# PDE Seminar 2010-2011

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

Date: **Friday, September 10, 2010**

- Speaker: Mathias Wilke, Martin-Luther Universität Halle-Wittenberg, Germany, and Vanderbilt University
- Title:
**Qualitative behaviour of the two-phase Navier-Stokes equations with surface tension I** - Abstract: In this talk we discuss the properties of solutions to the two-phase incompressible Navier-Stokes equations. We present results on local well-posedness and qualitative behaviour of the solutions. In particular the equilibria are stable and each solution which is initially close to an equilibrium converges to an equilibrium at an exponential rate.

Date: **Friday, September 17, 2010**

- Speaker:
**Mathias Wilke, Martin-Luther Universität Halle-Wittenberg, Germany, and Vanderbilt University** - Title:
**Qualitative behaviour of the two-phase Navier-Stokes equations with surface tension II** - Abstract: This talk is the continuation of my previous talk. We will discuss several techniques that are used in the proofs.

Date: **Friday, October 1, 2010**

- Speaker:
**Wolfgang Ruess, Universität Duisburg-Essen, Germany** - Title:
**Solution theory for nonlinear partial differential delay equations** - Abstract: AbstractRuess

Date: **Friday, October 22, 2010**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Blow up rates for indefinite parabolic problems I** - Abstract: We consider indefinite parabolic problems, that is, problems with a nonlinearity that depends on the space variable x and changes sign. Using the scaling method we show the connection between blow-up rates and Liouville theorems. Afterwards, using the method of moving hyperplanes, we prove Liouville theorems for indefinite problems on whole and half spaces.

Date: **Friday, October 29, 2010**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Blow up rates for indefinite parabolic problems II**

Date: **Friday, November 12, 2010**

- Speaker:
**Mathias Wilke, Martin-Luther Universität Halle-Wittenberg, Germany, and Vanderbilt University** - Title:
**Sectorial operators, H_\infty calculus, and applications I** - Abstract:

Date: **Friday, November 19, 2010**

- Speaker:
**Mathias Wilke, Martin-Luther Universität Halle-Wittenberg, Germany, and Vanderbilt University** - Title:
**Sectorial operators, H_\infty calculus, and applications II** - Abstract:

Date: **Friday, December 3, 2010**

- Speaker:
**Mathias Wilke, Martin-Luther Universität Halle-Wittenberg, Germany, and Vanderbilt University** - Title:
**Sectorial operators, H_\infty calculus, and applications III** - Abstract:

Date: **Friday, January 21, 2011**

- Speaker:
**Pierre Gabriel, University Pierre & Marie Curie, Paris** - Title:
**Long-time asymptotic behavior for nonlinear growth-fragmentation equations** - Abstract: In order to find the steady states of nonlinear growth-fragmentation equations, we investigate the dependency of the first eigenvalue of the linear growth-fragmentation operator on parameters. Adapting this method we also prove, in some cases, the existence of periodic solutions and nonlinear stability results.

Date: **Friday, February 4, 2011 Cancelled**

- Speaker:
**Gabriel Koch, University of Oxford, England** - Title:
**Profile decompositions and Navier-Stokes** - Abstract: We use the dispersive method of “critical elements” established by Kenig and Merle to give an alternative proof of a well-known Navier-Stokes regularity criterion due to Escauriaza, Seregin and Sverak. The key tool is a decomposition into profiles of bounded sequences in critical spaces. (Joint work with I. Gallagher and F. Planchon.)

Date: **Friday, February 11, 2011**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Convergence to a steady state for autonomous and asymptotically autonomous problems, I** - Abstract: This talk we will discuss existing results on convergence of solutions of autonomous parabolic equations to a unique steady state. We also show how the dynamic changes if we introduce a small perturbation that decays to zero as time approaches infinity.

Date: **Friday, February 18, 2011**

- Speaker:
**Juraj Foldes, Vanderbilt University** - Title:
**Convergence to a steady state for autonomous and asymptotically autonomous problems, II** - Abstract: We will discuss convergence results for higher dimensional parabolic equations and problems on the whole space. The techniques will be based on a careful analysis of flows on omega-limit sets and on the application of energy functionals. Approaches based on the Lojasiewicz inequality will be discussed as well.

