High Reynolds number flow is a classical research theme that retains its vitality at several levels, from real-world applications, through physical and computational modeling, up to rigorous mathematical analysis. There are two reasons for the continued relevance of this topic. The first is the ubiquity of such flows in situations of practical interest, such as blood flow in large caliber vessels, fluid-structure interaction, aerodynamics, geophysical and astrophysical flow modeling. The second issue is that, despite of half a century of vigorous efforts, there is still a lack of systematic understanding how different scales interact to form the inertial range from a smooth initial condition. The description of the behavior of solutions of the Navier-Stokes equations at high Reynolds number is at the heart of the problem, and surprisingly, mathematical analysis seems to be a promising route for gaining insight. Is singularity formation of incompressible flows at high Reynolds number necessary for the formation of the inertial range in a turbulent flow? or is the dynamical generation of extremely small but finite scales sufficient for this purpose? The choice of the singularity problem for the incompressible Navier-Stokes equation as one of the seven Millennium prize problems highlights the fundamental role that mathematical analysis may yet play in this subject, while attesting to the quality of the mathematical challenge posed by problems in this area.
This field has seen substantial progress in several independent directions. Let us cite a few prominent examples: the understanding of the interplay between the local geometric properties of the vorticity field and vortex stretching, the use of the Kato method applied to the Navier-Stokes equations in identifying critical spaces for well-posedness, the solution of the water wave problem and related research on interfacial dynamics, the mathematical understanding of the problem of boundary layers. A wide variety of methods have been employed, from classical functional analysis and operator theory to modern harmonic analysis and geometric measure theory.
Several interesting problems remain open. Beyond the singularity problem we highlight the uniqueness of weak solutions with p-th power integrable vorticity, existence of weak solutions with vortex sheet initial data for the two dimensional ideal flow equations and the convergence of the vanishing viscosity approximation in the presence of boundaries. This workshop is intended as a forum where the recent progress is examined from the point of view of understanding the large time behavior of the incompressible flows at high Reynolds number. The participants will include a representative sample of researchers active in the field of mathematical analysis of incompressible flows, together with a few specialists in fluid dynamics.
A limited amount of funding for participants at all levels is available, especially for researchers in the early stages of their career who want to attend the full program.
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