arXiv subjects

Jie Xin

Publications and source records attributed to Jie Xin.

7 recordsLinked to original sources

arXiv2026

Global regularity and general-coefficient singular limits for energy-critical complex Ginzburg--Landau equations

We study energy-critical complex Ginzburg--Landau equations with a linear damping term $Ru$, $R\geq 0$. For the undamped aligned equation in dimensions $d=3,4$, we treat the focusing and defocusing cases in a unified way and prove persistence of $H^1\cap C_0$ regularity and smoothness for positive times. In particular, this resolves the energy-critical cases of Cazenave's open problem. In dimensions $3\le d\le6$, we develop a coefficient-uniform critical stability framework for the zero-dispersion and inviscid limits. It applies to independent normalized complex coefficient paths in both the focusing and defocusing cases. From limiting data in the natural energy space $H^1$, we obtain convergence on every compact subinterval of the maximal lifespan of the limiting solution. Higher regularity is required only for explicit linear coefficient-error estimates, and the limits do not use global well-posedness, scattering, or global spacetime bounds for the limiting solution. At the inviscid limit, we establish coefficient-uniform homogeneous and retarded Strichartz estimates; a key technical ingredient is the retarded double-endpoint estimate required by the critical forcing space.

math.AP↗

arXiv2026

Analysis of the dynamics of Caputo fractional differential equations

It is known that a finite-dimensional Caputo fractional differential equation, though itself need not generate a semiflow, can be represented as a Volterra integral equation which generates an infinite-dimensional semiflow on the space $\mathfrak{C}=C([0,\infty); \mathbb{R}^d)$ under the standard compact-open topology. In this paper we construct a compact absorbing set and an attractor for this semiflow on $\mathfrak{C}$, and then prove that the attractor consists of equi globally H\"older continuous functions. This strengthens the previous work of Doan \& Kloeden \cite{DK21} where a bounded (with respect to a weighted norm) attractor was constructed.

math.DS↗

arXiv2025

$(H,H^2)$-smoothing effect of Navier-Stokes equations with additive white noise on two-dimensional torus

This paper is devoted to the regularity of Navier-Stokes (NS) equations with additive white noise on two-dimensional torus $\mathbb T^2$. Under the conditions that the external force $f(x)$ belongs to the phase space $ H$ and the noise intensity function $h(x)$ satisfies $\|\nabla h\|_{L^\infty} \leq \sqrt πνλ_1$, where $ ν$ is the kinematic viscosity of the fluid and $λ_1$ is the first eigenvalue of the Stokes operator, it was proved that the random NS equations possess a tempered $(H,H^2)$-random attractor whose (box-counting) fractal dimension in $H^2$ is finite. This was achieved by establishing, first, an $H^2$ bounded absorbing set and, second, an $(H,H^2)$-smoothing effect of the system which lifts the compactness and finite-dimensionality of the attractor in $H$ to that in $H^2$. Since the force $f$ belongs only to $H$, the $H^2$-regularity of solutions as well as the $H^2$-bounded absorbing set was constructed by an indirect approach of estimating the $H^2$-distance between the solution of the random NS equations and that of the corresponding deterministic equations.

math.AP↗

arXiv2025

$(H,H^3)$-smoothing effect and convergence of solutions of stochastic two-dimensional anisotropic Navier-Stokes equations driven by colored noise

This paper is devoted to the higher regularity and convergence of solutions of anisotropic Navier-Stokes (NS) equations with additive colored noise and white noise on two-dimensional torus $\mathbb T^2$. Under the conditions that the external force $f(\textbf{x})$ belongs to the phase space $ H$ and the noise intensity function $h(\textbf{x})$ satisfies $\|\nabla h\|_{L^\infty} \leq \sqrt{πδ} \frac{νλ_1}{2}$, it was proved that the random anisotropic NS equations possess a tempered $(H,H^2)$-random attractor whose (box-counting) fractal dimension in $H^2$ is finite. This was achieved by establishing, first, an $H^2$ bounded absorbing set and, second, an $(H,H^2)$-smoothing effect of the system which lifts the compactness and finite-dimensionality of the attractor in $H$ to that in $H^2$. Since the force $f$ belongs only to $H$, the $H^2$-regularity of solutions as well as the $H^2$-bounded absorbing set was constructed by an indirect approach of estimating the $H^2$-distance between the solution of the random anisotropic NS equations and that of the corresponding deterministic anisotropic NS equations. When the external force $f(\textbf{x})$ belongs to $H^2$ and the noise intensity function $h(\textbf{x})$ satisfies the Assumption 2, it was proved that the random anisotropic NS equations possess a tempered $(H,H^3)$-random attractor whose (box-counting) fractal dimension in $H^3$ is finite. Finally, we prove the upper semi-continuity of random attractors and the convergence of solutions of (8.3) as $δ\rightarrow0$ in the spaces $(H,H)$, $(H,H^1)$, $(H^1,H^2)$ and $(H^2,H^3)$, respectively.

