UC San Diego Analysis Seminar

In-person talks in AP & M 7321,

Remote talks on Zoom, contact organizers for Zoom details

Sebastian Herr

Bielefeld University

Global wellposedness of the Zakharov System below the ground state

The Zakharov system is a quadratically coupled system of a Schrödinger and a wave equation, which is related to the focussing cubic Schrödinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that it is globally well-posed in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schrödinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate. This is joint work with Timothy Candy and Kenji Nakanishi.

Dallas Albritton

Princeton

Kihyun Kim

IHES

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

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 solutions to (CSS) decomposes 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.

Jan Slovak

Masaryk U

Nearly invariant calculus for a few CR (and all parabolic) geometries

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Manuel Del Pino

University of Bath

Benoit Pausader

Brown University

Stability of a point charge for the repulsive Vlasov-Poisson system

We consider solutions of the repulsive Vlasov-Poisson system which are a combination of a point charge and a small density with respect to Liouville measure (a ``cloud''), and we show that these solutions exist globally, that the electric field decays at an optimal rate and that the particle distribution converges along a modified scattering dynamics. This follows by a Lagrangian study of the linearized equation, which is integrated by means of an asymptotic action-angle coordinate transformation, and an Eulerian study of the nonlinear dynamic which exhibits the ``mixing'' mechanism responsible for the asymptotic behavior. This is joint work with Klaus Widmayer (U. Zurich) and Jiaqi Yang (ICERM).

Nordine Mir

Texas A&M University at Qatar

Finite jet determination of CR maps into real-algebraic sets

We present recent results about finite jet determination of CR maps of positive codimension from real-analytic CR manifolds into real-algebraic subsets in complex space, or more generally Nash subsets. One instance of such results is the unique jet determination of germs of CR maps from minimal real-analytic CR submanifolds in $\C^N$ into Nash subsets in $\C^{N'}$ of D'Angelo finite type, for arbitrary $N,N'\geq 2$. This is joint work with B. Lamel.

Linfeng Lin

USC

On the local existence of solutions to the Navier-Stokes-wave system with a free interface

We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space $H^{2+\epsilon}$ and the initial structure velocity is in $H^{1.5+\epsilon}$, where $\epsilon\in(0,1/2)$.

Tadele Mengesha

University of Tennessee, Knoxville

Variational Analysis of some nonlocal functionals and associated function spaces

I will present a recent work on variational problems involving nonlocal energy functionals that appear in nonlocal mechanics. The well-posedness of variational problems is established via a careful study of the associated energy spaces, which are nonstandard. I will discuss some difficulties in proving fundamental structural properties of the function spaces such as compactness. For a sequence of parametrized nonlocal functionals in suitable form, we compute their variational limit and establish a rigorous connection with classical models.

Javier Cueto

University of Nebraska-Lincoln

A framework for variational problems based on nonlocal gradients on bounded domains inspired by Peridynamics

Inspired by the rise on the interest for nonlocal models, mainly Peridynamics, we decided to study a functional framework suitable for (variational) nonlocal models, such as that of nonlocal hyperelasticity. This has lead to a nonlocal framework based on truncated fractional gradients (i.e. nonlocal gradients with a fractional singularity defined over bounded domains), in which continuous and compact embeddings and, in particular, nonlocal Poincaré inqualities has been obtained thanks to a nonlocal version of the fundamental theorem of calculus. As a consequence, the existence of minimizers of nonlocal polyconvex vectorial functionals is obtained, and more recently quasiconvex functionals. Some of these last results have been obtained from a result that relates nonlocal gradients with classical ones and vice-versa. These results are accompanied by a study of the localization (recovering of the classical model) when s goes to 1 (actually, continuity on s with s being the fractional index of differentiability).

