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Wed Sep 27

PDE Seminar

3:35pm - Vincent Hall 570
The Calderón problem for variable coefficients nonlocal elliptic operators
Lin Yi-Hsuan

Abstract: We introduce an inverse problem of a Schrödinger type
variable nonlocal elliptic operator (-?. (A(x)?)) s + q, for 0 < s < 1.
We determine the unknown bounded potential q from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any space dimension n = 2. Our results generalize the recent initiative of introducing and solving inverse problem for fractional Schrödinger operator. We also prove some regularity results of the
direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.

Wed Oct 04

PDE Seminar

3:35pm - Vincent Hall 570
Regularity and asymptotic of Keller-Segel models coupled to fluid equations
Kyungkeun Kang, Yonsei University

We consider chemotaxis equations coupled to the Navier-Stokes
equations, which is a mathematical model describing the dynamics of oxygen,
swimming bacteria (Bacillus subtilis), and viscous incompressible fluids. It is not
known for parameters of general chemotactic sensitivity and consumption rate in
two and three dimensions whether or not regular solutions with sufficiently
smooth initial data exist globally in time or develop a singularity in a finite time.
We discuss existence of regular solutions and asymptotic as well as temporal
decays of solutions, under a certain type of conditions of parameters and initial
data, as time tends to infinity.

Wed Oct 18

PDE Seminar

3:35pm - Vincent Hall 570
Stochastic Models for Turbulent Convection
Nathan Glatt-Holtz, Tulane University

Buoyancy driven convection plays a fundamental role in diverse physical settings: from cloud formation to large scale oceanic and atmospheric circulation processes to the internal dynamics of planets and stars. Typically, such fluid systems are driven by heat fluxes acting both through boundaries (i.e. heating from below) and in the bulk (i.e. internal `volumetric' heating sources) both of which can have an essentially stochastic nature. In this talk I will discuss some recent mathematical developments concerning ergodicity, singular parameter limits and the onset of instability in the stochastic Boussinesq and Magnetohydrodynamics equations.

Wed Oct 25

PDE Seminar

3:35pm - Vincent Hall 570
Minimal initial data for potential Navier-Stokes singularities in a half-space
Tuan Pham, University of Minnesota

I will talk about the existence of potential minimal blow-up data for the Navier-Stokes equations in the half space. The problem in the whole space has been studied by Rusin - Sverak (2011), Gallagher - Koch - Planchon (2013) and Jia - Sverak (2013). The regularities of weak solutions near a flat boundary studied by Seregin (2002 - 2005) make it possible to reuse much of the work that has been done for the whole-space problem.

Wed Nov 01

PDE Seminar

3:35pm - Vincent Hall 570
Serrin Lecture: Spectral stability of inviscid columnar vortices
Thierry Gallay, Univ. de Grenoble Alpes, France

Columnar vortices are stationary solutions of the three-dimensional
Euler equations with axial symmetry, where the fluid particles spin
around the vertical axis while moving in horizontal planes. Stability
of such flows was first investigated by Rayleigh and by Kelvin in
1880, but until today the only analytical results that can be found in
the literature are necessary conditions for instability under either
planar or axisymmetric perturbations. The goal of this talk is to
show that columnar vortices are spectrally stable for a large class
of velocity profiles, including the most commonly used models in
atmospheric flows and engineering applications. This a joint work
with Didier Smets (Paris).

Wed Nov 08

PDE Seminar

3:35pm - Vincent Hall 570
Patterns and complex singularities of reaction diffusion equations
Hirokazu Ninomiya., Meiji University, Japan

In this talk I will extend the reaction diffusion equation into the complex domain.
Based on this extension, I will discuss the relation between patterns and the complex singularities. This is based on the joint work with K. Ikeda, M. Katsurada and M. Onodera.

Wed Nov 22

PDE Seminar

3:35pm - Vincent Hall 570
Stability of Planar Fronts of the Bidomain Equation
Yoichiro Mori, University of Minnesota

The bidomain model is the standard model describing electrical activity of the heart. In this talk, I will discuss the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain AllenCahn equation) in two spatial dimensions. In the bidomain Allen-Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen-Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. Our analysis reveals that planar fronts can be unstable in the bidomain Allen-Cahn equation in striking contrast to the classical or anisotropic Allen-Cahn equations. We identify two types of instabilities, one with respect to long-wavelength perturbations, the other with respect to medium-wavelength perturbations. Interestingly, whether the front is stable or unstable under long-wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate-wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate-wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions. This is joint work with Hiroshi Matano of the University of Tokyo.

