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Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe

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Science China(Physics,Mechanics & Astronomy) 2005年第4期48卷 485-495页

作者： TONG Dengke WANG Ruihe YANG Heshan Institute of Mathematics and Computional Science University of Petroleum Beijing 102249 China Institute of Mathematics and Computional Science University of Petroleum Beijing 102249 China Institute of Mathematics and Computional Science University of Petroleum Beijing 102249 China

This paper deals with some unsteady unidirectional transient flows of Oldroyd-B fluid in an annular pipe. The fractional calculus approach in the constitutive relationship model Oldroyd-B fluid is introduced and a generalized Jeffreys model with the fractional calculus has been built. Exact solutions of some unsteady flows of Oldroyd-B fluid in an annular pipe are obtained by using Hankel transform and Laplace transform for fractional calculus. The following four problems have been studied: (1) Poiseuille flow due to a constant pressure gradient; (2) axial Couette flow in an annulus; (3) axial Couette flow in an annulus due to a longitudinal constant shear; (4) Poiseuille flow due to a constant pressure gradient and a longitudinal constant shear. The well-known solutions for Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limited cases of our solutions.

关键词： fractional calculus, generalized Oldroyd-B fluid, exact solutions, velocity fields.

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

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Science China Mathematics 2006年第7期49卷 932-960页

作者： ZHANG Jianwen ZHAO Junning School of Mathematical Sciences Xiamen University Xiamen 361005 China School of Mathematical Sciences Xiamen University Xiamen 361005 China

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.

关键词： boundary layer, BV space, global weak solutions, stability and uniqueness.

Search-extension method for finding multiple solutions of nonlinear problem

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Science China Mathematics 2005年第6期48卷 721-734页

作者： CHEN Chuanmiao XIE Ziqing Institute of Computation Hunan Normal University Changsha 410081 China Institute of Computation Hunan Normal University Changsha 410081 China

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω, u = 0 on (?)Ω are approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces. Numerical experiments for f(u) = u3 in a square and L-shape domain are presented. The results show that there exist at least 3k - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -Δ (Conjecture 1).

关键词： nonlinear problem, multiple solutions, search-extension method, conjecture.

Singular solutions to Conformal Hessian Equations(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

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Chinese Annals of Mathematics,Series B 2017年第2期38卷 591-600页

作者： Nikolai NADIRASHVILI Serge VLADUT Aix Marseille Université CNRSCentrale MarseilleFrance IITP RAS B.Karetnyi9MoscowRussia

The authors show that for any ε∈]0, 1[, there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ? R^5 which belongs to C^(1,ε)(B) \ C^(1,ε+)(B).

关键词： Viscosity solutions, Conformal Hessian equation, Cartan's cubic

Quasi-periodic solutions of a semilinear Liénard equation at resonance

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Science China Mathematics 2005年第9期48卷 1234-1244页

作者： LIU Bin LMAM School of Mathematical Sciences Peking University Beijing 100871 China

We are concerned with the existence of quasi-periodic solutions for the following equation x" + Fx (x, t)x' + ω2x + φ(x,t) = 0,where F and φ are smooth functions and 2π-periodic in t, ω＞ 0 is a constant. Under some assumptions on the parities of F and φ, we show that the Dancer's function, which is used to study the existence of periodic solutions, also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e. all solutions are bounded).

关键词： quasi-periodic solutions, semilinear Liénard equations, boundedness of solutions, reversible systems.

Peaked wave solutions of CH-r equation

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Science China Mathematics 2003年第5期46卷 696-709页

作者：郭柏灵刘正荣 Institute of Applied Physics and Computational Mathematics. Beijing 100088 China Department of Mathematics Yunnan University and Institute of Applied Mathematics of Yunnan Province Kunming 650091 China

In this paper the qualitative analysis methods of planar autonomous systems and numerical simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wave solutions and peaked periodic wave solutions are obtained, and some of their relationships arerevealed. Why peaked points are generated is discussed.

关键词： CH-r equation, peaked wave solutions, bifurcation of phase portraits.

Structure of multiple solutions for nonlinear differential equations

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Science China Mathematics 2004年第Z1期47卷 172-180页

作者： CHEN Chuanmiao XIE Ziqing Institute of Computation Hunan Normal University Changsha 410081 China Institute of Computation Hunan Normal University Changsha 410081 China

Based on the eigensystem {λj,φjof -Δ, the multiple solutions for nonlinear problem Δu+f(u)=0 in Ω,u=0 onαΩ are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u)=u3,u2(u-p),u2(u2-p), are completed and some conjectures are presented.

关键词： nonlinear problem, multiple solutions, search-extension method, structure, conjecture.

On a quasilinear elliptic eigenvalue problem with constraint

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Science China Mathematics 2004年第4期47卷 523-537页

作者： CHEN Jianqing CHEN Shaowei LI Yongqing Department of Mathematics Fujian Normal UniversityFuzhou 350007China Institute of Mathematics AMSSChinese Academy of SciencesBeijing 100080China

Via construction of pseudo gradient vector field and descending flow argument, we prove the existence of one positive, one negative and one sign-changing solutions for a quasilinear elliptic eigenvalue problem with co... 详细信息

关键词： multiple solutions, sign-changing solution, critical point theory, quasilinear elliptic eigenvalue.

Monotone method for initial value problem for fractional diffusion equation

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Science China Mathematics 2006年第9期49卷 1223-1230页

作者： ZHANG Shuqin Department of Mathematics China University of Mining and Technology Beijing 100083 China

Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.

关键词： upper solutions and lower solutions, monotone iterative method, fractional diffusion equation, solution, existence and Uniqueness.