This article proves the logarithmically improved Serrin＇s criterion for solutions of the 3D generalized magneto-hydrodynamic equations in terms of the gradient of the velocity field, which can be regarded as improvem...This article proves the logarithmically improved Serrin＇s criterion for solutions of the 3D generalized magneto-hydrodynamic equations in terms of the gradient of the velocity field, which can be regarded as improvement of results in [10] （Luo Y W. On the regularity of generalized MHD equations. J Math Anal Appl, 2010, 365： 806-808） and [18] （Zhang Z J. Remarks on the regularity criteria for generalized MHD equations. J Math Anal Appl, 2011, 375：799 802）.展开更多
This note investigates the global regularity of 3D liquid crystal equations in terms of the vertical derivative of uh.More precisely,we prove that if the vertical derivative of the horizontal velocity component uh sat...This note investigates the global regularity of 3D liquid crystal equations in terms of the vertical derivative of uh.More precisely,we prove that if the vertical derivative of the horizontal velocity component uh satisfiesδ_(3uh)∈L^(p)(0,T;R^(3))with 2/p+3/q≤3/2,2≤p≤∞,then the local strong solution(u,d)can be smoothly extended beyond t=T.展开更多
In this paper, we study the initial boundary problem for 3D incompressible density-dependent Navier-Stokes-Allen-Cahn equations, and give a regularity criterion for local strong solutions. Our result refines the blow-...In this paper, we study the initial boundary problem for 3D incompressible density-dependent Navier-Stokes-Allen-Cahn equations, and give a regularity criterion for local strong solutions. Our result refines the blow-up criterion in [1].展开更多
In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space B ∞,∞ -r （R2）. The result shows that if 0 is a weak solutions satisfies ...In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space B ∞,∞ -r （R2）. The result shows that if 0 is a weak solutions satisfies ∫ 0 T ｜｜ θ （·,s）｜｜a/a-r B ∞,∞ -r /（1＋ln（e＋｜｜ ⊥（·,s）｜｜ L r2） ds〈∞ for some 0〈r〈a and 0〈a〈1,then θ is regular at t = T. In view of the embedding L 2/r M p 2/r B ∞,∞ -r with 2≤p〈2/r and 0≤r〈1, we see that our result extends the results due to [20] and [31].展开更多
Kernel-based methods work by embedding the data into a feature space and then searching linear hypothesis among the embedding data points. The performance is mostly affected by which kernel is used. A promising way is...Kernel-based methods work by embedding the data into a feature space and then searching linear hypothesis among the embedding data points. The performance is mostly affected by which kernel is used. A promising way is to learn the kernel from the data automatically. A general regularized risk functional （RRF） criterion for kernel matrix learning is proposed. Compared with the RRF criterion, general RRF criterion takes into account the geometric distributions of the embedding data points. It is proven that the distance between different geometric distdbutions can be estimated by their centroid distance in the reproducing kernel Hilbert space. Using this criterion for kernel matrix learning leads to a convex quadratically constrained quadratic programming （QCQP） problem. For several commonly used loss functions, their mathematical formulations are given. Experiment results on a collection of benchmark data sets demonstrate the effectiveness of the proposed method.展开更多
In this paper, we study the regularity criterion for the three-dimensional Boussinesq equations in Besov spaces. We show that the smooth solution(u,θ) is regular if the horizonal velocity u_h holds ■.
In this paper we consider the 3D ideal MHD system in a bounded domain.We first prove a regularity criterion and then use the bootstrap argument to show the well-posedness of global small solutions.
In this paper,we consider the density-dependent magnetohydrodynamic equations with vacuum,and provide a regularity criterion involving the velocity and magnetic fields in Besov space of negative order,which improves[J...In this paper,we consider the density-dependent magnetohydrodynamic equations with vacuum,and provide a regularity criterion involving the velocity and magnetic fields in Besov space of negative order,which improves[Jishan FAN,Fucai LI,G.NAKAMURA,Zhong TAN,Regularity criteria for the three-dimensional magnetohydrodynamic equations.J.Differential Equations,2014,256（8）：2858 2875]in some sense.The method is to establish a new bilinear estimate.展开更多
In this paper, we prove that suitable weak solution （u,b） of tne 3-D MHD equations can be extended beyond T if u E L∞（0,T; La（R3）） and the horizontal components bh of the magnetic field satisfies the well-known...In this paper, we prove that suitable weak solution （u,b） of tne 3-D MHD equations can be extended beyond T if u E L∞（0,T; La（R3）） and the horizontal components bh of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.展开更多
Regularity criteria in terms of bounds for the pressure are derived for the 3D MHD equations in a bounded domain with slip boundary conditions.A list of three regularity criteria is shown.
We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity.
In this note, a logarithmic improved regularity criteria for the micropolar fluid equations are established in terms of the velocity field or the pressure in the homogeneous Besov space.
Let u = （Uh,U3） be a smooth solution of the 3-D Navier-Stokes equations in R3 × [0, T）. It was proved that if u3 ∈ L^∞（0,T;Bp,q-1＋3/p（R3）） for 3 〈 p,q 〈 oe and uh ∈ L^∞（0, T;BMO-1（R3）） with uh（...Let u = （Uh,U3） be a smooth solution of the 3-D Navier-Stokes equations in R3 × [0, T）. It was proved that if u3 ∈ L^∞（0,T;Bp,q-1＋3/p（R3）） for 3 〈 p,q 〈 oe and uh ∈ L^∞（0, T;BMO-1（R3）） with uh（T） ∈ VMO-1（R3）, then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al. （2016）, which requires u ∈ L^∞（O,T;Bp,^-11＋3/P（R3））. Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.展开更多
文摘This article proves the logarithmically improved Serrin＇s criterion for solutions of the 3D generalized magneto-hydrodynamic equations in terms of the gradient of the velocity field, which can be regarded as improvement of results in [10] （Luo Y W. On the regularity of generalized MHD equations. J Math Anal Appl, 2010, 365： 806-808） and [18] （Zhang Z J. Remarks on the regularity criteria for generalized MHD equations. J Math Anal Appl, 2011, 375：799 802）.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1196103211971209)the Natural Science Foundation of Jiangxi Province(Grant No.20191BAB201003).
