The purpose of this project is to conduct international collaborative research, between two teams of scientists from the USA and Brazil, in the mathematical analysis and modeling of turbulent incompressible fluids. The topics to be investigated are: small viscosity regime of second grade fluids; uniqueness of weak solutions for certain linear perturbations of the two-dimensional Euler equations; the vanishing viscosity limit of the three-dimensional Navier-Stokes equations with helical symmetry; the search for hypotheses on the structure of invariant measures or stationary statistical solutions of the two and three-dimensional Navier-Stokes equations with anomalous dissipation; the search for energy cascade for flows in domains with physical boundaries; two-dimensional cascades with large gap bimodal forcing. Turbulence is a common phenomenon in fluid motion, in which macroscopic quantities (velocity, pressure, temperatures, etc.) no longer have a deterministic relation with global parameters of the flow. Direct numerical simulations of turbulent flows at large Reynolds numbers, that occur in practical applications, such as in geophysical modeling and mixing in industrial processes, are out of reach even for the state-of-the-art computer power. Therefore, there is an urgent need to pursue this challenging problem analytically, by developing rigorous mathematical and statistical tools to investigate it, and to test these tools computationally.
While turbulence is an everyday occurrence, our understanding is still lacking in many aspects. Quantifying the effect of small scales on the dynamics of large scales is fundamental in modern multiscale science. The goal of the project is to enable a predictive analytical study of turbulent flows. This study will impact wide-ranging applications, from geophysical modeling, such as dispersion of pollutants in the ocean, to biological and industrial modeling, such as design of polymeric materials. The project will consolidate the well-established collaborative efforts of the principal investigators with their Brazilian counterparts, and may lead to new collaboration, especially among the junior research personnel. International collaboration among scientists is a key to economic competitiveness in global markets. Four US and three Brazilian academic institutions are involved in the project. The international dimension of the project is further emphasized through two planned workshops. Training and supervision of at least six Ph.D. students and postdoctoral fellows is also achieved through planned summer schools and scientific workshops. Students and postdoctoral fellows from the US will travel to Brazil to participate in the workshops and the summer schools and interact with the US and Brazilian researchers.
This project is co-funded with the Americas Program of the Office of International Science and Engineering.
A fundamental part of many models of fluid flow is the Navier-Stokes equation. While one cannot expect to ever write down an explicit expression for the general solution to this system, and even accurate numerical simulation can be challenging under realistic circumstances, careful mathematical analysis can lead to more effective algorithms and a better understanding of crucial physical phenomena. This effort is aimed at several levels: from the averaged distribution of energy throughout different length scales as in turbulence, to the detailed long time dynamic behavior. The Navier-Stokes equation governs velocity and pressure which are key variables in weather forecasting. Among the numerous factors that make accurate forecasting a challenge is initialization. The finite number of data collecting instruments distributed about the globe and atmosphere give only an incomplete description of the state of the weather at any instant. Yet there is a wealth of such data over an extended time period in the past, which when combined with certain mathematical models, can provide a more complete description. The general procedure for this is called data assimilation. It provides more accurate starting values for computer simulations of the weather going into the future. This award was used to foster collaboration with Brazilian mathematicians on open questions in fluid flow. Broader impact was achieved by the participation of a postdoctoral researcher, and two graduate students to participate in two workshops in Brazil. The first workshop led to a follow-up solo visit by one graduate student to continue research, and sparked the project that became the thesis subject for another. The second workshop allowed the second student an opportunity to speak on his results before an international audience. Key work on data assimilation was done while the postdoc, the PI Jolly, and Titi (PI on collaborative grant) were together for the second workshop. In total, the award supported the work in 9 papers for publication in peer-reviewed journals, as well as two software packages. The PI Jolly gave a tutorial in the first workshop on how to develop a computer program to simulate the Navier-Stokes equation, which prompted him to write such a code in advance. This code continues to play a role in the research by several of his graduate students.
