ICERM Semester Program on "Singularities and Waves In Incompressible Fluids"
Incompressible fluids are an abundant source of mathematical and practical problems. The question of global-in-time regularity versus finite-time singularity formation for incompressible fluids, governed by the Navier-Stokes or Euler equations, has been one of the most challenging outstanding problems in applied PDE. There have also been new developments in the study of the onset of turbulence due to linear and nonlinear instabilities in incompressible fluids. Interfacial and surface water waves are physical phenomena that, in addition to the challenges outlined above, involve the evolution of free boundaries. These problems embody many of the mathematical challenges found in studies of nonlinear PDEs.
Progress on these topics is possible because of advances in analysis, numerical computations and physical experiments. In addition, ocean field observations provide a reality test to all conclusions and invite new problems to be addressed. In this program, we provide a venue for interaction among researchers engaged in all of these problem-solving techniques to focus on topics arising in incompressible fluids.
Topics of particular interest include: singularity formation, stability and bifurcation; the modeling and analysis of simplified phenomenological models for the description of coherent structures; and time-dependent and steady free boundary problems including water waves, vortex sheets, capillary problems with contact lines and viscous waves with boundary layers.