Singular solutions of a fully nonlinear 2x2 system of conservation laws

Original · EN

Existence and admissibility of δ-shock type solution is discussed for the following nonconvex strictly hyperbolic system arising in studues of plasmas: ₜ u + ₓ (u²+v²2) &=0 ₜ v +ₓ(v(u-1))&=0. The system is fully nonlinear, i.e. it is nonlinear with respect to both variables. The latter system does not admit the classical Lax-admissible solution to certain Riemann problems. By introducing complex valued corrections in the framework of the weak asymptotic method, we show that an compressive δ-shock type solution resolves such Riemann problems. By letting the approximation parameter to zero, the corrections become real valued and we obtain a δ-type solution concept. In the frame of that concept, we can show that every 2× 2 system of conservation laws admits δ-type solution.