Past Seminars: 2015
  Fall 2015
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September 15, 2015
Tarek M. Elgindi, Princeton University
Recent results on the transport equation
We discuss several recent results on the well/illposedness of transport equations with singular integral forcing in spaces at the same scaling as $L^\infty$. Such equations arise naturally in the study of fluid equations and other applications. As a byproduct of one of the theorems we will discuss, we prove the strong illposedness of the incompressible Euler equations in the class of C^1 velocity fields. Some of what will be discussed is joint work with N. Masmoudi and F. Bernicot-S.
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September 29, 2015
Tristan Buckmaster, Courant Institute, NYU
Onsager's Conjecture
In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to
Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions
belonging to any Hölder space with exponent less than 1/3 which dissipate energy.
The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994).
During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Philip Isett
and myself related to resolving the second component of Onsager's conjecture.
In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder
$1/3-\epsilon$ norm is Lebesgue integrable in time.
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October 20, 2015
Jerome Goddard, Auburn University Montgomery
Modeling the effects of habitat fragmentation via reaction diffusion equations
Two important aspects of habitat fragmentation are the size of fragmented patches of preferred habitat and the inferior habitat surrounding the patches,
called the matrix. Ecological field studies have indicated that an organism's survival in a patch is often linked to both the size of the patch and the quality of its surrounding matrix. In this talk, we will focus on modeling the effects of habitat fragmentation via the reaction diffusion framework. First, we will introduce the reaction diffusion framework and a specific reaction diffusion model with logistic growth and Robin boundary condition (which will model the negative effects of the patch matrix). Second, we will explore the dynamics of the model via some methods from nonlinear analysis and ultimately obtain a causal relationship between the size of the patch and the quality of the matrix versus the maximum population density sustainable by that patch.
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November 10, 2015
Michael Puls, John Jay College CUNY
The p-harmonic and p-Royden boundaries for metric measure spaces
In this talk we construct the p-Royden boundary for a metric measure space X.
We will also define a specialized subset of this boundary, known as the p-harmonic boundary.
A Dirichlet type problem at infinity for X will be discussed. A characterization of the p-parabolicity of X in terms of the cardinality
of the p-harmonic boundary will be given. This is joint work with Marcello Lucia and generalizes results
for Riemannian manifolds and graphs of uniformly bounded degree.
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November 24, 2015
Mythily Ramaswamy, TATA Institute, Bangalore
Control of compressible Navier-Stokes system
Compressible fluids are modelled through Navier Stokes equations for density and velocity.
In this talk I consider the model in a bounded interval and discuss null controllability
(steer the system to zero state in finite time) and stabilization (steer the system to a steady state as time goes to infinity). The control acts only on the velocity.
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December 8, 2015
Fabio Pusateri, Princeton University
The water waves problem
I will first introduce the water waves equations which
model the motion of waves such as those on the surface of the ocean.
I will then discuss in some generality the question of global
wellposedness for the Cauchy problem associated to this system.
Finally, I will present some recent results (joint with Deng, Ionescu
and Pausader) on the global regularity for the gravity-capillary
problem in three dimensions.
  Spring 2015:
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February 10, 2015
Luciano Medina, NYU Polytechnic
Vortex Equations Governing the Fractional Quantum Hall Effect
Governed by topological excitations in Chern-Simons gauge theories, an existence theory is established for a coupled non-linear elliptic system describing the fractional quantum Hall effect in 2-dimensional double-layered electron systems. Via variational methods, we prove the existence and uniqueness of multiple vortices over a doubly periodic domain and the full plane. In the doubly periodic situation, explicit sufficient and necessary conditions are obtained that relate the size of the domain and the vortex numbers. For the full plane case, vortex solutions are restricted to satisfy topological boundary conditions and exponential decay estimates are proved. Interestingly, quantization phenomena of the magnetic flux are found in both cases.
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February 24, 2015
Ovidiu Savin, Columbia University
A singular minimizer in the Calculus of Variations in low dimensions
I will discuss about a counterexample to C^1 regularity in the Calculus of Variations. We consider minimizers of smooth convex functionals depending only on the derivative of a map from R^n to R^m. Classical results of Morrey and De Giorgi-Nash state that such minimizing maps are smooth when $n=2$ or when $m=1$. On the other hand some examples due to Necas and Sverak-Yan show that the regularity of minimizers is not expected in general.
In my talk I will discuss an example of a singular minimizer in the lowest possible dimensions when $n=3$ and $m=2$.
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March 10, 2015
Chuan Xue, Ohio State University
Chemotaxis, velocity jump process and the Keller-Segel equations
Chemotaxis is the directed movement of cells or organisms towards external chemical signals
in the environment. Chemo-taxis of a single bacterium or a eukaryotic cell has been extensively
studied and a great deal is known on the molecular machinery involved in intracellular signaling
and cell movement. However, systematic methods to embed such information into continuum
PDE models for cell population dynamics are still in their infancy. In this talk, I will present our
recent results in this aspect for run-and-tumble bacteria whose movement, at the single cell
level, is usually modeled by velocity jump processes with internal dynamics. I will show that the
well-known Keller-Segel chemotaxis equation is valid when the external signal changes slowly,
but inadequate when the signal changes fast.
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March 31, 2015
Ratnasingham Shivaji, The University of North Carolina Greensboro
Existence Results for Classes of Steady State Reaction Diffusion Equations
Abstract
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April 14, 2015
Kyril Tintarev, Uppsala University, Sweden
TBA
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May 5, 2015
Xiangwen Zhang, Columbia University
Uniqueness Theorem for convex surfaces
A classical uniqueness problem of Alexandrov says that: a closed strictly convex twice differentiable surface in $R^3$ is uniquely determined to within a parallel translation when one gives a proper function of the principle curvatures. We will talk about a PDE proof for this theorem, by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is joint work with P. Guan and Z. Wang.