Many aspects of the theory of fluid dynamics have influenced mathematicians and the direction of mathematical research. In the last decades, it has been a source for numerous problems in nonlinear partial differential equations (PDE), spectral theory and dynamical systems. One theme that is strongly represented in the AIMS Laboratory is the study of the Navier - Stokes equations for a viscous fluid. Numerical simulations of the Navier-Stokes equations are at the heart of a number of major engineering modeling efforts, and there is a major role to be played by mathematicians to develop feasible full scale and efficient numerical methods for this work. Secondly, several major outstanding mathematical problems concern the regularity theory for the Navier-Stokes flow, and its statistical analog.

Another strongly represented field is the study of wave evolution in the surface, or in internal layers of the ocean. The oceanographer's modeling of surface water waves involves potential flow with free surface boundary conditions. This represents a Hamiltonian PDE, as was shown by V. E. Zakharov (1968). Its analysis, and the anlysis of mathematical models of water waves that have been derived over the period of the last one and one half centuries, is one of the focal points of the AIMS Lab. As well as presenting a challenging mathematical probelm for analysis, advances in numerical simulations and increasingly sophisticated approaches to modeling are relevant to physical oceanographers and coastal engineers. This topic is the focus of a major rearch collaboration between AIMS Lab members and a diverse group of experimentalists, theorests and mathematical modelers.

Philippe Guyenne

Slim Ibrahim

Vladislav Panferov

• Existence theory of three dimensional periodic traveling water waves (Craig and Nicholls, 2000, 2002)

• Nonexistence of solitary waves in three dimensions (Craig, 2003)

• ... some recent paper of Biryuk or Ibrahim