# Preprint 2007-031

# Multidimensional delta-shocks and the transportation and concentration processes

## V.M. Shelkovich

**Abstract:**
We introduce the definitions of a
*δ-shock wave type solution*
for the multidimensional system of conservation laws

*ρ*_{t} + ∇·(*ρF*(*U*))=0,
(*ρU*)_{t} + ∇·(*ρN*(*U*))=0,
*x*∈**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*.
The well-known particular cases of this system are
zero-pressure gas dynamics in a standard form

*ρ*_{t}+∇·(*ρU*)=0, (*ρU*)_{t} + ∇·(*ρU*⊗*U*)=0,

and in the relativistic form

*ρ*_{t} + ∇·(*ρC*(*U*))=0, (*ρU*)_{t} + ∇·(*ρU*⊗*C*(*U*))=0,

where *C*(*U*)=*c*_{0}*U*/√(*c*_{0}^2+|*U*|^2),
*c*_{0} 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.