Multidimensional Delta-shock Waves and the Transportation and Concentration Processes

δ-Shock wave type solutions in the multidimensional system of conservation laws ρt +∇ · (ρF (U)) = 0, (ρU)t +∇ · (ρN(U)) = 0, x ∈ R , are studied, where F = (Fj) is a given vector field, N = (Njk) is a given tensor field, Fj , Nkj : R n → R, j, k = 1, . . . , n; ρ(x, t) ∈ R, U(x, t) ∈ R. 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| 2 , c0 is the speed of light. We introduce the integral identities which constitute definition of δ-shocks for the above systems and using this definition derive the Rankine–Hugoniot conditions for curvilinear δshocks. We show that δ-shocks are connected with transportation processes and concentration processes and 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. We also prove that energy of the volume outside the wave front and total energy are nonincreasing quantities. The possibility of the effect of kinematic self-gravitation and the effect of dimensional bifurcations of δ-shock in zero-pressure gas dynamics are discussed.