arXiv subjects

Lifeng Zhao

Publications and source records attributed to Lifeng Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Linear Stability and Inviscid Damping of Monotone Shear Flows for 2D Compressible Euler Equations

We study 2D compressible Euler equations linearized around monotone shear flows $(U(y),0)$ on $\mathbb{T} \times \mathbb{R}$. The shear rate $U'$ is strictly positive, not necessarily close to any constant, and varies sufficiently slowly. For every fixed Mach number $M > 0$, we prove that the density and the irrotational velocity obey algebraic growth bounds, whereas the solenoidal velocity undergoes componentwise inviscid damping. Although a non-uniform shear couples the transverse Fourier frequencies and precludes the full Fourier reduction available for Couette flow, we are still able to recover the Couette rates without loss. The proof hinges on two new ingredients: a time-dependent pseudodifferential energy that restores a coercive structure for the variable-coefficient shear dynamics, and terminal-time-dependent higher- and lower-order weighted energies that capture the long-time effects of shear mixing.

math.AP↗

arXiv2026

Nonexistence of blow-up solutions with smooth radiation for energy-critical equivariant wave maps

We study $k$-equivariant energy critical wave maps $\mathbb{R}^{1+2} \to \mathbb{S}^2$, for any equivariance degree $k\ge 2$. We prove that the radiation associated with any finite-energy blow-up solution cannot satisfy a certain regularity condition; in particular, it cannot be smooth. The assumption $k \geq 2$ is necessary, since for $k = 1$ such solutions are known to exist. The starting point of our analysis is the soliton resolution theorem. The key ingredient is a novel application of the modulation method, in which we compare the effects of the radiation and inner bubbles to study the dynamic behavior of the widest bubble.

math.AP↗

arXiv2026

Invariant Gibbs measures and global dynamics for fractional cubic Schr\"odinger equations on the torus

We consider the defocusing Wick-ordered cubic fractional nonlinear Schr\"odinger equation on the two-dimensional torus with dispersion relation $\omega(k)=|k|^\alpha$. In the weakly dispersive regime $\frac{29}{15}<\alpha<2$, we construct global dynamics for almost every initial datum with respect to the associated Gibbs measure as the limit of the finite-dimensional truncated flows and prove invariance of the Gibbs measure. The core of the proof is an almost sure local theory based on the method of random averaging operators (arXiv:1910.08492v2). The main new ingredients are fractional lattice counting estimates and localized random tensor bounds, which exploit the geometric structure of the fractional phase in place of the classical number-theoretic tools available for quadratic dispersion.

math.AP↗

arXiv2026

Construction of multi-bubble solutions for the energy-critical wave equation in dimension four

For any $N\geq 2$, we construct a global solution of the energy-critical focusing wave equation in dimension four which blows up in infinite time at $N$ prescribed points $z_1,\ldots,z_N\in \mathbb R^4$, provided that the points form one orbit under a finite group of orthogonal symmetries. We denote by $c:=2\sum_{j\ne k}|z_j-z_k|^{-2}>0$ the corresponding interaction coefficient, which is independent of $k$. The common concentration scale satisfies \[ \log\frac{1}{\lambda(t)} = \left(\frac{9c}{4}\right)^{1/3}t^{2/3}+O(t^{1/3}) \qquad \text{as } t\to+\infty . \] This concentration rate comes from a genuinely four-dimensional effect: the borderline decay of the ground state makes the interaction between different bubbles enter the leading order parameter dynamics.

math.AP↗

arXiv2026

Rigidity of the multi-bubble solutions to the energy critical wave equation in dimension five

We study the asymptotic dynamics of multi-bubble solutions to the focusing energy-critical wave equation in five dimensions. Assuming that the solution asymptotically decomposes into a finite superposition of spatially separated bubbles with comparable scales, we prove a rigidity result that describes the precise long-time behavior of these scales. More precisely, we show that all scaling parameters are necessarily of order $t^{-2}$, and that the corresponding renormalized modulation vector converges to a connected component of a finite-dimensional algebraic set determined by the limiting spatial configuration of the bubbles. This algebraic system encodes the strong interactions between the polynomial tails of the bubbles and governs the effective asymptotic dynamics of the multi-bubble regime.

math.AP↗

arXiv2026

Global dynamics of a single vortex ring

We study the global-in-time dynamics of vortex rings for the three-dimensional incompressible Euler equations, under the assumption of axisymmetric flows without swirl. For a broad class of initial data sharing only the macroscopic invariants with a thin vortex ring, we prove that the vorticity remains sharply concentrated and propagates along the symmetry axis with leading-order speed given by the Kelvin--Hicks formula, providing the first global-in-time validation of the vortex filament conjecture for a single vortex ring arising from generic initial data. We further identify a universal filamentation mechanism driven by the competition between rapid core translation and slower local induction. This mechanism gives linear-in-time stretching of the vortex support under very general assumptions on the data, yielding dynamical instability of any thin vortex ring configurations in the $W^{2,\infty}$ norm.

