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

TL;DR

This paper introduces a novel $oldsymbol{ ext{delta}}$-shock solution framework for fully nonlinear 2x2 conservation laws, resolving Riemann problems where classical solutions fail, using complex corrections and weak asymptotics.

Contribution

It develops a $oldsymbol{ ext{delta}}$-shock solution concept for fully nonlinear systems, extending solution existence beyond classical admissible solutions.

Findings

Every 2x2 conservation law system admits a $oldsymbol{ ext{delta}}$-type solution.

Complex-valued corrections lead to real-valued $oldsymbol{ ext{delta}}$-solutions in the limit.

The method resolves Riemann problems lacking classical solutions.

Abstract

Existence and admissibility of $\delta$-shock type solution is discussed for the following nonconvex strictly hyperbolic system arising in studues of plasmas: \pa_t u + \pa_x \big(\Sfrac{u^2+v^2}{2} \big) &=0 \pa_t v +\pa_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 $\delta$-shock type solution resolves such Riemann problems. By letting the approximation parameter to zero, the corrections become real valued and we obtain a $\delta$-type solution concept. In the frame of that concept, we can show that every $2\times 2$ system of conservation laws admits $\delta$-type solution.