Hydrodynamics of Liquid Crystals and Extremum Problems for Eigenvalues

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This proposal is for a deep theoretical study on a class of complex fluids including both the liquid crystals which have been widely used in display devices and charged biological fluids which are extremely important in modern medicine and life sciences. Of particular interests are their intriguing and complex dynamical phenomena as well as formations of patterns and singularities. The modeling, analysis and simulations involved in understanding these complex fluids are theoretically very important and challenging. It needs new ideas and methods, and hence it would advance our knowledge which would be applicable to many other scientific problems as well.More specifically, the proposal consists of two parts. The first part is to study partial differential equations that describe the hydrodynamics of liquid crystals and related complex fluid models. The main focus of this part of the research will be to study global existence of suitable weak solutions of liquid crystal flows in the Ericksen-Leslie theory; the global existence of solutions of Oldroyd B-model of incompressible visco-elastic fluids; the inviscid incompressible magneto-hydrodynamic system and, in general, coupled nonlinear dynamics of fluids with other geometric objects. The second part is to study a large class of extremum problems of elliptic eigenvalues. Such problems also arise in optimal designs, pattern formations and other applications in material sciences and condense matter physics. The partial differential equations (PDE) that the PI plans to study involve nonlinear couplings between equations that describe transport, phase-field, mapping or geometric object's evolutions and that of Navier Stokes equations. They may be of both parabolic and hyperbolic nature and possess singularities or multiple scales. The variational problems for eigenvalues and eigenfunctions that linked with underlying domains are classical and fundamental. These are fascinating and challenging problems that require new ideas and methods, which could lead to new directions of research or programs in the analysis of PDE and calculus of variations. The proposed research activity is an important and integral part of the PI's training program of under-graduate, graduate, and post-doctoral students.