The APDE seminar on Monday, 10/11, will be given by Jacek Jendrej (Université Sorbonne Paris Nord) online via Zoom from 9.10am to 10.00am PST (note the time change). To participate, email Sung-Jin Oh ()
Title: Soliton resolution for energy-critical equivariant wave maps
Abstract: We consider wave maps R^(1+2) -> S^2, under the assumption of equivariant symmetry. We prove that every solution of finite energy resolves, as time passes, into a superposition of harmonic maps (solitons) and radiation. It was proved in works of Côte, and Jia and Kenig, that such a decomposition holds along a sequence of times. We show that the resolution holds continuously in time via a “no-return lemma” based on the virial identity. The proof combines a modulation analysis of solutions near a multi-soliton configuration with the concentration-compactness method. Joint work with Andrew Lawrie from MIT.
The APDE seminar on Monday, 10/4, will be given by Sanchit Chaturvedi (Stanford University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ()
Title: Stability of vacuum for the non-cut-off Boltzmann equation with moderately soft potentials.
Abstract: The vector field method developed by Klainerman has been widely successful in the study of wave equations and general relativity. Recently, the vector field approach has been adapted to understand the dispersion due to the transport operator in both collisionless and collisional kinetic models. As a proof of concept, I will discuss the stability of vacuum for Boltzmann equation with moderately soft potentials. The nonlocality of the Boltzmann operator poses a lot of difficulty and forces us to use a purely energy based approach. This is in contrast to the paper by Luk (Stability of vacuum for the Landau equation with moderately soft potentials) on Landau equation in a similar setting where a maximum principle is both proved and needed.
The APDE seminar on Monday, 9/27, will be given by Steve Zelditch (Northwestern University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ()
Title: Riemannian 3-manifolds whose eigenfunctions have just two nodal domains.
Abstract: In 1977, Hans Lewy published an article constructing a series $\phi_N$ of spherical harmonics of degree $N$ whose nodal (zero) sets cut the 2-sphere into just 2 nodal domains. It is an ingenious construction. Recently, Junehyuk Jung and I showed that there is a simple and canonical way to construct an infinite dimensional family of Riemannian metrics on certain 3 manifolds, all of whose eigenfunctions have this property (except for a certain trivial sequence). The construction generalizes to all odd dimensions. This unexpected behavior of 3D eigenfunctions is heuristically related to numerical computations of nodal sets of 3D spherical harmonics, where only one connected component is visible (the `giant component’). Sarnak conjectured that its genus is maximal among degree N polynomials . Jung and I showed that our result holds for random “equivariant” 3D spherical harmonics and showed that in a certain range the genus has maximal order of growth $N^3$. The purpose of my talk is to explain these phenomena of 3D nodal sets, which have no analogue for the much more studied 2D case
The APDE seminar on Monday, 9/20, will be given by Kenji Nakanishi (Kyoto University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh ()
Title: Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equation
Abstract: This is joint work with Kenjiro Ishizuka (Kyoto). We study global behavior of solutions for the nonlinear Klein-Gordon equation with a damping and a focusing nonlinearity on the Euclidean space. Recently, Cote, Martel and Yuan proved the soliton resolution conjecture for this equation completely in the one-dimensional case: every global solution in the energy space is asymptotic to superposition of solitons. Since the solitons are unstable, a natural question is which initial data evolve into each of the asymptotic forms. We consider the simplest setting in general space dimensions: the global behavior of solutions starting near a superposition of two ground states.
The main result is a complete classification of those solutions into 5 types of global behavior. Two of them are asymptotic to the positive ground state and the negative one respectively. They form two manifolds of codimension-1 that are joined at their boundary, which is the manifold of solutions asymptotic to superposition of two solitons. The connected union of those three manifolds separates the other solutions into the open set of global decaying solutions and that of blow-up. The manifold of 2-solitons was constructed by Cote, Martel, Yuan and Zhao. To get the classification, the main difficulty is in controlling the direction of instability attached to the two soliton components, because the soliton interactions are not integrable in time, breaking the simple linearized approximation. It is resolved by showing that the non-integrable interactions do not essentially affect the direction of instability, using the reflection symmetry of the equation and the 2-solitons.
I will also talk about a much harder difficulty in the 3-soliton case, which may be called soliton merger.
