Analysis of Some L-Infinity Variational Problems, Aronsson's Equation, Erickson-Leslie System Modeling Hydrodynamic Flow of Liquid Crystals

  • Wang, Changyou (PI)

Grants and Contracts Details


Analysis of some Loo-variational problems and Aronsson's equation, Ericksen-Leslie system modeling hydrodynamic flow of liquid crystals Intellectual Merit. The PI plans to continue his research on nonlinear partial differential equations and variational problems. This proposal focus on the following two areas. (I) Analysis of Loo-variational problems and the corresponding Euler-Lagrange equations, named the Aronsson equation. Calculus of variations in Loc studies the minimization problems of appropriate cost functionals that are supremum of integrand functions. The issues that we plan to address are: (a) Derivation of Aronsson's equation for absolute minimizer of quasiconvex C1-Hamiltonian functions. (b) Find sufficient conditions for viscosity solutions of Aronsson's equation to be absolute minimizers of Hamiltonian functions depending on x, z,p-variables. (c) Uniqueness problems of viscosity solutions to Aronsson's equations. (d) Homogenization problem in UJOunder the framework of f-convergence. (e) Loc-variational problems under the Dirichlet energy constraints. (f) UJO-design problems in dimensions two. (II) Analysis of the simplified Ericksen-Leslie system modeling hydrodynamic flow of liquid crystals. The system is a strong coupling between the Navier-Stokes equation for the fluid velocity field u and the transported heat flow of harmonic maps into S2 for the direction field n of the liquid crystals (n contributes the elastic stress tensor \7 . (\7 n (2) \7 n) to the equation of u and u induces the convection term u· \7n to the equation of n). \Ve are mainly interested in the existence of suitable weak solutions and their possible partial regularities in dimensions two or three, and the local and global well posedness in some critical LP-spaces. Broader Impact. The proposed problems in both areas have strong connections and profound applications to other fields such as engineering and material sciences. For example, the Loo-variational problem has found its applications in optimal controls, the image recovery engineering, and the random game theory; the Ericksen-Leslie system modeling the liquid crystal flows has its origination in engineering of materials. The proposed problems are very interesting and challenging mathematically. They either involve highly degenerate elliptic PDEs or system of PDEs with critical nonlinearities. Their resolutions will contribute new ideas and techniques that shall be useful for other problems as well. Furthermore, the PI will select carefully some of these proposed problems as the topics theme for several potential Ph.D. students. The PI has graduated two Ph.D. students who worked in related areas and is currently supervising three graduate students towards their Ph.D. The PI, joint with Dr. Y. Yu, is currently preparing a research monograph on Calculus of variations in UJO,which will include some research findings resulting from this proposal, that can serve as hoth an introduction to graduate students towards this fascinating field and a source of references of ideas and techniques for research scientists. The PI, joint with Professor F. H. Lin, has recently published a research monograph on the analysis of harmonic maps and their heat flows. The PI is in the discussion with Professor Robert Jensen from Loyola University of Chicago to prepare a workshop proposal to AIM for funds to support intensive workshops on the recent development on Loo-variational problems. 1
Effective start/end date8/1/107/31/14


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