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

u_{t}+(u^{2})_{x}=0,v_{t}+2(uv)_{x}=0,w_{t}+2(v^{2}+uw)_{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 foru(x,0),v(x,0),w(x,0))=(u_{−},v_{−},w_{−}), [x<0], (u_{+},v_{+},w_{+}), [x>0],t>0 its second componentvmay contain Dirac measures, and the third componentwmay contain a linear combination of Dirac measures and their derivatives, while the first componentuof 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"distributionalsolutions 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.

**Paper:**- Available as PDF (368 Kbytes), Postscript (672 Kbytes) or gzipped PostScript (280 Kbytes; uncompress using gunzip).
**Author(s):**- V. M. Shelkovich, <shelkv@vs1567.spb.edu>
**Publishing information:****Comments:****Submitted by:**- <shelkv@vs1567.spb.edu> December 13 2005.

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