Multidimensional Delta-shocks and the Transportation and Concentration Processes

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, where F = (Fj) is a given vector field, N = (Njk) is a given tensor field, Fj , Nkj : Rn → 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 q c0+|U| , c0 is the speed of light. Using this definition, the Rankine–Hugoniot conditions for δ-shocks are derived. We also derive the δshock 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.