Title: Existence and uniqueness for a system of conservation laws arising in magnetohydrodynamics
Speaker: Henrik Kalisch (U. Bergen, Norway)
Speaker's URL: https://www.uib.no/en/persons/Henrik.Kalisch
Abstract: The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics. It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear in the whole phase space. As a consequence, certain Riemann problems have no weak solutions in the traditional Lax admissible sense. It was argued in Hayes and LeFloch (1996 Nonlinearity 9 1547-63) that in order to solve the system, singular solutions containing Dirac delta distributions along the shock waves might have to be used. Solutions of this type were exhibited in Kalisch and Mitrovic (2012 Proc. Edinburgh Math. Soc. 55 711-29) and Sarrico (2015 Russ. J.Math. Phys. 22 518-27), but uniqueness was not obtained. In this lecture, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and we show that admissible solutions are unique in this framework.
URL associated with Seminar: https://research.math.osu.edu/pde/