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Jiahong Wu

Jiahong Wu appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.APComputer VisionArtificial IntelligenceMachine LearningCryptography and Securityeess.IVmath-phmath.MP

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Published work

31 published item(s)

preprint2026arXiv

DocOS: Towards Proactive Document-Guided Actions in GUI Agents

While Graphical User Interface (GUI) agents have shown promising performance in automated device interaction, they primarily depend on static parametric knowledge from pre-training or instruction tuning. This reliance fundamentally limits their ability to handle long-tailed tasks that require explicit procedural knowledge absent from model parameters, often forcing agents to resort to inefficient and brittle trial-and-error exploration. To mitigate this limitation, we introduce \textbf{Proactive Document-Guided Action} for GUI agents in dynamic, open-web environments, a novel paradigm that mirrors human problem-solving by enabling agents to autonomously search for relevant documentation to resolve long-tailed tasks. To evaluate agents' capability in this paradigm, we propose \textbf{DocOS}, a benchmark designed to assess document-guided problem solving in fully interactive environments. DocOS requires agents to autonomously navigate a web browser, locate relevant online documentation, comprehend procedural instructions, and faithfully ground them into executable GUI actions. Extensive experiments reveal that progress is strictly constrained by dual bottlenecks: agents struggle to reliably locate relevant information during proactive search and frequently fail to faithfully ground retrieved instructions into precise actions, pointing toward document-guided interaction as a crucial pathway for enabling self-evolving GUI agents in dynamic environments.

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preprint2026arXiv

Embedding-perturbed Exploration Preference Optimization for Flow Models

Recent advancements have established Reinforcement Learning (RL) as a pivotal paradigm for aligning generative models with human intent. However, group-based optimization frameworks (e.g., GRPO) face a critical limitation: the rapid decay of intra-group variance. As the distinctiveness among samples within a group diminishes, the variance approaches zero. This eliminates the very learning signal required for optimization, rendering the process unstable and forcing the policy into premature stagnation or reward hacking. Existing strategies, such as varying the initial noise or increasing group sizes, often fail to address this fundamental issue, resulting in training instability or diminishing returns. To overcome these challenges, we propose $\textbf{Embedding-perturbed Exploration Preference Optimization (}E^2\textbf{PO)}$, a novel framework that sustains optimization through embedding-level perturbation. Our method introduces structured, embedding-level perturbations within sample groups, guaranteeing a robust variance that preserves the discriminative signal throughout the training process. Extensive experiments demonstrate that our approach significantly outperforms state-of-the-art baselines, achieving a more faithful alignment with human preference.

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preprint2025arXiv

Taming Hallucinations: Boosting MLLMs' Video Understanding via Counterfactual Video Generation

Multimodal Large Language Models (MLLMs) have made remarkable progress in video understanding. However, they suffer from a critical vulnerability: an over-reliance on language priors, which can lead to visual ungrounded hallucinations, especially when processing counterfactual videos that defy common sense. This limitation, stemming from the intrinsic data imbalance between text and video, is challenging to address due to the substantial cost of collecting and annotating counterfactual data. To address this, we introduce DualityForge, a novel counterfactual data synthesis framework that employs controllable, diffusion-based video editing to transform real-world videos into counterfactual scenarios. By embedding structured contextual information into the video editing and QA generation processes, the framework automatically produces high-quality QA pairs together with original-edited video pairs for contrastive training. Based on this, we build DualityVidQA, a large-scale video dataset designed to reduce MLLM hallucinations. In addition, to fully exploit the contrastive nature of our paired data, we propose Duality-Normalized Advantage Training (DNA-Train), a two-stage SFT-RL training regime where the RL phase applies pair-wise $\ell_1$ advantage normalization, thereby enabling a more stable and efficient policy optimization. Experiments on DualityVidQA-Test demonstrate that our method substantially reduces model hallucinations on counterfactual videos, yielding a relative improvement of 24.0% over the Qwen2.5-VL-7B baseline. Moreover, our approach achieves significant gains across both hallucination and general-purpose benchmarks, indicating strong generalization capability. We will open-source our dataset and code.