文摘This note investigates the global regularity of 3D liquid crystal equations in terms of the vertical derivative of uh.More precisely,we prove that if the vertical derivative of the horizontal velocity component uh satisfiesδ_(3uh)∈L^(p)(0,T;R^(3))with 2/p+3/q≤3/2,2≤p≤∞,then the local strong solution(u,d)can be smoothly extended beyond t=T.
文摘In this paper, we study the initial boundary problem for 3D incompressible density-dependent Navier-Stokes-Allen-Cahn equations, and give a regularity criterion for local strong solutions. Our result refines the blow-up criterion in [1].
文摘In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space B ∞,∞ -r （R2）. The result shows that if 0 is a weak solutions satisfies ∫ 0 T ｜｜ θ （·,s）｜｜a/a-r B ∞,∞ -r /（1＋ln（e＋｜｜ ⊥（·,s）｜｜ L r2） ds〈∞ for some 0〈r〈a and 0〈a〈1,then θ is regular at t = T. In view of the embedding L 2/r M p 2/r B ∞,∞ -r with 2≤p〈2/r and 0≤r〈1, we see that our result extends the results due to [20] and [31].
基金supported by the National Natural Science Fundation of China (60736021)the Joint Funds of NSFC-Guangdong Province(U0735003)
文摘Kernel-based methods work by embedding the data into a feature space and then searching linear hypothesis among the embedding data points. The performance is mostly affected by which kernel is used. A promising way is to learn the kernel from the data automatically. A general regularized risk functional （RRF） criterion for kernel matrix learning is proposed. Compared with the RRF criterion, general RRF criterion takes into account the geometric distributions of the embedding data points. It is proven that the distance between different geometric distdbutions can be estimated by their centroid distance in the reproducing kernel Hilbert space. Using this criterion for kernel matrix learning leads to a convex quadratically constrained quadratic programming （QCQP） problem. For several commonly used loss functions, their mathematical formulations are given. Experiment results on a collection of benchmark data sets demonstrate the effectiveness of the proposed method.
基金Supported by Xinyang College Scientific Research Project(Grant Nos.2022-XJLYB-004 and 2022-XJLYB-018)。
文摘In this paper, we study the regularity criterion for the three-dimensional Boussinesq equations in Besov spaces. We show that the smooth solution(u,θ) is regular if the horizonal velocity u_h holds ■.
基金supported by the key project of university natural science of Anhui province(No.KJ2017A453)the University Teaching Research Foundation of Anhui province(No.2016jyxm0693)。
文摘In this paper we consider the 3D ideal MHD system in a bounded domain.We first prove a regularity criterion and then use the bootstrap argument to show the well-posedness of global small solutions.
基金Supported by the Natural Science Foundation of Jiangxi Province(Grant No.20151BAB201010)the National Natural Science Foundation of China(Grant Nos.1150112511361004)
文摘In this paper,we consider the density-dependent magnetohydrodynamic equations with vacuum,and provide a regularity criterion involving the velocity and magnetic fields in Besov space of negative order,which improves[Jishan FAN,Fucai LI,G.NAKAMURA,Zhong TAN,Regularity criteria for the three-dimensional magnetohydrodynamic equations.J.Differential Equations,2014,256（8）：2858 2875]in some sense.The method is to establish a new bilinear estimate.
基金Supported by NSFC(Grant Nos.11301048,11671067)the Fundamental Research Funds for the Central Universitiesthe Institute of Mathematical Sciences of CUHK
文摘In this paper, we prove that suitable weak solution （u,b） of tne 3-D MHD equations can be extended beyond T if u E L∞（0,T; La（R3）） and the horizontal components bh of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.
基金J.Fan is partially supported by NSFC(No.11171154)Ju is supported by NSFC(Grant Nos.12071044,12131007).
文摘Regularity criteria in terms of bounds for the pressure are derived for the 3D MHD equations in a bounded domain with slip boundary conditions.A list of three regularity criteria is shown.
基金Acknowledgements Fan was supported by the National Natural Science Foundation of China （Grant No. 11171154） Li was supported by the National Natural Science Foundation of China （Grant Nos. 11271184, 11671193） and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘We prove two new regularity criteria for the 3D incompressible Navier-Stokes equations in a bounded domain. Our results also hold for the 3D Boussinesq system with zero heat conductivity.
文摘In this note, a logarithmic improved regularity criteria for the micropolar fluid equations are established in terms of the velocity field or the pressure in the homogeneous Besov space.
基金supported by National Natural Science Foundation of China (Grant Nos. 11301048, 11371039 and 11425103)the Fundamental Research Funds for the Central Universities
文摘Let u = （Uh,U3） be a smooth solution of the 3-D Navier-Stokes equations in R3 × [0, T）. It was proved that if u3 ∈ L^∞（0,T;Bp,q-1＋3/p（R3）） for 3 〈 p,q 〈 oe and uh ∈ L^∞（0, T;BMO-1（R3）） with uh（T） ∈ VMO-1（R3）, then u can be extended beyond T. This result generalizes the recent result proved by Gallagher et al. （2016）, which requires u ∈ L^∞（O,T;Bp,^-11＋3/P（R3））. Our proof is based on a new interior regularity criterion in terms of one velocity component, which is independent of interest.
