Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation

We show that a solution to the supercritical surface quasi-geostrophic equation that is smooth up to a certain time must remain smooth forever.     

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.     

TheC∞ version of the edge of the wedge for nonsmooth edges

One discusses characterizations of traces of smooth functions on nonsmooth manifolds, in the spirit of the edge-of-the-wedge theorem. Application to the study of maximum modulus sets in smooth boundaries is mentioned.     

ITS示范工程对城市交通畅通性的贡献率研究

通过分析城市交通畅通性的影响因素及其与ITS(智能交通系统)的关系,挖掘ITS对城市交通畅通性的贡献率指标.搜集实证研究对象数据,采用灰色预测模型预测贡献率指标的数值,将ITS示范工程实施前后的数值进行对比,计算得到ITS示范工程对城市交通畅通性的贡献率.由贡献率结果判断ITS示范工程对城市交通畅通性的作用及ITS示范工程的实施效果如何,为未来ITS的发展提供相应的参考建议.     

六阶对流Cahn-Hilliard方程解的整体适定性

In this paper,we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data.We first construct a local smooth solution,then by combining some a priori estimates,continuity argument,the local smooth solution is extended step by step to all t>0 provided that the L1 norm of initial data is suitably small and the smooth nonlinear functions f(u)and g(u)satisfy certain local growth conditions at some fixed point■.     

Mathematical analysis of absorbing boundary conditions for the wave equation: the corner problem

Our goal in this work is to establish the existence and the uniqueness of a smooth solution to what we call in this paper the corner problem, that is to say, the wave equation together with absorbing conditions at two orthogonal boundaries. First we set the existence of a very smooth solution to this initial boundary value problem. Then we show the decay in time of energies of high order--higher than the order of the boundary conditions. This result shows that the corner problem is strongly well-posed in spaces smaller than in the half-plane case. Finally, specific corner conditions are derived to select the smooth solution among less regular solutions. These conditions are required to derive complete numerical schemes.

     

Geodesic flow on the diffeomorphism group of the circle

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.     

On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture

We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher-dimensional codimension-one Anosov flow is topologically transitive. Recently, Simić showed that any higher-dimensional codimension-one Anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an Anosov diffeomorphism. Therefore, our result gives a complete classification of codimension-one Anosov flows up to topological equivalence in higher dimensions.     

On the classification of the coadjoint orbits of the Sobolev Bott-Virasoro group

The purpose of this paper is to give the classification of the Bott-Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott-Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott-Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied.     

The Zero Mach Number Limit of the Three-Dimensional Compressible Viscous Magnetohydrodynamic Equations

This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.     

Almost complex rigidity of the complex projective plane

An isomorphism of symplectically tame smooth pseudocomplex structures on the complex projective plane which is a homeomorphism and differentiable of full rank at two points is smooth.

     

Singularities of the quad curl problem

We consider the quad curl problem in smooth and non smooth domains of the space. We first give an augmented variational formulation equivalent to the one from [25] if the datum is divergence free. We describe the singularities of the variational space which correspond to the ones of the Maxwell system with perfectly conducting boundary conditions. The edge and corner singularities of the solution of the corresponding boundary value problem with smooth data are also characterized. We finally obtain some regularity results of the variational solution.     

An application of the smooth variational principle to the existence of nontrivial invariant subspaces

We show that the consideration of Gâteaux smooth functions on Banach spaces which admit an equivalent Gâteaux smooth norm allows us to show that certain linear operators have nontrivial closed invariant subspaces. It is in particular the case of all operators on a real Banach space which admit a moment sequence.     

Blow-up of the Smooth Solution to the Compressible Nematic Liquid Crystal System

In this paper, we investigate the blow-up of the smooth solutions to a simplified Ericksen-Lesile system for compressible flows of nematic liquid crystals in different dimensional case. We prove that whether the smooth solution of the Cauchy problem or the initial-boundary problem to the nematic liquid crystal system will blow up in finite time. The main technique is the construction of the functional differential inequality.     

Convergence of the Implicit Filtering Method for Constrained Optimization of Noisy Functions

In this paper, we develop convergence theory of the implicit filtering method for solving the box constrained optimization whose objective function includes a smooth term and a noisy term. It is shown that under certain assumption on the noisy function, the sequence of projected gradients on the smooth function produced by the method goes to zero. Moreover, it is shown that if the smooth function is convex and the noisy function decays near optimality, the whole sequence of iterates converges to a solution of the concerned problem and possesses the finite identification for the optimal active set under the nondegenerate assumption. Finally, preliminary numerical results are reported.     

Bezier curve smoothing of the Kaplan-Meier estimator

Estimation of a survival function from randomly censored data is very important in survival analysis. The Kaplan-Meier estimator is a very popular choice, and kernel smoothing is a simple way of obtaining a smooth estimator. In this paper, we propose a new smooth version of the Kaplan-Meier estimator using a Bezier curve. We show that the proposed estimator is strongly consistent. Numerical results reveal the that proposed estimator outperforms the Kaplan-Meier estimator and its kernel weighted smooth version in the sense of mean integrated square error. This research is supported by the Korea Research Foundation (1998-015-d00047) made in the program year of 1998.     

Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

In [Xiang Zhang, The embedding flows of C^∞ hyperbolic diffeomorphisms, J. Differential Equations 250 (5) (2011) 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator.We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterion is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.     

A smooth invariant not a smooth function of the invariant polynomials

An explicit example is given of a smooth function invariant under a linear group action that is not a smooth function of the invariant polynomials.

     

A Characterization of Counterexamples to the Kodaira-Ramanujam Vanishing Theorem on Surfaces in Positive Characteristic

The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic. More precisely, it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic, then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type. Furthermore, it is proved that up to blow-ups, X admits a fibration to a smooth projective curve, such that each fiber is a singular curve.     

Cut-by-curves criterion for the log extendability of overconvergent isocrystals

In this paper, we prove a ??cut-by-curves criterion?? for an overconvergent isocrystal on a smooth variety over a field of characteristic p?>?0 to extend logarithmically to its smooth compactification whose complement is a simple normal crossing divisor, under certain assumption. This is a p-adic analogue of a version of cut-by-curves criterion for regular singularity of an integrable connection on a smooth variety over a field of characteristic 0. In the course of the proof, we also prove a kind of cut-by-curves criteria on solvability, highest ramification break and exponent of ?-modules.