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Register a new userClassical Solutions for a Class of Burgers Equation
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In this paper we consider a class of Burgers equation. We propose a new method of investigation for existence of classical solutions.
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We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $rightarrow$ +$infty$. MSC2010: 35F55, 35L65. Notations. We denote $times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its norm is denoted $times$ M. The Dirac mass at X $in$ R n is $delta$ X or $delta$ x=X. If $ u$ $in$ M (R m) and $mu$ $in$ M (R q), then $ u$ $otimes$ $mu$ is the measure over R m+q uniquely defined by $ u$ $otimes$ $mu$, $psi$ = $ u$, f $mu$, g whenever $psi$(x, y) $ otequiv$ f (x)g(y). The closed halves of the real line are denoted R + and R --. * U.M.P.A., UMR CNRS-ENSL # 5669. 46 all{e}e dItalie,
In this review we discuss the weak KPZ universality conjecture for a class of 1-d systems whose dynamics conserves one or more quantities. As a prototype example for the former case, we will focus on weakly asymmetric simple exclusion processes, for which the density is preserved and the equilibrium fluctuations are shown to cross from the Edwards-Wilkinson universality class to the KPZ universality class. The crossover depends on the strength of the asymmetry. For the latter case, we will present an exclusion process with three species of particles, known as the ABC model, for which we aim to prove the convergence to a system of coupled stochastic Burgers equations, i.e. gradien
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.
This work studies exact solvability of a class of fractional reaction-diffusion equation with the Riemann-Liouville fractional derivatives on the half-line in terms of the similarity solutions. We derived the conditions for the equation to possess scaling symmetry even with the fractional derivatives. Relations among the scaling exponents are determined, and the appropriate similarity variable introduced. With the similarity variable we reduced the stochastic partial differential equation to a fractional ordinary differential equation. Exactly solvable systems are then identified by matching the resulted ordinary differential equation with the known exactly solvable fractional ones. Several examples involving the three-parameter Mittag-Leffler function (Kilbas-Saigo function) are presented. The models discussed here turn out to correspond to superdiffusive systems.
Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.
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