On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ
Yong Yang Bingsheng Zhang

One particular metric that generates the weak topology on the weak global attractor $\mathcal{A}_w$ of three dimensional incompressible Navier-Stokes equations is introduced and used to obtain an upper bound for the Kolmogorov entropy of $\mathcal{A}_w$. This bound is expressed explicitly in terms of the physical parameters of the fluid flow.

keywords: 3D Navier-Stokes equations fluid flow weak global attractor Kolmogorov entropy functional dimension
Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type
Yue-Jun Peng Yong-Fu Yang
We study linearly degenerate hyperbolic systems of rich type in one space dimension. It is showed that such a system admits exact traveling wave solutions after a finite time, provided that the initial data are Riemann type outside a space interval. We prove the convergence of entropy solutions toward traveling waves in the $L^1$ norm as the time goes to infinity. The traveling waves are determined explicitly in terms of the initial data and the system. We also obtain the stability of entropy solutions in $L^1$. Applications concern physical models such as the generalized extremal surface equations, the Born-Infeld system and augmented Born-Infeld system.
keywords: Entropy solution long-time behavior rich system $L^1$ stability. linearly degeneracy
A new auxiliary function method for systems of nonlinear equations
Zhiyou Wu Fusheng Bai Guoquan Li Yongjian Yang
In this paper, we present a new global optimization method to solve nonlinear systems of equations. We reformulate given system of nonlinear equations as a global optimization problem and then give a new auxiliary function method to solve the reformulated global optimization problem. The new auxiliary function proposed in this paper can be a filled function, a quasi-filled function or a strict filled function with appropriately chosen parameters. Several numerical examples are presented to illustrate the efficiency of the present approach.
keywords: Auxiliary function method global optimization problems. nonlinear equations
A filled function method for constrained nonlinear integer programming
Yongjian Yang Zhiyou Wu Fusheng Bai
A filled function method is presented in this paper to solve constrained nonlinear integer programming problems. It is shown that for a given non-global local minimizer, a better local minimizer can be obtained by local search staring from an improved initial point which is obtained by locally solving a box-constrained integer programming problem. Several illustrative numerical examples are reported to show the efficiency of the present method.
keywords: Constrained nonlinear integer programming filled function method discrete global optimization.
On an exact penalty function method for semi-infinite programming problems
Cheng Ma Xun Li Ka-Fai Cedric Yiu Yongjian Yang Liansheng Zhang
In this paper, we study a new exact and smooth penalty function for semi-infinite programming problems with continuous inequality constraints. Through this exact penalty function, we can transform a semi-infinite programming problem into an unconstrained optimization problem. We find that, under some reasonable conditions when the penalty parameter is sufficiently large, the local minimizer of this penalty function is the local minimizer of the primal problem. Moreover, under some mild assumptions, the local exactness property is explored. The numerical results demonstrate that it is an effective and promising approach for solving constrained semi-infinite programming problems.
keywords: smooth and exact penalty function nonsmooth optimization Constrained semi-infinite programming problem extended Managasarian-Fromovitz constraint qualification.
Mechanism of the formation of singularities for diagonal systems with linearly degenerate characteristic fields
Yong-Fu Yang
For inhomogeneous diagonal system with distinct characteristics or with characteristics with constant multiplicity, under the assumption that the system is linearly degenerate and the $C^1$ norm of the initial data is bounded, we show that the mechanism of the formation of singularities of classical solution to its Cauchy problem must be of ODE type. Similar results are also obtained for corresponding mixed initial-boundary value problems on a semi-unbounded domain.
keywords: global $C^1$ solution quasilinear hyperbolic system of diagonal form Formation of singularities linearly degenerate characteristic Cauchy problem one-side mixed initial-boundary value problem

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