# PDE Seminar 2011-2012

### Fridays 4:10pm, Stevenson Center 1307

Date: **Friday, September 16, 2011**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**On Serrin’s symmetry result in non-smooth domains and its applications** - Abstract: In this talk we discuss a famous result of Serrin on overdetermined quasilinear elliptic problems on smooth domains. Using a new approach, we show that the result can be extended to fully nonlinear equations on symmetric non-smooth domains, provided one assumes specific overdetermined condition on the boundary. As an application we improve results on positivity of non-negative solutions. At the end possible generalizations and research directions will be mentioned.

Date: **Friday, September 30, 2011**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**On Serrin’s symmetry result in non-smooth domains and its applications, II** - Abstract: Continuation of the previous talk.

Date: **Friday, October 14, 2011**

- Speaker:
**Jeremy LeCrone, Vanderbilt University** - Title:
**Elliptic operators and the periodic inhomogeneous Cauchy problem, I** - Abstract: In this talk, I will discuss the behavior of higher order elliptic operators acting on spaces of periodic functions. I will introduce definitions of ellipticity for differential operators of order 2m with variable, periodic, coefficients. Then I will show how Fourier multipliers and perturbation results can be used to establish solutions to the inhomogeneous Cauchy problem with periodic boundary conditions.

Date: **Friday, October 21, 2011**

- Speaker:
**Jeremy LeCrone, Vanderbilt University** - Title:
**Elliptic operators and the periodic inhomogeneous Cauchy problem, II** - Abstract: Continuation of the previous talk.

Date: **Friday, November 4, 2011**

- Speaker:
**Tuoc Van Phan, University of Tennessee** - Title:
**Stationary Navier-Stokes equations with critically singular external forces: existence and stability results** - Abstract: We show the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces belonging to a critical space. To the best of our knowledge, this is the largest critical space that is available up to now for this kind of existence. This result can be viewed as the stationary counterpart of the existence result obtained by H. Koch and D. Tataru for the free non-stationary Navier-Stokes equations with initial data in $BMO^{-1}$. The stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup.

Date: **Wednesday, November 9, 2011 (SC 1308, 4:10 pm)**

- Speaker:
**Alain Miranville, Universite de Poitiers, France** - Title:
**The Cahn-Hilliard equation with a logarithmic potential** - Abstract: Our aim in this talk is to discuss the Cahn-Hilliard equation in phase separation with singular, and, in particular, the thermodynamically relevant logarithmic, potentials. We first consider the usual Neumann boundary conditions and then dynamic boundary conditions recently proposed to account for the interactions with the walls in confined systems.

Date: **Friday, November 11, 2011**

- Speaker:
**Mathias Wilke, Martin-Luther Universität Halle-Wittenberg, Germany** - Title:
**On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations with surface tension in a capillary** - Abstract: In this talk we consider the two-phase Navier-Stokes equations with surface tension in a capillary. We present results on well-posedness and qualitative behaviour as time tends to infinity. It turns out that stability of the trivial solution depends on the radius R>0 of the capillary. To be precise, there exists a critical radius R*>0 such that the trivial solution is exponentially stable if R<R* and unstable if R>R*. This phenomenon is known as the Rayleigh-Taylor instability. At R=R* a bifurcation of Crandall-Rabinowitz type occurs. The critical radius R* can be explicitely computed; it depends on the jump of the densities of the fluids, the surface tension, the gravity constant and on the first nontrivial eigenvalue of the negative Neumann-Laplacian on the unit disk.

Date: **Friday, November 18, 2011**

- Speaker:
**Pierre Magal, Universite de Bordeaux Segalen, France** - Title:
**P-gp transfer and acquired multi-drug resistance in tumors cells** - Abstract: Multi-Drug resistance for cancer cells has been a serious issue for several decades. In the past, many models have been proposed to describe this problem. These models use a discrete structure for the cancer cell population, and they may include some classes of resistant, non resistant, and acquired resistant cells. Recently, this problem has received a more detailed biological description, and it turns out that the resistance in treatment is due in 40% of cancers to a protein called P-glycoprotein (P-gp). Moreover some new biological experiments show that transfers can occur by means of tunneling nano tubes built in between cells (direct transfers). Transfers can also occur through microparticles (containing P-pg) released by over-expressing cells into the liquid surrounding these cells. These microparticles can then diffuse and can be recaptured by the cells (indirect transfers).These transfers turn out to be responsible for the acquired resistance of sensitive cells. The goal of this talk is to introduce this problem, and to present a cell population dynamic model with continuous P-gp structure.