Wed Nov 29

PDE Seminar

3:35pm - Vincent Hall 570
Regularity Results for a Penalized Boundary Obstacle Problem
Donatella Danielli, Purdue University

n this talk we will discuss a two-penalty boundary obstacle problem of interest in thermics and fluid dynamics. Specifically, our goal is to establish existence, uniqueness and optimal regularity of the solutions, as well as structural properties of the free boundary. The study hinges on the monotone character of a perturbed frequency function of Almgren's? type, and the analysis of the associated blow-ups. This is joint work with Thomas Backing and Rohit Jain.

Wed Dec 06

PDE Seminar

3:35pm - Vincent Hall 570
Two-Bubble Dynamics for the Equivariant Wave Maps Equation
Jacek Jendrej, University of Chicago

I will consider the energy-critical wave maps equation with values in the two-dimensional sphere in the equivariant case, that is for symmetric initial data. It is known that if the initial data has small energy, then the corresponding solution scatters. Moreover, the initial data of any scattering solution has topological degree 0. I try to answer the following question: what are the non-scattering solutions of topological
degree 0 and the least possible energy? Such "threshold" solutions would have to decompose asymptotically into a superposition of two ground states at different scales, with no radiation. It turns out that one can construct non-scattering threshold solutions. I will also describe the dynamical behavior of any threshold solution. Joint work with Andrew Lawrie (MIT).

Wed Dec 13

PDE Seminar

3:35pm - Vincent Hall 570
On boundary estimates for solutions to second order elliptic and parabolic equations
Mikhail Safonov, University of Minnesota

I am going to discuss results and techniques by different authors related to interior and boundary regularity of solutions to second order elliptic and parabolic equations.

Wed Jan 24

PDE Seminar

3:35pm - Vincent Hall 570
Channel of energy inequality and Null concentration of energy for wave maps
Hao Jia, University of Minnesota

The channel of energy inequality for linear wave equations was introduced by Duyckaerts-Kenig-Merle to study soliton resolution for the focusing energy critical wave equations. It has been very useful for understanding long time behavior of semilinear wave equations, in the non-perturbative regime. Sometimes, it is the only available tool to understand dispersion of energy in the presence of solitons. Recently we found a new form of this type of inequality for outgoing waves, that turns out to be useful for studying energy critical wave maps, especially in ruling out the so-called ``null concentration of energy". In this talk we will give an outline of the inequality, the history of wave maps, explain why the absence of null energy is important, and why channel of energy inequality seems to be uniquely good for ruling out this type of energy concentration. Joint work with Duyckaerts, Kenig and Merle.

Wed Feb 07

PDE Seminar

3:35pm - Vincent Hall 570
On positive solutions of semi-linear elliptic inequalities on Riemannian manifolds
Alexander Grigor'yan, Bielefeld University, Germany

Plan of the talk:
1. Laplace-Beltrami operator and its Green function on a Riemannian manifold.
2. Semi-linear inequality $\Delta u+ F(x) u^\sigma \le 0$.
Pointwise lower bound for positive solutions $u$ via the Green function.
3. Approach to the proof.
4. Existence of positive solutions of $\Delta u+ u^\sigma \le 0$.
Volume test for non-existence. Green function test.
5. Examples of application.
6. Approach to the proof.

Wed Feb 21

PDE Seminar

3:35pm - Vincent Hall 570
Elasticity and curvature: the elastic energy of non-Euclidean thin bodies
Cy Maor, University of Toronto, Canada

Non-Euclidean, or incompatible elasticity is an elastic theory for bodies that do not have a reference, stress-free configuration. It applies to many systems, in which the elastic body undergoes inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional. Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them. In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results on the relations between their elastic behavior and their curvature.
Based on a joint work with Asaf Shachar.