Date: **Thursday, February 24, 2011, Colloquium Talk**

- Speaker:
**Jan Prüss, Martin-Luther Universität Halle-Wittenberg** - Title:
**Evolution equations, maximal regularity, and free boundary problems** - Abstract: In this survey talk I will explain the basic ideas of the theory of abstract evolution equations as well as present applications to partial differential equations. I intend to show how the concept of maximal regularity naturally comes into play for quasilinear parabolic problems. An outline of its impact on the analysis of free boundary problems will be given.

Date: **Friday, March 4, 2011**

- Speaker:
**Ugo Gianazza, University of Pavia, Italy** - Title:
**Boundary estimates for quasilinear parabolic equationss** - Abstract: Gianazza_abstract

Date: **March, 25, 2011**

- Speaker:
**Mehdi Lejmi, Vanderbilt University** - Title:
**Elliptic PDE’s and local symplectic property** - Abstract: An almost-complex structure has a local symplectic property if it is compatible locally with a symplectic form. In this talk, we will show that in dimension 4, any almost-complex structure has the local symplectic property using the Malgrange existence theorem of local solutions of elliptic systems of PDE’s. This fact is not true in dimension 6 for example.

Date: **Friday, April 1, 2011**

- Speaker:
**Todd W. Troyer, University of Texas, San Antonio** - Title:
**Noise contributions to response dynamics in one dimensional model neurons** - Abstract: The most basic model of neuronal activity posits that the internal state of nerve cells can be approximated by a single variable, and that action potentials are generated when this variable crosses a threshold value. It is also common to assume that the intuitions developed with deterministic models generalize directly to the corresponding stochastic models. I will discuss two situations where this is not the case, focusing on ongoing work related to the experimental determination of the phase response curve for oscillatory neurons. I will also sketch a possible framework for connecting the analysis of oscillatory and non-oscillatory models.

Date: **Friday, April 8, 2011**

- Speaker:
**Richard Kollar, Comenius University, Bratislava** - Title:
**New methods for spectral problems with applications** - Abstract: Nonlinear and non-self-adjoint spectral problems appear in a wide variety of problems in mathematics: they determine stability properties of nonlinear waves, rates of decays of waves in fluids or a location of spectra in scattering problems in integrable systems. The traditional way to study these problems analytically is to find an algebraic transformation that maps the problem to a self-adjoint problem. There are multiple disadvantages of this approach as it requires to discover a hidden transformation and it often leads to a problem posed in an indefinite metric space. We will present two different methods for studying these problems: a generalized Krein signature and a homotopy method. Both are based on analytical ideas and do not use special algebraic structure of the problem. We will also present how the ideas from nonlinear problems can be used to study linear problems, particularly we show how to determine the Krein signature from the Evans function.The Evans function is a robust analytic and numerical mathematical tool for finding eigenvalues of finite and infinite-dimensional spectral problems with applications in stability of nonlinear waves, vibration analysis of damped waves, integrable systems, and nonlinear optimization. The central question in all these problems is existence of eigenvalues of a particular operator with a positive real part. Any numerical simulation is limited to search for spectra of these operators only inside a bounded part of complex plane. Krein signature allows to use a global information to check whether all such eigenvalues were find together with possible sources of a parameter change instability that are located on the imaginary axis. Hence, the relation between the Krein signature and Evans function is an important open problem.

Date: **Wednesday, April 13, 4:10 pm, SC 1431**

- Speaker:
**Jürgen Saal, Center of Smart Interfaces, TU Darmstadt** - Title:
**A vector measure approach to rotating boundary layers** - Abstract: Rotating boundary layer flows usually display an oscillating behavior, i.e., they are nondecaying at space infinity. Moreover, there is some significance in uniformness of such flows w.r.t. related parameters such as angular velocity of rotation. These requirements i.g. cannot be satisfied simultaneously by a treatment of boundary layer problems in standard function spaces. In my talk, I will present a new approach in spaces of Fourier transformed vector Radon measures. Besides giving account to the nature of boundary layer problems, this approach offers a couple of further nice outcomes:

– the computations are rather elementary and as a consequence we can find explicit dependence of the solution on related parameters;

– the eigenvalues producing unstable eigenmodes belong to the point spectrum of the linearized operator;

This, for instance, leads to a short proof of linear and nonlinear instability of the Ekman spiral for large Reynolds numbers. The talk is based on joint works with Yoshikazu Giga and Andre Fischer.

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