Related papers: Microscopic shape of shocks in a domain growth mod…
In inviscid solutions of the forced Burgers equation the matter accumulates in the shock discontinuities. We describe the limit motion of particles everywhere including the shocks as the trajectories of a discontinuous velocity field being…
A phenomenological model of the evolution of an ensemble of interacting dislocations in an isotropic elastic medium is formulated. The line-defect microstructure is described in terms of a spatially coarse-grained order parameter, the…
We conjecture the exact shock statistics in the inviscid decaying Burgers equation in D>1 dimensions, with a special class of correlated initial velocities, which reduce to Brownian for D=1. The prediction is based on a field-theory…
We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the…
Shocks due to hyperbolic partial differential equations (PDEs) appear throughout mathematics and science. The canonical example is shock formation in the inviscid Burgers' equation $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial…
The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of…
Growth processes in many living organisms create thin, soft materials with an intrinsically hyperbolic geometry. These objects support novel types of mesoscopic defects - discontinuity lines for the second derivative and branch points -…
Given a first-order nonlinear hyperbolic system of conservation laws endowed with a convex entropy-entropy flux pair, we can consider the class of weak solutions containing shock waves depending upon some small scale parameters. In this…
The shock discontinuities, generically present in inviscid solutions of the forced Burgers equation, and their bifurcations happening in the course of time (perestroikas) are classified in two and three dimensions -- the one-dimensional…
Viscous shocks are a particular type of extreme events in nonlinear multiscale systems, and their representation requires small scales. Model reduction can thus play an important role in reducing the computational cost for an efficient…
We analyze the stochastic scaling laws arising in the invicid limit of the decaying solutions of the Burgers equation. The linear scaling of the velocity structure functions is shown to reflect the domination by shocks of the long-time…
We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit…
The microscopic mechanism by which amorphous solids yield plastically under an externally applied stress or deformation has remained elusive in spite of enormous research activity in recent years. Most approaches have attempted to identify…
We consider one-dimensional, locally finite interacting particle systems with two conservation laws. The models have a family of stationary measures with product structure and we assume the existence of a uniform bound on the inverse of the…
Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable…
We study driven 1d lattice gas models with two types of particles and nearest neighbor hopping. We find the most general case when there is a shock solution with a product measure which has a density-profile of a step function for both…
We combine experiments, large scale simulations and continuum models to study the emergence of coherent structures in a suspension of magnetically driven microrollers sedimented near a floor. Collective hydrodynamic effects are predominant…
We investigate the presence of static solutions in generalized models described by a real scalar field in four-dimensional space-time. We study models in which the scalar field engenders higher-order derivatives and spontaneous symmetry…
The subject of this work is the shock development problem in fluid mechanics. A shock originates from an acoustically spacelike surface in spacetime at which the 1st derivatives of the physical variables blow up. The solution requires the…
We study the inviscid Burgers equation in the presence of spatially periodic potential force. We prove that for foliated initial value problem there are always solutions developing shocks in a finite time. We give an application of this…