Abstract
Mathematical models of physical, biological and atmospheric phenomena form a bridge between the real world and theoretical sciences, they link the applications and the abstraction. Such models are interesting for the engineers and physicists but also for pure mathematicians. The most important among such models are those described by partial differential equations (PDEs) and their generalizations containing fractional derivatives. We plan to study the following subjects:* Partial differential equations with fractional-order derivatives,* Long time behavior of equations with fractional derivatives. One of the natural origin for equations including fractional-order derivative is the stochastic analysis; if the driven process is a jumps process (Levy process), then the corresponding Fokker-Planck equation will contain a fractional Laplacian. Recent decade, in the study off luid mechanics, finances, molecular biology and many other fields, it was discovered that their draught of random factors can bring many new phenomenon and features which are more realistic than the deterministic approach alone. Hence, it would be natural to include stochastic terms, and so the fractional-order operators, when we establish the mathematical models. Moreover, the problems containing such fractional-order derivative terms becomes more challenging and many classical PDEs methods are hardly applicable directly to them, so that new ideas and theories are required. Comparing with the usual PDEs' theory, we can even say that the fractional-order PDEs are only on their initial stage of development. For example, to the best of our knowledge, the first systematical results about fully nonlinear fractional-order PDEs belongs to L. Caffarelli, L. Silvestre [C-S]. Also, in fluid mechanics, it was found in [C], that the solution of 2D critical (the order is 12 ) Quasi-Geostrophic equation (Q-G equation) resembles the evolution of the vorticity in the three-dimensional Euler equation, however, its global regularity was settled only recently by A. Kiselev et al. [K] and L. Caffarelli, A. Vasseur in [K-V], and the supercritical case (the order less than 12 ) remains open up to now. One of the major task in the studies of evolution equations of mathematical physics is to investigate the behavior of its solutions when time is large or tends to infinity. However, in general, for most nonlinear PDEs (especially the PDEs arising from physics and atmospheric science) we cannot get their solutions explicitly. Although scientists can get some information via numerical methods using super computers, such methods cannot give valuable approximations of solutions as time goes to infinite. Hence, it would be important to get information without solving the PDEs; for example for the atmospheric equations in weather forecast. The theory of dynamical systems has been successfully used for a few decades to analyze qualitative properties of many differential equations. Usually the concept of an attractor plays a crucial role in such considerations, since focusing our attention on the attractors significant simplification can be achieved. Consequently, it is important to study possible attractors and to characterize their properties. There are many famous mathematicians working in this field, like M. Vishik, R. Temam, C. Foias, P. Constantin, J.M. Ball, and others. However, due to complexity of PDEs, the understanding in the existing literature of geometrical properties of attractors is still very limited. Especially, it is almost empty concerning the attractors for fractional-order PDEs. Our research will focus on the study of the local and global well posedness and asymptotic behavior for models with fractional order derivatives. (AU)