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Changxing Miao

Publications and source records attributed to Changxing Miao.

At least 19 recordsLinked to original sources
arXiv2026

Bochner-Riesz means for critical magnetic Schrödinger operators in ${\mathbb R^2}$

We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schrödinger operators $\LL_{\A}$ in ${\mathbb R^2}$, which involve the {physical} Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\not= 2$, the Bochner-Riesz operator $S_λ^δ(\LL_{\A})$ of order $δ$ is bounded on $L^p(\R^2)$ if and only if $δ>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient {in} the proof is to obtain the localized $L^4(\R^2)$ estimate of $S_λ^δ(\LL_{\A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_λ^δ(Δ)$ for the Laplacian $Δ$ in ${\mathbb R}^2$.

math.AP↗
arXiv2026

Rough log-log blowup solutions to mass-critical NLS in higher dimensions $d\geq 3$

We study the stability of the log--log blow-up regime for the focusing mass-critical nonlinear Schrödinger equation under small $H^s$ perturbations. Previously, stability was established for every $0 1/(1+\min\{1,4/d\})$ by Sun and the fourth author [J. Math. Pures Appl. (2020)]. We remove this restriction and establish stability throughout the full subcritical range $0<s<1$.

math.AP↗
arXiv2026

Global well-posedness and scattering for mass-critical Hartree equation

In this paper, we prove that the defocusing mass-critical Hartree equation is global well-posed and scatters for any initial data $u_0\in L^2$, and prove the analogous result in the focusing case, provided the mass of $u_0$ is strictly less than that of the ground state $Q$. This result removes the radial restriction from [C. Miao, G. Xu and L. Zhao, J. Math. Pures Appl. (9), 91(2009), 49-79.], thereby proving the scattering conjecture for the Hartree equation at the scaling-critical regularity.

math.AP↗
arXiv2026

Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators

We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-Δ)$ of two-dimensional Schrödinger operators $H=-Δ+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón--Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind---an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

math.AP↗
arXiv2026

Refined $L^p$ restriction estimate for eigenfunctions on Riemannian surfaces

We refine the $L^p$ restriction estimates for Laplace eigenfunctions on a Riemannian surface, originally established by Burq, Gérard, and Tzvetkov. First, we establish estimates for the restriction of eigenfunctions to arbitrary Borel sets on the surface, following the formulation of Eswarathasan and Pramanik. We achieve this by proving a variable coefficient version of a weighted Fourier extension estimate of Du and Zhang. Our results naturally unify the $L^p(M)$ estimates of Sogge and the $L^p(γ)$ restriction bounds of Burq, Gérard, and Tzvetkov, and are sharp for all $p \geq 2$, up to a $λ^\varepsilon$ loss. Second, we derive sharp estimates for the restriction of eigenfunctions to tubular neighborhoods of a curve with nonvanishing geodesic curvature. These estimates are closely related to a variable-coefficient version of the Mizohata--Takeuchi conjecture, providing new insights into eigenfunction concentration phenomena.

math.AP↗
arXiv2026

Sharp non-uniqueness of weak solutions to 2D magnetohydrodynamic equations

In this paper, we prove that weak solutions to the 2D viscous and resistive magnetohydrodynamic (MHD) equations are non-unique in $L^2_t L^p(\mathbb{R}^2) \cap L^1_t W^{1,p}(\mathbb{R}^2)$ for given any $1\le p<\infty$, showing the sharpness of the Ladyzhenskaya--Prodi--Serrin condition at the endpoint $(2,\infty)$ and the solutions live on the borderline of the Beale--Kato--Majda criterion. To the best of our knowledge, this is the first non-uniqueness result for the 2D viscous and resistive MHD system. As byproducts, we also obtain non-uniqueness for the Navier--Stokes equations in $L^2_t L^p$ with $1\le p<\infty$, and for the MHD system with large $\mathrm{BMO}^{-1}$ initial data.

math.AP↗
arXiv2026

Geometric uncertainty principles for Schrödinger evolutions on negatively curved manifolds