March 14

Zoom

Jacek Jendrej

Sorbonne Paris Nord

Bubble decomposition for the harmonic map heat flow in the equivariant case

I will present a recent joint work with Andrew Lawrie from MIT. We consider the harmonic map heat flow for maps from the plane R^2 to the sphere S^2, under the so-called equivariant symmetry. It is known that solutions to the initial value problem exhibit bubbling along a sequence of times - the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval motivated by our recent work on the soliton resolution problem for equivariant wave maps.

Shi-Zhuo Looi

UC Berkeley

Asymptotics of odd- and even-dimensional waves

In this talk, I will give a survey of recent and upcoming results on various linear, semilinear and quasilinear wave equations on a wide class of dynamical spacetimes in various even and odd spatial dimensions. These results include asymptotics for a wide range of nonlinearities. For many of these results, the spacetimes under consideration have only weak asymptotic flatness conditions and are allowed to be large perturbations of the Minkowski spacetime, provided that an integrated local energy decay estimate holds. We explain the dichotomy between even- and odd-dimensional wave behaviour. Part of this work is joint with Mihai Tohaneanu and Jared Wunsch.

Animesh Biswas

University of Nebraska-Lincoln

Extension equation for fractional power of operator defined on Banach spaces.

In this talk, we show the extension (in spirit of Caffarelli-Silvestre) of fractional power of operators defined on Banach spaces. Starting with the Balakrishnan definition, we use semigroup method to prove the extension. This is a joint work with Pablo Raul Stinga.

Hong Wang

UCLA

Shukun Wu

Caltech

An improvement on the three-dimensional restriction problem.

Stein's restriction conjecture is one of the central topics in Fourier analysis. It is closely related to other areas of math, for example, number theory, PDEs. In this talk, I will discuss a recent improvement of this conjecture in R^3, based on the joint work with Hong Wang. Our proof is built upon the framework of polynomial partitioning, and, among other things, it uses the refined decoupling theorem, a two-ends Kakeya estimate. The two-ends estimate captures some information from the method of induction on scales.

Koffi Enakoutsa

UCLA

The Morrey Conjecture: Examining an Unsolved Problem with Numerical Simulations

The Morrey Conjecture pertains to the properties of quasi-convexity and rank-one convexity in functions, where the former implies the latter, but the converse relationship is not yet established. While Sverak has proven the conjecture in three dimensions, it remains unresolved in the two-dimensional case. Analyzing these properties analytically is a formidable task, particu- larly for vector-valued functions. Consequently, to investigate the validity of the Morrey Conjecture, we conducted numerical simulations using a set of example functions by Dacorogna and Marcellini. Based on our results, the Morrey Conjecture appears to hold true for these functions.

Yuming Paul Zhang

Auburn U.

Regularity of Hele-Shaw flow with source and drift: Flat free boundaries are Lipschitz

The classical Hele-Shaw flow describes the motion of incompressible viscous fluid, which occupies part of the space between two parallel, nearby plates. With source and drift, the equation is used in models of tumor growth where cells evolve with contact inhibition, and congested population dynamics. We consider the flow with Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. This is joint work with Inwon Kim.

Jacob Bedrossian

UCLA

Chaos and turbulence in stochastic fluid mechanics: What is it and how could we study it?

In this survey-style talk I discuss the (old) idea of studying turbulence in stochastically-forced fluid equations. I will discuss definitions of chaos, anomalous dissipation, and various other predictions by physicists that can be phrased as mathematically precise conjectures in this context. Then, I will discuss some recent work by my collaborators and I on various aspects, namely (1) a straightforward characterization of anomalous dissipation that implies the classical Kolmogorov 4/5 law for 3d NSE (joint with Michele Coti Zelati, Sam Punshon-Smith, and Franziska Weber); (2) the study of "Lagrangian chaos" and exponential mixing of scalars and how it leads to a proof of anomalous dissipation and of the power spectrum predicted by Batchelor in 1959 for the simpler problem of Batchelor-regime passive scalar turbulence (joint with Alex Blumenthal and Sam Punshon-Smith); (3) the more recent proof of "Eulerian chaos" for Galerkin truncations of the Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith).