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Xiaojing Xu

Publications and source records attributed to Xiaojing Xu.

At least 19 recordsLinked to original sources
arXiv2026

Global Stability of 3D Compressible Non-Resistive MHD: Hidden Damping and Rational Background Fields

We prove the global well-posedness and nonlinear stability of classical solutions to the three-dimensional compressible viscous, non-resistive MHD system on $\mathbb{T}^3$ near the equilibrium $(1,\mathbf{0},\mathbf{e}_3)$, for small $x_3$-symmetric perturbations and without any Diophantine condition on the background magnetic field. The central obstruction is the $x_3$-independent sector, in which the density and the magnetic field possess no dissipation, no damping, and no decay. We overcome it by exhibiting a hidden wave structure for the exact total pressure $\mathcal{D}=P(1+a)+B_3+\frac12|\mathbf{B}|^2$, the averaged pair $(\overline{\mathbf{u}},\overline{\mathcal{D}})$ obeys a closed system, and $\overline{\mathcal{D}}$ satisfies a strongly damped wave equation whose principal wave part propagates at the fast magnetosonic speed and whose non-parabolic branch damps at a rate that involves no gain of derivatives. This hidden damping substitutes for the missing magnetic dissipation, and simultaneously absorbs the magnetic pressure $\frac12\nabla|\mathbf{B}|^2$, the main obstruction created by compressibility. Together with a damped wave structure for the oscillatory sector and space-time weighted energy functionals with shifted time weights, this yields global existence, uniform stability, and explicit polynomial decay rates.

math.AP↗
arXiv2026

Time-asymptotic stability of generic Riemann solutions for the system of heat-conductive ideal gas without viscosity

This paper is concerned with the time-asymptotic stability of the generic Riemann solution for the one-dimensional system of heat-conductive ideal gas without viscosity, where the generic Riemann solution consists of a shock, a contact discontinuity, and a rarefaction wave. We prove that, as time tends to infinity, the solution of the non-viscous and heat-conductive ideal gas system converges uniformly to a composite wave composed of rarefaction wave, viscous contact wave, and viscous shock wave with a time-dependent shift. Motivated by the recent work of Kang-Vasseur-Wang [Arch. Ration. Mech. Anal. 249: 42 (2025)], we overcome the difficulties arising from the concurrence of shock and rarefaction waves for the partially dissipative hyperbolic-parabolic system with dissipation acting only on a single variable. More notably, the absence of velocity dissipation gives rise to new and intrinsic difficulties when handling the terms associated with the density and velocity. To resolve this, we exploit the precise structure of the governing equations and the additional properties of shock waves. Furthermore, we utilize the wave structure of the system without viscosity and perform separate space-time estimates for the density and velocity.

math.AP↗
arXiv2026

Global solutions to an initial-boundary value problem for a model of convection driven by surface tension

This paper establishes the global existence of non-negative weak solutions to a two-dimensional, fourth-order nonlinear degenerate parabolic equation modeling surface-tension-driven convection in thin fluid films. First, we construct a regularized approximate problem and prove its solvability via the Galerkin method. Utilizing energy and entropy functionals alongside a singular entropy condition $1/h_0 \in L^1(Ω)$, we secure uniform a priori bounds for higher-order spatial and time derivatives. These bounds enable the use of the Aubin-Lions lemma and Gagliardo-Nirenberg inequalities to achieve strong compactness and essential $L^6$-integrability. Furthermore, we adopt the Alber-Zhu framework to rigorously define higher-order local weak derivatives and pass to the limit. Finally, we prove the limit function is non-negative, confirming it as a global weak solution to the original problem.

math.AP↗
arXiv2026

Global Solutions to a Fourth-order Degenerate Model for Surface-Tension-Driven Convection

