Pathological Behavior of Solutions to Fluid Equations (23w5110)

Organizers

Mimi Dai (Institute for Advanced Study)

(University of Chicago Illinois)

(CUNY Hunter College & Graduate Center)

Description

The Casa Matemática Oaxaca (CMO) will host the "Pathological Behavior of Solutions to Fluid Equations" workshop in Oaxaca, from August 6 to August 11, 2023.



The equations of fluid motion, derived over a century ago, are a source of enormous challenges and exciting problems for mathematicians. At large velocities, fluid flows become seemingly chaotic, but nevertheless possess some structure. It is widely believed that such flows, called turbulent and often observed in nature, still satisfy the equations of motion (Navier-Stokes equations), but are hard to construct or study mathematically.



In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit what one might call “pathological” or “wild" behavior. These include non uniqueness, singularity formation, and the loss of energy balance. While these constructions are interesting from the mathematical point of view, as they provide counterexamples to various well-posedness (uniqueness, regularity, stability) results, they are becoming more and more relevant from the physical point of view as well. Indeed, an important physical property exhibited by turbulent flows is the existence of energy cascades. This was conjectured by Kolmogorov and has been observed both experimentally and numerically, but had been difficult to produce analytically. The technique of convex integration, however, introduced into the mathematical fluids community in the early 2000s, allows one to explicitly construct solutions to the fluid equations that do exhibit cascades of energy. There has also been tremendous progress in another exciting and related direction — the search for singularity formations in fluid equations. Such phenomenon is also based on the transfer of energy to small scale structures (in some stable manner), which is an important component to the formation of hurricanes and tornados found in nature.



This workshop on pathological behaviors of solutions to fluid equations brings together experts in mathematical fluid mechanics to discuss recent exciting developments in this direction from the mathematical point of view, as well as their relevance to turbulence.



The Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT). The research station in Oaxaca is funded by CONACYT