### The 5th seminar

We will have the 5th student seminar on

Tuesday, Nov 17th, 5:00pm~6:00pm, at 106 McAllister

Speaker: Xiang Xu

Title: Global existence and asymptotic behavior of the Ericksen-Leslie system

Abstract: In this talk we will discuss the classical solutions of a hydrodynamic system modeling the nematic liquid crystal materials. This system is a coupled system of Navier-Stokes equations and kinematic transport equations for the molecular orientations. First using different energetic variational approaches, we can recover the system in different ways and distinguish the conservative and dissipative parts of the induced stress terms. Next based on a modified Galerkin method and the study of higher order energy law, we can prove the existence of global classical solutions in both 2D and 3D cases, with some extra assumptions on those viscosity coefficients. Then by a suitable Lojasiewicz-Simon type inequality, we get the convergence of solutions to a steady state solution as time goes to infinity. Moreover, an estimation of convergence rate is provided. Finally, we reveal the relation between Parodi's condition and certain stability of the system.

### The 4th seminar

We will have the 4th student seminar on

Tuesday, Nov 3rd, 5:00pm~6:00pm, at 106 McAllister

Speaker: Brian Haines

Title: Effective Viscosity and Dynamics of Dilute Bacterial Suspensions: A Three-Dimensional Model

Abstract: We present a Stochastic PDE model for dilute suspensions of bacteria in a three-dimensional Stokesian fluid. This model is used to calculate the statistically-stationary bulk deviatoric stress and effective viscosity of the suspension from the microscopic details of the interaction of an elongated body with the background flow. A bacterium is modeled as a prolate spheroid with self-propulsion provided by a point force, which shows up in the model as an inhomogeneous delta function in the PDE. The bacterium is also subject to a stochastic torque in order to model tumbling (random reorientation). Due to a bacterium's asymmetric shape, interactions with a prescribed generic background flow, such as pure straining or planar shear, cause the bacterium to preferentially align in certain directions. Due to the stochastic torque, the steady-state distribution of orientations is unique for a given background flow. Under this distribution of orientations, self-propulsion produces a

reduction in the effective viscosity. For sufficiently weak background flows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to an effective viscosity of the suspension that is lower than the viscosity of the ambient fluid. This is in agreement with recent experiments on suspensions of Bacillus subtilis.

### The third student seminar

The third seminar will be on

Tuesday, Oct 27th, 5:00pm~6:00pm, at 106 McAllister

Speaker: Wen Cheng

Topic: Closed form approximation to parabolic equations with applications to finance

Abstract: For parabolic equations with coefficients depending both on time and space variables, generally speaking there are no closed form solutions. In the literature, there are several methods approximating the true solution, but they are either very complicated in practice or not very accurate. In this talk, I will introduce a new method which gives asymptotic solutions to the above mentioned equations. Our method is very simple from the practical point of view, and numerical tests show that our method is also very accurate even for low order approximations. I will also give some applications of our closed form asymptotics in finance, specifically, option pricing.

### Future plan

Our future seminars are planned to be on Tuesday 5:00pm~6:00pm, every other week.

### The second student seminar

The second seminar will be on

Thursday, Oct 8th, 4:00pm~5:00pm, at 216 McAllister

Speaker: Andong He

Topic: Some Geometrical Aspects of Nonlinear Waves

Abstract: Nonlinear and dispersive waves over finite depth of water can be described by Korteweg-de Vries (KdV) equation. If the bottom is uneven, the KdV equation is of variable coefficients. The ray method originating from linear theory is used to derive a variable coefficient KdV equation. Behavior

of wave amplitude far away from the origin of waves is also studied.

### The first student seminar

Thursday, Sept 24th, 4:00pm~5:00pm, Room 216 McAllister

Speaker: Yao Chen

Topic: Fourier Analysis on Iterative Methods

Abstract: For some certain iterative methods, (local) Fourier analysis could describe the convergence rates quantitatively. Therefore it is possible to construct several iterative methods, which can converge slowly or even diverge. By applying them in correct orders, we can get a method that converges fast.