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preprint2023arXiv

Calmed 3D Navier-Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

We propose a modification to the nonlinear term of the three-dimensional incompressible Navier-Stokes equations (NSE) in either advective or rotational form which "calms" the system in the sense that the algebraic degree of the nonlinearity is effectively reduced. This system, the calmed Navier-Stokes Equations (calmed NSE), utilizes a "calming function" in the nonlinear term to locally constrain large advective velocities. Notably, this approach avoids the direct smoothing or filtering of derivatives, thus we make no modifications to the boundary conditions. Under suitable conditions on the calming function, we are able to prove global well-posedness of calmed NSE and show the convergence of calmed NSE solutions to NSE solutions on the time interval of existence for the latter. In addition, we prove that the dynamical system generated by the calmed NSE in the rotational form possesses both an energy identity and a global attractor. Moreover, we show that strong solutions to the calmed equations converge to strong solutions of the NSE without assuming their existence, providing a new proof of the existence of strong solutions to the 3D Navier-Stokes equations.

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preprint2022arXiv

Global well-posedness for 2D non-resistive compressible MHD system in periodic domain

This paper focuses on the 2D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion in a periodic domain. We present a systematic approach to establishing the global existence of smooth solutions when the initial data is close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. When there is no magnetic diffusion, the magnetic field and the density are governed by forced transport equations and the problem considered here is difficult. This paper implements several key observations and ideas to maximize the enhanced dissipation due to hidden structures and interactions. In particular, the weak smoothing and stabilization generated by the background magnetic field and the extra regularization in the divergence part of the velocity field are fully exploited. Compared with the previous works, this paper appears to be the first to investigate such system on bounded domains and the first to solve this problem by pure energy estimates, which help reduce the complexity in other approaches. In addition, this paper combines the well-posedness with the precise large-time behavior, a strategy that can be extended to higher dimensions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

AinnoSeg: Panoramic Segmentation with High Perfomance

Panoramic segmentation is a scene where image segmentation tasks is more difficult. With the development of CNN networks, panoramic segmentation tasks have been sufficiently developed.However, the current panoramic segmentation algorithms are more concerned with context semantics, but the details of image are not processed enough. Moreover, they cannot solve the problems which contains the accuracy of occluded object segmentation,little object segmentation,boundary pixel in object segmentation etc. Aiming to address these issues, this paper presents some useful tricks. (a) By changing the basic segmentation model, the model can take into account the large objects and the boundary pixel classification of image details. (b) Modify the loss function so that it can take into account the boundary pixels of multiple objects in the image. (c) Use a semi-supervised approach to regain control of the training process. (d) Using multi-scale training and reasoning. All these operations named AinnoSeg, AinnoSeg can achieve state-of-art performance on the well-known dataset ADE20K.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Global regularity of the three-dimensional fractional micropolar equations

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Δ)^αu$ and $(-Δ)^βw$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $α\geq \frac{5}{4}$, $β\geq 0$ and $α+β\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Provenance-based Classification Policy based on Encrypted Search

As an important type of cloud data, digital provenance is arousing increasing attention on improving system performance. Currently, provenance has been employed to provide cues regarding access control and to estimate data quality. However, provenance itself might also be sensitive information. Therefore, provenance might be encrypted and stored in the Cloud. In this paper, we provide a mechanism to classify cloud documents by searching specific keywords from their encrypted provenance, and we prove our scheme achieves semantic security. In term of application of the proposed techniques, considering that files are classified to store separately in the cloud, in order to facilitate the regulation and security protection for the files, the classification policies can use provenance as conditions to determine the category of a document. Such as the easiest sample policy goes like: the documents have been reviewed twice can be classified as "public accessible", which can be accessed by the public.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Stability of Couette flow for 2D Boussinesq system with vertical dissipation

This paper establishes the nonlinear stability of the Couette flow for the 2D Boussinesq equations with only vertical dissipation. The Boussinesq equations concerned here model buoyancy-driven fluids such as atmospheric and oceanographic flows. Due to the presence of the buoyancy forcing, the energy of the standard Boussinesq equations could grow in time. It is the enhanced dissipation created by the linear non-self-adjoint operator $y\partial_x -ν\partial_{yy}$ in the perturbation equation that makes the nonlinear stability possible. When the initial perturbation from the Couette flow $(y, 0)$ is no more than the viscosity to a suitable power (in the Sobolev space $H^b$ with $b>\frac43$), we prove that the solution of the 2D Boussnesq system with only vertical dissipation on $\mathbb T\times \mathbb R$ remains close to the Couette at the same order. A special consequence of this result is the stability of the Couette for the 2D Navier-Stokes equations with only vertical dissipation.

