Department of


Seminar Calendar
for Harmonic Analysis and Differential Equations events the next 12 months of Sunday, January 1, 2012.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 17, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, January 17, 2012

The Rigorous Derivation of the 2d Cubic NLS with Anisotropic Switchable Quadratic Traps

Xuwen Chen (University of Maryland)

Abstract: In this talk, we will explain how the nonlinear Schroedinger equations arises from an experimentally observed phenomenon called Bose-Einstein condensation. We will also present a rigorous derivation of the 2d cubic nonlinear Schroedinger equation with anisotropic switchable quadratic traps from a N-body linear Schroedinger equation.

Tuesday, January 31, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, January 31, 2012

The defocusing energy-supercritical cubic nonlinear wave equation

Aynur Bulut (Institute for Advanced Study)

Abstract: In this talk , we will discuss some recent results on global well-posedness and scattering for the defocusing cubic nonlinear wave equation (NLW) treating the energy-supercritical regime, that is dimensions five and higher. More precisely in a series of works (first in dimensions six and higher for general data, in dimension five with radial data and, very recently, in dimension five for general data) we prove global well-posedness under an a priori uniform in time control of the critical Sobolev norm by establishing strong integrability and regularity properties for a particular class of solutions to NLW. In particular, we will focus on our recent work in dimension five in the case of the general data.

Friday, February 17, 2012

1:00 pm in 343 Altgeld Hall,Friday, February 17, 2012

Stability of Periodic Wave Trains in a Kuramoto-Sivashinsky Equation

Mat Johnson (University of Kansas)

Abstract: (NOTE UNUSUAL DATE AND TIME) In this talk, we consider the spectral and nonlinear stability of periodic traveling wave solutions of a dispersion modified Kuramoto-Sivashinsky equation modeling viscous thin film flow down an inclined plane. In special cases, it has been known (with varying levels of rigor) since 1976 that, when subject to weak localized perturbations, spectrally stable solutions of this form exist. Although numerical time evolution studies indicate that these waves should also be nonlinearly stable to such perturbations, an analytical verification of this result has only recently been provided. In this talk, I will discuss a nonlinear stability theory for such spectral stability periodic wave trains, as well as the numerical and analytical verification of the required spectral stability and structural hypothesis of this theorem in particular canonical limits of dispersion/dissipation. This is joint work with Blake Barker and Kevin Zumbrun (Indiana University), as well as Pascal Noble and L. Miguel Rodrigues (University of Lyon I).

Tuesday, February 21, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, February 21, 2012

Dispersive estimates for Schrodinger operators in dimension two

William Green (Eastern Illinois University)

Tuesday, February 28, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, February 28, 2012

Scattering for wave maps exterior to a ball

Andrew Lawrie (U Chicago Math)

Abstract: In this talk I will discuss some recent work that was completed in collaboration with Professor Wilhelm Schlag. We consider $1$-equivariant wave maps from $\mathbb{R}_t\times (\mathbb{R}^3_x\setminus B) \to S^3$ where $B$ is a ball centered at $0$, and $\partial B$ gets mapped to a fixed point on~$S^3$. We show that $1$-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree.

Tuesday, March 6, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, March 6, 2012

Unusual properties of numerical instability of the split-step method applied to NLS soliton

Taras Lakoba, (University of Vermont, Math)

Abstract: The split-step method (SSM) is widely used for numerical solution of nonlinear evolution equations. Its idea and implementation are simple. Namely, it is common that the evolution of variable $u$ is governed by: $u_t = A(u,t) + B(u,t)$ where both ``individual'' evolutions $u_t = A(u,t) \qquad \mbox{and} \qquad u_t = B(u,t) $ can be solved exactly (or at least ``easily''). Then the numerical approximation of the full solution is sought in steps that alternatingly solve each equation. The SSM has long been used to simulate the NLS: $$i \, u_t - \beta u_{xx} + \gamma u|u|^2 = 0$$ ( so here $A=-\beta u_{xx}$ and $B=\gamma u|u|^2$). However, until recently, a possible development of numerical instability of the SSM has been studied only in one simplest case, which does not include the soliton or multi-soliton solutions of the NLS. In this talk I will present recent results concerning the development of the numerical instability of the SSM when it is used to simulated a near-soliton solution of NLS. Properties of this instability are stunningly different from instability properties of most other numerical schemes. I will not assume prior familiarity of the audience with instabilities of numerical methods and therefore will first review a couple of basic examples of such instabilities. This will set a benchmark for the subsequent exposition of the instability properties of the SSM. I will show how those properties, and --- more importantly --- their analysis, are different from the instability properties and analysis for most other numerical schemes.

