the evolution of the multivalued solution of for this initial data are shown in Figure \[fig:numerics2\].

![*The multivalued solution (black) of and the signed sum of branches (red) corresponding to $u_0(x)=2(1+x^2)^{-1}$. Left: $t=0$. Middle: $t=1$. Right: $t=2$. Before the breaking time as well as afterwards but outside the oscillation interval there is only one solution branch and hence no difference between the red and black curves.*[]{data-label="fig:numerics2"}](BOFig2.pdf)

Our result is then the following.

\[MainTheorem\] Let $u^{\mathrm{B}}_{0}(x,t) <
u^{\mathrm{B}}_{1}(x,t)< \cdots < u^{\mathrm{B}}_{2P(x,t)}(x,t)$ be the branches of the multivalued (method of characteristics) solution of the inviscid Burgers equation subject to an admissible initial condition $u^\mathrm{B}(x,0) = u_{0}(x)$. Then, the weak $L^2(\mathbb{R})$ (in $x$) limit of ${\tilde{u}_\epsilon}(x,t)$ is given by $$\label{eq:MainResult}
\mathop{\mathrm{w}_x\mathrm{-lim}}_{\epsilon \downarrow
0}{\tilde{u}_\epsilon}(x,t)=\sum_{n=0}^{2P(x,t)}(-1)^{n}u^{\mathrm{B}}_{n}(x,t),$$ uniformly for $t$ in arbitrary bounded intervals. Note that the right-hand side extends by continuity to the caustic curves.

The signed sum of branches that is the weak limit is illustrated with red curves in Figure \[fig:numerics2\] for the same initial data as in Figure \[fig:numerics1\]. Of course convergence in the weak $L^2(\mathbb{R})$ (in $x$) topology means that for every $v\in L^2(\mathbb{R})$, we have $$\lim_{\epsilon\downarrow 0}\int_{\mathbb{R}}{\tilde{u}_\epsilon}(x,t)v(x)\,dx = 
\int_{\mathbb{R}}
\left[\sum_{n=0}^{2P(x,t)}(-1)^{n}u^{\mathrm{B}}_{n}(x,t)\right]v(x)\,dx$$ with the limit being uniform with respect to $t$ in arbitrary 