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When the velocity depends on $u$, different initial values travel at different speeds. The resulting characteristic crossing can destroy a classical solution in finite time, even though the initial function is smooth and bounded. For the Inviscid Burgers equation, take $u_0(a)=-\tanh a$. Before crossing,andAt $a=0$ this tends to $-\infty$ as $t\uparrow 1$, while $|u|\le 1$. Thus $\boxed{\text{bounded states can undergo finite-time gradient blow-up and shock formation}.}$ The bounded-derivative Burgers flux agrees with this equation on the whole state range, if a global bounded-speed flux is desired. After crossing, arbitrary weak solutions need not be unique; an entropy solution condition is needed to select the admissible continuation.

A bounded weak solution satisfiesfor every compactly supported $C^1$ test function, of either sign, allowed to meet $t=0$. Across a moving jump $x=s(t)$ with distinct left/right constant states, the distributional delta coefficient vanishes exactly when the Rankine-Hugoniot condition holds:It is required on every interface of a piecewise constant solution; the initial trace must also agree with the data.

For nonuniqueness respecting the paper's global bound on $f^{\prime }$, use the bounded-derivative Burgers flux $f(z)=z^2/2$ on $|z|\le 1$ and $f(z)=|z|-1/2$ outside. It is $C^1$ and $|f^{\prime }|\le 1$. Take data $-1$ for $x<0$, $1$ for $x>0$. A stationary jump obeys the condition because $f(-1)=f(1)$. Another weak solution is the rarefaction waveThe fan solves the equation inside each smooth region, is continuous at its edges, and has the same local $L^1$ initial trace. Thus weak solutions are not unique. The stationary expansion shock violates the entropy inequality, whereas the fan is admissible; using the unmodified quadratic flux alone would miss the printed global bounded-derivative hypothesis.