Statistical Methods in Fluid Dynamics

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

Most natural phenomena involve interactions of many particles and exhibit chaotic dynamics and strong sensitivity to parameters as well as to exterior influences such as thermal fluctuations. Additionally, in many vital areas of application, notably in geology or astrophysics, direct detailed observations are not available. The chaotic microscopic behavior can give rise, however, to very robust macroscopic structures, for example, the great red spot on Jupiter or pattern formation in chemical reactions, which can be studied by methods of statistical physics. A rigorous derivation of the existence and properties of such coherent patterns from the fundamental laws of physics is a notoriously difficult problem that requires additional assumptions that select typical solutions. These assumptions frequently include an addition of randomness, which models unknown influences, or a condition that the system equally explores all possible states. Then the existence and properties of statistical steady states of complex systems is accessible mathematically. This research project explores and further develops this important class of mathematical models. The project branches in two directions. The first part focuses on properties of statistical invariant states for stochastically forced systems, especially their stability, dependence on parameters, and distribution on the state space. Special attention is given to the structure of the introduced forcing, as it is crucial for application, where the noise might act only in parts of the physical space or be localized is some scales. Although mainly problems arising from fluid dynamics are explored, equations originating in biology and chemistry are also considered. The second branch aims to gain more insight into the problems for which the addition of random forcing is not well motivated or it causes undesired behavior of the model, such as blow-up. Assuming the ergodic hypothesis, the large deviation principle implies that the system evolves toward thermodynamic equilibrium, and statistical solutions can be viewed as maximizers of an entropy, which effectively transforms evolution equations to variational problems. The analysis of stability, observability, and qualitative properties of these so-called Gibbs states relies on variational methods, bifurcation theory, and investigation of associated Euler-Lagrange equations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.