Wave breaking in Burgers' equation

Question

My understanding is that Burgers' equation is a prototype for wave breaking: Solution $u(x,t)$ stays bounded as its derivative $u_x(x,t)\to\infty$. I solved $$u_t+uu_x=0,\,\,\, u(x,0)=-x,\,\,\, x\in[0,1],$$ and got $$u(x,t)=-\frac{x}{1-t},\,\,\, u_x(x,t)=-\frac{1}{1-t}.$$ Surely $u_x(x,t)\to-\infty$ everywhere in $[0,1]$, but $u$ also does at any $x\in(0,1]$. I know $u$ along characteristics is constant, in this case it equals $-x$, and at the singularity time, $t^*=1$, all characteristics are converging to $x=0$, where $u$ is zero. So there is steepening happening in $u$ at $x=0$, but $u$ is not bounded for $t\in[0,t^*)$.

Answer 1

Since the Burgers equation is nonlinear, there can be infinitely many solutions in the sense that you are considering (just taking the derivatives and plugging them in the equation).

In order to select the solution to the equation, one has to be more restrictive on what can be considered a solution. In that sense, shocks are defined, mass conservation is imposed... and rarefaction waves appear. In your case, a rarefaction wave arises in $(x=1,t=0)$, opening to the left, which controls the height of the peak you are mentioning. In the graph below (characteristics in $x$ vs. $t$), you can see that, while the non-zero part of the initial data moves into a shock at $t=1$ (in red), a rarefaction wave appears in the empty space that it appears behind it.

I really like how this is explained in Partial Differential Equations by Lawrence C. Evans, but you can find lots of notes about the topic in Google.

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EDIT: In that book, you can find a theorem that describes the behavior of solutions to the Burgers equation in the whole spatial domain. It says that for bounded and summable initial data (hence, vanishing at $\pm\infty$), there exists $C>0$ such that $|u(x,t)|\le Ct^{-1/2}$ for all $x\in\mathbb{R}$ and $t>0$.

Moreover, the dynamics of the Burgers equation push all such initial data (whatever their shape is) towards a function called N-wave, which IMHO is fascinating.