This paper investigates the global existence and non-negativity of weak solutions to an initial-boundary value problem for a one-dimensional fourth-order nonlinear degenerate parabolic equation. This model governs the convection phenomena in thin films driven by surface tension. Our analytical approach begins with the formulation of a regularized problem and a corresponding Galerkin approximating scheme. We first establish the existence of solutions to the approximate problem. Subsequently, by constructing specialized energy and entropy functionals, we derive uniform a priori estimates for the approximating solutions. Leveraging the Aubin-Lions compactness lemma, we pass to the limit and establish the non-negativity of the limit function. Finally, we demonstrate that this limit is indeed a global weak solution to the original initial-boundary value problem.

math.AP↗
arXiv2026

Global regularity of the 2D fractional Boussinesq equations with subcritical dissipation

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by $(-Δ)^{\frac\alpha2}u$ and $(-Δ)^{\frac\beta2} θ$. Attention is focused on the subcritical regime where $α+ β>1$. The case $α>\frac23$ was recently settled in a joint work of the authors [Math. Ann., \textbf{391} (2025), 5965-6012], which established global regularity under this condition. This paper addresses the remaining case $α\leq \frac23$. We obtain the sharpest regularity result by minimizing assumptions on $α$ and $β$. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.

math.AP↗
arXiv2026

Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and stratified temperature

This paper examines the stability threshold at high Reynolds numbers $\textbf{Re}$ for the three-dimensional Boussinesq equations with rotation on the domain $Ω=\{(x,\,y,\,z)\in \mathbb{T} \times \mathbb{R} \times \mathbb{T}\}$ around the Couette flow $(y,0,0)$ and the vertically stratified temperature $Θ_s=1+α^2 z$. For the linear system without rotation, stratification not only suppresses the lift-up effect but also exhibits certain dispersion effects, except for some points where degradation occurs, which will bring essential difficulties to nonlinear estimates. In contrast, when rotation is taken into account, we observe that this degeneracy in dispersion effects disappears; furthermore, we can derive dispersive estimates for the second and third components of the simple-zero mode within the velocity field. Additionally, we develop three good unknowns to minimize linear coupling terms as much as possible while mitigating growth induced by linear stretching terms; through constructing a series of multipliers, we achieve enhanced dissipation and inviscid damping effects. In our analysis of the nonlinear system aimed at establishing an improved stability threshold, we utilize quasi-linearization methods to rectify deficiencies in dispersive estimates related to both the first component of velocity and temperature, as well as address regularity issues along vertical directions caused by buoyancy forces and stratification. Consequently, we demonstrate that if initial perturbations in velocity and temperature satisfy $\left\|u_{\mathrm{in}}\right\|_{H^{N+2}\cap W^{N+3,1}}+\left\|θ_{\mathrm{in}}\right\|_{H^{N+1}\cap W^{N+3,1}}<δ\mathbf{Re}^{-\frac{14}{15}}$, for any $N\geq 11$ and some $δ>0$ independent of $\mathbf{Re}$, then the solution to the 3D Boussinesq equations with rotation is nonlinearly stable without transitioning away from the steady state.

math.AP↗
arXiv2025

FlagEval Findings Report: A Preliminary Evaluation of Large Reasoning Models on Automatically Verifiable Textual and Visual Questions

We conduct a moderate-scale contamination-free (to some extent) evaluation of current large reasoning models (LRMs) with some preliminary findings. We also release ROME, our evaluation benchmark for vision language models intended to test reasoning from visual clues. We attach links to the benchmark, evaluation data, and other updates on this website: https://flageval-baai.github.io/LRM-Eval/

