Russian Journal of Mathematical Physics

, Volume 22, Issue 4, pp 518–527 | Cite as

The Riemann problem for the Brio system: a solution containing a Dirac mass obtained via a distributional product

Article

Abstract

The system of conservation laws \({u_t} + {\left( {\frac{{{u^2} + {v^2}}}{2}} \right)_x} = 0\), v t + (uvv)x = 0 with the initial conditions u(x, 0) = l 0 + b 0 H(x), v(x, 0) = k 0 + c 0 H(x), where H is the Heaviside function is studied. This strictly hyperbolic system was introduced by M. Brio in 1988 and provides a simplified model for the magnetohydrodynamics equations. Under certain compatibility conditions for the constants l 0, b 0, k 0, c 0, an explicit solution containing a Dirac mass is given and we prove the uniqueness of this solution within a convenient class of distributions which includes Dirac-delta measures. Our concept of solution is defined within the framework of a distributional product, and it is a consistent extension of the concept of a classical solution. This direct method seems considerably simpler than the weak asymptotic method usually used in the study of delta-shocks emergence in nonlinear conservation laws.

Keywords

Conservation Laws Distributional Product Riemann Problem Delta Wave Magnetohydrodynamics Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.CMAFCIOFaculdade de Cincias da Universidade de LisboaLisboaPortugal

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