The paper is concerned with Lp error analysis of semi-discrete Galerkin FEMs for nonlinear parabolic equations. The classical energy approach relies heavily on the strong regularity assumption of the diffusion coefficient, which may not be satisfied in many physical applications. Here we focus our attention on a general nonlinear parabolic equation (or system) in a convex polygon or polyhedron ...

Under special conditions an approximate wave front solution to a system of nonlinear reaction-diffusion type equations is obtained. An assumption of fast kinetics permits the system to be reformulated as a single nonlinear differential-integral equation of convolution type. If the convolution kernel has the form of a mollifier then the convolution is localized to obtain an infinite system of di...

A characteristics finite volume element discretization technique based on two subspaces is presented for a nonlinear convection-dominated diffusion equations. The solution of a nonlinear system on the fine space is composed of solving one small (nonlinear) system on the coarse space and a linear system on the fine space. Error estimates are derived and numerical experiments are performed to val...

A novel nonlinear scale space framework is proposed for the purpose of multi-scale image representation. The scale space decomposition problem is formulated as a general Bayesian least-squares estimation problem. A quasi-random density estimation approach is introduced for estimating the posterior distribution between consecutive scale space realizations. In addition, the application of the pro...

In the article, a one-dimensional bipolar hydrodynamic model (Euler-Poisson system) in the quarter plane is considered. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. The global existence of smooth small solutions for the corresponding initial-boundary value problem is firstly shown. Next, the asymptotic behavior of the so...

Multiscale methods have proved to be successful tools in image denoising, edge enhancement and shape recovery. They are based on the numerical solution of a nonlinear diffusion problem where a noisy or damaged image which has to be smoothed or restorated is considered as initial data. Here a novel approach is presented which will soon be capable to ensure real time performance of these methods....

This is a study of a class of nonlocal nonlinear diffusion equations (NNDEs). We present several new qualitative results for nonlocal Dirichlet problems. It is shown that solutions with positive initial data remain positive through time, even for nonlinear problems; in addition, we prove that solutions to these equations obey a strong maximum principle. A striking result shows that nonlocal sol...

We review some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian. To Haim, with respect and admiration.

In this paper, two models for curvelet-regularized nonlinear diffusion are proposed for discontinuity-preserving denoising, using a new tight frame of curvelets and a nonlinear diffusion scheme. Numerical experiments show their good performance to recover the shape of edges and some important detailed components of the image while we use less iterations, in comparison to traditional methods. Ma...

– The porous media equation has been proposed as a phenomenological “nonextensive” generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous medi...