math.AP↗

arXiv2025

Regularity of random attractor and fractal dimension of fractional stochastic Navier-Stokes equations on three-dimensional torus

In this paper we will study the asymptotic dynamics of fractional Navier-Stokes (NS) equations with additive white noise on three-dimensional torus $\mathbb T^3$. Under the conditions that the external forces $f(x)$ belong to the phase space $ H$ and the noise intensity function $h(x)$ satisfies $\|\nabla h\|_{L^\infty} < \sqrt πνλ_1^\frac{5}{4}$, where $ ν$ is the kinematic viscosity of the fluid and $λ_1$ is the first eigenvalue of the Stokes operator, we shown that the random fractional three-dimensional NS equations possess a tempered $(H,H^\frac{5}{2})$-random attractor whose fractal dimension in $H^\frac{5}{2}$ is finite. This was proved by establishing, first, an $H^\frac{5}{2}$ bounded absorbing set and, second, a local $(H,H^\frac{5}{2})$-Lipschitz continuity in initial values from which the $(H,H^\frac{5}{2})$-asymptotic compactness of the system follows. Since the forces $f$ belong only to $H$, the $H^\frac{5}{2}$ bounded absorbing set was constructed by an indirect approach of estimating the $H^\frac{5}{2}$-distance between the solutions of the random fractional three-dimensional NS equations and that of the corresponding deterministic equations. Furthermore, under the conditions that the external forces $f(x)$ belong to the $ H^{k-\frac{5}{4}}$ and the noise intensity function $h(x)$ belong to $H^{k+\frac{5}{4}}$ for $k\geq\frac{5}{2}$, we shown that the random fractional three-dimensional NS equations possess a tempered $(H,H^k)$-random attractor whose fractal dimension in $H^k$ is finite. This was proved by using iterative methods and establishing, first, an $H^k$ bounded absorbing set and, second, a local $(H,H^k)$-Lipschitz continuity in initial values from which the $(H,H^k)$-asymptotic compactness of the system follows.

math.AP↗

arXiv2024

ScaleFold: Reducing AlphaFold Initial Training Time to 10 Hours

AlphaFold2 has been hailed as a breakthrough in protein folding. It can rapidly predict protein structures with lab-grade accuracy. However, its implementation does not include the necessary training code. OpenFold is the first trainable public reimplementation of AlphaFold. AlphaFold training procedure is prohibitively time-consuming, and gets diminishing benefits from scaling to more compute resources. In this work, we conducted a comprehensive analysis on the AlphaFold training procedure based on Openfold, identified that inefficient communications and overhead-dominated computations were the key factors that prevented the AlphaFold training from effective scaling. We introduced ScaleFold, a systematic training method that incorporated optimizations specifically for these factors. ScaleFold successfully scaled the AlphaFold training to 2080 NVIDIA H100 GPUs with high resource utilization. In the MLPerf HPC v3.0 benchmark, ScaleFold finished the OpenFold benchmark in 7.51 minutes, shown over $6\times$ speedup than the baseline. For training the AlphaFold model from scratch, ScaleFold completed the pretraining in 10 hours, a significant improvement over the seven days required by the original AlphaFold pretraining baseline.

cs.LG↗

arXiv2023

Lattice effect on the superexchange interaction in antiferromagnetic Bi$_2$Sr$_2$CaCu$_2$O$_8$

The in-plane superexchange interaction $J$ of cuprate superconductors has long been suggested to be an important parameter for exploring their high-temperature superconductivity. The bilayer Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$ is the most studied system with high-quality single crystals in the wide doping range with the the same structure and phase. So far, the lattice parameter dependence of $J$ in its antiferromagnetic parent compound Bi$_2$Sr$_2$CaCu$_2$O$_{8}$ has not been established. By combining Raman scattering and x-ray diffraction techniques on the same sample in the same pressure environment, we obtain the evolution of both the two-magnon spectrum and the structural parameters with pressure up to nearly 30 GPa, The relationship between pressure or the in-plane lattice parameter and $J$ is thus established for Bi$_2$Sr$_2$CaCu$_2$O$_{8}$. Over the studied pressure range, superconductivity does not appear in this parent compound based on a sensitive magnetic measurement technique. The effects of pressure and chemical doping on the superexchange interaction and structure and their implications for superconductivity are discussed from the comparison of the obtained experimental data with the existing experiments. The results and findings provide valuable information for the understanding of superconductivity and the future theory developments for superconductivity in cuprates.

cond-mat.supr-con↗