Searcharxiv⌕ Search

arXiv subjects

Yifu Zhou

Publications and source records attributed to Yifu Zhou.

16 recordsLinked to original sources
arXiv2026

Collapsing-tube type II blow-up for the energy-supercritical heat equation

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation \[ u_t=\Delta u+u^3, \qquad n\geq 5. \] The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As $t\nearrow T$, the solution concentrates in a thin tubular region around an $(n-4)$-dimensional sphere whose radius shrinks to zero at the self-similar scale \[ \xi_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ \lambda(t)\sim \kappa_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some $\kappa_*>0$. More precisely, in cylindrical coordinates $r=|x'|$, $z\in\mathbb R^3$, the leading profile is \[ u(x,t) \sim \frac{1}{\lambda(t)} U\left( \frac{r-\xi_r(t)}{\lambda(t)}, \frac{z}{\lambda(t)} \right), \] where $U$ is the Aubin--Talenti bubble in $\mathbb R^4$. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale $\sqrt{T-t}$, whereas its transverse thickness is governed by the much smaller type II scale $\lambda(t)$. The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent $p=3$ is energy-supercritical in dimensions $n\geq5$, but lies below the Joseph--Lundgren exponent for $5\leq n\leq 12$, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.

math.AP↗
arXiv2026

Global oscillatory solutions for the Yang-Mills heat flow

We investigate the long-time dynamics for the global solution of the $SO(4)$-equivariant Yang-Mills heat flow (YMHF) with structure group $SU(2)$ in space dimension $4$. For a class of initial data with specific decay at spatial infinity, we prove that the long-time dynamics of YMHF can be described by the initial data in a unified manner. As a consequence, the global solutions can exhibit blow-up, blow-down, and more exotically, {\it oscillatory} asymptotic behavior at time infinity. This seems to be the first example of Yang-Mills heat flows with oscillatory behavior as $t\to \infty$.

math.AP↗
arXiv2026

Liquid crystals and topological vorticity: smoothness of mild solutions

We introduce several new models whose common feature is to take into account effects from topological vorticity. The macroscopic unknown is driven by a dissipative anomalous diffusion (of SQG-type) and is coupled with the orientation of the crystal, moving by the gradient flow of the energy of maps. The main idea of such models is to have a better insight on the vorticity formulation of the Liquid Crystal Flow and to tackle some regularity issues in the associated conserved geometric motions. One of the advantage of the present PDEs is to capture features of the Navier-Stokes equations (or Euler) through a {\sl scalar} unknown, keeping the advection-diffusion structure of the orientation field. We obtain regularity for mild solutions under natural assumptions for the initial data, which are actually near-optimal. Along the way, we also draw some links with natural models of (anti-)ferromagnets previously investigated.

math.AP↗
arXiv2025

NFIG: Multi-Scale Autoregressive Image Generation via Frequency Ordering

Autoregressive models have achieved significant success in image generation. However, unlike the inherent hierarchical structure of image information in the spectral domain, standard autoregressive methods typically generate pixels sequentially in a fixed spatial order. To better leverage this spectral hierarchy, we introduce NextFrequency Image Generation (NFIG). NFIG is a novel framework that decomposes the image generation process into multiple frequency-guided stages. NFIG aligns the generation process with the natural image structure. It does this by first generating low-frequency components, which efficiently capture global structure with significantly fewer tokens, and then progressively adding higher-frequency details. This frequency-aware paradigm offers substantial advantages: it not only improves the quality of generated images but crucially reduces inference cost by efficiently establishing global structure early on. Extensive experiments on the ImageNet-256 benchmark validate NFIG's effectiveness, demonstrating superior performance (FID: 2.81) and a notable 1.25x speedup compared to the strong baseline VAR-d20. The source code is available at https://github.com/Pride-Huang/NFIG.

cs.CV↗
arXiv2025

Multi-bubble solutions for the Dirichlet problem of the $H$-system with higher degree

We consider a Dirichlet problem of the $H$-system \begin{equation*} \begin{cases} Δv = 2v_x\wedge v_y ~& \text{ in }\mathcal{D},\\ v=\varepsilon \tilde g ~& \text{ on }\partial{\mathcal{D}}, \end{cases} \end{equation*} where $\mathcal D\subset \mathbb{R}^2$ is the unit disk, $v:\mathcal D\to \mathbb{R}^3$, and $\tilde g:\partial \mathcal D\to \mathbb{R}^3$ is a given smooth map. As $\varepsilon\to 0^+$, we construct multi-bubble solutions concentrating at distinct points, taking around each point the profile of degree 2 $H$-bubble. This gives a partial answer to a conjecture due to Brezis-Coron and Chanillo-Malchiodi concerning the limiting configuration in the case of higher degrees. This seems to be the first construction in employing higher-degree harmonic maps as the primary configurations.