In this paper, we study the uncertainty principle for Schrödinger equations with a bounded time-independent potentials on certain Cartan-Hadamard manifolds endowed with an asymptotic hyperbolic metric in dimensions $n\geq2$. The classical Hardy uncertainty principle in Euclidean space, as developed in the works of Escauriaza-Kenig-Ponce-Vega (JEMS, 2008; Duke Math. J., 2010), reveals a rigidity phenomenon for solution $u$ to Schrödinger equations: sufficiently strong Gaussian decay at two distinct times yields $u\equiv0$. In this work, we show that a similar rigidity persists in the setting of hyperbolic geometry, despite the absence of translation invariance and Fourier representation. Our approach follows a general strategy of Escauriaza-Kenig-Ponce-Vega, where the underlying geometry brings an essential change. This enables us to establish new Carleman estimates and logarithmic convexity. Unlike the Euclidean setting, the hyperbolic geometry exhibits exponential volume growth and nontrivial geodesic escape at infinity, which fundamentally alters the propagation mechanism of Schrödinger evolutions. Based on the newly-built virial identities and an approximation argument, we derive the logarithmic convexity. The main difficulty in proving the logarithmic convexity is the lack of convolution structure on general manifolds. By making use of the exponential map and Jacobi field, we define a new mollifier on curved geometry. Meanwhile, to establish the Carleman estimate adapted to hyperbolic space, we introduce a new weight function adapted to the curved manifold. Our results highlight the role of curvature in shaping quantitative uniqueness properties for dispersive equations.

math.AP↗
arXiv2026

A multidimensional Szemerédi theorem in integers

For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N]^n$ of density at least $(\log N)^{-c}$ contains a nontrivial configuration of the form \begin{equation*} \boldsymbol{x},\boldsymbol{x}+r^{m_{1}}\boldsymbol{e_{1}},\ldots,\boldsymbol{x}+r^{m_{n}}\boldsymbol{e_{n}}, \end{equation*} where $c=c(n,m_{1},\ldots,m_{n} )$ is a positive constant. This quantitative multidimensional Szemerédi theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),\left(x,y+r^{2}\right)$. The theorem is obtained as a consequence of an effective ``popular'' version.

math.NT↗
arXiv2026

Sharp restriction estimates for some degenerate higher codimensional quadratic surfaces

The Fourier restriction conjecture is a fundamental problem in harmonic analysis. In this paper, we investigate restriction estimates for degenerate higher codimensional quadratic surfaces and obtain sharp results for some types of degenerate cases. A major obstacle in establishing sharp restriction estimates is the failure of rescaling invariance, which is crucial for induction on scale to be effective. Motivated by the work of Guo and Oh (2022), we introduce a method, building on an iterative variant of the broad-narrow analysis, that does not heavily rely on induction on scale. To obtain suitable transversality conditions for this analysis and to derive desirable bounds for the broad part, we define a generalized notion of Jacobian, and establish its structural properties. These properties are proved using tools and techniques from both algebra and graph theory.

math.CA↗
arXiv2026

Nonuniqueness for high-dimensional ideal MHD equations via differential inclusion

In this paper, we establish the non-uniqueness of solutions to the ideal magnetohydrodynamics equations in any dimension greater than three by proving the existence of infinitely many compactly supported weak solutions. In particular, these solutions fail to conserve the total energy. Our proof relies on the differential inclusion framework tailored to the geometry of ideal MHD system, which enables the simultaneous use of Baire category method and convex integration scheme.

math.AP↗
arXiv2026

Non-uniqueness of smooth solutions to the Navier-Stokes equations on torus $\TTT^2$

The local well-posedness theory for the incompressible Navier-Stokes equations in $\BMO^{-1}$ has attracted considerable attention over the past two decades. In a recent breakthrough, Coiculescu and Palasek (Invent. Math., 2025) settled the three-dimensional case by demonstrating the existence of two distinct global solutions, both smooth for $t>0$, evolving from a common initial datum in ${\rm BMO}^{-1}(\mathbb{T}^3)$. However, the two-dimensional case remains open. In this paper, we solve the two-dimensional problem. Unlike its three-dimensional counterpart, the two-dimensional setting presents additional difficulties stemming from the geometric intersections of two-dimensional Mikado flows. To overcome these difficulties, we develop a heat-dominated Fourier mode flow built upon steady two-dimensional Euler flows, and present the proof using a new iterative scheme.

math.AP↗
arXiv2025

Improvement of Pólya's conjecture for balls and cylinders

Pólya's conjecture on the eigenvalues of the Laplacian has been one of the core problems in spectral geometry. Building upon the recent breakthrough works on Pólya's conjecture for balls and annuli by Filonov, Levitin, Polterovich and Sher, we study several aspects of Pólya's conjecture for balls and cylinders: by refining the purely analytical portion of the proof in [2] for the Neumann Pólya's conjecture for the disk, we extend the regime of the spectral parameter that can be established without computer assistance; we obtain improvement of Pólya's conjecture for disks and balls; we obtain improvement of Pólya's conjecture for cylinders and confirm the Neumann Pólya's conjecture for cylinders in $\mathbb{R}^3$. As a supplementary effort, we study Weyl's law for cylinders.

