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| Products of Distributions, Conservation Laws and the Propagation of δ -Shock Waves |
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Citation： |
Carlos Orlando R. SARRICO.Products of Distributions, Conservation Laws and the Propagation of δ -Shock Waves[J].Chinese Annals of Mathematics B,2012,33(3):367~384 |
| Page view： 1891
Net amount： 1337 |
Authors： |
Carlos Orlando R. SARRICO; |
Foundation： |
Funda\c c\~ao para a Ci\^encia e a Tecnologia, PEst OE/MAT/UI0209/2011. |
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| Abstract： |
This paper contains a study of propagation of singular travelling waves $ u(x,t)$ for conservation laws $u_{t}+[\phi (u)]_{x}=\psi(u)$, where $\phi ,\psi $ are entire functions taking real values on the real axis. Conditions for the propagation of wave profiles $\beta +m\delta $ and $\beta +m\delta '$ are presented ($\beta $ is a real continuous function, $m\neq 0$ is a real number and $\delta'$ is the derivative of the Dirac measure $\delta $). These results are obtained with a consistent concept of solution based on our theory of distributional products. Burgers equation $u_{t}+(\frac{u^{2}}{2})_{x}=0$, the diffusionless Burgers-Fischer equation $ u_{t}+a(\frac{u^{2}}{2})_{x}=ru(1-\frac{u}{k})$ with $a,r,k$ being positive numbers, Leveque and Yee equation $u_{t}+u_{x}=\mu u(1-u)(u-\frac{1}{2})$ with $\mu \neq 0$, and some other examples are studied within such a setting. A "tool box" survey of the distributional products is also included for the sake of completeness. |
Keywords： |
Conservations laws, Travelling waves, $\delta '$-shock waves,$\delta $-shock waves, $\delta $-solitons, Propagation of distributional wave profiles |
Classification： |
46F10, 35D |
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