math.AP↗
arXiv2025

Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two

We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from ${\mathbb R}^2$ into $S^2$ \begin{equation*} \begin{cases} u_t= a(Δu+|\nabla u|^2u) -b u\wedge Δu &\ \mbox{ in }\ {\mathbb R}^2\times(0,T), u(\cdot,0) = u_0\in S^2 &\ \mbox{ in }\ {\mathbb R}^2, \end{cases} \end{equation*} where $a^2+b^2=1,~a > 0,~ b\in {\mathbb R}$. Given any prescribed $N$ points in $\mathbb{R}^2$ and small $T>0$, we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time $t=T$, taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speed \begin{equation*} \| \nabla u\|_{L^\infty } \sim \frac{|\ln(T-t)|^2}{ T-t } \ \mbox{ as } \ t\to T. \end{equation*} The proof is based on the {\em parabolic inner-outer gluing method}, developed in \cite{17HMF} for Harmonic Map Flow (HMF). However, a direct consequence of the presence of dispersion is the {\em lack of maximum principle} for suitable quantities, which makes the analysis more delicate even at the linearized level. To overcome this difficulty, we make use of two key technical ingredients: first, for the inner problem we employ the tool of {\em distorted Fourier transform}, as developed by Krieger, Miao, Schlag and Tataru \cite{Krieger09Duke,KMS20WM}. Second, the linear theory for the outer problem is achieved by means of the sub-Gaussian estimate for the fundamental solution of parabolic system in non-divergence form with coefficients of Dini mean oscillation in space ($\mathsf{DMO_x}$), which was proved by Dong, Kim and Lee \cite{dong22-non-divergence} recently.

math.AP↗
arXiv2023

Finite-time singularity formation for the heat flow of the $H$-system

We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = Δu - 2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\ &u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right. \end{equation*} where $u$: $\mathbb{R}^2\times\mathbb{R}_+\to \mathbb{R}^3$. The singularity at finite time forms as a scaled least energy $H$-bubble, denoted as $W$, exhibiting type II blow-up speed. One key observation is that the linearized operators around $W$ projected onto $W^\perp$ and in the $W$-direction are in fact decoupled. On $W^\perp$, the linearization is the linearized harmonic map heat flow, while in the $W$-direction, it is the linearized Liouville-type flow. Based on this, we also prove the non-degeneracy of the $H$-bubbles with any degree.

math.AP↗
arXiv2023

Long-time dynamics for the energy critical heat equation in $R^5$

We investigate the long-time behavior of global solutions to the energy critical heat equation in $R^5$ \begin{equation*} \begin{cases} \pp_t u=Δu+|u|^{\frac{4}{3}} u ~&\mbox{ in }~ R^5 \times (t_0,\infty), u(\cdot,t_0)=u_0~&\mbox{ in }~ R^5. \end{cases} \end{equation*} For $t_0$ sufficiently large, we show the existence of positive solutions for a class of initial value $u_0(x)\sim |x|^{-γ}$ as $|x|\rightarrow \infty$ with $γ>\frac32$ such that the global solutions behave asymptotically \begin{equation*} \| u(\cdot,t) \|_{L^\infty (\R^5)} \sim \begin{cases} t^{-\frac{3(2-γ)}{2}} ~&\mbox{ if }~ \frac32<γ<2 (\ln t)^{-3} ~&\mbox{ if }~ γ=2 1 ~&\mbox{ if }~ γ>2 \end{cases} \mbox{ \ for \ } t >t_0, \end{equation*} which is slower than the self-similar time decay $t^{-\frac{3}{4}}$. These rates are inspired by Fila-King \cite[Conjecture 1.1]{FilaKing12}.

