A diffusion process associated to the Prandtl equation

A Diffusion Equation with Exponential Nonlinearity Recant Developments

The purpose of this paper is to analyze in detail a special nonlinear partial differential equation (nPDE) of the second order which is important in physical, chemical and technical applications. The present nPDE describes nonlinear diffusion and is of interest in several parts of physics, chemistry and engineering problems alike. Since nature is not linear intrinsically the nonlinear case is t...

On the ill-posedness of the Prandtl equation

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data [13, 10], or for data with monotonicity properties [11, 15]. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solution...

Remarks on the ill-posedness of the Prandtl equation

In the lines of the recent paper [5], we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some C∞ initial data, local in time C∞ solutions do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz cont...

Convergence of a Kinetic Equation to a Fractional Diffusion Equation

The understanding of thermal conductance in both classical and quantum mechanical systems is one of the fundamental problems of non-equilibrium statistical mechanics. A particular aspect that has attracted much interest is the observation that autonomous translation invariant systems in dimensions one and two exhibit anomalously large conductivity. The canonical example here is a chain of anhar...

Gevrey Class Smoothing Effect for the Prandtl Equation