Quasi-linear first order pde: find shock time

by Riccardo Orlando   Last Updated September 14, 2018 15:20 PM

Using the characteristics method, show that the Cauchy Problem for the quasi-linear equation $$u_t + x u u_x = 0 \qquad u(0, x) = \phi(x) = > \frac \pi 2 - \arctan(x)$$ has two shock times, $t^*_\pm $, one in the future and one in the past. Determine $t^*_\pm $.

The characteristic curves are the solutions to the ODE system $$ \dot t = 1 \qquad t(0) = 0$$ $$ \dot x = ux \qquad x(0) = x_0$$ $$ \dot u = 0 \qquad u(0) = \phi (x_0)$$ This yields the solutions $$x(t) = \exp(t\phi(x_0))+x_0+1$$ If this expression can be inverted into a function $x_0(t, x)$ then there is no shock, but if that is the case then $x(t)$ must be monotone as function of $x_0$. Thus, define $F_t(x) = \exp(t\phi(x))+x+1$ : I need to find the values of $t$ for which $F' >0$ for all $x$. This reduces to: $$ \frac s {1+x^2} \exp\left (s\left (\frac \pi 2-\arctan(x)\right )\right ) < 1$$ At which point I am stuck. I cannot solve this inequality, but also I have the nagging feeling that I made a mistake somewhere earlier. Did I? And what are the shock times?

Tags : pde


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