Clemson Analysis and PDE Seminar

Abstract: We will start the talk with a very brief introduction on the theory of limit operators on the sequence space \ell^2(\mathbb Z). After having done so, we will pass to the space L^2(\mathbb R) and show that the theory of limit operators, at least when copied verbatim from the sequence space, fails to provide any interesting results in this setting.

The main part of the talk will focus on the connection of band-dominated operators and uniformly continuous operators (in the sense of Werner's quantum harmonic analysis) on L^2(\mathbb R). Once this connection has been discussed, we will see how the theory from the sequence space \ell^2(\mathbb Z) indeed does carry over to L^2(\mathbb R), at least when the correct operator algebras are considered. In the last part of the talk, we will try to give some details on how limit operators can be used to obtain a reasonable description of Fredholm property of operators on L^2(\mathbb R).

The talk will be based on joint work with Raffael Hagger.

Abstract: This talk will address some recent developments in the mathematical analysis of state and parameter reconstruction in nonlinear PDEs, specifically in the context of hydrodynamic equations, with the 2D Navier-Stokes equations serving as a paradigm. We will attempt to present unifying perspectives and insights into the problem of state and parameter reconstruction mapped out through several recently obtained analytical results.

Abstract: The Bergman projection is a fundamental operator in complex analysis with connections to singular integral theory, and it is of interest to study the commutator operator of the Bergman projection with multiplication by a measurable function b.  In particular, we study the boundedness and compactness of the Bergman projection commutators in two weighted settings via weighted BMO (bounded mean oscillation) and VMO (vanishing mean oscillation) spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. Complete characterizations of two weight boundedness and compactness are obtained in the analytic case, which parallel results of S. Bloom for the Hilbert transform. Our work initiates a study of the commutators acting on complex function spaces with different symbols. In this talk, we will discuss our main results, as well as the principal ideas of the proofs. This talk is based on joint work with Bingyang Hu and Ji Li. 

Abstract: We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \phi^4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes. We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. 

This is a joint work with Jonas Lührmann (Texas A&M).

Abstract: TBA

Abstract: TBA