math.AP↗
arXiv2023

Trichotomy dynamics of the 1-equivariant harmonic map flow

For the 1-equivariant harmonic map flow from $ R^2$ into $S^2$ \begin{equation*} \left\{ \begin{aligned} &v_t=v_{rr}+\frac{v_r}{r} - \frac{\sin(2v)}{2r^2} , ~\quad(r,t)\in R_+\times (t_0,+\infty),\\ &v(r,t_0)=v_0, \qquad\qquad\qquad\quad r\in R_+, \end{aligned} \right. \end{equation*} we construct global growing, bounded and decaying solutions with the initial data $v_0(r)$ satisfying $$v_0(0)=π~\mbox{ and }~ v_0(r)\sim r^{1-γ} ~\mbox{ as }~ r\to+\infty, \quad γ>1.$$ These global solutions exhibit the following trichotomy long-time asymptotic behavior \begin{equation*} \| v_r(\cdot,t) \|_{L^\infty ([0,\infty))} \sim \begin{cases} t^{\frac{γ-2}{2}}\ln t ~&\mbox{ if }~ 1<γ<2,\\ 1 ~&\mbox{ if }~ γ=2,\\ \ln t ~&\mbox{ if }~ γ>2,\\ \end{cases} ~\mbox{ as }~ t\to +\infty. \end{equation*}

math.AP↗
arXiv2022

On Fila-King Conjecture in Dimension Four

We consider the following Cauchy problem for the four-dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^{3} ~&\mbox{ in }~ {\mathbb R}^4 \times (0,\infty),\\ u(x,0)=u_0(x) ~&\mbox{ in }~ {\mathbb R}^4. \end{cases} \end{equation*} We construct a positive infinite time blow-up solution $u(x,t)$ with the blow-up rate $ \| u(\cdot,t)\|_{L^\infty({\mathbb R}^4)} \sim \ln t$ as $t\to \infty$ and show the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King \cite[Conjecture 1.1]{filaking12}.

math.AP↗
arXiv2022

Nematic liquid crystal flow with partially free boundary

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.

math.AP↗
arXiv2021

Global existence of free-energy solutions to the 2D Patlak--Keller--Segel--Navier--Stokes system with critical and subcritical mass

We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $\mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.

math.AP↗
arXiv2021

Parallel Extraction of Long-term Trends and Short-term Fluctuation Framework for Multivariate Time Series Forecasting

Multivariate time series forecasting is widely used in various fields. Reasonable prediction results can assist people in planning and decision-making, generate benefits and avoid risks. Normally, there are two characteristics of time series, that is, long-term trend and short-term fluctuation. For example, stock prices will have a long-term upward trend with the market, but there may be a small decline in the short term. These two characteristics are often relatively independent of each other. However, the existing prediction methods often do not distinguish between them, which reduces the accuracy of the prediction model. In this paper, a MTS forecasting framework that can capture the long-term trends and short-term fluctuations of time series in parallel is proposed. This method uses the original time series and its first difference to characterize long-term trends and short-term fluctuations. Three prediction sub-networks are constructed to predict long-term trends, short-term fluctuations and the final value to be predicted. In the overall optimization goal, the idea of multi-task learning is used for reference, which is to make the prediction results of long-term trends and short-term fluctuations as close to the real values as possible while requiring to approximate the values to be predicted. In this way, the proposed method uses more supervision information and can more accurately capture the changing trend of the time series, thereby improving the forecasting performance.

cs.LG↗
arXiv2020

New type II Finite time blow-up for the energy supercritical heat equation

We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=Δu+u^{3},~&\mbox{ in } Ω\times (0,T),\\ u(x,t)=u|_{\partialΩ},~&\mbox{ on } \partialΩ\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $5\leq n\leq 7$, $Ω=\R^n$ or $Ω\subset \R^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {\em shrinking sphere} in $Ω$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(ξ_r(t),ξ_z(t))}{\la(t)}\right|^2}$$ where $r=\sqrt{x_1^2+\cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,\cdots,x_n)\inΩ$. Moreover, the singularity location $$(ξ_r(t),ξ_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$ for some fixed $z_0$, and the blow-up rate $$\la(t)\sim \frac{T-t}{|\log(T-t)|^2}~\mbox{ as }~t\nearrow T.$$ This is a completely new phenomenon in the parabolic setting.

math.AP↗
arXiv2019

Finite time blow-up for the nematic liquid crystal flow in dimension two

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $Ω\subset \mathbb R^2$. Given any $k$ distinct points in the domain, we develop a new {\em inner--outer gluing method} to construct solutions which blow up exactly at those $k$ points as $t$ goes to a finite time $T$. Moreover, we obtain a precise description of the blow-up.

math.AP↗