The APDE seminar on Monday, 11/06, will be given by Mihaela Ifrim (UW Madison) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: The small data global well-posedness conjecture for 1D defocusing dispersive flows

Abstract: I will present a very recent conjecture which broadly asserts that small data should yield global solutions for 1D defocusing dispersive flows with cubic nonlinearities, in both semilinear and quasilinear settings. This conjecture was recently proved in several settings in joint work with Daniel Tataru.

The APDE seminar on Monday, 10/23, will be given by Kevin Ren (Princeton) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Pinned Distances in R^d

Abstract: Given a set E in R^d with Hausdorff dimension > d/2, Falconer conjectured that the set of distances between any two points in E has positive Lebesgue measure. This conjecture remains open in all dimensions, despite significant progress in the last 30 years. Building upon this progress, we show that if d >= 3 and dim_H (E) > d/2 + 1/4 – 1/(8d+4), then the distance set of E has positive Lebesgue measure. The proof uses a new radial projection theorem in R^d applied to a variant of a decoupling framework of Guth-Iosevich-Ou-Wang. Joint work with Xiumin Du, Yumeng Ou, and Ruixiang Zhang.

The APDE seminar on Monday, 10/30, will be given by Yuzhou (Joey) Zou (Northwestern) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Weighted X-ray mapping properties on the Euclidean and Hyperbolic Disks

Abstract: We discuss recent works studying the sharp mapping properties of weighted X-ray transforms and weighted normal operators. These include a C^\infty isomorphism result for certain weighted normal operators on the Euclidean disk, whose proof involves studying the spectrum of a distinguished Keldysh-type degenerate elliptic differential operator. We also discuss further work studying additional self-adjoint realizations of this operator, using the machinery of boundary triplets. In addition, we discuss ongoing work which applies these results to the X-ray transform on the hyperbolic disk by using a projective equivalence between the Euclidean and hyperbolic disks. Joint works with N. Eptaminitakis, R. K. Mishra, and F. Monard.

The APDE seminar on Monday, 9/25, will be given by Benjamin Pineau (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 Federico Pasqualotto () or Mengxuan Yang ().

Title: Sharp Hadamard well-posedness for the incompressible free boundary Euler equations

Abstract: I will talk about a recent preprint in which we establish an optimal local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations on a connected fluid domain. Some components of this result include: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: A uniqueness result which holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove essentially scale invariant energy estimates for solutions, relying on a newly constructed family of refined elliptic estimates; (v) Continuation criterion: We give the first proof of a continuation criterion at the same scale as the classical Beale-Kato-Majda criterion for the Euler equation on the whole space. Roughly speaking, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of the construction of regular solutions.

Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is based on joint work with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.

The APDE seminar on Monday, 9/11, will be given by Zhongkai Tao (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 Federico Pasqualotto () or Mengxuan Yang ().

Title: Solution operators for divergence type equations and relativistic initial data gluing

Abstract: Given two solutions of the Einstein vacuum equation, can you glue them together along a spacelike hypersurface? Since the pioneering work of Corvino and Corvino–Schoen, we know it is possible to glue two initial data on an annulus in the asymptotically flat regime, modulo a 10-parameter obstruction, given by the energy, momentum, center of mass and angular momentum. Recently, Czimek–Rodnianski showed that the 10-parameter obstruction can be removed: instead they only need certain positivity assumptions on the energy-momentum tensor! Their proof of the obstruction-free gluing involves the null-gluing technique developed recently by Aretakis–Czimek–Rodnianski. We develop a new, simple, spacelike method to obtain the above gluing results, which also optimizes the positivity, regularity and decay assumptions. It is based on solution operators for divergence type equations with nice support properties. I will explain the construction of such solution operators, and how the underlying positivity in the nonlinear part of scalar curvature enters the story. This talk is based on joint work with Yuchen Mao and Sung-Jin Oh.

The APDE seminar on Monday, 8/28, will be given by Ruixiang Zhang (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 Federico Pasqualotto () or Mengxuan Yang ().

Title: A new conjecture to unify Fourier restriction and Bochner-Riesz

Abstract: The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\”{o}rmander asked if a more general class of operators (known as H\”{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\”{o}rmander’s question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\”{o}rmander’s question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a “positive curvature” assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it “Bourgain’s condition”. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\”{o}rmander type operator satisfying Bourgain’s condition should have the same $L^p$-boundedness as in those two conjectures. As evidences, we prove that the failure of Bourgain’s condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about history and results, leaving comments on proof techniques mainly to my HADES talk.

The APDE seminar on Monday, 9/18, will be given by Junehyuk Jung (Brown) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Nodal domains of equivariant eigenfunctions on Kaluza-Klein 3-folds.

Abstract: In this talk, I’m going to present my work with Steve Zelditch, where we prove that, when M is a principle $S^1$-bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically 2, independent of the eigenvalues. Note that principle S1-bundle equipped with a Kaluza-Klein metric never admits ergodic geodesic flow. This, for instance, contrasts the case when (M,g) is a surface with non-empty boundary with ergodic geodesic flow (billiard flow), in which case the number of nodal domains of typical eigenfunctions tends to +∞. I will also present an orthonormal eigenbasis of Laplacian on a flat 3-torus, where every non-constant eigenfunction has exactly two nodal domains. In particular, this tells us that the number of nodal domain could be uniformly bounded independent of the eigenvalue.

The APDE seminar on Monday, 4/24, will be given by Perry Kleinhenz (MSU) in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Energy decay for the damped wave equation

Abstract: The damped wave equation describes the motion of a vibrating system exposed to a damping force. For the standard damped wave equation, exponential energy decay is equivalent to the Geometric Control Condition (GCC). The GCC requires every geodesic to meet the positive set of the damping coefficient in finite time. A natural generalization is to allow the damping coefficient to depend on time, as well as position. I will give an overview of the classical results and discuss how a time dependent generalization of the GCC implies exponential energy decay. I will also mention some results for unbounded damping when the GCC is not satisfied.

The APDE seminar on Monday, 4/17, will be given by Kihyun Kim in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Rigidity of long-term dynamics for the self-dual Chern-Simons-Schrödinger equation within equivariance

Abstract: We consider the long time dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. Being a gauged 2D cubic nonlinear Schrödinger equation (NLS), (CSS) is L2-critical and has pseudoconformal invariance and solitons. However, there are two distinguished features of (CSS), the self-duality and non-locality, which make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one(!) modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem.

The APDE seminar on Monday, 4/10, will be given by Jian Wang (UNC) in-person in Evans 732, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Federico Pasqualotto () or Mengxuan Yang ().

Title: Damping for Fractional Wave Equations

Abstract: Motivated by highly successful numerical methods for damping the surface water wave equations proposed by Clamond et al. (2005), we study the following leading order linear model for damped gravity water waves \[ \partial_t^2 U + |D| U + \chi \partial_tU = 0 \] We show that the energy of the solution has polynomial decay by proving a resolvent estimate. Joint work with Thomas Alazard and Jeremy L. Marzuola.