arXiv subjects

Changxing Miao

Publications and source records attributed to Changxing Miao.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2025

On Onsager-type conjecture for the Elsässer energies of the ideal MHD equations

In this paper, we investigate the ideal magnetohydrodynamics (MHD) equations on tours $\TTT^d$. For $d=3$, we resolve the flexible part of Onsager-type conjecture for Elsässer energies of the ideal MHD equations. More precisely, for $β< 1/3$, we construct weak solutions $(u, b) \in C^β([0,T] \times \mathbb{T}^3)$ with both the total energy dissipation and failure of cross helicity conservation. The key idea of the proof relies on a symmetry reduction that embeds the ideal MHD system into a 2$\frac{1}{2}$D Euler flow and the Newton-Nash iteration technique recently developed in \cite{GR}. For $d=2$, we show the non-uniqueness of Hölder-continuous weak solutions with non-trivial magnetic fields. Specifically, for $β< 1/5$, there exist infinitely many solutions $(u, b) \in C^β([0,T] \times \mathbb{T}^2)$ with the same initial data while satisfying the total energy dissipation with non-vanishing velocity and magnetic fields. The new ingredient is developing a spatial-separation-driven iterative scheme that incorporates the magnetic field as a controlled perturbation within the convex integration framework for the velocity field, thereby providing sufficient oscillatory freedom for Nash-type perturbations in the 2D setting. As a byproduct, we prove that any Hölder-continuous Euler solution can be approximated by a sequence of $C^β$-weak solutions for the ideal MHD equations in the $L^p$-topology for $1\le p<\infty$.

math.AP↗

arXiv2025

Step-Video-T2V Technical Report: The Practice, Challenges, and Future of Video Foundation Model

We present Step-Video-T2V, a state-of-the-art text-to-video pre-trained model with 30B parameters and the ability to generate videos up to 204 frames in length. A deep compression Variational Autoencoder, Video-VAE, is designed for video generation tasks, achieving 16x16 spatial and 8x temporal compression ratios, while maintaining exceptional video reconstruction quality. User prompts are encoded using two bilingual text encoders to handle both English and Chinese. A DiT with 3D full attention is trained using Flow Matching and is employed to denoise input noise into latent frames. A video-based DPO approach, Video-DPO, is applied to reduce artifacts and improve the visual quality of the generated videos. We also detail our training strategies and share key observations and insights. Step-Video-T2V's performance is evaluated on a novel video generation benchmark, Step-Video-T2V-Eval, demonstrating its state-of-the-art text-to-video quality when compared with both open-source and commercial engines. Additionally, we discuss the limitations of current diffusion-based model paradigm and outline future directions for video foundation models. We make both Step-Video-T2V and Step-Video-T2V-Eval available at https://github.com/stepfun-ai/Step-Video-T2V. The online version can be accessed from https://yuewen.cn/videos as well. Our goal is to accelerate the innovation of video foundation models and empower video content creators.

cs.CV↗

arXiv2025

Nonuniqueness analysis on the Navier-Stokes equation in $C_{t}L^{q}$ space

In the presence of any prescribed kinetic energy, we implement the intermittent convex integration scheme with $L^{q}$-normalized intermittent jets to give a direct proof for the existence of solution to the Navier-Stokes equation in $C_{t}L^{q}$ for some uniform $2<q\ll3$ without the help of interpolation inequality. The result shows the sharp nonuniqueness that there evolve infinite nontrivial weak solutions of the Navier-Stokes equation starting from zero initial data. Furthermore, we improve the regularity of solution to be of $C_{t}W^{α,q}$ in virtue of the fractional Gagliardo-Nirenberg inequalities with some $0<α\ll1$. More importantly, the proof framework provides a stepping stone for future progress on the method of intermittent convex integration due to the fact that $L^{q}$-normalized building blocks carry the threshold effect of the exponent $q$ arbitrarily close to the critical value $3$.

math.AP↗

arXiv2024

Sharp non-uniqueness for the Navier-Stokes equations in R^3

In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the incompressible Navier-Stokes equations in $\R^3$. To be more precise, we exhibit the non-uniqueness result in a strong sense, that is, any weak solution is non-unique in L^p([0,T];L^\infty(\R^3)) with 1\le p<2. Moreover, this non-uniqueness result is sharp with regard to the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint (2, \infty), which extends the sharp nonuniqueness for the Navier-Stokes equations on torus $\TTT^3$ in the recent groundbreaking work (Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987-1054) to the setting of the whole space. The key ingredient is developing a new iterative scheme that balances the compact support of the Reynolds stress error with the non-compact support of the solution via introducing incompressible perturbation fluid.

math.AP↗

arXiv2024

Non-uniqueness of weak solutions to the Navier-Stokes equations in R^3

To our knowledge, the convex integration method has been widely applied to the study of non-uniqueness of solutions to the Naiver-Stokes equations in the periodic region, but there are few works on applying this method to the corresponding problems in the whole space or other regions. In this paper, we prove that weak solutions of the Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy in the whole space, which extends the non uniqueness result for the Navier-Stokes equations on torus T3in the groundbreaking work (Buckmaster and Vicol, Ann. of Math., 189 (2019), pp.101-144) to R3. The critical ingredients of the proof include developing an iterative scheme in which the approximation solution is refined by decomposing it into local and non-local parts. For the non-local part, we introduce the localized corrector which plays a crucial role in balancing the compact support of the Reynolds stress error with the non-compact support of the solution. As applications of this argument, we first prove that there exist infinitely many weak solutions that dissipate the kinetic energy in smooth bounded domain. Moreover, we show the instability of the Navier-Stokes equations near Couette flow in L2(R3).

