Partial compactness for Landau-Lifshitz maxwell equation in two-dimension

We study the partial regularity of weak solutions to the 2-dimensional LandauLifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C ∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.     

On partial regularity of suitable weak solutions to the stationary fractional Navier–Stokes equations in dimension four and five

In this paper, we investigate the partial regularity of suitable weak solutions to the multidimensional stationary Navier Stokes equations with fractional power of the Laplacian (-△)~α 1 and α≠ 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 e-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].     

Remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations

In this paper, we are concerned with the partial regularity for suitable weak solutions of the tri-dimensional magnetohydrodynamic equations. With the help of the De Giorgi iteration method, we obtain the results proved by He and Xin (C. He, Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal. 227 (2005) 113–152), namely, the one dimensional parabolic Hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero.     

Partially regular solution to Landau-Lifshitz-Maxwell equations in two dimensions

The present paper is concerned with the existence, uniqueness and singularities of the Landau-Lifshitz equation coupled with Maxwell equations on Riemannian manifold in two dimensions. It is shown that there exists a weak solution which is regular with exception of at most finitely many singular points.     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

Boundary partial regularity for the high dimensional Navier–Stokes equations

We consider suitable weak solutions of the incompressible Navier–Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero in both cases.     

非线性差分方程的多重周期解

用变分方法得到一类非线性差分方程多重周期解的存在性．我们的结果推广了Cai，Yu和Guo[Comput．Math．Appl．，52（2006），1630-1647]的结果，并且这里给出的证明显著地简化了．     

A note on the iterative solutions of general coupled matrix equation

Recently, Ding and Chen [F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269-2284] developed a gradient-based iterative method for solving a class of coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified, so that the iterative solutions are obtained by applying hierarchical identification principle. In this note, by considering the coupled Sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm. We also propose some faster algorithms and present some numerical results.     

On the Geometric Measure of Nodal Sets of Solutions

We present some general methods for the estimation of the local Hausdorff measure of nodal sets of solutions to elliptic and parabolic equations. Our main results (Theorems 3.1 and 4.1) improve previous results of Lin Fanghua in [1].     

A Note on Three Classes of Nonlinear Parabolic Equations with Measurable Coefficients

In this note a note and simple technique is presented to replace the complicated one in [1] to obtain Hölder continuity of the weak solutions for a class of nonlinear parabolic equations with measurable coefficients, whose prototype is the singular p-Laplacian. This new approach is also applied to two other classes of nonlinear parabolic equations with measurable coefficients, whose weak solutions exhibit the similar property to those of equations mentioned above.     

Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces

We consider a Kolmogorov operator L₀ in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drift , defined on a suitable space of regular functions. We show that L₀ is essentially m-dissipative in the space Lp([0,T]×H;ν), p?1, where and the family (νt)_t∈[0,T] is a solution of the Fokker-Planck equation given by L₀. As a consequence, the closure of L₀ generates a Markov C₀-semigroup. We also prove uniqueness of solutions to the Fokker-Planck equation for singular drifts F. Applications to reaction-diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753-774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397-418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631-640] to infinite dimensions.     

Regularity for the porous medium equation with variable exponent: The singular case

We extend to the singular case the results of [E. Henriques, J.M. Urbano, Intrinsic scaling for PDEs with an exponential nonlinearity, Indiana Univ. Math. J. 55 (5) (2006) 1701-1721] concerning the regularity of weak solutions of the porous medium equation with variable exponent. The method of intrinsic scaling is used to show that local weak solutions are locally continuous.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

动态规划中提出的一类泛函方程组公共解和重合解的存在性定理

本文讨论了动态规划中提出的一类更一般的泛函方程组公共解和重合解的存在性问题.本文的结果不仅包含引文[6,7]中相应结果为特例,而且也对引文[2～5]在讨论动态规划的原理和模型时所提出的一类新型的泛函方程给出解的存在性条件.     

A generalized finite type condition for iterated function systems

We study iterated function systems (IFSs) of contractive similitudes on Rd with overlaps. We introduce a generalized finite type condition which extends a more restrictive condition in [S.-M. Ngai, Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2) 63 (3) (2001) 655-672] and allows us to include some IFSs of contractive similitudes whose contraction ratios are not exponentially commensurable. We show that the generalized finite type condition implies the weak separation property. Under this condition, we can identify the attractor of the IFS with that of a graph-directed IFS, and by modifying a setup of Mauldin and Williams [R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811-829], we can compute the Hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix.     

Existence theorems for perturbed abstract measure differential equations

In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [B.C. Dhage, On some nonlinear alternatives of Leray-Schauder type and functional integral equations, Arch. Math. (Brno) 42 (2006) 11-23] under the mixed generalized Lipschitz and Carathéodory conditions. The existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage given in the above-mentioned reference, on ordered Banach spaces. Our existence results include the existence results of Sharma [R.R. Sharma, An abstract measure differential equation, Proc. Amer. Math. Soc. 32 (1972) 503-510], Joshi [S.R. Joshi, A system of abstract measure delay differential equations, J. Math. Phy. Sci. 13 (1979) 497-506] and Shendge and Joshi [G.R. Shendge, S.R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hung. 41 (1983) 53-54] as special cases under weaker continuity condition.     

Initial-boundary value problem for higher dimensional landau–lifshitz systems

This article is concerned with the partial regularity of the weak solutions satisfying energy inequality and monotonicity inequality to the initial-boundary value problem of Landau–Lifshitz system of ferromagnetic spin chain with Gilbert damping in higher dimensions. A generalized monotonicity inequality is derived. The singular set and its Hausdorff dimension are discussed.     

Global Existence of Large Data Weak Solutions for a Simplified Compressible Oldroyd-B Model Without Stress Diffusion

We start with the compressible Oldroyd-B model derived in [2] (J. W. Barrett, Y. Lu, and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15 (2017), 1265-1323), where the existence of global-in-time finite-energy weak solutions was shown in two dimensional setting with stress diffusion. In the paper, we investigate the case without stress diffusion. We first restrict ourselves to the corotational setting as in [28] (P. L. Lions, and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math., Ser. B, 21(2) (2000), 131-146). We further assume the extra stress tensor is a scalar matrix and we derive a simplified model which takes a similar form as the multi-component compressible Navier-Stokes equations, where, however, the pressure term related to the scalar extra stress tensor has the opposite sign. By employing the techniques developed in [30,35], we can still prove the global-in-time existence of finite energy weak solutions in two or three dimensions, without the presence of stress diffusion.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields

Let X = {X(t):t ∈ R~N} be an anisotropic random field with values in R~d.Under certain conditions on X,we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel-Riesz capacity.We also obtain the Hausdorff dimension of its inverse image,and the Hausdorff and packing dimensions of its level sets.These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.