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Project Title:

Interconnected Methods to Analysis of Deterministic and Stochastic Partial Differential Equations

Project Code:

PN-II-CT-ERC-2012-1

Project Number:

1ERC/02.07.2012

Principal Investigator:

Acad. Viorel BARBU

Funding Authority:

Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI)

Project's Host Institution:

Institutuin Name:   "Alexandru Ioan Cuza" University, Iaşi
Faculty:   Mathematics
Address:   Carol I Blvd., No. 11, Iaşi
Telephone:   +40232 20 10 60
Fax:   +40232 20 11 60
E-mail:   barbu_erc@yahoo.com

Keywords:

1. Navier-Stokes equations
2. Porous media equations
3. Probabilistic methods
 

Summary:

     The present project which focuses on interplay between stochastic methods in deterministic partial differential equations (PDEs) and methods of nonlinear analysis in infinite dimensional stochastic differential equations (SDEs), refer to two detailed research problems.
     Classical topics in the theory of deterministic differential equations are now studied for stochastic PDEs which arise naturally as stochastic perturbations of classical mathematical models in physics and applied sciences. The analysis of these stochastic models is a combination of analytic methods in deterministic PDEs and of probabilistic methods; the role of the analytic tool in the recent development of stochastic PDEs is quite obvious and cannot be under-estimated. On the other hand, the stochastic analysis provides sharp instruments for the study of deterministic PDEs viewed as Gaussian perturbations of PDEs and ordinary infinite dimensional equations.
     A research direction to be followed is that of feedback stabilization of Navier-Stokes equations by internal and boundary stochastic controllers. Another remarkable aspect is that of using analytic method of PDE theory (semigroup approach, nonlinear analysis) in order to study the existence problems, and long-time behaviour, for the infinite dimensional stochastic systems (with emphasis on stochastic porous media equation).
     The methods of stochastic analysis are used as a powerful tool in the existence theory for general parabolic PDEs, and in this context, a related problem is that of constructions of Kolmogorov operator corresponding to processes generated by stochastic differential equations.