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preprint2020arXiv

The stabilizing effect of the temperature on buoyancy-driven fluids

The Boussinesq system for buoyancy driven fluids couples the momentum equation forced by the buoyancy with the convection-diffusion equation for the temperature. One fundamental issue on the Boussinesq system is the stability problem on perturbations near the hydrostatic balance. This problem can be extremely difficult when the system lacks full dissipation. This paper solves the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion. We establish the stability for the nonlinear system and derive precise large-time behavior for the linearized system. The results presented in this paper reveal a remarkable phenomenon for buoyancy driven fluids. That is, the temperature actually smooths and stabilizes the fluids. If the temperature were not present, the fluid is governed by the 2D Navier-Stokes with only vertical dissipation and its stability remains open. It is the coupling and interaction between the temperature and the velocity in the Boussinesq system that makes the stability problem studied here possible. Mathematically the system can be reduced to degenerate and damped wave equations that fuel the stabilization.

0 citations0 reviewsNo code linkedOpen access
preprint2018arXiv

A Large-scale Attribute Dataset for Zero-shot Learning

Zero-Shot Learning (ZSL) has attracted huge research attention over the past few years; it aims to learn the new concepts that have never been seen before. In classical ZSL algorithms, attributes are introduced as the intermediate semantic representation to realize the knowledge transfer from seen classes to unseen classes. Previous ZSL algorithms are tested on several benchmark datasets annotated with attributes. However, these datasets are defective in terms of the image distribution and attribute diversity. In addition, we argue that the "co-occurrence bias problem" of existing datasets, which is caused by the biased co-occurrence of objects, significantly hinders models from correctly learning the concept. To overcome these problems, we propose a Large-scale Attribute Dataset (LAD). Our dataset has 78,017 images of 5 super-classes, 230 classes. The image number of LAD is larger than the sum of the four most popular attribute datasets. 359 attributes of visual, semantic and subjective properties are defined and annotated in instance-level. We analyze our dataset by conducting both supervised learning and zero-shot learning tasks. Seven state-of-the-art ZSL algorithms are tested on this new dataset. The experimental results reveal the challenge of implementing zero-shot learning on our dataset.

0 citations0 reviewsNo code linkedOpen access
preprint2017arXiv

AI Challenger : A Large-scale Dataset for Going Deeper in Image Understanding

Significant progress has been achieved in Computer Vision by leveraging large-scale image datasets. However, large-scale datasets for complex Computer Vision tasks beyond classification are still limited. This paper proposed a large-scale dataset named AIC (AI Challenger) with three sub-datasets, human keypoint detection (HKD), large-scale attribute dataset (LAD) and image Chinese captioning (ICC). In this dataset, we annotate class labels (LAD), keypoint coordinate (HKD), bounding box (HKD and LAD), attribute (LAD) and caption (ICC). These rich annotations bridge the semantic gap between low-level images and high-level concepts. The proposed dataset is an effective benchmark to evaluate and improve different computational methods. In addition, for related tasks, others can also use our dataset as a new resource to pre-train their models.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

A global regularity result for the 2D Boussinesq equations with critical dissipation

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $Λ^αu$ in the velocity equation and by $Λ^βθ$ in the temperature equation, where $Λ=\sqrt{-Δ}$ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when $(α,β)$ is in the critical range: $α>\frac{\sqrt{1777}-23}{24} =0.798103..$, $β>0$ and $α+ β=1$. This result improves the previous work of Jiu, Miao, Wu and Zhang \cite{JMWZ} which obtained the global regularity for $α> \frac{23-\sqrt{145}}{12} \approx 0.9132$, $β>0$ and $α+ β=1$.