Tuesday, March 13, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, March 13, 2012

High frequency interactions in nonlinear Schroedinger equations and applications

Christof Sparber (UIC Math)

Abstract: We consider the cubic nonlinear Schroedinger equation in a weakly nonlinear semiclassical scaling and analyze the interaction of highly oscillatory waves within this context. An extension to the Davey-Stewartson system will be discussed, as well as applications in proving ill-posedness of NLS in Sobolev spaces of negative order. This is based joint works with R. Carles and E. Dumas.

Thursday, March 15, 2012

1:00 pm in 345 Altgeld Hall,Thursday, March 15, 2012

Essential self-adjointness criteria for Schroedinger operators on bounded domains

Irina Nenciu (UIC Math)

Abstract: We consider a Schroedinger operator on a bounded domain in R^n, and search for optimal growth criteria for the potential close to the boundary of the domain insuring essential self-adjointness of the associated operator. We find an abstract integral criterion for the potential, from which we prove that one can add optimal logarithmic type corrections to the classical criteria. As a consequence of our method, we study the question of confinement of spinless and spin 1/2 quantum particles on the unit disk in R^2, and achieve magnetic confinement solely by means of the growth of the magnetic field.

Tuesday, April 10, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, April 10, 2012

Backward Stochastic Integral Equations

Yufeng Shi (University of Central Florida and Shandong University )

Abstract: In this talk we introduce a Volterra type of backward stochastic integral equations, i.e. so called backward stochastic Volterra integral equations (BSVIEs in short), which are natural generalization of backward stochastic differential equations (BSDEs in short). We will present some survey of old and introductory results, followed by some most recent developments, including M-solutions, S-solutions, C-solutions, Lp solutions, multi-dimensional comparison theorem, and mean-field BSVIEs. Main motivations of studying such kind of equations are as follows: (i) in studying optimal controls of (forward) stochastic Volterra integral equations, such kind of equations are needed when a Pontryagin type maximum principle is to be stated; (ii) in measuring dynamic risk for a position process in continuous time, such an equation seems to be suitable; (iii) when a differential utility needs to be considered with possible time-inconsistent preferences, one might want to use such equations.

Tuesday, April 24, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, April 24, 2012

Cavitation instabilities in nonlinear elastic solids: a defect-growth formulation based on iterated homogenization

Oscar Lopez-Pamies   [email] (UIUC Civil and Environmental Engineering)

Abstract: I will introduce a new formulation to study cavitation instabilities in nonlinear elasticity. The basic idea is to first cast cavitation as a homogenization problem of nonlinear elastic solids containing random distributions of zero-volume cavities, or defects. This problem is then addressed by means of a novel iterated homogenization procedure. Ultimately, the relevant calculations amount to solving Hamilton-Jacobi equations, in which the initial size of the defects plays the role of ``time'' and the applied load plays the role of ``space''. When specialized to the case of isotropic loading conditions, isotropic solids, and vacuous defects, the proposed formulation recovers the classical result of John Ball (1982) for radially symmetric cavitation. I will discuss the nature and implications of this remarkable connection.

Thursday, April 26, 2012

1:00 pm in 345 Altgeld Hall,Thursday, April 26, 2012

Synchronization in Power Networks and Coupled Oscillators

Florian Dorfler (UCSB Engineering)

Abstract: We discuss the synchronization and transient stability problem in power networks. We exploit the relationship between the power network model considered in transient stability analysis and the well-known Kuramoto model of coupled phase oscillators. A tight connection between these two models can be rigorously established by means of topological conjugacy arguments. In particular, we show the equivalence of local synchronization conditions in both models. Furthermore, we present novel algebraic conditions for synchronization of coupled Kuramoto oscillators. Our synchronization conditions are necessary and sufficient for particular interconnection topologies and network parameters, they are sufficient in the general case, and they improve upon previously-available tests for the Kuramoto model. In the end, we are able to state concise and purely algebraic conditions that relate synchronization in a power network to certain graph-theoretical properties of the underlying electric network. The results reveal elegant connections between the transient stability problem in power networks and the theory of coupled oscillators and multi-agent dynamical systems.