The APDE seminar on Monday, 9/13, will be given by Viorel Barbu (AI. I. Cuza) online via Zoom from 9.10am to 10.00am PST (note the time change). To participate, email Sung-Jin Oh ()
Title: Nonlinear Fokker-Planck equations and trend to equilibrium in statistical mechanics
Abstract: We survey a few recent results on the H-theorem for nonlinear Fokker-Planck equations and the existence of compact attractors in $L^1$ for the solutions.
The APDE seminar on Monday, 5/10, will be given by Jeffrey Galkowski online via Zoom from 11.10am to 12.00pm PT (note the time change). To participate, email Georgios Moschidis () or Federico Pasqualotto ().
Title: Exponential accuracy for the method of perfectly matched layers.
Abstract: The method of perfectly matched layers (PML) is used to compute solutions to time harmonic wave scattering problems. This method can be seen as a numerical adaptation of the method of complex scaling in which the infinite exterior region is replaced by a Dirichlet condition on a finite region. In this talk, we recall the methods of complex scaling and PML and study the error produced by replacing the genuine scattering problem with the PML truncation. We show that this error decays exponentially as a function of the scaling angle, the scaling width, and the frequency. Based on joint work with E. Spence and D. Lafontaine.
The APDE seminar on Monday, 5/3, will be given by Elena Giorgi online via Zoom from 4.10pm to 5.00pm PT. To participate, email Georgios Moschidis () or Federico Pasqualotto ().
Title: The stability of charged black holes.
Abstract: Black holes solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.
The APDE seminar on Monday, 4/26, will be given by Rita Teixeira da Costa online via Zoom from 12.10pm to 1.00pm PT (note the time change). To participate, email Georgios Moschidis () or Federico Pasqualotto ().
Title: Oscillations in wave map systems and an application to General Relativity
Abstract: Due to their nonlinear nature, the Einstein equations are not closed under weak convergence. Compactness singulaties associated to highly oscillatory solutions may be identified with some non-trivial matter. In 1989, Burnett conjectured that, for vacuum sequences, this matter produced in the limit is captured by the Einstein-massless Vlasov model. In this talk, we give a proof of Burnett’s conjecture under some gauge and symmetry assumptions, improving previous work by Huneau—Luk from 2019. Our methods are more general, and apply to oscillating sequences of solutions to the wave maps equation in (1+2)-dimensions. This is joint work with André Guerra (University of Oxford).
The APDE seminar on Monday, 4/19, will be given by Namaluba Malawo online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().
Title: Resonances for thin barriers on the half-line.
Abstract: The analysis of scattering by thin barriers is important for many physical problems, including quantum corrals. To model such a barrier, we use a delta function potential on the half-line. Our main results compute decay rates for particles confined by this barrier. The decay rates are given by imaginary parts of resonances. We show that they energy dependence of the decay rates is logarithmic when the barrier is weaker and polynomial when the barrier is stronger. To compute them, we derive a formula for resonances in terms of the Lambert W function and apply a series expansion. Joint work with Kiril Datchev.
The APDE seminar on Monday, 4/12, will be given by Jeffrey Kuan online via Zoom from 4:10 to 5:00pm. To participate, email Georgios Moschidis () or Federico Pasqualotto ().
Title: A stochastic fluid-structure interaction model given by a stochastic viscous wave equation
Abstract: We consider a stochastic fluid-structure interaction (FSI) model, given by a stochastic viscous wave equation perturbed by spacetime white noise. The wave equation part of the model describes the elastodynamics of a thin structure, such as an elastic membrane, while the viscous part, which is in the form of the Dirichlet-to-Neumann operator, describes the impact of a viscous, incompressible fluid in a two-way coupled fluid-structure interaction problem. The stochastic perturbation describes random deviations observed in real-life data. We prove that this stochastic viscous wave equation has a mild solution in dimension one, and also in dimension two, which is the physical dimension of the FSI problem (thin 2D membrane). This behavior contrasts that of the stochastic heat and the stochastic wave equations, which do not have function valued mild solutions in dimensions two and higher. This means that in the two dimensional model, unlike the heat and wave equations, dissipation due to fluid viscosity in the viscous wave equation, keeps the stochastically perturbed solution “in control”. We also consider Hölder continuity path properties of solutions and show that the solution is Hölder continuous up to Hölder exponent 1/2 in both space and time, after stochastic modification. This is joint work with Suncica Canic.