366 research outputs found
Asymptotic properties of entropy solutions to fractal Burgers equation
We study properties of solutions of the initial value problem for the
nonlinear and nonlocal equation u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0 with
alpha in (0,1], supplemented with an initial datum approaching the constant
states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was
shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for
alpha in (1,2), the large time asymptotics of solutions is described by
rarefaction waves. The goal of this paper is to show that the asymptotic
profile of solutions changes for alpha \leq 1. If alpha=1, there exists a
self-similar solution to the equation which describes the large time
asymptotics of other solutions. In the case alpha \in (0,1), we show that the
nonlinearity of the equation is negligible in the large time asymptotic
expansion of solutions.Comment: 23 pages. To appear to SIMA. This version contains details that are
skipped in the published versio
On convergence of solutions of fractal Burgers equation toward rarefaction waves
In the paper, the large time behavior of solutions of the Cauchy problem for
the one dimensional fractal Burgers equation with is studied. It is shown that if the
nondecreasing initial datum approaches the constant states ()
as , respectively, then the corresponding solution converges
toward the rarefaction wave, {\it i.e.} the unique entropy solution of the
Riemann problem for the nonviscous Burgers equation.Comment: 15 page
Asymptotic behaviour of solutions to fractional diffusion-convection equations
We consider a convection-diffusion model with linear fractional diffusion in
the sub-critical range. We prove that the large time asymptotic behavior of the
solution is given by the unique entropy solution of the convective part of the
equation. The proof is based on suitable a-priori estimates, among which
proving an Oleinik type inequality plays a key role.Comment: 24
Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates
We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller
type such as the fractal Burgers equation. The existence of traveling wave
solutions with monotone decreasing profile has been established recently (in
special cases). We show the local asymptotic stability of these traveling wave
solutions in a Sobolev space setting by constructing a Lyapunov functional.
Most importantly, we derive the algebraic-in-time decay of the norm of such
perturbations with explicit algebraic-in-time decay rates
Continuous dependence estimates for nonlinear fractional convection-diffusion equations
We develop a general framework for finding error estimates for
convection-diffusion equations with nonlocal, nonlinear, and possibly
degenerate diffusion terms. The equations are nonlocal because they involve
fractional diffusion operators that are generators of pure jump Levy processes
(e.g. the fractional Laplacian). As an application, we derive continuous
dependence estimates on the nonlinearities and on the Levy measure of the
diffusion term. Estimates of the rates of convergence for general nonlinear
nonlocal vanishing viscosity approximations of scalar conservation laws then
follow as a corollary. Our results both cover, and extend to new equations, a
large part of the known error estimates in the literature.Comment: In this version we have corrected Example 3.4 explaining the link
with the results in [51,59
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
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