Related papers: Shock statistics in higher-dimensional Burgers tur…
We consider a planar viscous shock of moderate strength for a scalar viscous conservation law in multi-D. We consider a strictly convex flux, as a small perturbation of the Burgers flux, along the normal direction to the shock front.…
Considering the hydrodynamical limit of some interacting particle systems leads to hyperbolic differential equation for the conserved quantities, e.g. the inviscid Burgers equation for the simple exclusion process. The physical solutions of…
We revisit the one-dimensional Burgers equation in the inviscid limit for white-noise initial velocity. We derive the probability distributions of velocity and Lagrangian increments, measured on intervals of any length $x$. This also gives…
We study the dynamics and clustering of dust particles with inertia in shock-dominated compressible turbulence using the two-dimensional, stochastically forced Burgers equation. At small Stokes numbers, shock trapping leads to extreme…
We construct a discrete shell-model for two-dimensional turbulence that takes into account local and nonlocal interactions between velocity modes in Fourier space. In real space, its continuous limit is described by the one-dimensional…
This paper proves the existence of unstable shocks of the Burgers-Hilbert equation conjectured in arXiv:2006.05568. More precisely, we construct smooth initial data with finite $H^9$-norm such that the solution in self-similar coordinates…
We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of $\textit{interval collapse…
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…
An exact analysis is performed for the two-point correlation function C(r,t) in dissipative Burgers turbulence with bounded initial data, in arbitrary spatial dimension d. Contrary to the usual scaling hypothesis of a single dynamic length…
Gathering together some existing results, we show that the solutions to the one-dimensional Burgers equation converge for long times towards the stationary solutions to the steady Burgers equation, whose Fourier spectrum is not integrable.…
A putative powerlaw range of the probability density of velocity gradient in high-Reynolds-number forced Burgers turbulence is studied. In the absence of information about shock locations, elementary conservation and stationarity relations…
We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation…
We present results for the 1 dimensional stochastically forced Burgers equation when the spatial range of the forcing varies. As the range of forcing moves from small scales to large scales, the system goes from a chaotic, structureless…
We investigate the stability of large-scale structures in Burgers' equation under the perturbation of high wave-number noise in the initial conditions. Analytical estimates are obtained for random initial data with spatial spectral density…
We use the mapping between Burgers' equation and the problem of a directed polymer in a random medium in order to study the fully developped turbulence in the $N$ dimensional forced Burgers' equation. The stirring force corresponds to a…
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…
We study viscous-dispersive shock waves with infinite oscillations of the Korteweg-de Vries-Burgers (KdVB) equation. First, we establish detail structures of the shock waves, including the rates at which the local extrema converge to the…
Lagrangian pair dispersion provides insights into mixing in turbulent flows. By direct numerical simulations (DNS) we show that the statistics of pair dispersion in the randomly forced two-dimensional Burgers equation, which is a typical…
We establish necessary and sufficient conditions for the shock statistics to approach self-similar form in Burgers turbulence with L\'{e}vy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and…
We study the statistical properties of solutions to Burgers' equation, $v_t + vv_x = \nu v_{xx}$, for large times, when the initial velocity and its potential are stationary Gaussian processes. The initial power spectral density at small…