math.AP↗

arXiv2026

Developing a Machine-Learning Interatomic Potential for Non-Covalent Interactions in Proteins

Machine learning interatomic potentials (MLIPs) enable efficient modeling of molecular interactions with quantum mechanical (QM) accuracy. However, constructing robust and representative training datasets that capture subtle, system-specific interaction motifs remains challenging. We introduce PANIP (PAirwise Non-covalent Interaction Potential), an ensemble MLIP model built upon the NequIP framework and trained on non-covalent interactions (NCIs) between protein-derived fragments. PANIP is trained using an automated multi-fidelity active learning (MFAL) workflow, in which a representative training subset, termed PDB-FRAGID (PDB Fragment Interaction Dataset), was distilled from an otherwise prohibitively large pool of fragment dimers extracted from the Protein Data Bank (PDB). PANIP retains $\omega$B97X-D3BJ/def2-TZVPP-level accuracy and achieves mean absolute errors below 0.2 kcal/mol on out-of-distribution systems, demonstrating excellent transferability across diverse NCI motifs. Compared to the widely used ANI-2x potential, PANIP delivers substantially lower errors, particularly for charged and strongly interacting dimers. Coupled with a fragmentation-based energy decomposition scheme, PANIP estimates protein-ligand binding energies at near force-field computational cost yet QM-level accuracy, enabling its use as a fragment-based scoring function that rivals specialized docking scoring functions.

physics.chem-ph↗

arXiv2025

AnchorFormer: Differentiable Anchor Attention for Efficient Vision Transformer

Recently, vision transformers (ViTs) have achieved excellent performance on vision tasks by measuring the global self-attention among the image patches. Given $n$ patches, they will have quadratic complexity such as $\mathcal{O}(n^2)$ and the time cost is high when splitting the input image with a small granularity. Meanwhile, the pivotal information is often randomly gathered in a few regions of an input image, some tokens may not be helpful for the downstream tasks. To handle this problem, we introduce an anchor-based efficient vision transformer (AnchorFormer), which employs the anchor tokens to learn the pivotal information and accelerate the inference. Firstly, by estimating the bipartite attention between the anchors and tokens, the complexity will be reduced from $\mathcal{O}(n^2)$ to $\mathcal{O}(mn)$, where $m$ is an anchor number and $m < n$. Notably, by representing the anchors with the neurons in a neural layer, we can differentiably learn these anchors and approximate global self-attention through the Markov process. It avoids the burden caused by non-differentiable operations and further speeds up the approximate attention. Moreover, we extend the proposed model to three downstream tasks including classification, detection, and segmentation. Extensive experiments show the effectiveness of our AnchorFormer, e.g., achieving up to a 9.0% higher accuracy or 46.7% FLOPs reduction on ImageNet classification, 81.3% higher mAP on COCO detection under comparable FLOPs, as compared to the current baselines.

cs.CV↗

arXiv2024

Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in $\mathbb{R}^3$

We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t Ω+U \cdot \nabla Ω+Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_0(x) \end{aligned}\right. \end{equation*} in the whole space $\mathbb{R}^3$. Under the assumption that the initial velocity is helical and without swirl, we prove the global well-posedness of weak solutions in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^3)$. The vortex transport formula is also obtained in our article.

math.AP↗

arXiv2024

Long time dynamics for helical vortex filament in Euler flows

We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t Ω+U \cdot \nabla Ω-Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_0(x) \end{aligned}\right. \end{equation*} under the assumption that $Ω^z$ is helical and in the absence of vorticity stretching. Assuming that the initial vorticity $Ω_0$ is primarily concentrated within an $ε$ neighborhood of a helix $Γ_0$, we prove that its solution $Ω(\cdot,t)$ remain concentrated near a helix $Γ(t)$ for any $t \in [0,T)$, where $Γ(t)$ can be interpreted as $Γ_0$ rotating around the $x_3$ axis with a speed $V=C\log \frac{1}ε+O(1)$. It should be emphasized that the dynamics for the helical vortex filament are exhibited on the time interval $[0,T)$, which is longer than $\left[0, \frac{T}{\log\frac{1}ε}\right)$.

math.AP↗

arXiv2024

Asymptotic behavior of 3-D evolutionary model of Magnetoelasticity for small data

In this article, we consider the evolutionary model for magnetoelasticity with vanishing viscosity/damping, which is a nonlinear dispersive system. The global regularity and scattering of the evolutionary model for magnetoelasticity under small size of initial data is proved. Our proof relies on the idea of vector-field method due to the quasilinearity and the presence of convective term. A key observation is that we construct a suitable energy functional including the mass quantity, which enable us to provide a good decay estimates for Schrödinger flow. In particular, we establish the asymptotic behavior in both mass and energy spaces for Schrödinger map, not only for gauged equation.

math.AP↗

arXiv2024

Dynamics for the corotational energy-critical wave map equation with quantized blow-up rates