Tuesday, May 1, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, May 1, 2012

Thin Liquid Films with Driving

Mary Pugh   [email]

Abstract: We present two thin liquid film problems with driving. The first problem is experimentally motivated and considers questions such as steady states and the existence of dynamic solutions. The second problem is more PDE-motivated and considers questions such as the presence (or absence) of finite-time blow-up. In the first problem, we consider a horizontal cylinder, rotating about its center. A viscous fluid is on the outside of the cylinder, coating the cylinder as it rotates. We consider a lubrication approximation of the Navier Stokes equations for the regime in which the fluid film is relatively thin and the surface tension is relatively large. The resulting lubrication model may have no steady state, a unique steady state, or more than one steady state. Using both numerics and analysis, we consider the dynamics of this flow, including whether or not solutions can become singular in finite time. In the second problem, we consider a long-wave unstable thin film problem $u_t = - (u^n u_{xxx})_x - B (u^m u_x)_x. The dynamics are strongly affected by the balance between the exponents n and m. We discuss the subcritical, critical, and supercritical regimes for the equation and present new results for finite-time blow-up for the problem on the line. This is joint work with Marina Chugunova (University of Toronto) and Roman Taranets (Nottingham).

Tuesday, September 11, 2012

1:00 pm in 347 Altgeld Hal,Tuesday, September 11, 2012

Quasilinear Schroedinger Equations

Jeremy Louis Marzoula   [email] (UNC Math)

Abstract: In this talk, we will discuss joint works with Jason Metcalfe and Daniel Tataru on short time local well-posedness in low-regularity Sobolev spaces for quasilinear Schrödinger equations. Such results are refinements of the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove existence of solutions in very high regularity spaces. Our arguments however are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the same authors related to local smoothing estimates.

Tuesday, November 13, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, November 13, 2012

Inverse boundary problems for systems in two dimensions

Pierre Albin   [email] (UIUC Math)

Abstract: Inverse problems typically consist in trying to recover data about the interior of a domain from observations made at the boundary. For instance, if you know the boundary values of solutions to a Schrodinger equation or a Dirac equation on a surface, can you recover the operators in the interior? I will describe joint work with Colin Guillarmou, Leo Tzou, and Gunther Uhlmann in which we show that, except for an obvious gauge obstruction, the answer is yes.

Tuesday, November 27, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, November 27, 2012

Ludicrous speed of decay of solitons in Hertzian chains

Dirk Hundertmark   [email] (Karlsruhe Institute of Technology)

Abstract: A Hertzian chain is a model for granular matter. It describes a chain of beads which are just touching each other. The system is modeled by an advance delay equation and the discrete nature of this equation makes is somewhat harder to study than the usual continuum models. It turns out, both experimentally and theoretically, that there are solitary pulses in this system, which are highly localized. The existence of such pulses was also shown rigorously and it was also shown that there are solitary pulses which decay at a double exponential rate, i.e., the asymptotic profile of the pulse goes to zero like exp(-exp(x)). We give an argument that every solitary pulse must decay at this ludicrously fast rate. (Which reminds us of... )

Tuesday, December 4, 2012

1:00 pm in 347 Altgeld Hall,Tuesday, December 4, 2012

Sharp Spectral Bounds on Starlike Domains

Richard Laugesen   [email] (UIUC Math)

Abstract: If one knows geometric properties of a domain, such as its area or perimeter, then what inequalities can one deduce on the eigenvalues of the Laplacian for that domain? For example, the fundamental tone (first eigenvalue) and spectral functionals such as the spectral zeta function and heat trace have long been known to be extremal when the domain is a ball, provided the volume of the domain is fixed. We prove complementary bounds in the opposite direction (again sharp for the ball) by introducing an additional geometric quantity that measures the "deviation of the domain from roundness". An intriguing role in the proof is played by volume-preserving diffeomorphisms that are not the identity map. [Joint work with B. Siudeja, U. of Oregon]