Department of


Seminar Calendar
for Harmonic Analysis and Differential Equations events the year of Tuesday, March 26, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 22, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, January 22, 2019

Singular limits of sign-changing weighted eigenproblems

Derek Kielty   [email] (Illinois Math)

Abstract: Eigenvalue problems with positive weights are related to heat flow and wave propagation in inhomogeneous media. Sign-changing weights have ecological interpretations, and generate spectra that accumulate at both positive and negative infinity. This talk will discuss recent results on limits of such eigenvalue problems when a negative portion of the weight is made arbitrarily large.

Tuesday, January 29, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, January 29, 2019

Traveling waves in an inclined channel and their stability

Zhao Yang (Indiana University Bloomington)

Abstract: The inviscid Saint-Venant equations are commonly used to model fluid flow in a dam or spillway. To classify known traveling wave solutions to the St. Venant equations, the condition of hydrodynamic stability introduces a dichotomy on the parameter F (Froude number): Namely, the constant flow solution is stable for F < 2 where one expect persistent asymptotically-constant traveling wave solutions and unstable for F > 2 where one expect rather complex pattern formation. We will discuss for F>2 Dressler's construction of the inviscid roll wave solution and for F<2 Yang-Zumbrun's construction of the smooth/discontinuous hydraulic shock profiles. We will then present recent stability results of these traveling waves. That is a complete spectral stability diagram for F>2 roll wave case obtained in [JNRYZ18] and spectral, linear orbital, and nonlinear orbital stability of all the hydraulic shock profiles obtained in [YZ18] and [SYZ18].

Tuesday, March 5, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 5, 2019

Some recent progress on the Falconer distance conjecture and applications

Alex Iosevich (U. Rochester)

Abstract: We are going to discuss some recent results related to the Falconer distance conjecture and applications of some of these methods to the theory of exponential bases and frames.

Tuesday, March 12, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 12, 2019

The lattice bump multiplier problem

Loukas Grafakos (University of Missouri-Columbia)

Abstract: Given a smooth bump supported in a ball centered at the origin in $R^n$, we consider the multiplier formed by adding the translations of this bump by $N$ distinct lattice points. We investigate the behavior as $N$ tends to infinity of the $L^p$ norm of the multiplier operators associated with this finite sum of $N$ bumps.

Tuesday, March 19, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 19, 2019

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Alexandre Girouard   [email] (Université Laval)

Abstract: The Dirichlet-to-Neumann map is a first order pseudodifferential operator acting on the smooth functions of the boundary of a compact Riemannian manifold M. Its spectrum is known as the Steklov spectrum of M. The asymptotic behaviour (as j tends to infinity) of the Steklov eigenvalues s_j is determined by the geometry of the boundary of M. Neverthless, each individual eigenvalue can become arbitrarily big if the Riemannian metric is perturbed adequately. This can be achieved while keeping the geometry of the boundary unchanged, but it requires wild perturbations in arbitrarily small neighborhoods of the boundary. In recent work with Bruno Colbois and Asma Hassannezhad, we impose constraints on the geometry of M on and near its boundary. This allows the comparison of each Steklov eigenvalue s_j with the corresponding eigenvalues l_j of the Laplace operator acting on the boundary. This control is uniform in the index j. The proof is based on a generalized Pohozaev identity and on comparison results for the principal curvatures of hypersurfaces that are parallel to the boundary.

Tuesday, April 2, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 2, 2019

Direct Scattering and Small Dispersion for the Benjamin-Ono Equation with Rational Initial Data

Alfredo Wetzel (Wisconsin-Madison)

Abstract: The Benjamin-Ono (BO) equation describes the weakly nonlinear evolution of one-dimensional interface waves in a dispersive medium. It is an integrable equation, with a known Lax pair and inverse scattering transform, that may be viewed as a prototypical problem for the study of multi-dimensional integrable equations and Riemann-Hilbert problems with a non-local jump condition. In this talk, we propose explicit formulas for the scattering data of the BO equation with a rational initial condition. For this class of initial conditions, the recovery of the scattering data can be done directly by exploiting the analyticity properties of the Lax pair solutions. Our procedure validates previous well-known formal results and provides new details concerning the leading order behavior of the scattering data in the small dispersion limit. In the small dispersion limit, we are able to derive formulas for the location and density of the eigenvalues, magnitude and phase of the reflection coefficient, and density of the phase constants.

Tuesday, April 9, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 9, 2019

Convexity of Whitham's wave of extreme form

Bruno Vergara (ICMAT, Spain)

Abstract: In this talk I will discuss a conjecture of Ehrnström and Wahlén concerning travelling wave solutions of greatest height to Whitham's non-local model of water waves. We will see that there exists a cusped periodic solution whose profile is convex between consecutive peaks of $C^{1/2}$-regularity. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.

Tuesday, April 23, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 23, 2019

On Hardy-Rellich-type inequalities

Fritz Gesztesy (Baylor University)

Abstract: We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities, as special cases. Actually, other special cases yield additional and apparently less well-known inequalities. We will indicate that our method is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators. If time permits, we might illustrate a very recent new and most elementary proof in the one-dimensional context. This talk will be accessible to students. This is based on joint work with Lance Littlejohn, Isaac Michael, and Michael Pang.

Tuesday, April 30, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 30, 2019

Blow-up and Soliton Stability in KdV-type equations

Svetlana Roudenko (Florida International University)

Abstract: While the KdV equation and its generalizations with higher power nonlinearities (gKdV) have been long studied, a question about existence of blow-up solutions for higher power nonlinearities has posed lots of challenges and far from being answered. One of the main obstacles is that unlike other dispersive models such as the nonlinear Schrodinger or wave equations, the gKdV equation does not have a suitable virial quantity which is the key to prove the finite time blow-up. Partially, the question of existence and formation of singularities intertwines with the soliton stability or actually the instability, which may lead to a blowup. Only at the dawn of this century the groundbreaking works of Martel and Merle showed the existence of finite-time blow-up solutions for the quintic (critical) gKdV equation, as well as the asymptotic stability of solitons in the subcritical gKdV equation. We consider a higher dimensional extension of the gKdV equation, called generalized Zakharov-Kuznetsov (gZK) equation (the gKdV is limited as a spatially one-dimensional model), and investigate stability of solitons and the existence of blow-up solutions. We positively answer the question of existence of blowup in the two dimensional version of critical Zakharov-Kuznetsov equation and also obtain the asymptotic stability in the subcritical setting. We will discuss some of the important ingredients to obtain these results, including the Liouville-type theorem, which uses time-decay estimates, a la virial type quantity and spectral properties associated to it (this is a joint work with Luiz Farah, Justin Holmer and Kai Yang).

Tuesday, August 27, 2019

1:00 pm in Altgeld Hall,Tuesday, August 27, 2019

To Be Announced

Stathis Charalampidis