Multidimensional delta-shocks and the transportation and concentration processes
Abstract: We introduce the definitions of a δ-shock wave type solution for the multidimensional system of conservation laws
ρt + ∇·(ρF(U))=0, (ρU)t + ∇·(ρN(U))=0, x∈R^n,
where F=(Fj) is a given vector field, N=(Njk) is a given tensor field, Fj, Nkj:R^n → R, j,k=1,…,n. The well-known particular cases of this system are zero-pressure gas dynamics in a standard form
ρt+∇·(ρU)=0, (ρU)t + ∇·(ρU⊗U)=0,
and in the relativistic form
ρt + ∇·(ρC(U))=0, (ρU)t + ∇·(ρU⊗C(U))=0,
where C(U)=c0U/√(c0^2+|U|^2), c0 is the speed of light. Using this definition, the Rankine–Hugoniot conditions for δ-shocks are derived. We also derive the δ-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.