Objective
This INTAS project brings together the leading West European and NIS experts in differential equations. It includes different generations from senior scientists K. Kirchgässner, R. Temam, and M.I. Vishik down to excellent Postdocs and Dr students. The scientific cooperation of the teams of this project is long going and has materialized in more than 10 joint publications. The present project is largely the product and the development of this cooperation. The backbone of the project is the development of the theory of infinite dimensional dissipative dynamical systems with main interest being focused on the qualitative and quantitative description of the invariant sets that carry all the essential information on the long time behaviour of the solutions of the problem. These are global, exponential and trajectory attractors for autonomous and non-autonomous equations, inertial and essential manifolds and so on. Theoretical predictions will be of crucial importance for the interpretation of numerical experiments in such areas as turbulent flows of viscous fluid. The work will be done in a tight cooperation between the NIS and INTAS teams. The main objectives of the project and the expected results are as follows.
Global attractors and trajectory attractors for non-autonomous equations of mathematical physics. Kolmogorov e-entropy of global attractors. Global averaging and global homogenisation of equations with rapidly oscillating terms. The expected results are: optimal estimates for the Kolmogorov e-entropy of the attractors; convergence of the attractors and inertial manifolds of equations with rapidly oscillating terms to the attractors and inertial manifolds of the averaged (homogenized) equations and the corresponding explicit proximity estimates.
Dynamical systems associated with fluid mechanics, geophysical fluid mechanics, and material sciences. The expected results are: physically realistic explicit estimates for the dimension of the attractors for the equations in fluid mechanics, geophysical fluid mechanics, explicit lower bounds for the instability index of the generalized Kolmogorov flows on the sphere; existence, uniqueness, stability, and attractors for generalizations of the Cahn-Hilliard equation (with internal micro forces, deformations, anisotropy, and heat transfer); generalization of the construction of exponential attractors to Banach spaces.
Determining functionals and analyticity of global attractors of non-linear evolution equations. Long-time behaviour of composite systems. Approximate inertial manifolds for retarded equations. Expected results are: proof of the analyticity of global attractors and realistic upper bounds for the determining functionals; attractors and determining functionals for two-component composite systems of non-linear parabolic equations; construction of finite dimensional approximate inertial manifolds for retarded evolution equations.
Local dynamics in the phase space of first-order quasi-linear hyperbolic systems. The expected results are: asymptotic stability and exponential dichotomy for first-order hyperbolic systems; estimates of the Lyapunov exponents; construction of integral manifolds; linearization theorems of Grobman-Hartman type.
Equations in unbounded domains. The expected results are: upper bounds for the Kolmogorov e-entropy of the attractors of equations in unbounded domains; analysis of the local dynamics in a vicinity of spatially localized (multi-pulse) solutions of generalized complex Ginzburg-Landau equations in unbounded (strip-like) domains; derivation of the conditions for the finite-dimensionality of the essential sets of non-linear elliptic equations and methods for estimating their dimension; generalization of the theory of spatial dynamics for non-linear elliptic equations to the case of several unbounded spatial directions. Analysis of the multi-dimensional translation group; analysis of hydrodynamic problems in cylindrical domains with specified integral characteristics of the solutions in function spaces without decay at infinity.
For all teams INTAS provides an ideal framework for scientific cooperation.
Call for proposal
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91405 Orsay Cedex
France