cs.CL↗
arXiv2024

Stability of the Couette flow for 3D Navier-Stokes equations with rotation

Rotation significantly influences the stability characteristics of both laminar and turbulent shear flows. This study examines the stability threshold of the three-dimensional Navier-Stokes equations with rotation, in the vicinity of the Couette flow at high Reynolds numbers ($\mathbf{Re}$) in the periodical domain $\mathbb{T} \times \mathbb{R} \times \mathbb{T}$, where the rotational strength is equivalent to the Couette flow. Compared to the classical Navier-Stokes equations, rotation term brings us more two primary difficulties: the linear coupling term involving in the equation of $u^2$ and the lift-up effect in two directions. To address these difficulties, we introduce two new good unknowns that effectively capture the phenomena of enhanced dissipation and inviscid damping to suppress the lift-up effect. Moreover, we establish the stability threshold for initial perturbation $\left\|u_{\mathrm{in}}\right\|_{H^σ} < δ\mathbf{Re}^{-2}$ for any $σ> \frac{9}{2}$ and some $δ=δ(σ)>0$ depending only on $σ$.

math.AP↗
arXiv2024

On the Sobolev stability threshold for 3D Navier-Stokes equations with rotation near the Couette flow

Rotation is a crucial characteristic of fluid flow in the atmosphere and oceans, which is present in nearly all meteorological and geophysical models. The global existence of solutions to the 3D Navier-Stokes equations with large rotation has been established through the dispersion effect resulting from Coriolis force (i.e., rotation). In this paper, we investigate the dynamic stability of periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number $\mathbf{Re}$. Our aim is to determine the stability threshold index on $\mathbf{Re}$: the maximum range of perturbations within which the solution remains stable. Initially, we examine the linear stability effects of a linearized perturbed system. Comparing our results with those obtained by Bedrossian, Germain, and Masmoudi [Ann. Math. 185(2): 541--608 (2017)], we observe that mixing effects (which correspond to enhanced dissipation and inviscid damping) arise from Couette flow while Coriolis force acts as a restoring force inducing a dispersion mechanism for inertial waves that cancels out lift-up effects occurred at zero frequency velocity. This dispersion mechanism exhibits favorable algebraic decay properties distinct from those observed in classical 3D Navier-Stokes equations. Consequently, we demonstrate that if initial data satisfies $\left\|u_{\mathrm{in}}\right\|_{H^σ}<δ\mathbf{Re}^{-1}$ for any $σ>\frac{9}{2}$ and some $δ=δ(σ)>0$ depending only on $σ$, then the solution to the 3D Navier-Stokes equations with rotation is global in time without transitioning away from Couette flow. In this sense, Coriolis force contributes as a factor enhancing fluid stability by improving its threshold from $\frac{3}{2}$ to 1.

math.AP↗
arXiv2024

Large-Time Behavior of Solutions to Compressible Navier-Stokes System in Unbounded Domains with Degenerate Heat-Conductivity and Large Data

We are concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system with degenerate heat-conductivity describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. Both the specific volume and temperature are proved to be bounded from below and above independently of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity.

math.AP↗
arXiv2023

Spatial second-order positive and asymptotic preserving filtered $P_N$ schemes for nonlinear radiative transfer equations

A spatial second-order scheme for the nonlinear radiative transfer equations is introduced in this paper. The discretization scheme is based on the filtered spherical harmonics ($FP_N$) method for the angular variable and the unified gas kinetic scheme (UGKS) framework for the spatial and temporal variables respectively. In order to keep the scheme positive and second-order accuracy, firstly, we use the implicit Monte Carlo linearization method [6] in the construction of the UGKS numerical boundary fluxes. Then, by carefully analyzing the constructed second-order fluxes involved in the macro-micro decomposition, which is induced by the $FP_N$ angular discretization, we establish the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Finally, we employ linear scaling limiters for the angular variable in the $P_N$ reconstruction and for the spatial variable in the piecewise linear slopes reconstruction respectively, which are shown to be realizable and reasonable to enforce the sufficient conditions holding. Thus, the desired scheme, called the $PPFP_N$-based UGKS, is obtained. Furthermore, in the regime $ε\ll 1$ and the regime $ε=O(1)$, a simplified spatial second-order scheme, called the $PPFP_N$-based SUGKS, is presented, which possesses all the properties of the non-simplified one. Inheriting the merit of UGKS, the proposed schemes are asymptotic preserving. By employing the $FP_N$ method for the angular variable, the proposed schemes are almost free of ray effects. To our best knowledge, this is the first time that spatial second-order, positive, asymptotic preserving and almost free of ray effects schemes are constructed for the nonlinear radiative transfer equations without operator splitting. Various numerical experiments are included to validate the properties of the proposed schemes.

