## Grants and Contracts Details

### Description

Status | Finished |
---|---|

Effective start/end date | 8/1/10 → 7/31/14 |

- Wang, Changyou (PI)

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.
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Status | Finished |
---|---|

Effective start/end date | 8/1/10 → 7/31/14 |

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