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preprint2015arXiv

Blowup in Stagnation-point Form Solutions of the Inviscid 2d Boussinesq Equations

The 2d Boussinesq equations model large scale atmospheric and oceanic flows. Whether its solutions develop a singularity in finite-time remains a classical open problem in mathematical fluid dynamics. In this work, blowup from smooth nontrivial initial velocities in stagnation-point form solutions of this system is established. On an infinite strip $Ω=\{(x,y)\in[0,1]\times\mathbb{R}^+\}$, we consider velocities of the form $u=(f(t,x),-yf_x(t,x))$, with scalar temperature\, $θ=yρ(t,x)$. Assuming $f_x(0,x)$ attains its global maximum only at points $x_i^*$ located on the boundary of $[0,1]$, general criteria for finite-time blowup of the vorticity $-yf_{xx}(t,x_i^*)$ and the time integral of $f_x(t,x_i^*)$ are presented. Briefly, for blowup to occur it is sufficient that $ρ(0,x)\geq0$ and $f(t,x_i^*)=ρ(0,x_i^*)=0$, while $-yf_{xx}(0,x_i^*)\neq0$. To illustrate how vorticity may suppress blowup, we also construct a family of global exact solutions. A local-existence result and additional regularity criteria in terms of the time integral of $\left\|f_x(t,\cdot)\right\|_{L^\infty([0,1])}$ are also provided.

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preprint2015arXiv

Local well-posedness for the Hall-MHD equations with fractional magnetic diffusion

The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, $(-Δ)^α$. Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need $α\ge 1$ in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any $α>\frac12$. The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane

This paper focuses on the two-dimensional Benjamin-Bona-Mahony and Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is at the global (in time) well-posedness of the initial-and boundary-value problem for these equations defined in the upper half-plane. Under suitable growth conditions on the flux function, we are able to establish the global well-posedness in a Sobolev class. When the initial- and boundary-data become more regular, the corresponding solutions are shown to be classical. In addition, the continuous dependence on the data is also obtained.

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preprint2014arXiv

An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations

We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations.

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preprint2014arXiv

Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models

We discuss general incompressible inviscid models, including the Euler equations, the surface quasi-geostrophic equation, incompressible porous medium equation, and Boussinesq equations. All these models have classical unique solutions, at least for short time. We show that they have real analytic Lagrangian paths. More precisely, we show that as long as a solution of any of these equations is in a class of regularity that assures Hölder continuous gradients of velocity, the corresponding Lagrangian paths are real analytic functions of time. The method of proof is conceptually straightforward and general, and we address the combinatorial issues head-on.

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preprint2013arXiv

Small global solutions to the damped two-dimensional Boussinesq equations

The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the "vortex stretching", we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption. By positioning the solutions in a suitable functional setting (more precisely the homogeneous Besov space $\mathring{B}^1_{\infty,1}$), we are able to obtain a unique global solution under a minimal smallness assumption.

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preprint2013arXiv

The 2D Incompressible Magnetohydrodynamics Equations with only Magnetic Diffusion

This paper examines the global (in time) regularity of classical solutions to the 2D incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator $(-Δ)^β$. We establish the global regularity for the case when $β>1$. This result significantly improves previous work which requires $β>\frac32$ and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely the case when $β=1$.

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preprint2012arXiv

The 2D incompressible Boussinesq equations with general critical dissipation

This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $α+β=1$ when $Λ^α\equiv (-Δ)^{\fracα{2}}$ and $Λ^β$ represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. We establish the global regularity for the general case with $α+β=1$ and $0.9132\approx α_0<α<1$. The cases when $α=1$ and when $α=0$ were previously resolved by Hmidi, Keraani and Rousset \cite{HKR1,HKR2}. The global existence and uniqueness is achieved here by exploiting the global regularity of a generalized critical surface quasi-gesotrophic equation as well as the regularity of a combined quantity of the vorticity and the temperature.

