Seminar Calendar
for Harmonic Analysis and Differential Equations events the year of Friday, April 21, 2017.

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Tuesday, February 14, 2017

Harmonic Analysis and Differential Equations (HADES)
1:00 pm   in 347 Altgeld Hall,  Tuesday, February 14, 2017
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Submitted by vzh.
Tobias Ried (Karlsruhe Institute of Technology)
Gevrey smoothing of weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules
Abstract: We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with finite mass, energy and entropy, that is, $f_0 \in L^1_2({\mathbb R}^d) \cap L \log L({\mathbb R}^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity. This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers. (Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)

Tuesday, April 4, 2017

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, April 4, 2017
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Submitted by berdogan.
Michael Goldberg (U. Cincinnati)
Pointwise bounds for the 3-dimensional wave equation and spectral multipliers
Abstract: The sine propagator for the wave equation in three dimensions, $\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}$, has an integral kernel $K(t,x,y)$ with the property $\int_{\mathbb R} |K(t, x, y)|dt = (2\pi|x-y|)^{-1}$. Finiteness comes from the sharp Huygens principle and power-law decay comes from dispersion. Estimates of this type are useful for proving ``reversed Strichartz" inequalities that bound a solution in $L^p_x L^q_t$ for admissible pairs $(p,q)$. We examine the propagator $\frac{\sin(t\sqrt{H})}{\sqrt{H}}P_{ac}(H)$ for operators $H = -\Delta + V$ with the potential $V$ belonging to the Kato-norm closure of test functions. Assuming zero is not an eigenvalue or resonance, the bound $\int_{\mathbb R} |K(t,x,y) \leq C|x-y|^{-1}$ continues to be true. Combined with a Huygens principle for the perturbed wave equation, this estimate suggests pointwise bounds for spectral multipliers of fractional integral or H\"ormander-Mikhlin type. This is joint work with Marius Beceanu (SUNY - Albany).

Tuesday, April 18, 2017

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, April 18, 2017
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Submitted by tzirakis.
Jeremy Marzuola (University of North Carolina at Chapel Hill)
Ground states for nonlinear Schr\"odinger equation on a dumbbell graph
Abstract: With Dmitry Pelinovsky, we describe families of standing waves on a closed quantum graph in the shape of a dumbbell, namely having two loops connected by a link with Kirchhoff boundary conditions. We describe symmetry breaking bifurcations and prove a remarkable asymptotic property that is a bit surprising in terms of the energy minimizing solutions at a given mass.

Tuesday, April 25, 2017

Harmonic Analysis and Differential Equations
1:00 pm   in 347 Altgeld Hall,  Tuesday, April 25, 2017
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Submitted by berdogan.
Gong Chen (U Chicago)
Strichartz estimates for linear wave equations with moving potentials
Abstract: We will discuss Strichartz estimates for linear wave equations with several moving potentials in $\mathbb{R}^{3}$ (a.k.a. charge transfer Hamiltonians) which appear naturally in the study of nonlinear multisoliton systems. We show that local decay estimates systematically imply Strichartz estimates. To study local decay estimates, we introduce novel reversed Strichartz estimates along slanted lines and energy comparison under Lorentz transformations. As applications, we will also discuss related scattering problems and a construction of multisoliton in $\mathbb{R}^{3}$ with strong interactions.