Consider the initial value problem for $x \in \mathbb R~\text{and}~t>0$: $$u_t + uu_x=2$$ where $u(x,0)=x$.
Are there any shock forming with this initial condition? I'm not quite sure how to show this, though I can definitely solve the equation by method of characteristics. Thanks.
The solution is $u(x,t)= \frac{x+t^2 + 2t}{t+1}$
$\frac{dx}{dr}=u$,$\frac{dt}{dr}=1$ and $\frac{du}{dr}=2$ with $\Gamma: (s,0,s)$. So $\frac{dx}{ds}=r+1$, $\frac{dt}{ds}=0$ and we can calculate the Jacobian of $x(r,s)$ and $t(r,s)$: -r-1=0, which means $t$ needs to be -1, so, it's impossible to have a shock, right?