The Øresund Seminar on Analysis, PDE & Related Topics

Abstract: 

PDE systems encountered in many fields of physics and applied sciences can become quite complex. However near a constant steady state, the dynamics of these nonlinear systems can very often be effectively approximated by a small set of scalar equations called amplitude (or modulation) equations. For dissipative systems close to the onset of an instability, the Ginzubrg-Landau equation is one such amplitude equation. For dispersive systems with frequency-localized perturbations, the corresponding amplitude equation is the NLS equation. The rigorous justification of these amplitude equations can provide important information about the long-time dynamics of the original system. In this talk, we look at systems with one unbounded spatial direction and discuss new approximation results for quasilinear pattern-forming systems and for a nonlocal Klein-Gordon equation. This talk is based on two joint works with Guido Schneider and with Anna Logioti and Nils Thorin. 

Abstract: 

We derive an upper bound for the ground-state energy per unit volume of a dilute Bose gas in the thermodynamic limit which resolves the third order term predicted by Wu, Hugenholtz and Pines, and Sawada. Our result applies to radial, compactly supported potentials V ∈ L2(R3) that have positive scattering length, are stable, and do not admit two-body bound states. In particular, this includes the case V ≥ 0, and thus extends the class of potentials covered by the Lee-Huang-Yang upper bound established by Yau and Yin, as well as the one by Basti, Cenatiempo, and Schlein. The main novelty of our proof is the introduction of a cutoff that constrains the local number of excitations and permits the construction of a trial state with the correct energy density at any length scale below the thermodynamic one.

Abstract: 

We consider degenerate non-autonomous parabolic equations in $L^p$ spaces for all $1\le p\le\infty$, and prove integrated Gaussian space-time upper bounds on their solutions. Our bounds cover both linear and nonlinear equations on general Riemannian manifolds.

Abstract: 

Universality describes the phenomena that the local behavior of zeros of orthogonal polynomial depends only on general characteristics of the system, rather than on its precise properties. From a modern perspective, such local behavior is analyzed through local scaling limits of the Christoffel–Darboux kernel. Recent works have established necessary and sufficient conditions for the most studied universality classes, including bulk universality and hard-edge universality. In all these settings, the rescaled Christoffel–Darboux kernel converges to a single limit kernel.

In this talk, we consider scaling limits in which there is not a single limit kernel, but rather an entire cycle of limit kernels. We show that this type of behavior arises naturally when studying scaling limits for orthogonal polynomials associated with the equilibrium measure of the real Julia set of an expanding polynomial or with the Cantor measure on the classical middle-third Cantor set.

This talk is based on joint work with Alexander Kheifets, Milivoje Lukić and Peter Yuditskii.