Wed Feb 28

PDE Seminar

3:35pm - Vincent Hall 570
Uniform Rectifiability and Elliptic Partial Differential Equations
Simon Bortz, University of Minnesota

In geometric measure theory, the notion of rectifiability plays a central role. Roughly speaking, a set is rectifiable if and only if it possesses "measure theoretic tangents". Around 1990, Guy David and Stephen Semmes introduced a quantitative notion of rectifiability, uniform rectifiability. This turns out to be the appropriate rough setting for several `standard' harmonic analysis tools to work (L^2 boundedness of singular integrals, Littlewood-Paley, etc.). Recently, there has been a significant interest in providing (elliptic) PDE characterizations of uniformly rectifiable sets. Several such characterizations have been furnished by combining the work of Hofmann-Martell-Mayboroda ('14) and Azzam-Garnett-Mourgoglou-Tolsa ('16). I will survey these results and report on some of my own work in this area, which characterizes uniform rectifiability of a set by control on the oscillation of bounded harmonic functions on the complement of the set.

Wed Mar 28

PDE Seminar

3:35pm - Vincent Hall 570
The oblique derivative problem in Lipschitz domains
Hongjie Dong, Brown University

I will present a recent work about the W^2_p estimate for the oblique derivative problem for nondivergence form elliptic equations with VMO coefficients in Lipschitz domains with locally small Lipschitz constant. This improves an earlier result by G. Lieberman, where the domains are required to be in C^{1,a} for a > 1-1/p. An extension to fully nonlinear equations will also be discussed. Joint work with Zongyuan Li (Brown).

Wed Apr 04

PDE Seminar

3:35pm - Vincent Hall 570
The Aviles-Giga functional - A history, a survey and some new results
Andrew Lorent, University of Cincinnati
Wed Apr 11

PDE Seminar

3:35pm - Vincent Hall 570
Asymptotic limit of fractional Allen-Cahn equations and nonlocal minimal surfaces.
Yannick Sire, Johns Hopkins University

I will describe recent results on the singular perturbation limit of a fractional Allen-Cahn Equation modeling nonlocal phase transitions. The limiting interface appears to be a stationary nonlocal minimal surface. New nonlocal phenomena appear and the convergence is actually strong, something not happening in the local case. The basic tool is a deep GMT theorem due to Marstrand.

Wed Apr 18

PDE Seminar

3:35pm - Vincent Hall 570
The Landis conjecture for elliptic operators
Luca Rossi, EHESS, Paris

The Landis conjecture, proposed in the 80s, states that if a solution of an elliptic equation decays faster than a suitable exponential then it must be identically equal to zero.This conjecture has been disproved by Meshkov in the case of complex-valued functions, but it remains open in the real case. In this talk, I will recall some partial results obtained by Kenig and collaborators. Next, I will present the proof of the conjecture in dimension one and its application to the radial case. Finally, I will consider the restriction to positive solutions and to operators with positive generalized principal eigenvalue.

Wed Apr 25

PDE Seminar

3:35pm - Vincent Hall 570
Scattering below the ground state for nonlinear Schrödinger equations
Jason Murphy

The ground state solution to the nonlinear Schrödinger equation (NLS) is a global, non-scattering solution that often provides a threshold between scattering and blowup. In this talk, we will discuss new, simplified proofs of scattering below the ground state threshold (joint with B. Dodson), as well as some extensions to other models of NLS (joint with R. Killip, M. Visan, J. Zheng, as well as with C. Miao and J. Lu).

Wed May 02

PDE Seminar

3:35pm - Vincent Hall 570
A Phase Field Model for Thin Elastic Structures with Topological Constraint
Patrick Dondl, University of Freiburg, Germany

With applications in the area of biological membranes in mind, we consider the problem of minimizing Willmore’s energy among the class of closed, connected surfaces with given surface area that are confined to a fixed container. Based on a phase field model for Willmore’s energy originally introduced by de Giorgi, we develop a technique to incorporate the connectedness constraint into a diffuse interface model of elastic membranes. Our approach uses a geodesic distance function associated to the phase field to discern different connected components of the support of the limiting mass measure. We obtain both a suitable compactness property for finite energy sequences as well as a Gamma-convergence result. Furthermore, we present computational evidence for the effectiveness of our technique. The main argument in our proof is based on a new, natural notion to describe convergence of phase fields.