# 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.

## Keywords

Products of distributions Brio’s system*δ*-Shock waves

*δ*′-Shock waves Riemann problem

## 2000 MR Subject Classification

46F10 35D## Preview

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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.MathSciNetCrossRefMATHGoogle Scholar - [2]Brenier, Y. and Grenier, E., Sticky particles and scalar conservation laws,
*SIAM J. Numer. Anal.*,**35**, 1998, 2317–2328.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar - [6]Colombeau, J. F. and Le Roux, A., Multiplication of distributions in elasticity and hydrodynamics,
*J. Math. Phys.*,**29**, 1988, 315–319.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetGoogle Scholar - [11]Huang, F., Weak solutions to pressureless type system,
*Comm. PDE*,**30**, 2005, 283–304.CrossRefMATHGoogle Scholar - [12]Joseph, K. T., A Riemann problem whose viscosity solutions contains
*δ*-measures,*Asymptot. Analysis*,**7**, 1993, 105–120.MATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar - [16]Maslov, V. P., Nonstandard characteristics in asymptotical problems,
*Russian Math. Surveys*,**38**(6), 1983, 1–42.CrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle Scholar - [20]Nedeljkov, M., Delta and singular delta locus for one-dimensional systems of conservation laws,
*Math. Method. Appl. Sci.*,**27**, 2004, 931–955.MathSciNetCrossRefMATHGoogle Scholar - [21]Nedeljkov, M., Shadow waves: Entropies and interactions of delta and singular shocks,
*Archives Rat. Mech. Anal.*,**197**, 2010, 489–537.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefGoogle Scholar - [23]Sarrico, C. O. R., About a family of distributional products important in the applications,
*Port. Math.*,**45**, 1988, 295–316.MathSciNetMATHGoogle Scholar - [24]Sarrico, C. O. R., Distributional products and global solutions for nonconservative inviscid Burgers equation,
*J. Math. Anal. Appl.*,**281**, 2003, 641–656.MathSciNetCrossRefMATHGoogle Scholar - [25]Sarrico, C. O. R., New solutions for the one-dimensional nonconservative inviscid Burgers equation,
*J. Math. Anal. Appl.*,**317**, 2006, 496–509.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefGoogle 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.Google Scholar
- [31]Sheng, W. and Zhang, T., The Riemann problem for transportation equations in gas dynamics,
*Mem. Amer. Math. Soc.*,**137**, 1999, 1–77.MathSciNetGoogle 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.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar

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