We review recent developments in the rigorous derivation of hydrodynamic-type macroscopic equations from simple microscopic models: continuous time stochastic cellular automata. The deterministic evolution of hydrodynamic variables emerges as the "law of large numbe~s," which holds with probability one in the limit in which the ratio of the microscopic to the macroscopic spatial and temporal sc...

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves, and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, w...

Analytic solutions for Burgers equations with source terms, possibly stiff, represent an important element to assess numerical schemes. Here we present a procedure, based on the characteristic technique to obtain analytic solutions for these equations with smooth initial conditions.

We study shock statistics in the scalar conservation law ∂tu+∂xf(u) = 0, x ∈ R, t > 0, with a convex flux f and random initial data. We show that a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps) is preserved by the entropy solution to the conservation law and we derive kinetic equations that describe the evolution ...

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wacławczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extend...

We study Lagrangian trajectories and scalar transport statistics in decaying Burgers turbulence. We choose velocity fields solutions of the inviscid Burgers equation whose probability distributions are specified by Kida’s statistics. They are time-correlated, not time-reversal invariant and not Gaussian. We discuss in some details the effect of shocks on trajectories and transport equations. We...

For Burgers equations with real data and complex forcing terms, Lerner, Morimoto and Xu [Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems, Amer. J. Math. 2010] proved that only analytical data generate local C solutions. The corresponding instabilities are however not observed numerically; rather, numerical simulations show an exponential growth only after a del...

It is pointed out that for the case of (compressible) magnetohydrodynamics (MHD) with the fields v y (y, t) and B x (y, t) one can have equations of the Burgers type which are integrable. We discuss the solutions. It turns out that the propagation of the non-linear effects is governed by the initial velocity (as in the Burgers case) as well as by the initial Alfvén velocity. Many results previo...