math.AP↗

arXiv2024

Global well-posedness and scattering for mass-critical inhomogeneous NLS when $d\ge3$

We prove global well-posedness and scattering for solutions to the mass-critical inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=\pm |x|^{-b}|u|^{\frac{4-2b}{d}}u$ for large $L^2(\mathbb{R} ^d)$ initial data with $d\ge3,0 0$) are that the presence of the inhomogeneity $|x|^{-b}$ creates a nontrivial singularity at the origin, and breaks the translation symmetry as well as the Galilean invariance of the equation, which makes the establishment of the profile decomposition and long time Strichartz estimates more difficult. To overcome these difficulties, we perform the concentration compactness/rigidity methods of [Kenig and Merle, Invent. Math. (2006)] in the Lorentz space framework, and reduces the problem to the exclusion of almost periodic solutions. The exclusion of these solutions will utilize fractional estimates and long time Strichartz estimates in Lorentz spaces. In our study, we obseve that the decay of the inhomogeneity $|x|^{-b}$ at infinity prevents the concentration of the almost periodic solution at infinity in either physical or frequency space. Therefore, we can use classical Morawetz estimates, rather than interaction Morawetz estimates, to exclude the existence of the quasi-soliton.

math.AP↗

arXiv2024

On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations

In this paper, a class of anisotropic weights having the form of $|x'|^{θ_{1}}|x|^{θ_{2}}|x_{n}|^{θ_{3}}$ in dimensions $n\geq2$ is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find the optimal range of $(θ_{1},θ_{2},θ_{3})$ such that this type of weights belongs to the Muckenhoupt class $A_{p}$. Then we further study its doubling property, which shows that it provides an example of a doubling measure but is not in $A_{p}$. As a consequence, we obtain anisotropic weighted Poincaré and Sobolev inequalities, which are used to study the local behavior for solutions to non-homogeneous weighted $p$-Laplace equations.

math.AP↗

arXiv2024

Two-point polynomial patterns in subsets of positive density in $\mathbb{R}^n$

Let $γ(t)=(P_1(t),\ldots,P_n(t))$ where $P_i$ is a real polynomial with zero constant term for each $1\leq i\leq n$. We will show the existence of the configuration $\{x,x+γ(t)\}$ in sets of positive density $ε$ in $[0,1]^n$ with a gap estimate $t\geq δ(ε)$ when $P_i$'s are arbitrary, and in $[0,N]^n$ with a gap estimate $t\geq δ(ε)N^n$ when $P_i$'s are of distinct degrees where $δ(ε)=\exp\left(-\exp\left(cε^{-4}\right)\right)$ and $c$ only depends on $γ$. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain's reduction are primarily utilised. For the first result, dimension-reducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the corner-type Roth theorem previously proven by the first author and Guo.

math.CA↗

arXiv2024

$L^p$ maximal estimates for Weyl sums with $k\ge3$ on $\mathbb{T}$

In this paper, we study the $L^p$ maximal estimates for the Weyl sums $\sum_{n=1}^{N}e^{2πi(nx + n^{k}t)}$ with higher-order $k\ge3$ on $\mathbb{T}$, and obtain the positive and negative results. Especially for the case $k=3$, our result is sharp up to the endpoint. The main idea is to investigate the structure of the set where large values of Weyl sums are achieved by making use of the rational approximation and the refined estimate for the exponential sums.

math.NT↗

arXiv2024

On Onsager's type conjecture for the inviscid Boussinesq equations

In this paper, we investigate the Cauchy problem for the three dimensional inviscid Boussinesq system in the periodic setting. For $1\le p\le \infty$, we show that the threshold regularity exponent for $L^p$-norm conservation of temperature of this system is $1/3$, consistent with Onsager exponent. More precisely, for $1\le p\le\infty$, every weak solution $(v,θ)\in C_tC^β_x$ to the inviscid Boussinesq equations satisfies that $\|θ(t)\|_{L^p(\mathbb{T}^3)}=\|θ_0\|_{L^p(\mathbb{T}^3)}$ if $β>\frac{1}{3}$, while if $β<\frac{1}{3}$, there exist infinitely many weak solutions $(v,θ)\in C_tC^β_x$ such that the $L^p$-norm of temperature is not conserved. As a byproduct, we are able to construct many weak solutions in $C_tC^β_x$ for $β<\frac{1}{3}$ displaying wild behavior, such as fast kinetic energy dissipation and high oscillation of velocity. Moreover, we also show that if a weak solution $(v, θ)$ of this system has at least one interval of regularity, then this weak solution $(v,θ)$ is not unique in $C_tC^β_x$ for $β<\frac{1}{3}$.

math.AP↗

arXiv2024

Local regularity for solutions to quasi-linear singular parabolic equations with anisotropic weights

This paper develops a concise procedure for the study on local behavior of solutions to anisotropically weighted quasi-linear singular parabolic equations of $p$-Laplacian type, which is realized by improving the energy inequalities and applying intrinsic scaling factor to the De Giorgi truncation method. In particular, it also presents a new proof for local Hölder continuity of the solution in the unweighted case.

math.AP↗

arXiv2024

Scattering theory for subcritical wave equation with inverse square potential

We consider the scattering theory for the defocusing energy subcritical wave equations with an inverse square potential. By employing the energy flux method we establish energy flux estimates on the light cone. Then by the characteristic line method and radiation theorem, we show that the radial finite-energy solutions scatter to free waves outside of light cones. Using Morawetz estimates we then obtain the scattering theory for radial solutions with finite weighted energy initial data.

math.AP↗