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

TL;DR

This paper analyzes the energy dissipation and steep gradient resolution in 1D Burgers flows, deriving bounds for energy dissipation rate and degrees of freedom, and confirms the $k^{-2}$ spectral scaling through analysis and numerics.

Contribution

It provides new analytical bounds on energy dissipation and degrees of freedom in Burgers flows, confirming the $k^{-2}$ spectral scaling and demonstrating its realizability.

Findings

Energy spectrum remains no steeper than $k^{-2}$ as viscosity approaches zero.

Derived bounds for energy dissipation rate $$ and degrees of freedom $N$.

Numerical simulations confirm the analytical predictions and spectral scaling.

Abstract

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.