Multidimensional delta-shock waves and the transportation and concentration processes

Abstract

{\it $\delta$-Shock wave type solutions} in the multidimensional system of conservation laws $$ \rho_t + \nabla\cdot(\rho F(U))=0, \qquad (\rho U)_t + \nabla\cdot(\rho N(U))=0, \quad x\in \bR^n, $$ are studied, where $F=(F_j)$ is a given vector field, $N=(N_{jk})$ is a given tensor field, $F_j, N_{kj}:\bR^n \to \bR$, $j,k=1,...,n$; $\rho(x,t)\in \bR$, $U(x,t)\in \bR^n$. The well-known particular cases of this system are zero-pressure gas dynamics in a standard form $$ \rho_t + \nabla\cdot(\rho U)=0, \quad (\rho U)_t + \nabla\cdot(\rho U\otimes U)=0, $$ and in the relativistic form $$ \rho_t + \nabla\cdot(\rho C(U))=0, \quad (\rho U)_t + \nabla\cdot(\rho U\otimes C(U))=0, $$ where $C(U)=\frac{c_0U}{\sqrt{c_0^2+|U|^2}}$, $c_0$ is the speed of light. We introduce the integral identities which constitute definition of $\delta$-shocks for the above systems and using this definition derive the Rankine--Hugoniot conditions for curvilinear $\delta$-shocks. We show that $\delta$-shocks are connected with {\em transportation processes and concentration processes} and derive the $\delta$-shock balance laws describing mass and momentum transportation between the volume outside the wave front and the wave front. In the case of zero-pressure gas dynamics the transportation process is the concentration process. We also prove that energy of the volume outside the wave front and total energy are {\em nonincreasing quantities}. The possibility of the {\em effect of kinematic self-gravitation} and the {\em effect of dimensional bifurcations of $\delta$-shock} in zero-pressure gas dynamics are discussed.