Project Title: Collaborative Research: Deterministic and Statistical Relations Between the Navier-Stokes Equations and Its Determining Forms
Dr. Animikh Biswas has been awarded a NSF grant (DMS-1517027) in the amount of $173,121 for the period 2015–2018 by the Division of Mathematical Sciences (Applied Mathematics Program). The award includes summer support for a graduate research assistant for the three year period.
Dr. Biswas and his collaborators plan to study certain key open problems concerning the equations governing the atmosphere and ocean with a view towards applications in meteorology and weather forecasting. The equations describe fluid motions for gases and liquids under quite general conditions: from laminar to turbulent flows; on scales ranging from below a millimeter to astronomical lengths. Consequently, their study has wide ranging applications in aeronautical sciences, in meteorology, in the petroleum industry, in plasma physics, and more recently, in biophysical fluid dynamics.
The inherent nonlinear and multi-scale nature of the equations make the problem of weather forecasting and understanding climate evolution challenging. The investigators plan to develop analytical and statistical tools to study the long-term behavior of the fundamental governing equations of the atmosphere and climate. Among other things, the project will connect recently developed techniques of data assimilation with the theory of statistical solutions of these equations. This will in turn facilitate the incorporation of vast amounts of weather/climate data, collected over a long period of time, into mathematical models of weather and climate, potentially leading to an improved statistical prediction.
Additionally, Dr. Biswas and his collaborators plan to study several aspects of turbulent flows, including the time of existence of smooth solutions, statistical properties and the long term dynamics on the strong or weak global attractor in a new class of functional spaces that allows dampening of the nonlinear term in the equations, thus allowing for an improved analysis, which is nearer to the well-understood linear case. This new class of functional spaces will be employed to study the finite dimensional properties of the statistical solutions of the Navier-Stokes equations, via the determining modes, nodes and determining forms. Furthermore, the investigators will develop data assimilation techniques for statistical solutions of hydrodynamic equations. Statistical solutions are directly related to the average physical quantities that engineers are mostly concerned with. They provide a bridge between time average of physical observables such as energy, enstrophy and heat transfer to ensemble measures on the phase space. Thus, the techniques developed in this project should have immediate engineering and scientific applications.
Dr. Animikh Biswas has been awarded a NSF grant (DMS-1517027) in the amount of $173,121 for the period 2015–2018 by the Division of Mathematical Sciences (Applied Mathematics Program). The award includes summer support for a graduate research assistant for the three year period.
Dr. Biswas and his collaborators plan to study certain key open problems concerning the equations governing the atmosphere and ocean with a view towards applications in meteorology and weather forecasting. The equations describe fluid motions for gases and liquids under quite general conditions: from laminar to turbulent flows; on scales ranging from below a millimeter to astronomical lengths. Consequently, their study has wide ranging applications in aeronautical sciences, in meteorology, in the petroleum industry, in plasma physics, and more recently, in biophysical fluid dynamics.
The inherent nonlinear and multi-scale nature of the equations make the problem of weather forecasting and understanding climate evolution challenging. The investigators plan to develop analytical and statistical tools to study the long-term behavior of the fundamental governing equations of the atmosphere and climate. Among other things, the project will connect recently developed techniques of data assimilation with the theory of statistical solutions of these equations. This will in turn facilitate the incorporation of vast amounts of weather/climate data, collected over a long period of time, into mathematical models of weather and climate, potentially leading to an improved statistical prediction.
Additionally, Dr. Biswas and his collaborators plan to study several aspects of turbulent flows, including the time of existence of smooth solutions, statistical properties and the long term dynamics on the strong or weak global attractor in a new class of functional spaces that allows dampening of the nonlinear term in the equations, thus allowing for an improved analysis, which is nearer to the well-understood linear case. This new class of functional spaces will be employed to study the finite dimensional properties of the statistical solutions of the Navier-Stokes equations, via the determining modes, nodes and determining forms. Furthermore, the investigators will develop data assimilation techniques for statistical solutions of hydrodynamic equations. Statistical solutions are directly related to the average physical quantities that engineers are mostly concerned with. They provide a bridge between time average of physical observables such as energy, enstrophy and heat transfer to ensemble measures on the phase space. Thus, the techniques developed in this project should have immediate engineering and scientific applications.
