We establish global well-posedness for both the defocusing and
focusing complex-valued modified Korteweg–de Vries equations on the real line
in modulation spaces $\small M^{s,2}_p(\mathbb{R})$, for all $\small 1 \leq p <\infty$ and $\small 0 \leq s < 3/2 − 1/p$. We will also show that such solutions admit global-in-time bounds in these spaces
and that equicontinuous sets of initial data lead to equicontinuous ensembles
of orbits. Indeed, such information forms a crucial part of our well-posedness
argument.
Bounds of restriction of characters to submanifolds
[]
[ArXiv]
An important approach to the study of concentration of Laplacian eigenfunctions on a compact manifold as the eigenvalue grows to infinity is to bound their restriction to submanifolds. In this paper we take this approach in the setting of a compact Lie group, and provide sharp restriction bounds of general Laplacian eigenfunctions as well as important special ones such as sums of matrix coefficients and in particular characters of irreducible representations of the group. We deal with two classes of submanifolds, namely, maximal flats and all of their submanifolds, and the conjugation-invariant submanifolds. We prove conjecturally sharp asymptotic $\small L^p$ bounds of restriction of general Laplacian eigenfunctions to maximal flats and all of their submanifolds for all $\small p\geq 2$. We also prove sharp asymptotic $\small L^p$ bounds of restriction of characters to maximal tori and all of their submanifolds for all $\small p>0$ and to the conjugation-invariant submanifolds for all $\small p\geq 2$, and of general sums of matrix coefficients to maximal flats and all of their submanifolds for all $\small p\geq 2$.
(with Hanlong Fang and Xiaocheng Li)
Harmonic analysis on the fourfold cover of the space of ordered triangles I: the invariant differentials
[] [ArXiv]
Denote by $\small SL_3(\mathbb R)$ the special linear group of degree 3 over the real numbers, $\small A$ the subgroup consisting of the diagonal matrices with positive entries. In this paper we study the algebraic and analytic properties of the invariant differential operators on the homogeneous space $\small SL_3(\mathbb R)/A$. Firstly, we specify the noncommutative algebra of invariant differential operators in terms of generators and their relations. Secondly, we describe the center of this algebra and prove that all of its symmetric elements are essentially self-adjoint. Thirdly, for the first time on homogeneous spaces, we identify several essentially self-adjoint invariant differential operators which do not lie in the center of the algebra of invariant differential operators.
Analysis on compact symmetric spaces: eigenfunctions and nonlinear Schrödinger equations
In: Methusalem Lectures, Trends in Mathematics vol. 3 (2024), 235–240, Birkhäuser, Cham.
[] [Book]
We discuss several open problems on harmonic analysis on compact globally symmetric spaces, and their applications towards nonlinear Schrödinger equations.
In this article, we obtain new results for Fourier restriction type problems on compact Lie groups. We first provide a sharp form of $\small L^p$ estimates of irreducible characters in terms of their Laplace–Beltrami eigenvalue and as a consequence provide some sharp $\small L^p$ estimates of joint eigenfunctions for the ring of bi-invariant differential operators. Then we improve upon the previous range of exponent for scale-invariant Strichartz estimates for the Schrödinger equation, and provide new $\small L^p$ bounds of Laplace–Beltrami eigenfunctions in terms of their eigenvalue similar to known bounds on tori. A key ingredient in our proof of these results is a barycentric-semiclassical subdivision of the Weyl alcove in a maximal torus. On each component of this subdivision we carry out the analysis of characters and exponential sums, and the circle method of Hardy–Littlewood and Kloosterman.
In this article, we establish scale-invariant Strichartz estimates for the Schrödinger equation on
arbitrary compact globally symmetric spaces and some bilinear Strichartz estimates on products of rank-one
spaces. As applications, we provide local well-posedness results for nonlinear Schrödinger equations on such
spaces in both subcritical and critical regularities.
Strichartz estimates for the Schrödinger
equation on products of odd-dimensional spheres Nonlinear Analysis 199 (2020), 112052, 21 pp.
[] [Journal] [ArXiv]
We prove Strichartz estimates for the Schrödinger equation which are scale-invariant up to an $\small \varepsilon$-loss on products of odd-dimensional spheres. Namely, for any product of odd-dimensional spheres $\small M=\mathbb{S}^{d_1}\times\cdots\times\mathbb{S}^{d_r}$ (so that $\small M$ is of dimension $\small d=d_1+\cdots+d_r$ and rank $\small r$) equipped with rational metrics, the following Strichartz estimate
$$\small
\|e^{it\Delta}f\|_{L^p(I\times M)}\leq C_\varepsilon\|f\|_{H^{\frac{d}{2}-\frac{d+2}{p}+\varepsilon}(M)}
$$
holds for any $\small p\geq 2+\frac{8(s-1)}{sr}$,
where
$$\small s=\max\left\{\frac{2d_i}{d_i-1}, i=1,\ldots,r\right\}.$$
Strichartz estimates for the Schrödinger
flow on compact Lie groups Analysis & PDE 13 (2020), no. 4, 1173–1219.
[] [Journal]
[ArXiv]
We establish scale-invariant Strichartz estimates for the Schrödinger flow on any
compact Lie group equipped with canonical metrics. The highlights of this paper include an estimate for some
Weyl type sums defined on rational lattices, the different decompositions of the Schrödinger kernel determined by how close the points inside the maximal torus are to the cell walls, and an
application of the BGG–Demazure operators or Harish-Chandra's integral formula to the estimate of the
difference between characters.
Strichartz estimates for the Schrödinger
flow on compact symmetric spaces Thesis (Ph.D.)–University of California, Los Angeles (2018), 103 pp.
[] [Full Text]
This thesis studies scaling critical Strichartz estimates for the Schrödinger flow on compact
symmetric spaces. A general scaling critical Strichartz estimate (with an $\small\varepsilon$-loss, respectively)
is given conditional on a conjectured dispersive estimate (with an $\small\varepsilon$-loss, respectively) on
general compact symmetric spaces. The dispersive estimate is then proved for the special
case of connected compact Lie groups. Slightly more generally, for products of connected
compact Lie groups and spheres of odd dimension, the dispersive estimate is proved with an
$\small\varepsilon$-loss.
Teaching
I am teaching MATH 1021, College Algebra in Fall 2024.
Contact
Address:
Department of Mathematical Sciences
University of Cincinnati
Cincinnati, OH 45221-0025
Office:
5305 French Hall
Phone:
(513) 556-4088
Email:
yunfengzhang108 at gmail dot com
or
zhang8y7 at ucmail dot uc dot edu
Last updated: 1 Aug. 2024
Study hard what interests you the most in the most undisciplined, irreverent and original manner possible. —RICHARD FEYNMANN
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