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preprint2011arXiv

Generalized surface quasi-geostrophic equations with singular velocities

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi-geostrophic (SQG) equation with the velocity field $u$ related to the scalar $θ$ by $u=\nabla^\perpΛ^{β-2}θ$, where $1<β\le 2$ and $Λ=(-Δ)^{1/2}$ is the Zygmund operator. The borderline case $β=1$ corresponds to the SQG equation and the situation is more singular for $β>1$. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions and the local existence of patch type solutions. The second family is a dissipative active scalar equation with $u=\nabla^\perp (\log(I-Δ))^μθ$ for $μ>0$, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani \cite{Oh}.

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preprint2011arXiv

Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation

This paper establishes the global in time existence of classical solutions to the 2D anisotropic Boussinesq equations with vertical dissipation. When only the vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the $L^\infty$-norm of the vertical velocity $v$ and prove that $\|v\|_{L^{r}}$ with $2\le r<\infty$ at any time does not grow faster than $\sqrt{r \log r}$ as $r$ increases. A delicate interpolation inequality connecting $\|v\|_{L^\infty}$ and $\|v\|_{L^r}$ then yields the desired global regularity.

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preprint2011arXiv

The 2D Boussinesq equations with logarithmically supercritical velocities

This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq-Navier-Stokes equations. The velocity $u$ in this system is related to the vorticity $ω$ through the relations $u=\nabla^\perp ψ$ and $Δψ= Λ^σ(\log(I-Δ))^γω$, which reduces to the standard velocity-vorticity relation when $σ=γ=0$. When either $σ>0$ or $γ>0$, the velocity $u$ is more singular. The "quasi-velocity" $v$ determined by $\nabla\times v =ω$ satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when $σ=0$ and $γ\ge 0$. In addition, the vorticity $ω$ is shown to be globally bounded in several functional settings such as $L^2$ for $σ>0$ in a suitable range.

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preprint2011arXiv

Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions

The paper is concerned with the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space $L^q$ with $2<q\le\infty$, we show the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In the special case when the vorticity is in $L^\infty$, an explicit convergence rate is obtained.

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preprint2010arXiv

Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations

This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field $u$ is determined by the active scalar $θ$ through $\mathcal{R} Λ^{-1} P(Λ) θ$ where $\mathcal{R}$ denotes a Riesz transform, $Λ=(-Δ)^{1/2}$ and $P(Λ)$ represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case $P(Λ)=I$ while the surface quasi-geostrophic (SQG) equation to $P(Λ) =Λ$. We obtain the global regularity for a class of equations for which $P(Λ)$ and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with $P(Λ)= (\log(I-Δ))^γ$ for any $γ>0$ are globally regular.

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preprint2010arXiv

Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations

Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component $u_j$ of the velocity field $u$ is determined by the scalar $θ$ through $u_j =\mathcal{R} Λ^{-1} P(Λ) θ$ where $\mathcal{R}$ is a Riesz transform and $Λ=(-Δ)^{1/2}$. The 2D Euler vorticity equation corresponds to the special case $P(Λ)=I$ while the SQG equation to the case $P(Λ) =Λ$. We develop tools to bound $\|\nabla u||_{L^\infty}$ for a general class of operators $P$ and establish the global regularity for the Loglog-Euler equation for which $P(Λ)= (\log(I+\log(I-Δ)))^γ$ with $0\le γ\le 1$. In addition, a regularity criterion for the model corresponding to $P(Λ)=Λ^β$ with $0\le β\le 1$ is also obtained.

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preprint2009arXiv

Complex-valued Burgers and KdV-Burgers equations

Spatially periodic complex-valued solutions of the Burgers and KdV-Burgers equations are studied in this paper. It is shown that for any sufficiently large time T, there exists an explicit initial data such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of series solutions is established for initial data satisfying mild conditions.

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preprint2008arXiv

Global regularity for a modified critical dissipative quasi-geostrophic equation

In this paper, we consider the modified quasi-geostrophic equation \begin{gather*} \del_t θ+ (u \cdot \grad) θ+ κΛ^αθ= 0 u = Λ^{α- 1} R^{\perp}θ. \end{gather*} with $κ> 0$, $α\in (0,1]$ and $θ_0 \in \lp{2}(\R^2)$. We remark that the extra $Λ^{α- 1}$ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system.

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