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Travelling-wave solutions of the inviscid Burgers equation having smoothinitial wave profiles of suitable shapes are known to develop shocks (infinitegradients) in finite times. Such singular solutions are characterized by energyspectra that scale with the wave number $k$ as $k^{-2}$. ****  In this study, we carry out an analysis which verifies the dynamical featuresdescribed above and derive upper bounds for $\epsilon$ and $N$. It is foundthat $\epsilon$ 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 thevelocity field at $t=t_*$. Given $\epsilon>0$ in the limit $\nu\to0$, thisimplies that the energy spectrum remains no steeper than $k^{-2}$ in thatlimit. 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 thelower 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 isshown to coincide with a rigorous estimate for the number of degrees of freedomdefined in terms of local Lyapunov exponents. We demonstrate both analyticallyand numerically an instance where the $k^{-2}$ scaling is uniquely realizable.The numerics also return $\epsilon$ and $t_*$, consistent with analytic valuesderived from the corresponding limiting weak solution.

Energy dissipation and resolution of steep gradients in one-dimensional Burgers flows