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partial regularity FOR STATIONARY NAVIER-STOKES SYSTEMS BY THE METHOD OF A-HARMONIC APPROXIMATION

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Acta Mathematica Scientia 2020年第3期40卷 835-854页

作者： Yichen DAI Zhong TAN School of Mathematical Sciences Xiamen UniversityFujian 361005China School of Mathematical Science and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityFujian 361005China

In this article,we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth *** proof is based on the A-harmonic approximation *** this article,we extend the result of Shuhong Chen and Zhong Tan[7]and Giaquinta and Modica[18]to the stationary Navier-Stokes system with subquadratic growth.

关键词： Stationary Navier-Stokes systems controllable growth condition partial regularity A-harmonic approximation

partial regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow

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Acta Mathematicae Applicatae Sinica 2021年第2期37卷 348-363页

作者： Ya-zhou CHEN Hai-liang LI Xiao-ding SHI College of Mathematics and Physics Beiing University of Chemical TechnologyBejing 100029China School of Mathematical Sciences Capital Normal UniversityBejing 100048China

This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domainΩ■R^(n),n≥*** is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain(for n≥2).For 3 D model,it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.

关键词： partial regularity suitable weak solutions shear-thinning flow

partial regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy

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Acta Mathematica Sinica,English Series 2023年第11期39卷 2219-2246页

作者： Yan Qing WANG Yi Ke HUANG Gang WU Dao Guo ZHOU College of Mathematics and Information Science Zhengzhou University of Light IndustryZhengzhou 450002P.R.China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049P.R.China School of Mathematics Hangzhou Normal UniversityHangzhou 311121P.R.China

In this paper,we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set S of suitable weak solutions and the parameterαin the nonlinear term in the following parabolic equationh_(t)+h_(xxxx)+∂_(xx)|h_(x)|^(α)=*** is shown that when 5/3≤α<7/3,the 3α-5/α−1-dimensional parabolic Hausdorff measure of S is zero,which generalizes the recent corresponding work of Ozánski and Robinson in[SIAM ***.,51,228–255(2019)]forα=2 and f=*** same result is valid for a 3D modified Navier–Stokes system.

关键词： Surface growth model modified Navier–Stokes equations partial regularity Hausdorff dimension

partial regularity FOR THE NAVIER-STOKES-FOURIER SYSTEM

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Acta Mathematica Scientia 2011年第5期31卷 1653-1670页

作者： Luisa Consiglieri Independent Researcher Painel da Lameira Cx 8 7300-405 Portalegre Portugal

This paper addresses a nonstationary flow of heat-conductive incompressible Newtonian fluid with temperature-dependent viscosity coupled with linear heat transfer with advection and a viscous heat source term, under Navier/Dirichlet boundary conditions. The partial regularity for the velocity of the fluid is proved for each proper weak solution, that is, for such weak solutions which satisfy some local energy estimates in a similar way to the suitable weak solutions of the Navier-Stokes system. Finally, we study the nature of the set of points in space and time upon which proper weak solutions could be singular.

关键词： partial regularity Navier-Stokes-Fourier system Joule effect suitable weak solutions

partial regularity FOR OPTIMAL DESIGN PROBLEMS INVOLVING BOTH BULK AND SURFACE ENERGIES

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Chinese Annals of Mathematics,Series B 1999年第2期20卷 137-158页

作者： F.H.LIN R.V.KOHN CourantInstituteofMathematicalSciences USA projectsupportedbytheNationalNaturalScienceFoundationofUSA

This paper studies a class of variational problems which involving both bulk and surfaceenergies. The bulk energy is of Dirichlet type though it can be in very general forms allowingunknowns to be scalar or *** surface energy is an arbitrary elliptic parametric integralwhich is defined on a free interface. One also allows other constraints such as volumes of partitioning sets. One establishes the existence and regularity theory, in particular, the regularityof the free interface of such problems.

关键词： partial regularity Optimal design problem Nonlinear variational problems

partial regularity for the 3D Magneto-hydrodynamics System with Hyper-dissipation

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Acta Mathematica Sinica,English Series 2015年第7期31卷 1097-1112页

作者： Wei REN Gang WU Department of Mathematics and Systems Science BeiHang University School of Mathematical Science University of Chinese Academy of Science

We prove that for the 3D MHD equations with hyper-dissipations （-△）^α （ 1 〈 α 〈 5/4） the Hausdorff dimension of singular set at the first blowing up time is at most 5 - 4α, by means of physical and frequency... 详细信息

关键词： 3D MHD system fractal dissipation partial regularity physical and frequency localization Bony's paraproduct decomposition

partial regularity of weak solutions for Landau-Lifshitz system with potential

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Science China Mathematics 2009年第5期52卷 981-990页

作者： CHU YuMing LIU XianGao Department of Mathematics Huzhou Teachers CollegeHuzhou 313000China Institute of Mathematics Fudan UniversityShanghai 200433China

In this note, we prove the partial regularity of stationary weak solutions for the Landau- Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey's energy to avoid the difficulties by blowing up.

关键词： partial regularity potential Landau-Lifshitz system 35K55 35D10

On partial regularity of Suitable Weak Solutions to the Stationary Fractional Navier-Stokes Equations in Dimension Four and Five

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Acta Mathematica Sinica,English Series 2017年第12期33卷 1632-1646页

作者： Xiao Li GUO Yue Yang MEN Department of Mathematics and Information Science Zhengzhou University of Light Industry Zhengzhou 450002 P. R. China Institute of Applied Physics and Computational Mathematics Beijing 100088 P. R. China

In this paper, we investigate the partial regularity of suitable weak solutions to the multi- dimensional stationary Navier-Stokes equations with fractional power of the Laplacian （-△）^α （n/6 ≤α〈1 and a ≠ 1/2）. It is shown that the n ＋ 2 - 6α （3 ≤ n ≤5） dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu [19] to four and five dimension. Moreover, the pressure in ε-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].

关键词： Stationary Navier-Stokes equations suitable weak solutions partial regularity

C^(1,α)-partial regularity FOR NONLINEAR ELLIPTIC SYSTEMS

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Acta Mathematica Scientia 1995年第3期15卷 254-263页

作者：谭忠 Department of Mathematics Xiamen University Xiamen 361005 China

We prove C1.α-partial regularity of weak solutions of nonlinear elliptic systems under the main assumption that Aia and Bi satisfy the controllable growth condition or the natural growth condition.

关键词： nonlinear elliptic systems controllable growth condition natural growth condition partial regularity