## 7th Pacific Northwest PDE Meeting

- Date: 09/27/2008

University of Victoria

**Point singularities of very weak solutions of 3D stationary Navier-Stokes equations - Tai-Peng Tsai**

I will talk about a joint work with Hideyuki Miura, which characterizes the singularities of very weak solutions bounded by $C_*|x|^{-1}$ for $C_*$ sufficiently small, of 3D stationary Navier-Stokes equations with Dirac $\delta$ functions as external forces.

**Optimal Gagliardo-Nirenberg, Sobolev and Nash inequalities - Martial Agueh**

It is known that the sharp constants and optimal functions of many geometric inequalities can be obtained via Optimal transportation.

But so far, this approach has been successful for a special subclass of the Gagliardo-Nirenberg inequalities, namely those proved by Del Pino and Dolbeault, while the other cases still remained unknown. Here, we investigate the sharp constants and optimal functions of all the Gagliardo-Nirenberg inequalties by studying a p-Laplacian type equation.

By introducing a change of the unknown function in the equation, we show that the optimal functions can be explicitly derived from a nonlinear ODE, which becomes linear for all these inequalities in dimension 1, and for a subclass of them in higher dimensions. In these cases, we find the optimal functions along with the sharp constants of the corresponding Gagliardo-Nirenberg inequalties. Our analysis also covers the sharp Sobolev and Nash inequalities.

**Solitary waves in periodic structures - Randall J. LeVeque (University of Washington)**

One-dimensional plane waves in an elastic material can be modelled by a hyperbolic system of partial differential equations. In a homogeneous material, a nonlinear stress-strain relation leads to the formation of shock waves. Instead consider a laminated medium that consists of alternating layers of two different nonlinear materials. In this case the wave is partially reflected at each material interface, leading to dispersion and more complicated wave behavior. This dispersion, coupled with the nonlinearity, sets the stage for the appearance of solitary waves that behave like solitons in many respects. I will present some nonlinear homogenization results that yield effective equations containing dispersive terms and some recent computational results in two space dimensions. This is joint work with Darryl Yong and David Ketcheson.

**Leray-type regularizations of the Burgers and the isentropic Euler equations - Razvan Fetecau**

We start from the Burgers equation $u_t+uu_x=0$ and investigate a smoothing mechanism that replaces the convective velocity $u$ in the nonlinear term by a smoother velocity field $u^\alpha$, obtained from the former by a convolution with a molifier ($u^\alpha = \varphi^\alpha \ast u$). This type of regularization was first proposed in 1934 by Leray, who applied it in the context of the incompressible Navier-Stokes equations. We study the existence and uniqueness of solutions, the Riemann problem, the stability of traveling waves, the limit $\alpha \to 0$ of the solutions to the regularized equation. We also investigate whether the Leray smoothing procedure yields a valid regularization of the Burgers equation. Finally, we apply the Leray regularization to the isentropic Euler equations and use the weakly nonlinear geometrical optics (WNGO) asymptotic theory to analyze the resulting system. As it turns out, the Leray procedure regularizes the Euler equations only in special cases. We further investigate these cases using Riemann invariants techniques.

**The Janossy effect and hybrid variational principles - David Kinderlehrer (Center for Nonlinear Analysis and Department of Mathematical Sciences Carnegie Mellon University)**

Light can change the orientation of a liquid crystal. This is the optical Freedericksz transition, discovered by Saupe. In the Janossy effect, the threshold intensity for the optical Freedericksz transition is dramatically reduced by the additon of a small amount of dye to the sample. This has been interpreted as an optically pumped orientational rachet mechanism, similar to the rachet mechanism in biological molecular motors. To interpret the evolution system proposed for this effect requires an innovative gradient flow. Here we introduce this gradient flow and illustrate how it also provides the boundary conditions, some unusual coupling conditions, between the liquid crystal and the dye. We describe briefly existence and asymptotic stability. The proposed model correctly predicts the onset of the Janossy effect, but we do not have complete results in this direction. This is joint work with Michal Kowalczyk.

**Population spread and the dynamics of biological invasions - Mark Lewis**

Classical models for the growth and spread of introduced species track the front of an expanding wave of population density. Underlying equations are typically parabolic partial differential equations and related integral formulations. One method to infer the speed of the expanding wave is to equate the speed of spread of the nonlinear system with the speed of spread of a related linear system. When these two speeds coincide we say that the spread rate is linearly predictable. In this talk I will discuss linear predictability in multispecies models. In particular I will show how some competitive models are not linearly predictable.

In the talk I will the connect spread rates analysis to classical ideas in travelling wave theory. Lastly I will apply some of the results to real biological problems, including species competition, spread of disease and population dynamics of stream ecosystems.

**9:30 a.m. **Coffee and snacks**
10:00 a.m. to 10:50 a.m. **Razvan Fetecau

**David Kinderlehrer**

10:50 a.m. to 11:40 a.m.

10:50 a.m. to 11:40 a.m.

**Break**

11:40 a.m.

11:40 a.m.

**Tai-Peng Tsai**

11:50 a.m. to 12:40 p.m.

11:50 a.m. to 12:40 p.m.

**Lunch**

12:40 p.m.

12:40 p.m.

**Mark Lewis**

2:00 p.m. to 2:50 p.m.

2:00 p.m. to 2:50 p.m.

**Randy LeVeque**

2:50 p.m. to 3:40 p.m.

2:50 p.m. to 3:40 p.m.

**Coffee and cake break**

3:40 p.m.

3:40 p.m.

**Martial Agueh**

4:10 p.m. to 5:00 p.m.

4:10 p.m. to 5:00 p.m.

**Location
**

Workshop to be held in the Arbutus/Queenswood Room at the University of Victoria.

**Registration
**

To confirm your attendance please register by clicking here. Registration is free.

**Accomodation
**

To better arrange travel please contact one of the many accommodations listed throughout Victoria at www.tourismvictoria.com.