  Travelling-wave solutions of the inviscid Burgers equation having smooth
initial wave profiles of suitable shapes are known to develop shocks (infinite
gradients) in finite times. Such singular solutions are characterized by energy
spectra that scale with the wave number $k$ as $k^{-2}$. ****
  In this study, we carry out an analysis which verifies the dynamical features
described above and derive upper bounds for $\epsilon$ and $N$. It is found
that $\epsilon$ satisfies $\epsilon \le
\nu^{2\alpha-1}\norm{u_*}_\infty^{2(1-\alpha)}
\norm{(-\Delta)^{\alpha/2}u_*}^2$, where $\alpha<1$ and $u_*=u(x,t_*)$ is the
velocity field at $t=t_*$. Given $\epsilon>0$ in the limit $\nu\to0$, this
implies that the energy spectrum remains no steeper than $k^{-2}$ in that
limit. For the critical $k^{-2}$ scaling, the bound for $\epsilon$ reduces to
$\epsilon\le\sqrt{3}k_0\norm{u_0}_\infty\norm{u_0}^2$, where $k_0$ marks the
lower end of the inertial range and $u_0=u(x,0)$. This implies
$N\le\sqrt{3}L\norm{u_0}_\infty/\nu$, where $L$ is the domain size, which is
shown to coincide with a rigorous estimate for the number of degrees of freedom
defined in terms of local Lyapunov exponents. We demonstrate both analytically
and numerically an instance where the $k^{-2}$ scaling is uniquely realizable.
The numerics also return $\epsilon$ and $t_*$, consistent with analytic values
derived from the corresponding limiting weak solution.