We consider the wave maps from $\mathbb{R}^{1+2}$ into $\mathbb{S}^2\subset \mathbb{R}^3.$ Under an additional assumption of $k$-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation: \begin{equation*} \partial_t^2 u-\partial_r^2 u-\frac{\partial_r u}{r}+k^2 \frac{\sin(2u)}{2r^2}=0. \end{equation*} Given any integer $k\ge 1$ and any integer $m\ge 2k,$ we exhibit a set of initial data $(u_0,u_1)$ with energy arbitrarily close to that of the ground state solution $Q$, such that the corresponding solution $u$ blows up in finite time by concentrating its energy. To be precise, the solution $u$ satisfies \begin{equation*} \lim\limits_{t\rightarrow T} \left\|\left(u(t,r)-Q\left(\frac{r}{λ(t)}\right)-u_1^*(r), \partial_t u-u_2^*(r)\right)\right\|_{H\times L^2}=0 \end{equation*} with a quantized speed \begin{equation*} λ(t)=c(u_0,u_1)(1+o_{t\to T}(1))\frac{(T-t)^{\frac{m}{k}}}{|\log(T-t)|^{\frac{m}{k(m-k)}}}, \end{equation*} where $\|u\|_{H}:=\int_{\mathbb{R}^2}\left(|\partial_r u|^2+\frac{|u|^2}{r^2}\right).$

math.AP↗

arXiv2023

Soliton resolution for the energy critical damped wave equations in the radial case

We consider energy-critical damped wave equation \begin{equation*} \partial_{tt}u-\Delta u+\alpha \partial_t u=\left|u\right|^{\frac{4}{D-2}}u \end{equation*} with radial initial data in dimensions $D\geq 4$. The equation has a nontrivial radial stationary solution $W$, called the ground state, which is unique up to sign and scale. We prove that any bounded energy norm solution behaves asymptotically as a superposition of the modulated ground states and a radiation term. In the global case, particularly, the solution converges to a pure multi-bubble due to the damping effect.

math.AP↗

arXiv2023

Almost global well-posedness of 2-D Ericksen-Leslie's hyperbolic liquid crystal model for small data

This article is devoted to the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction $d$ of liquid crystal molecules satisfies a wave map equation with an acoustical metric. We established the almost global well-posedness for small and smooth initial data near the constant equilibrium. Our proof relies on the idea of vector-field method and ghost weight method. There are two key ingredients in our proof: (i) Inspired by the gauge theory in Tataru \cite{Tataru,Tataru05}, we reformulate the wave map equation into a free wave equation with acoustical metric, where the nonlinearity is annihilated due to the geometry of $\mathbb S^1$; (ii) Motivated by the ghost weight method in Alinhac \cite{A01}, we introduce a new and important ``good unknown", the velocity $u$, which provides the additional dissipation $u/\langle t-r\rangle\in L^2_tL^2_x$. These new observations turn out to be extremely crucial in resolving the system in low dimensions.

math.AP↗

arXiv2023

Asymptotic behaviors of incompressible Schrödinger flow for small data in three dimensions

The incompressible Schrödinger flow is a Madelung's hydrodynamical form of quantum mechanics, which can simulate classical fluids with particular advantage in its simplicity and its ability of capturing thin vortex dynamics. This model enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments. In this article, we prove the global regularity and asymptotic behaviors for incompressible Schrödinger flow with small and localized data in three dimensions. We choose a suitable gauge to rewrite the system, and then use Fourier analysis and vector fields method to prove global existence and asymptotic behaviors.

math.AP↗

arXiv2022

The viscous damping of three dimensional spherical gas bubble inside unbounded compressible liquid

The present paper considers a homogeneous bubble inside an unbounded polytropic compressible liquid with viscosity. The system is governed by the Navier-Stokes equation with free boundary which is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. The global existence of solution is proved, and the $\dot{H}^1$ asymptotic stability of the spherical equilibrium in terms of viscous damping together with a explicit decay rate is given in bare energy methods.

math.AP↗

arXiv2022

Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow

In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping $\mathbb{R}^2$ to $\mathbb{S}^2$ with arbitrary given coefficients $ρ_1\in \mathbb{R}$, $ρ_2>0$. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schrödinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functional. However, the blowup rate is independent of the coefficients.

math.AP↗

arXiv2022

Soliton resolution for the energy critical wave equation with inverse-square potential in the radial case

In this paper, we establish the soliton resolution for the energy critical wave equation with inverse square potential in the radial case and in all dimensions $N\geq3$. The structure of the radial linear operator $\mathcal{L}_a :=-Δ+\frac{a}{|x|^2}=A^*A$, is essential for the channel of energy, where $A$ is a first order differential operator and $A^*$ is its adjoint operator. Modulation and analysis of the multi-solitons are performed in the function spaces $\dot{H}^1_a(\Bbb R^N)\times L^2(\Bbb R^N)$ associated with $\mathcal{L}_a$.

math.AP↗