math.CA↗
arXiv2025

The optimal transition threshold for the 2D Couette flow in the infinite channel

We investigate the stability of the 2-D Navier-Stokes equations in the infinite channel $\mathbb{R}\times [-1,1]$ with the Navier-slip boundary condition. We show that if the initial perturbations $ω^{in}$ around the Couette flow satisfy $\|ω^{in}\|_{H^3_{x,y}\cap L^1_x H^3_y}\leq cν^{\frac13}$, the solution admits enhanced dissipation at $x$-frequencies $|k|\gg ν$ and inviscid damping effect. The key contributions lie in two parts: (1) we adopt the new decomposition of the vorticity $ω=ω_{L}+ω_e$, where $ω_L$ effectively captures a ``weak" enhanced dissipation $(1+ν^{\frac13} t)^{-\frac14}e^{-νt}$ and the corresponding velocity exhibits the inviscid damping effect; (2) we introduce the dyadic decomposition for the long time scale $t\geq ν^{-\frac16}$ and apply the ``infinite superposition principle" to the equation for $ω_e$ in order to control the growth induced by echo cascades, which appears to be novel and may hold independent significance.

math.AP↗
arXiv2025

Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition

In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover the key division point at $10ν$ in the frequency interval $(0,1)$ by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval $(0,1)$, and then we establish the space-time estimates on the low-frequency $0\leq |k|\leq 10 ν$ and the intermediate-frequency $ 10 ν\leq |k|<1$, respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be $γ\leq\frac{1}{2}$.Meanwhile, we also show that when the frequencies $|k|\geq ν^{1-}$, the enhanced dissipation effect occurs for the linearized Navier-Stokes equations.

math.AP↗
arXiv2025

Dynamics of subcritical threshold solutions for the 4d energy-critical NLS

We study dynamics of the 4$d$ energy-critical nonlinear Schrödinger equation at the ground state energy. Previously, Duyckaerts and Merle [Geom. Funct. Anal. (2009)] proved that any radial solution with kinetic energy less than that of the ground state either scatters in both time directions or coincides (modulo symmetries) with a heteroclinic orbit, which scatters in one time direction and converges to the ground state in the other. We extend this result to the non-radial setting.

math.AP↗
arXiv2025

Weighted $L^2$ restriction and comparison of nondegeneracy conditions for quadratic manifolds of arbitrary codimensions

We systematically study weighted $L^2$ restriction for quadratic manifolds of arbitrary codimensions by sharp uniform Fourier decay estimates and a refinement of the Du-Zhang method. Comparison with prior results is also discussed. In addition,we obtain an almost complete relation diagram for all existing nondegeneracy conditions for quadratic manifolds of arbitrary codimensions. These conditions come from various topics in harmonic analysis related to "curvature": Fourier restriction, decoupling, Fourier decay, Fourier dimension, weighted restriction, and Radon-like transforms. The diagram has many implications, such as "best possible Stein-Tomas implies best possible $\ell^pL^p$ decoupling". The proof of the diagram requires a combination of ideas from Fourier analysis, complex analysis, convex geometry, geometric invariant theory, combinatorics, and matrix analysis.

math.CA↗
arXiv2025

Hörmander oscillatory integral operators: a revisit

In this paper, we present new proofs for both the sharp $L^p$ estimate and the decoupling theorem for the Hörmander oscillatory integral operator. The sharp $L^p$ estimate was previously obtained by Stein\;\cite{stein1} and Bourgain-Guth \cite{BG} via the $TT^\ast$ and multilinear methods, respectively. We provide a unified proof based on the bilinear method for both odd and even dimensions. The strategy is inspired by Barron's work \cite{Bar} on the restriction problem. The decoupling theorem for the Hörmander oscillatory integral operator can be obtained by the approach in \cite{BHS}, where the key observation can be roughly formulated as follows: in a physical space of sufficiently small scale, the variable setting can be essentially viewed as translation-invariant. In contrast, we reprove the decoupling theorem for the Hörmander oscillatory integral operator through the Pramanik-Seeger approximation approach \cite{PS}. Both proofs rely on a scale-dependent induction argument, which can be used to deal with perturbation terms in the phase function.

math.AP↗