Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 31B (Partial Differential Equations)

Characteristics beginning at $\xi$ have $x=\xi +t{u}_I(\xi )$ and carry the constant value $u_I(\xi )$. In the sloping interval, $x=t+(1-t)\xi$. For $0\le t<1$ the resulting single-valued solution isIt is continuous and piecewise smooth, locally Lipschitz for $t<1$, but not a classical $C^1$ solution across the two derivative corners. Its weak derivatives satisfy the equation, since the function itself has no jumps there. The sloping gradient is $-1/(1-t)$ and blows up at $t=1$, when those characteristics coalesce.

Thus the continuous characteristic description has maximal interval $0\le t<1$. If “weak solution” is taken literally, it does not cease to exist then: the entropy continuation is a shock with left/right states 1 and 0 and Rankine-Hugoniot speed $1/2$, situated at $x=(t+1)/2$ for $t\ge 1$. It is a global entropy weak solution, with the step at $t=1$ as its limiting trace. This distinguishes breakdown of the characteristic parametrization from nonexistence of a weak continuation.

The exterior characteristics give $u=0$ for $x<0$ and $u=1$ for $x>t$. In the gap put $\xi =x/t$ and $u=f(\xi )$. The Inviscid Burgers equation becomes $(f-\xi )f^{\prime }=0$. Matching the increasing states selects the rarefaction wave $f(\xi )=\xi$, soIt is continuous and piecewise smooth at positive times and satisfies the entropy condition. A discontinuous shock joining the same increasing states would be nonentropic.

For any fixed $t>0$, the characteristic map $X_t(\xi )=\xi +t{u}_I(\xi )$ has derivative $1+t{u}_I^{\prime }(\xi )>1$. It is strictly increasing and onto: for $\xi \ge 0$ it is at least $\xi +t{u}_I(0)$, and for $\xi \le 0$ at most that quantity. Hence the inverse function theorem supplies a global $C^1$ inverse. Define $u(x,t)=u_I(X_t^{-1}(x))$. Differentiating givesThe derivatives are continuous, and $u_x\le 1/t$ for positive times. Thus a classical solution exists for all $t>0$, without any characteristic intersection or shock formation.