[2] 6/DEC/11

(as of December 6, 2011)

10:30     Opening

10:45 -- 11:45   Saleh Tanveer (Ohio State University)

lunch break

13:30 -- 14:30   Saleh Tanveer (Ohio State University)

coffee break

15:00 -- 16:00   Hirokazu Ninomiya (Meiji University)

16:00 -- 17:00   Discussion

10:30 -- 11:30   Saleh Tanveer (Ohio State University)

lunch break

13:30 -- 14:30   Izumi Takagi (Tohoku University)

coffee break

15:00 -- 16:00   Xuming Xie (Morgan State University)

16:00 -- 17:00    Discussion

10:30 -- 11:30   Xuming Xie (Morgan State University)

Local smoothing effect and existence for a needle crystal growth problem with anisotropic surface tension.

lunch break

13:30 -- 14:30   Harunori Monobe (Tohoku University)

coffee break

15:00 -- 16:00   Xuming Xie (Morgan State University)

16:15   Closing

10:45 -- 11:45   Saleh Tanveer (Ohio State University)

It is very common in science and engineering to neglect terms to simplify the mathematics. The neglect of such terms is justified usually by estimating size of the neglected terms based on the solution to the simplified model. This assumes small errors in satisfying the equation will result in small errors in the solution. This is not the case for structurally unstable system, when solutions lose continuity with respect to change of parameters, as is the case for viscous fingering when a less viscous fluid displaces a more viscous fluid, either in porous media or Hele-Shaw call and a class of mathematically similar free boundary problems arising in different applications when regularization is ignored. We will show that for such systems, even with small regularization, the structure of regularization can change unexpectedly when other terms are introduced in the equations, because of the effect of exponentially small terms.

13:30 -- 14:30   Saleh Tanveer (Ohio State University)

Conformal mapping and boundary integral formulations of Laplacian growth problems in 2-D will be discussed, including in particular viscous fingering problem. We will then discuss how further analysis of steady traveling fingers and bubbles can be formulated conveniently in a conformal mapping approach, and how exponential asymptotics arise in such problems. (The details of the exponential asymptotic analysis will be left for subsequent lecture by Professor Xuming Xie's). We will also discuss how conformal mapping approach gives rise to a whole class of exact solutions for zero surface tension, though ill-posedness of this problem necessarily requires introduction of regularization for physical relevance.

coffee break

15:00 -- 16:00   Hirokazu Ninomiya (Meiji University)

Various patterns are observed in nature and simulations. For example, we observe the spiral waves and target patterns in Belousov-Zhabotinskii reaction. In photosensitive BZ reaction with feedback, the locallized patterns are also observed. Many researchers have regarded these patterns by the curves without thickness and have treated such patterns mathematically.
In this talk, using the wave front interaction model proposed by Zykov and Showalter, we can treat the existence of traveling spots in the plane and the rotating waves in the disk as two dimensional patterns and show the existence of such solutions. I will also explain the existence of the rotating spirals and rotating spots of this system. This talk is based on several join works with Professors J.S. Guo, J.-C. Tsai, C.C. Wu and my Ph.D student Y.Y. Chen.

16:00 -- 17:00   Discussion

10:30 -- 11:30   Saleh Tanveer (Ohio State University)

We will give detailed analysis for global existence and nonlinear stability for translating bubbles in a Hele-Shaw cell based on a boundary integral approach. We will also outline how analysis of a suitable linear stability operator, available through exponential asymptotics, gives a general framework for nonlinear stability of more general translating shapes.

lunch break

13:30 -- 14:30   Izumi Takagi (Tohoku University)

15:00 -- 16:00   Xuming Xie (Morgan State University)

There are a class of physical problems that involve selection. In this class of problems, when some parameter $\epsilon$(such as surface tension) is zero, there are a continuum set of solutions. However, as $\epsilon $ is NOT zero , only a discrete set of solutions exist. In these talks, we are going to discuss rigorous results in existence and selection for some free boundary problems such as viscous fingering and dendritic crystal growth.
We are also going to talk about well-posedness in Sobolev space for an unsteady needle crystal problem. Some results are joint work with Professor Saleh Tanveer.

16:00 -- 17:00    Discussion

10:30 -- 11:30   Xuming Xie (Morgan State University)

Local smoothing effect and existence for a needle crystal growth problem with anisotropic surface tension.

lunch break

13:30 -- 14:30   Harunori Monobe (Tohoku University)

We deal with a mathematical model related to cell crawling, which is one of cellular motions, e.g., white blood cell, keratocyte, cancer cell and so on. The mathematical model is represented as a free boundary problem with a non-local term. The validity of the model was confirmed numerically by Tamiki Umeda, but there are no mathematical results about the solvability of the model equation and properties of solutions (if they exist).
In this talk, we consider the existence and behavior of spherically symmetric solutions for the free boundary problem.

15:00 -- 16:00   Xuming Xie (Morgan State University)

Supported in part by JSPS Grant-in-Aid for challenging Exploratory Research "Mathematical Analysis of Dendritic Crystal Growth"

created on 4/DEC/11