[Paper Review] A quasi-potential for conservation laws with boundary conditions

Christophe Bahadoran|arXiv (Cornell University)|Oct 18, 2010

Navier-Stokes equation solutions39 references16 citations

TL;DR

This paper computes the quasi-potential and minimizing paths for scalar conservation laws with boundary conditions in the sense of Bardos et al. (BLN), using a flux function $f(\rho)$, strictly convex entropy $h(\rho)$, and boundary densities $\rho_l, \rho_r$. The key result is a generalization of the stationary large deviation functional from the asymmetric exclusion process, revealing non-local, shock-based fluctuation paths that depend on boundary data and flux convexity.

ABSTRACT

We compute the quasi-potential and determine minimizing paths for an action functional related to scalar conservation laws on an interval with boundary conditions in the sense of Bardos et al. (1979). Taking as input an exclusion-like flux function, a strictly convex entropy, and boundary data, we obtain a generalization of the functional derived by Derrida, Lebowtiz and Speer (2003) for the stationary large deviations of the asymmetric exclusion process.

Motivation & Objective

• To derive the quasi-potential $V[\rho(\cdot)]$ for scalar conservation laws on an interval with boundary conditions in the BLN sense.
• To identify the minimizing paths (fluctuation paths) that realize the quasi-potential, corresponding to rare events in nonequilibrium stationary states.
• To generalize the stationary large deviation functional previously derived for the asymmetric exclusion process to a broader class of flux functions and boundary conditions.
• To establish a connection between dynamical large deviations and static large deviations via the quasi-potential in boundary-driven systems.

Proposed method

• Formulates an action functional $I[\rho(\cdot,\cdot)]$ quantifying deviation from a BLN entropy solution, motivated by large deviation principles in exclusion processes.
• Defines the quasi-potential $V[\rho(\cdot)]$ as the infimum of $I$ over all paths connecting a stationary solution to a given profile $\rho(\cdot)$.
• Uses the method of characteristics and viscosity solution techniques to analyze the Hamilton-Jacobi equation associated with the quasi-potential.
• Derives explicit fluctuation paths by solving the time-reversed Hamiltonian system, identifying shock and rarefaction wave structures depending on boundary data and flux convexity.
• Applies the Onsager-Machlup principle in a nonequilibrium setting, showing that minimizing paths are time-reversed relaxation paths under suitable conditions.
• Classifies minimizing paths into three cases based on the relative position of $\rho$ with respect to $\rho^*$, the flux maximizer, and boundary values $\rho_l, \rho_r$.

Experimental results

Research questions

• RQ1 What is the explicit form of the quasi-potential for scalar conservation laws with BLN boundary conditions?
• RQ2 How do minimizing paths (fluctuation paths) behave in nonequilibrium stationary states of boundary-driven systems?
• RQ3 Under what conditions does the quasi-potential exhibit non-locality, and how is this related to shock formation?
• RQ4 How does the structure of the minimizing path depend on the relative values of $\rho_l$, $\rho_r$, and the flux maximizer $\rho^*$?
• RQ5 Can the quasi-potential be derived from a dynamical large deviation principle and linked to the static large deviation function in exclusion-like models?

Key findings

• The quasi-potential is computed explicitly for a general class of flux functions $f(\rho)$ satisfying $f \in C^2$, $f(0)=f(K)=0$, and $f''<0$, with strictly convex entropy $h(\rho)$.
• Minimizing paths are shown to consist of rarefaction waves and shocks, depending on the boundary data and the position of $\rho$ relative to $\rho^*$, the flux maximizer.
• For $\rho_l < \rho_r$, the minimizing path involves a rarefaction wave from $\rho_l$ and a shock connecting to $\rho_r$, with the shock speed determined by the Rankine-Hugoniot condition.
• In the case $\rho_r < \rho^* < \rho_l$, the path structure depends on whether $\rho \geq \rho^*$ or $\rho \leq \rho^*$, leading to different wave configurations involving $\rho^\prime = \varphi(\rho)$.
• For $\rho^* < \rho_r \leq \rho_l$, the system exhibits symmetric behavior with respect to boundaries, and the path includes a shock propagating from $x=1$ to $x=0$ in finite time.
• The time-reversed relaxation path is shown to be the unique minimizer, confirming the nonequilibrium generalization of the Onsager-Machlup principle.