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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

ρt + ∇(ρF(U))=0, (ρU)t + ∇(ρN(U))=0, xR^n,

where F=(Fj) is a given vector field, N=(Njk) is a given tensor field, Fj, Nkj:R^nR, 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 + ∇(ρUU)=0,

and in the relativistic form

ρt + ∇(ρC(U))=0,(ρU)t + ∇(ρUC(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.

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V.M. Shelkovich,
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; 2007-12-27.