We describe an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators we consider do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we...

This paper deals with the Burgers equation which is most common model used in nonlinear conservation laws. Here theoretical aspect of law discussed by using inviscid equation. At first, we introduce general non-linear as a partial differential and its solution procedure method characteristic. Next, present weak problem entropy condition. Taking into account shock wave rarefaction wave, Riemann ...

We investigate the non-perturbative results of multi-dimensional forced Burgers equation coupled to the continuity equation. In the inviscid limit, we derive the exact exponents of two-point density correlation functions in the universal region in arbitrary dimensions. We then find the universal generating function and the tails of the probability density function (PDF) for the longitudinal vel...

Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with part equation. Altogether, this analysis complemented by unique continuation principles that determine optimal spatial decay. This work intended to estab...

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 these partial differential equations develop discontinuities, called shocks. The microscopic structure of these shocks is of much interest and far from bein...

In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the convergence rate is O(1= p N + p t) for the L 1 (IR)-norm of the error. To obtain that result, we link the PDE and the algorithm to a system of weakly interacting stochastic particles; the diiculty of the analysis comes from the discontinuity of the interaction k...