BMO and the regularity criterion for weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity criterion for weak solutions to the incompressible magnetohydrodynamic equations. We derive the regularity of weak solutions in the marginal class. Moreover, our result demonstrates that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations.     

Gevrey class regularity for quasi-linear sub-elliptic equations of second order

In this work we study the Gevrey regularity of solutions to a general class of second order quasi-linear equations. Under some kind of sub-ellipticity conditions, we obtain the Gevrey regularity of weak solutions to these equations.     

Regularity criteria for the three-dimensional MHD equations

In this paper,we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions.By using some classical inequalities,we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.     

Boussinesq方程解的部分正则性

本文考虑Boussinesq方程一类合适弱解的部分正则性.我们先运用广义能量不等式和奇异积分理论得到一些无维量的估计;再通过合适弱解满足的等式,运用迭代技巧,推导出温度场的小性估计;最后由尺度分析(scaling arguments)得到了一类合适弱解的部分正则性.     

一类趋化性生物模型行波解的存在性与正则性

研究了一类趋化性生物模型行波解的存在性和正则性.通过直接计算得到了其行波解存在的充分必要条件;在一定条件下,研究了行波解的正则与非正则的性质;在特殊情形下给出了行波解的显式解.     

Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation

In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the regularity of densities of such solutions. We also apply these results to construct a classical solution to the stochastic transport equation when the drift is Lipschitz.     

Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations

We first prove the existence and regularity of the trajectory attractor for a three-dimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.     

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

In this paper we establish interior regularity for weak solutions and partial regularity for suitable weak solutions of the perturbed Navier-Stokes system, which can be regarded as generalizations of the results in L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure. Appl. Math. 35 (1982), 771–831, and S. Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscr. Math. 69 (1990), 237–254.     

Metric Regularity of the Sum of Multifunctions and Applications

The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported.     

Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.     

Regularity theory for the Isaacs equation through approximation methods

In this paper, we propose an approximation method to study the regularity of solutions to the Isaacs equation. This class of problems plays a paramount role in the regularity theory for fully nonlinear elliptic equations. First, it is a model-problem of a non-convex operator. In addition, the usual mechanisms to access regularity of solutions fall short in addressing these equations. We approximate an Isaacs equation by a Bellman one, and make assumptions on the latter to recover information for the former. Our techniques produce results in Sobolev and Hölder spaces; we also examine a few consequences of our main findings.     

The method of p-harmonic approximation and optimal interior partial regularity for energy minimizing p-harmonic maps under the controllable growth condition

In this paper, we are concerned with the partial regularity for the weak solutions of energy minimizing p-harmonic maps under the controllable growth condition. We get the interior partial regularity by the p-harmonic approximation method together with the technique used to get the decay estimation on some Degenerate elliptic equations and the obstacle problem by Tan and Yan. In particular, we directly get the optimal regularity.     

Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control

In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the H¹-energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in H²-sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data.     

完全非线性混合可积微分算子解的正则性——献给陆善镇教授75华诞

本文介绍一类带有非对称正核的完全非线性混合可积微分算子并研究其解的正则性.具体地,本文建立关于该算子非局部A-B-P（Alexandroff-Bakelman-Pucci）估计、Harnack不等式以及解的Holder和C1,α正则性.     

On the regularity of the axisymmetric solutions of the Navier-Stokes equations

We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations. Received: 11 April 2000; in final firm: 26 November 2000 / Published online: 28 February 2002     

On Regularity Property of Retarded Ornstein–Uhlenbeck Processes in Hilbert Spaces

In this work, some regularity properties of mild solutions for a class of stochastic linear functional differential equations driven by infinite-dimensional Wiener processes are considered. In terms of retarded fundamental solutions, we introduce a class of stochastic convolutions which naturally arise in the solutions and investigate their Yosida approximants. By means of the retarded fundamental solutions, we find conditions under which each mild solution permits a continuous modification. With the aid of Yosida approximation, we study two kinds of regularity properties, temporal and spatial ones, for the retarded solution processes. By employing a factorization method, we establish a retarded version of the Burkholder–Davis–Gundy inequality for stochastic convolutions.     

Semilinear hypoelliptic differential operators with multiple characteristics

In this paper we consider the regularity of solutions of semilinear differential equations principal parts of which consist of linear polynomial operators constructed from real vector fields. Based on the study of fine properties of the principal linear parts we then obtain the regularity of solutions of the nonlinear equations. Some results obtained in this article are also new in the frame of linear theory.

     

Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries

Here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems. The limiting problems involve multiple valued harmonic functions or, in general, harmonic maps to singular spaces and free interfaces between supports of various components of the maps. The main results of the paper are the uniform Lipschitz regularity of solutions as well as the regularity of free interfaces.

     

Time analyticity and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow

We prove that solutions of the Navier-Stokes equations of three-dimensional compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. As consequences we derive backward uniqueness of solutions as well as sharp rates of smoothing for higher-order Lagrangean time derivatives. The solutions under consideration are in a reasonably broad regularity class corresponding to small-energy initial data with a small degree of regularity, the latter being required for conversion to the Lagrangean coordinate system in which the analysis is carried out.     

A mathematical analysis of a nonisothermal Allen–Cahn type system: error estimates

In this article, under certain conditions, we prove the regularity for the solutions of an Allen–Cahn phase‐field type system obtained as limits of approximate solutions constructed by using a semidiscrete spectral Galerkin method. With the help of this improved regularity, as one compares to previous results, we then derive error estimates for the approximate solutions in terms of the inverse of the eigenvalues of the Laplacian operator. The system under investigation may model the evolution of solidification or melting of certain binary alloys. Copyright © 2012 John Wiley & Sons, Ltd.