Complete Integrability of Shock Clustering and Burgers Turbulence

Abstract

We consider scalar conservation laws with convex flux and random initial data. The Hopf-Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial data is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group that is in analogy with geodesic flow on SO(N ); (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for N -dimensional approximations of the Lax equations, and yield formulas for the limit N → ∞. The main observation is that the Lax equations are a N → ∞ limit of a Markovian variant of the N -wave model. This allows us to introduce a variety of methods from the theory of integrable systems.MSC classification: 35R60, 37K10, 60J35, 60H99, 82C99, 35L67

Scalar conservation laws with monotone pure-jump Markov initial conditions

“…for each fixed t > 0 has x = 0 marginal given by ℓ(t, dρ 0 ) and for 0 < x < ∞ evolves according to rate kernel f (t, ρ − , dρ + ). The Lax pair and integrable systems approach (in the case of finitely many states, where the generator is a triangular matrix) have been further explored by Menon [22] and in a forthcoming work by Li [20].…”

Hierarchies of N-point functions for nonlinear conservation laws with random initial data

“…Moreover, Burgers' equation can be used to study the statistics of solutions that develop from the initial data, whether Markov properties are conserved, and whether there are universality classes for Burgers' equation, as in [17]. In Menon [16] and Menon and Srinivasan [18], further analysis on these equations was performed to gain deeper understanding of particle dynamics and the evolution of the system starting with random initial data. The analysis went beyond Burgers' equation to the more general case of a C 1 , convex flux.…”

A minimization approach to conservation laws with random initial conditions and non-smooth, non-strictly convex flux

“…The conservation law in the form have a wide array of applications among fluid mechanics, shocks, and turbulence. An interesting feature is that even for a smooth flux function H and smooth initial data, discontinuous solutions to the conservation law (1.1) may occur [4,6,10,11,12,14,15,17,18,19,21,22,23,24,25,31,32]. In particular, this is observed in the prototypical case of Burgers' equation, for which H (w) = w 2 /2.…”