Date: **Friday, December 2, 2011**

- Speaker:
**Guillermo Reyes, Polytechnical University of Madrid, Spain** - Title:
**The Cauchy problem for the porous medium equation with variable density. Asymptotic behavior of solutions.** - Abstract: AbstractPresNHGReyes

Date: **Friday, January 27, 2012**

- Speaker:
**Daniele Andreucci, Universita di Roma “La Sapienza”, Italy** - Title:
**Modeling alternating pores in biological membranes: analytical and numerical approach** - Abstract: Channels, or pores, in cell membranes may alternate between a closed and an open state. This kind of gating has been connected to the feature of selectivity: possibly by means of sharp changes in affinity, the channel is able to select ions of preferred species for permeation.

We present a preliminary analytical study of permeation through a membrane with many alternating pores, via the theory of homogenization, obtaining the explicit limiting interface condition in terms of the spatiotemporal properties of the microstructure. Introducing then selection in the model, we report on numerical Montecarlo simulations investigating the efficiency of the filter.

Date: **Friday, February 10, 2012**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Qualitative properties of positive solutions of perturbed parabolic equations** - Abstract: Positive solutions of nonlinear parabolic problems can have very complex behavior. However, assuming certain symmetry conditions, it is possible to prove that the solutions converge to the space of symmetric functions. We show that this property is `stable’. More specifically if the symmetry conditions are replaced by asymptotically symmetric ones, the solutions still approach the space of symmetric functions. We discuss problems on bounded and unbounded domains and, by possibly suprising counterexamples, we show optimality of our assumptions. As an application, we formulate new results on convergence of solutions to a single equilibrium.

Date: **Friday, February 17, 2012**

- Speaker:
**Georgi Kapitanov, Vanderbilt University** - Title:
**A mathematical model of cancer stem cell lineage population dynamics with mutation accumulation and telomere length hierarchies** - Abstract: There is evidence that cancer develops when cells acquire a sequence of mutations that alter normal cell characteristics. This sequence determines a hierarchy among the cells, based on how many more mutations they need to accumulate in order to become cancerous. When cells divide, they exhibit telomere loss and differentiate, which defines another cell hierarchy, on top of which is the stem cell. We propose a mutation-generation model, which combines the mutation-accumulation hierarchy with the differentiation hierarchy of the cells, allowing us to take a step further in examining cancer development and growth. The results of the model support the hypothesis of the cancer stem cell’s role in cancer pathogenesis: a very small fraction of the cancer cell population is responsible for the cancer growth and development. Also, according to the model, the nature of mutation accumulation is sufficient to explain the faster growth of the cancer cell population. However, numerical results show that in order for a cancer to develop within a reasonable time frame, cancer cells need to exhibit a higher proliferation rate than normal cells.

Date: **Friday, March 16, 2012**

- Speaker:
**Peter Hinow, University of Wisconsin, Milwaukee** - Title:
**Size-structured populations with distributed states at birth** - Abstract: Age-structured models have been employed successfully in population dynamics for a long time and are considerably well understood. In contrast to such models where every individual is born at the same age 0, size-structured models allow to take into account different, distributed birth sizes. “Size” here can be a quite general concept, for example mass, energy content or pathogen load in a disease model. This introduces a birth operator that takes values in an infinite-dimensional Banach space and complicates greatly the mathematical analysis. In this survey, we will describe some examples of models that we recently investigated in a series of joint papers with Jozsef Farkas (University of Stirling, United Kingdom). The emphasis will be on questions such as asymptotic growth for linear models and the existence and stability of steady states for nonlinear models.

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