Transport of mass, momentum and energy in zero-pressure gas dynamics

We introduce integral identities to define delta-shock wave type solutions for the multidimensional system of conservation laws rho(t) + V . (rho F(U)) = 0, (rho U)(t) + del . (rho N(U)) = 0, x is an element of R(n), where F = (F(j)) is a given vector field, N = (N(jk)) is a given tensor field, F(j), N(kj) : R(n) -> R, j, k = 1, ... , n; rho(x, t) is an element of R, U(x, t) is an element of R(n). We show that delta-shocks are connected with transport and concentration processes and derive the volume-delta-shock wave front balance laws. A well-known particular case of the above system is zero-pressure gas dynamics. In the case of zero-pressure gas dynamics we derive the balance laws describing mass, momentum, and energy transport between the volume outside of the delta-shock wave front and the delta-shock moving wave front. We prove that these processes are going on in such a way that the mass of the S-shock wave front is an increasing quantity, while the energy of the volume (outside of the delta-shock wave front) and the total energy are nonincreasing quantities. These results can be used in modeling of mediums which can be treated as a pressureless continuum (dusty gases, two-phase flows with solid particles or droplets, granular gases).