**STARTPAGE**
HUMAN RESOURCES AND MOBILITY (HRM)
ACTIVITY
MARIE CURIE ACTIONS
Marie Curie Host Fellowships for the Transfer of Knowledge (ToK)
Development Host Scheme
PART B
## **SPADE2**
♠♠
## **DETERMINISTIC AND STOCHASTIC DYNAMICS, FRACTALS, TURBULENCE**
**B1. Scientific Quality of the Project**
**B1.1. Research topic (general description)**
The research topics include: mathematical modelling of various processes (physical, biological, societal, economical), by ordinary and partial differential equations (ODE's, PDE's), deterministic and stochastic, long time behaviour of trajectories, scaling and aggregation limits, studies of local geometric (fractal) structure of arising attractors. The challenges concern the complexity of phenomena and their models as well as the structural richness of mathematical methods used. The research content of SPADE has horizontal and vertical structure. It is a coherent package of areas and methods in mathematics and mathematical physics, of great importance in applications.
__Areas (horizontal) include__
**a)** Finite (low dimensional) dynamical systems (flows and iteration), in particular: 1D-dynamics, real and complex Iterated Function Systems (IFS) as a kernel of higher dimension phenomena (hyperbolic, non-uniformly hyperbolic, dominated splitting). A spectacular progress in 1D iteration has been achieved by D.Sullivan, J.-Ch.Yoccoz (Fields medal in 1994), C.McMullen (Fields medal in 1998) in renormalization and J.Graczyk, G.Swiatek, M. Lyubich and very recently O.Kozlovski, W. Shen, S. van Strien in density of hyperbolicity. This development is leading to better understanding of dimensions and measures on metric attractors and basins.
**b)** Infinite-dimensional systems. We will focus on turbulence in 2D (Navier-Stokes Eq.), attractors, Sinai-Ruelle-Bowen (SRB) measures and multifractal spectra (J.Bricmont, A.Kupiainen, S. Kuksin), turbulence in 3D thin domains, with engineering applications in lubrication (P.Constantin, C.Foias, R.Temam).
**c)** Scaling limits in physical processes. Since eighties theoretical physicists predicted exact values of many dimensions and scaling exponents for the 2D models like percolation, Ising model, self avoiding random walk, etc. Recently works on Schramm-Loewner evolution led to rigorous mathematical confirming of these predictions. We expect advances in the models like diffusion limited aggregation (a generic model of fractal growth) or random matrices (of major importance in studying disordered media).
__Methods include__
**1.** Ergodic Theory and Information: time averages and statistics along trajectories, equilibrium measures, mixing, etc.
**2. **Thermodynamical Formalism and Multifractal Spectra of Lyapunov exponents, entropy, dimensions. The subject was started in the sixties and seventies (SRB), and it was further
developped in the eighties by Procaccia, Kadanoff, et al. It remains in a close relation with geometric measure theory. We want to learn the applications at INRIA Rocquencourt “projet fractale”.
**3.** Evolution Stochastic Processes (also as areas of research). Evolution Stochastic ODE’s and PDE’s.
**4.** Methods of Function Spaces: Sobolev spaces, functions of bounded variation, analysis of singularities of measures, singular integrals and harmonic analysis. Wavelet methods.
Some of these fields are strong in Poland, in particular at IMPAN, but have become too specialised, and suffer somewhat from isolation and brain drain. Other fields, of primary importance, are at the beginner stage, but we do want to develop them. We have started to integrate researchers working in the fields by organising joint seminars. Acquiring fresh ideas and methods from leading research groups in the world should give us new strength. It will also mobilise Polish researchers to conduct interdisciplinary investigations and to apply the mathematical tools in other fields. The ToK programme would help to maintain IMPAN as an international research centre.
**B1.2. Project objectives**
We describe the scientific objectives according to the tasks (see B1.4). Partner organisations and/or names of some external researchers involved will be listed in B1.4 and further on.
Our overall objective is to increase the transfer of ideas and modern mathematical tools to Poland and (through the Banach Center) to Central and East Europe, by intensifying the contacts with the strongest centres in Europe and outside. We want to concentrate on the following scientific objectives.
**1. Dynamical Systems. **A progress in description of classes of rigid and stable chaotic (weakly hyperbolic) and quasi-periodic dynamics and deformations, modelling various processes. Exploration of function theory (quasi-conformal), topological-combinatorial methods, geometric measure theory (thermodynamical and multifractal formalisms) tools. Development of applications in population dynamics (blood cells, etc), small scale (fractal) structures. Some areas of applications (e.g. multiscale phenomena analysis, biomedical problems) are already investigated at the ICM of Warsaw University (IMPAN partner).
**1a. Applications of fractals and IFS.** We want to learn at INRIA “projet fractale”. An objective is to become involved in their “fractals and engineering” programmes, 1D-signals (computer network traffic analysis, speech synthesis), 2D-signals (image analysis, denoising and segmentation, image compression).
**2. Partial Differential Equations, turbulence and asymptotics**
Many features of the physical phenomenon of turbulence in fluid flows can be explained rigorously in the frame of the theory of infinite dimensional dynamical systems. We are interested in finite dimensionality of the flow of two-dimensional Navier-Stokes (NS) fluid (e.g. on two dimensional torus) and three dimensional NS fluid in thin 3-D domains (Constantin, Foias, Temam, Robinson, Sell, Raugel, etc.). The number of degrees of freedom of such flows is connected to the Hausdorff dimension of the global attractor. Dimension estimates given in terms of the Grashof number correspond to the Kraichnan and Kolmogorov dissipation lengths that concern classical descriptions of turbulence. We are also interested in such close problems as long time behaviour of solutions of semi-linear evolution equations, regularity of solutions in Gevrey classes, the study of models of pattern formation, blood motions and plane motions.
**2a. Turbulent transport**. Transport of pollutants, heat, chemical agents or magnetic fields plays an important role in environmental issues, meteorology, engineering or astrophysics. Rough models of passive turbulent transport based on random dynamical systems permit to understand the basic phenomena like occurrence of cascades of conserved quantities, and presence of intermittency characterized by anomalous scaling laws and multifractality. Such models provide a bridge between hydrodynamics and the dynamical systems theory stretching the latter to the situations (fully developed turbulence) where the usual tools break down.
Exploration IFS approximations for transport phenomena of practical importance: high Reynolds number flow and porous medium flow in multiscale materials.
**3. Stochastic Processes. **The main aim of stochastic PDEs, is the study of qualitative properties of solutions. In particular long time behaviour, existence of invariant measures and attractors, ergodicity, large deviation estimates, exit probabilities and large noise asymptotics. The stochastic turbulence applied to systems of interacting particles, stochastic Navier-Stokes, Euler and Burgers equations and Anderson models in two or three dimensional regions is
of special interest. These equations appear naturally in fluid mechanics as a description of the dynamics of a fluid under an influence of a random external force. We plan also to learn and start working in Growth Models (exhibiting fractal phenomena) and Stochastic Loewner equation, SLEκ. The need for the study of Stochastic Processes appears in the investigation of chaotic ODE (deterministic chaos).
**4. Methods of Function Spaces. ** Isomorphic properties of function spaces and their dependence on smoothness and domain; singular integral operators acting on the Sobolev spaces; the use of interpolation in the study of PDEs; multidimensional wavelet bases, approximations by splines in the spaces of smooth functions, Sobolev spaces on metric spaces (e.g. Carnot-Caratheodory spaces).
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