Grant PN-II-RU-RP-2007-1-3

Project sumarry

This project is devoted to the study of PDEs from the point of view of treir numerical approximations, asymptotic behaviour and control. More precisely we will analyze the following problems:

   1. Numerical approximation schemes for dispersive equations.
We will establish to which extent the classical numerical approximation schemes (or variants of them introduced trough the project) fulfill uniform (with respect to the mesh-size) dispersive properties. When needed, new adapted numerical schemes will be introduced to make these properties uniform. This analysis will be used to introduce  convergent approximations for  nonlinear initial value problems.

   2. Numerical approximations for control problems.
We analyze whether the observability inequalities used in the context of the control problems are uniform with respect to the mesh size. This study will allow us to prove the convergence of the discrete controls towards the continuous one.     

   3. Asymptotic properties of nonlocal diffusion equations.
For this class of problems  we  analyze the long time behaviour of the solutions as well as their shape. We will perform a theoretical and  numerical analysis.

Obtained resutls

  1. A nonlocal convection-diffusion equation

We have analyzed a nonlocal equation that takes into account convective and diffusive effects:
in, where  and  and  are nonnegative and verify . In this case we have a diffusion operator  and a nonlinear convective part given by . If  is not symmetric then individuals have greater probability of jumping in one direction than in others, provoking a convective effect.
Our goal is to obtain the long time behavior and the shape of solutions under some grow condition on, for example of the type. For the local convection-diffusion equation this analysis was performed by Schonbek in [53], [54] and under less restrictive hypothesis by Escobedo and Zuazua in [19] and by Carlen and Loss in [9]. However in the last two references energy estimates were used together with Sobolev inequalities to obtain the long time behavior of the solutions. These Sobolev inequalities are not available for the nonlocal model, since the linear part does not have any regularizing effect. For the nonlocal problem discussed above we havel adapted the Fourier Splitting method introduced by Schonbek in [53] and refined in various latter works [55], [34], to obtain the long time behavior of the solutions.

Articles supported by the grant

Liviu I. Ignat, Julio D. Rossi, A nonlocal convection–diffusion equation, Journal of Functional Analysis 251 (2007) 399–437, PDF