bounded intervals. Thus, the weak limit essentially smooths out the rapid oscillations seen in Figure \[fig:numerics1\] and (if we think of $v$ as the indicator function of a mesoscale interval) represents a kind of local average in $x$. What it means for an initial condition to be admissible will be explained later (see Definition \[def:admissible\]). Here ${\tilde{u}_\epsilon}(x,t)$ is not exactly the solution of the Cauchy problem for the BO equation with fixed initial data $u_0(x)$, but it is for every $\epsilon>0$ an exact solution of that satisfies an $\epsilon$-dependent initial condition that converges (in the strong $L^2$ sense, see Corollary \[corr:strong\] below) to $u_0(\cdot)$ as $\epsilon\downarrow 0$. See Definition \[def:modified\] for more details. This modification of the initial data is an analogue of the replacement of the true scattering data by its reflectionless WKB approximation in the Lax-Levermore theory.

For $t$ before the breaking time $T$ for Burgers’ equation, the weak limit guaranteed by Theorem \[MainTheorem\] may be strengthened as follows.

Suppose that $0\le t<T$, so that $P(x,t)=0$ for all $x\in\mathbb{R}$ (that is, the solution $u^\mathrm{B}=u^\mathrm{B}_0(x,t)$ of Burgers’ equation with initial data $u_0(x)$ is classical). Then $$\lim_{\epsilon\downarrow 0}{\tilde{u}_\epsilon}(x,t) = u^\mathrm{B}_0(x,t)$$ with the limit being in the (strong) $L^2(\mathbb{R}_x)$ topology. \[corr:strong\]

