyperbolic systems, Proc. Int. Conf. on Hyperbolic Problems, Springer-Verlag, Berlin, 1988.

    Google Scholar 

  5. Chen, G. Q. and Liu, H., Formation of delta shocks and vacuum states in the vanishing pressure limit of solutions to Euler equations for isentropic fluids, SIAM J. Math. Anal., 34, 2003, 925–938.

    Article  MathSciNet  MATH  Google Scholar 

  6. Colombeau, J. F. and Le Roux, A., Multiplication of distributions in elasticity and hydrodynamics, J. Math. Phys., 29, 1988, 315–319.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dal Maso, G., LeFloch, P. and Murat, F., Definitions and weak stability of nonconservative products, J. Math. Pures Appl., 74, 1995, 483–548.

    MathSciNet  MATH  Google Scholar 

  8. Danilov, V. G. and Mitrovic, D., Weak asymptotic of shock wave formation process, Nonlinear Analysis: Theory, Methods and Applications, 61, 2005, 613–635.

    Article  MathSciNet  MATH  Google Scholar 

  9. Danilov, V. G. and Shelkovich, V. M., Dynamics of propagation and interaction of shock waves in conservation law systems, J. Differential Equations, 211, 2005, 333–381.

    Article  MathSciNet  MATH  Google Scholar 

  10. Danilov, V. G. and Shelkovich, V. M., Delta shock wave type solution of hyperbolic systems of conservation laws, Q. Appl. Math., 29, 2005, 401–427.

    MathSciNet  Google Scholar 

  11. Huang, F., Weak solutions to pressureless type system, Comm. PDE, 30, 2005, 283–304.

    Article  MATH  Google Scholar 

  12. Joseph, K. T., A Riemann problem whose viscosity solutions contains δ-measures, Asymptot. Analysis, 7, 1993, 105–120.

    MATH  Google Scholar 

  13. Keyfitz, B. L. and Kranzer, H. C., Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations, 118, 1995, 420–451.

    Article  MathSciNet  MATH  Google Scholar 

  14. Korchinski, C., Solution of a Riemann problem for a 2 × 2 system of conservation laws possessing no classical weak solutions, Ph.D. Thesis, Adelphi University, 1977.

    Google Scholar 

  15. LeVeque, R. J., The dynamics of pressureless dust clouds and delta waves, J. Hyperbolic Diff. Eq., 1, 2004, 315–327.

    Article  MathSciNet  MATH  Google Scholar 

  16. Maslov, V. P., Nonstandard characteristics in asymptotical problems, Russian Math. Surveys, 38(6), 1983, 1–42.

    Article  MATH  Google Scholar 

  17. Maslov, V. P. and Omel’yanov, G. A., Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys, 36(3), 1981, 73–149.

    Article  MathSciNet  MATH  Google Scholar 

  18. Maslov, V. P. and Tsupin, V. A., Necessary conditions for existence of infinitely narrow solitons in gas dynamics, Soviet Phys. Dock, 24(5), 1979, 354–356.

    Google Scholar 

  19. Nedeljkov, M., Unbounded solutions to some systems of conservation laws-split delta shock waves, Mat. Ves., 54, 2002, 145–149.

    MathSciNet  MATH  Google Scholar 

  20. Nedeljkov, M., Delta and singular delta locus for one-dimensional systems of conservation laws, Math. Method. Appl. Sci., 27, 2004, 931–955.

    Article  MathSciNet  MATH  Google Scholar 

  21. Nedeljkov, M., Shadow waves: Entropies and interactions of delta and singular shocks, Archives Rat. Mech. Anal., 197, 2010, 489–537.

    Article  MathSciNet  MATH  Google Scholar 

  22. Nedeljkov, M. and Oberguggenberger, M., Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl., 334, 2008, 1143–1157.

    Article  MathSciNet  Google Scholar 

  23. Sarrico, C. O. R., About a family of distributional products important in the applications, Port. Math., 45, 1988, 295–316.

    MathSciNet  MATH  Google Scholar 

  24. Sarrico, C. O. R., Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl., 281, 2003, 641–656.

    Article  MathSciNet  MATH  Google Scholar 

  25. Sarrico, C. O. R., New solutions for the one-dimensional nonconservative inviscid Burgers equation, J. Math. Anal. Appl., 317, 2006, 496–509.

    Article  MathSciNet  MATH  Google Scholar 

  26. Sarrico, C. O. R., Collision of delta-waves in a turbulent model studied via a distribution product, Nonlinear Analysis, 73, 2010, 2868–2875.

    Article  MathSciNet  MATH  Google Scholar 

  27. Sarrico, C. O. R., The multiplication of distributions and the Tsodyks model of synapses dynamics, Int. J. of Math. Analysis, 6(21), 2012, 999–1014.

    MathSciNet  MATH  Google Scholar 

  28. Sarrico, C. O. R., Products of distributions and singular travelling waves as solutions of advection-reaction equations, Russian J. of Math. Phys., 19(2), 2012, 244–255.

    Article  MathSciNet  MATH  Google Scholar 

  29. Sarrico, C. O. R., Products of distributions, conservation laws and the propagation of δ′-shock waves, Chin. Ann. Math., 33B(3), 2012, 367–384.

    Article  MathSciNet  Google Scholar 

  30. Sarrico, C. O. R., The Riemann problem for the Brio system: A solution containing a Dirac mass obtained via a distributional product, submitted.

  31. Sheng, W. and Zhang, T., The Riemann problem for transportation equations in gas dynamics, Mem. Amer. Math. Soc., 137, 1999, 1–77.

    MathSciNet  Google Scholar 

  32. Tan, D., Zhang, T. and Zheng, Y., Delta shock waves as a limits of vanishing viscosity for a system of conservation laws, J. Differential Equations, 112, 1994, 1–32.

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, H. and Zhang, Y., New developments of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252, 2012, 5951–5993.

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to C. O. R. Sarrico.

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This work was supported by FCT, PEst-OE/MAT/UI0209/2013.

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Sarrico, C.O.R. The Brio system with initial conditions involving Dirac masses: A result afforded by a distributional product. Chin. Ann. Math. Ser. B 35, 941–954 (2014). https://doi.org/10.1007/s11401-014-0862-8

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  • DOI: https://doi.org/10.1007/s11401-014-0862-8

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