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

Abstract

The system of conservation laws \({u_t} + {\left( {\frac{{{u^2} + {v^2}}}{2}} \right)_x} = 0\), vt + (uv − v)x = 0 with the initial conditions u(x, 0) = l0 + b0H(x), v(x, 0) = k0 + c0H(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 l0, b0, k0, c0, 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.