The APDE seminar on Monday, 3/16, will be given by Yuefeng Song (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Weak null singularity for the Einstein–Euler system

Abstract: We study the behavior of a self-gravitating perfect relativistic fluid satisfying the Einstein–Euler system in the presence of a weak null terminal spacetime singularity. This type of singularities is expected in the interior of generic dynamical black holes. In the vacuum case, weak null singularities have been constructed locally by Luk, where the metrics extend continuously to the singularities while the Christoffel symbols fail to be square integrable in any neighborhood of any point on the singular boundaries. We prove that this type of singularities persists in the presence of a self-gravitating fluid. Moreover, using the fact that the speed of sound is strictly less than the speed of light, we prove that the fluid variables also extend continuously to the singularity.

The APDE seminar on Monday, 3/9, will be given by Jesús Oliver (Cal State East Bay) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Global existence for a Fritz John equation in expanding FLRW spacetimes

Abstract: We study the semilinear wave equation

\[\square_{\mathbf g_p}\phi = (\partial_t \phi)^2\]

on expanding FLRW spacetimes with spatial slices $\mathbb{R}^3$ and power-law scale factor $a(t)=t^p$, where $0 < p \le 1$. This equation extends the classical Fritz John blow-up model on Minkowski space (the case $p=0$) to a non-stationary cosmological background.

In Minkowski space, nontrivial solutions arising from smooth, compactly supported data blow up in finite time. In contrast, we prove that for $0 < p \le 1$, sufficiently small, smooth, compactly supported initial data generate global-in-time solutions toward the future.

Earlier joint work with Costa and Franzen treated the accelerated regime $p>1$, where global existence follows from the integrability of the inverse scale factor. In the present setting, this mechanism is unavailable. Instead, we develop a vector field method adapted to FLRW geometry that exploits the interaction between dispersion and spacetime expansion to suppress the nonlinear blow-up mechanism. The argument relies on commuting the Laplace–Beltrami operator with a boosts-free subset of the Poincaré algebra and establishing Klainerman–Sideris type estimates adapted to the non-stationary background.

The approach provides a robust framework for quantifying the regularizing effect of cosmological expansion and is expected to extend to a broader class of nonlinear wave equations.

The APDE seminar on Monday, 2/23, will be given by Richard Bamler (Berkeley) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

Abstract: We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons.

Our approach relies on two new techniques. The first, called the PDE-ODI principle, converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This framework bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. The second combines a new leading mode condition combined with an “induction over thresholds” argument” to obtain even finer asymptotic estimates.

This is joint work with Yi Lai.

The APDE seminar on Monday, 2/9, will be given by Federico Franceschini (Stanford) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: The dimension and behaviour of singularities of stable solutions to semilinear elliptic equations

Abstract: Let f(t) be a convex, positive, increasing nonlinearity. It is known that stable solutions of -\Delta u =f(u) can be singular (i.e., unbounded) if the dimension n>9.

Brezis asked wether, if x=0 is such a singular point, then in general f'(u(x)) blows-up like ~|x|^{2-n}, as it happens in the model cases f(u)=u^p, f(u)=e^u.

In this talk I will show the answer to this question and the interesting consequences it entails. This is a joint work with Alessio Figalli.

The APDE seminar on Monday, 2/2, will be given by Patrick Flynn (UCLA) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Linear decay of the beta-plane equation near Couette flow on the plane

Abstract: We prove time decay for the linearized beta-plane equation near shear flow on the plane. Specifically, we show that the profiles of the velocity field components decay polynomially on any compact set, and identify specific rates of decay. Our proof entails the analysis of oscillatory integrals with homogeneous phase and multipliers that diverge in the infinite time limit. To handle this singular limit, we prove a Van der Corput type estimate, followed by a multi-scale asymptotic analysis of the phase and multipliers. This is joint work with Jacob Bedrossian and Sameer Iyer.

The APDE seminar on Monday, 1/26, will be given by our own Sung-Jin Oh (UC Berkeley) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: A physical-space approach to global asymptotics for variable-coefficient Schrödinger equations

Abstract: In this talk, I will discuss a new physical-space approach to establish the time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in (3+1)-dimensions. A key innovation in our methodology is the concept of a “good commutator,” which extends the classical commuting vector field method, and which combines well with Ifrim-Tataru’s testing by wave packets. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles. This talk is based on an upcoming work with F. Pasqualotto (UCSD) and N. Tang (UC Berkeley).

The APDE seminar on Monday, 12/1, will be given by Mingfeng Chen (UW Madison) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Rational points near space curves

Abstract: How many rational points with a bounded denominator lie in a small neighborhood of a given manifold? This fundamental question in Diophantine approximation connects to dynamics, number theory, and harmonic analysis, with applications to problems like Khinchin’s theorem for manifolds and the dimension growth conjecture.
In this talk, I will present new results that establish the main conjecture for the case of space curves. This is joint work with A. Seeger, R. Srivastava and N. Technau.

The APDE seminar on Monday, 11/24, will be given by Shukun Wu  (IU Bloomington) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Weighted L^2 estimates and applications to L^p problems.

Abstract: We will discuss some weighted L^2 estimates in the plane and their applications to a couple of L^p problems. These include the almost everywhere convergence of the planar Bochner-Riesz means, decay of circular L^p-means of Fourier transform of fractal measures, estimates for the maximal Schrödinger operator and the maximal extension operator, and an L^p analogue of the Mizohata–Takeuchi conjecture.

The APDE seminar on Monday, 11/17, will be given by Serban Cicortas (Princeton) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Critical collapse in 2+1 gravity

Abstract: Starting with the work of Choptuik ’92, numerical relativity predicts that naked singularity spacetimes arise on the threshold of dispersion and black hole formation, a phenomenon referred to as critical collapse. In this talk, I will present for 2+1 gravity the first rigorous construction of threshold naked singularities in general relativity. Joint work with Igor Rodnianski (Princeton University).

The APDE seminar on Monday, 11/10, will be given by Katya Krupchyk (UC Irvine) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PDT. To participate, please email Adam Black (adamblack@berkeley.edu).

Title: Fractional Anisotropic Calderón Problem

Abstract: The classical anisotropic Calderón problem, in its geometric formulation, asks whether a Riemannian metric, or more generally a compact Riemannian manifold with boundary, can be recovered from the Dirichlet-to-Neumann map for the Laplace–Beltrami operator, given on the boundary of the manifold. The problem remains open in general for smooth metrics in dimensions three and higher.
In this talk, we will present uniqueness results for the fractional anisotropic Calderón problem, a nonlocal analogue of the classical anisotropic Calderón problem, in dimensions two and higher, in two settings: on smooth closed Riemannian manifolds with source-to-solution data, and on domains in Euclidean space with external measurements. Specifically, we will show that the source-to-solution map for the fractional Laplace–Beltrami operator, known on an arbitrary open subset of a smooth closed Riemannian manifold, determines the manifold up to isometry. In the Euclidean case, for smooth Riemannian metrics that coincide with the Euclidean metric outside a compact set, we will demonstrate that the partial exterior Dirichlet-to-Neumann map for the fractional Laplace–Beltrami operator, known on arbitrary open subsets in the exterior of the domain, determines the Riemannian metric up to diffeomorphism fixing the exterior. The talk is based on joint works with Ali Feizmohammadi, Tuhin Ghosh, Angkana Rüland, Johannes Sjöstrand, and Gunther Uhlmann.