Preprint 2005-053

The Riemann problem admitting δ′-shock wave type solution (the vanishing viscosity approach)

V. M. Shelkovich

Abstract: In this paper, using the vanishing viscosity method, a solution of the Riemann problem for the system of conservation laws

ut+(u2)x=0,     vt+2(uv)x=0,      wt+2(v2+uw)x=0

with the initial data

(u(x,0),v(x,0),w(x,0))=(u,v,w), [x<0],   (u+,v+,w+), [x>0],

is constructed. This problem admits a δ′-shock wave type solution, which is a new type of singular solutions to systems of conservation laws first introduced in [25]. Roughly speaking, it is a solution of the above system such that for t>0 its second component v may contain Dirac measures, and the third component w may contain a linear combination of Dirac measures and their derivatives, while the first component u of the solution has bounded variation. Using the above mentioned results, we solve the δ-shock Cauchy problem for the first two equations of the above system. Since δ′-shocks can be constructed by the vanishing viscosity method, these solutions are "natural" distributional solutions to systems of conservation laws. The results of this paper as well as those of the paper [25] show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well.

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V. M. Shelkovich, <>
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<> December 13 2005.

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