math.NA↗
arXiv2021

Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq systems helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is $\mathbb R^2$, the stability problem in a Sobolev setting remains open. When the spatial domain is $\mathbb T\times \mathbb R$, this paper solves the stability problem and specifies the precise large-time behavior of the perturbation. By decomposing the velocity $u$ and temperature $θ$ into the horizontal average $(\bar u, \barθ)$ and the corresponding oscillation $(\widetilde u, \widetilde θ)$, and deriving various anisotropic inequalities, we are able to establish the global stability in the Sobolev space $H^2$. In addition, we prove that the oscillation $(\widetilde u, \widetilde θ)$ decays exponentially to zero in $H^1$ and $(u, θ)$ converges to $(\bar u, \barθ)$. This result reflects the stratification phenomenon of buoyancy-driven fluids.

math.AP↗
arXiv2018

Global Existence of Strong Solutions to Compressible Navier-Stokes System with Degenerate Heat Conductivity in Unbounded Domains

In one-dimensional unbounded domains, we prove global existence of strong solutions to the compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas, when the viscosity is constant and the heat conductivity $κ$ depends on the temperature $θ$ according to $κ=\barκθ^β(β>0)$. Note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case ([Kazhikhov, A. V. Siberian Math. J. 23 (1982), 44-49]) and can be arbitrarily large. Therefore, our result generalizes Kazhikhov's result for the constant heat conductivity case to the degenerate and nonlinear one.

math.AP↗
arXiv2017

ApproxDBN: Approximate Computing for Discriminative Deep Belief Networks

Probabilistic generative neural networks are useful for many applications, such as image classification, speech recognition and occlusion removal. However, the power budget for hardware implementations of neural networks can be extremely tight. To address this challenge we describe a design methodology for using approximate computing methods to implement Approximate Deep Belief Networks (ApproxDBNs) by systematically exploring the use of (1) limited precision of variables; (2) criticality analysis to identify the nodes in the network which can operate with such limited precision while allowing the network to maintain target accuracy levels; and (3) a greedy search methodology with incremental retraining to determine the optimal reduction in precision to enable maximize power savings under user-specified accuracy constraints. Experimental results show that significant bit-length reduction can be achieved by our ApproxDBN with constrained accuracy loss.

cs.NE↗
arXiv2015

Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation

As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40].

math.AP↗
arXiv2015

Global regularity for the 2D Oldroyd-B model in the corotational case

This paper is dedicated to the Oldroyd-B model with fractional dissipation $(-Δ)^ατ$ for any $α>0$. We establish the global smooth solutions to the Oldroyd-B model in the corotational case with arbitrarily small fractional powers of the Laplacian in two spatial dimensions. The methods described here are quite different from the tedious iterative approach used in recent paper \cite{XY}. Moreover, in the Appendix we provide some a priori estimates to the Oldroyd-B model in the critical case which may be useful and of interest for future improvement. Finally, the global regularity to to the Oldroyd-B model in the corotational case with $-Δu$ replaced by $(-Δ)^γu$ for $γ>1$ are also collected in the Appendix. Therefore our result is more closer to the resolution of the well-known global regularity issue on the critical 2D Oldroyd-B model.

math.AP↗
arXiv2015

Hybrid stress quadrilateral finite element approximation for stochastic plane elasticity equations

This paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen-Lo$\grave{e}$ve expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by $k-$version/$p-$version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We show that the derived a priori error estimates are uniform with respect to the Lam$\acute{e}$ constant $λ\in (0, +\infty)$. Finally, we provide some numerical results.

math.NA↗