[Paper Review] Estimates for Solutions of a Low-Viscosity Kick-Forced Generalised Burgers Equation

Alexandre Boritchev|arXiv (Cornell University)|Jul 25, 2011

Advanced Mathematical Physics Problems3 references3 citations

TL;DR

This paper studies a low-viscosity, kick-forced generalized Burgers equation on the periodic domain $S^1$, establishing that after a transient time $t \geq 2$, the ensemble-averaged Sobolev norms of solutions scale as $C\nu^{-\gamma}$ with $\gamma = \max(0, m - 1/p)$, both in upper and lower bounds. The estimates are independent of initial conditions and depend only on the forcing and nonlinearity, confirming a quasi-stationary regime with optimal power-law scaling in the viscous limit.

ABSTRACT

We consider a non-homogeneous generalised Burgers equation: $$ \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - ν\frac{\partial^2 u}{\partial x^2} = η^ω,\quad t \in \R,\ x \in S^1. $$ Here, νis small and positive, f is strongly convex and satisfies a growth assumption, while η^ω is a space-smooth random "kicked" forcing term. For any solution $u$ of this equation, we consider the quasi-stationary regime, corresponding to t>=2. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of $u$. These estimates are of the form C ν^{-β} with the same values of $β$ for bounds from above and from below. They depend on ηand f, but do not depend on the time t or the initial condition.

Motivation & Objective

• To analyze the long-time behavior of solutions to a generalized Burgers equation with small viscosity and random kick forcing.
• To establish rigorous upper and lower bounds for ensemble-averaged Sobolev norms of solutions in the quasi-stationary regime.
• To remove dependence on initial conditions by introducing random kick forcing and ensemble averaging.
• To characterize the energy spectrum and intermittency properties of solutions in the small-$\nu$ limit.
• To confirm the optimality of the scaling $\nu^{-\gamma}$ for Sobolev norms via matching upper and lower estimates.

Proposed method

• Use of the Gagliardo-Nirenberg inequality to control higher-order Sobolev norms in terms of lower-order $L_p$ norms.
• Application of maximum principle and energy estimates to control the $W^{1,1}$ norm after a damping time.
• Introduction of a random kick forcing term $\eta^\omega$ with smooth spatial structure to model stochastic energy injection.
• Employment of ensemble averaging to eliminate initial condition dependence and establish time-independent bounds.
• Derivation of upper bounds via energy and moment estimates in the framework of Kuksin and Biryuk’s methods.
• Use of time-averaged estimates and Hölder’s inequality to derive lower bounds matching the upper bounds in power-law scaling.

Experimental results

Research questions

• RQ1 What is the asymptotic scaling of Sobolev norms of solutions to the generalized Burgers equation as viscosity $\nu \to 0^+$, under random kick forcing?
• RQ2 Can the dependence on initial conditions be removed in the long-time behavior of solutions through ensemble averaging?
• RQ3 What is the structure of the energy spectrum in the inertial and dissipation ranges for this equation in the small-$\nu$ limit?
• RQ4 Does the solution exhibit intermittency in the quasi-stationary regime, and how is this reflected in the norm scaling?
• RQ5 Are the upper and lower bounds for Sobolev norms of the same order in $\nu$, confirming optimality of the scaling?

Key findings

• For $t \geq 2$, the ensemble-averaged $W^{m,p}$ norm of the solution satisfies $\left(\mathbb{E} \sup_{t \in [k,k+1)} |u|_{m,p}^n \right)^{1/n} \leq C(m,p,n) \nu^{-\gamma}$ with $\gamma = \max(0, m - 1/p)$, independent of initial data.
• Time-averaged lower bounds satisfy $\left(\frac{1}{T}\int_t^{t+T} \mathbb{E} |u(s)|_{m,p}^n \right)^{1/n} \geq C(m,p) \nu^{-\gamma}$ for $t \geq 2$, $T \geq N'$, confirming matching scaling.
• The energy spectrum is supported in the interval $(0, \nu^0]$, with the dissipation range $k \in (\nu^{-1}, \infty)$ and inertial range $k \sim \nu^{-1}$, where $E_k \sim k^{-2}$ is suggested.
• The solution exhibits intermittency: $\max_x |u_x| \sim 1$ while $\int_{S^1} u_x^2 \, dx \sim \nu^{-1}$, indicating large negative gradients on small sets.
• The power-law scaling $\nu^{-\gamma}$ is optimal, as upper and lower bounds match exactly in exponent for all $m \geq 0$, $p \in [1, \infty]$, except possibly for $m \geq 2$, $p=1$.
• The damping time to reach the quasi-stationary regime is bounded independently of the initial condition, with $t \geq 2$ sufficient for all estimates to hold.