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$\delta ^{\prime }$-shock waves as a new type of solutions to systems of conservation laws. (English) Zbl 1108.35116
Summary: A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called $\delta^{(n)}$-shock wave, where $\delta^{(n)}$ is $n$th derivative of the Dirac delta function $(n=1,2,\dots)$. In this paper the case $n=1$ is studied in details. We introduce a definition of $\delta'$-shock wave type solution for the system $$u_t+\bigl(f(u)\bigr)_x=0,\quad v_t+\bigl(f' (u)v\bigr)_x=0, \quad w_t+\bigl(f''(u)v^2+f'(u)w\bigr)_x=0.$$ Within the framework of this definition, the Rankine-Hugoniot conditions for $\delta'$-shock are derived and analyzed from geometrical point of view. We prove $\delta'$-shock balance relations connected with area transportation. Finally, a solitary $\delta'$-shock wave type solution to the Cauchy problem of the system of conservation laws $u_t+(u^2)_x= 0$, $v_t+2(uv)_x =0$, $w_t+2(v^2+uw)_x=0$ with piecewise continuous initial data is constructed. These results first show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of $\delta$-shocks) but their derivatives as well.

MSC:
35L67Shocks and singularities
35L65Conservation laws
76L05Shock waves; blast waves (fluid mechanics)
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References:
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