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Existence and Uniqueness of Stationary Solutions of Non-Newtonian Viscous Incompressible Fluids

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Communications in Nonlinear Science and Numerical Simulation 1999年第1期4卷 63-68页

作者： Boling GUO* and guoguang lin**,+ (Institute of Applied Physics and Computational Mathematics, Beijing 100088, China) guoguang lin (Graduate School of China Academy of Engineering Physics, Beijing 100088, China) + (Department of Mathematics, Yunnan Unive Inst. Appl. Phys. and Compl. Math. Beijing 100088 China Graduate School China Academy of Engineering Physics Beijing 100088 China Department of Mathematics Yunnan University Kunming 650091 China

The existence and uniqueness of stationary solution a bipolar incompressible viscous fluids is *** is also obtained that the every solution of the system converges to the stationary solution as time t → ∞.

关键词： non-Newtonian fluid stationary solution, asymptotic behavior bipolar fluid

Uniqueness results for monopolar fluids

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Communications in Nonlinear Science and Numerical Simulation 2000年第1期5卷 21-23页

作者： guoguang lin Department of Mathematics, Yunnan University, Kunming 650091, China, e-mail: gglin@ynu. edu. cn Department of Mathematics Yunnan University Kunming 650091 China

We prove a unique property for weak solutions of monopolar fluid equations.

关键词： monopolar fluid equations non-Newtonian fluids uniqueness of weak solutions

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

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Advances in Pure Mathematics 2021年第4期11卷 317-333页

作者： guoguang lin Lujiao Yang Department of Mathematics Yunnan University Kunming China

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

关键词： Generalized Kirchhoff Equation The Existence and Uniqueness of Solution A Family of the Global Attractor Dimension Estimation

A Family of the Random Attractors for a Class of Generalized Kirchhoff-Type Equations

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Journal of Applied Mathematics and Physics 2021年第11期9卷 2966-2978页

作者： guoguang lin Lujiao Yang Department of Mathematics Yunnan University Kunming China

In this paper, we studied the existence of a family of the random attractor for a class of generalized Kirchhoff-type equations with a strong dissipation term. Firstly, according to Ornstein-Uhlenbeck process, we transformed the equation into a stochastic equation with random variables and multiplicative white noise. Secondly, we proved the existence of a bounded random absorbing set. Finally, by using the isomorphic mapping method and the compact embedding theorem, we get the stochastic dynamical system with a family of random attractors.

关键词： Stochastic Kirchhoff Equation A Family of the Random Attractors Multiplicative White Noise Ornstein-Uhlenbeck Process

A Family of Inertial Manifolds of Coupled Kirchhoff Equations

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Journal of Applied Mathematics and Physics 2022年第6期10卷 2074-2085页

作者： guoguang lin Fumei Chen Department of Mathematics Yunnan University Kunming China

In this paper, we study the long-time behavior of the solution of the initial boundary value problem of the coupled Kirchhoff equations. Based on the relevant assumptions, the equivalent norm on Ek is obtained by using the Hadamard graph transformation method, and the Lipschitz constant lF of F is further estimated. Finally, a family of inertial manifolds satisfying the spectral interval condition is obtained.

关键词： Kirchhoff Equation the Family of Inertial Manifolds Hadamard Graph Transformation Spectral Interval Condition

The Family of Global Attractors for Kirchhoff-Type Coupled Equations

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Journal of Applied Mathematics and Physics 2022年第6期10卷 2040-2060页

作者： guoguang lin Jiaying Zhou Department of Mathematics Yunnan University Kunming China

This paper mainly studies the initial value problems of Kirchhoff-type coupled equations. Firstly, by giving the hypothesis of Kirchhoff stress term , the Galerkin’s method obtains the existence uniqueness of the overall solution of the above problem by using a priori estimates in the spaces of E0 and Ek, and secondly, it proves that there is a family of global attractors for the above problem, and finally estimates the Hausdorff dimension and the Fractal dimension of the family of global attractors.

关键词： Kirchhoff Equation the Existence and Uniqueness of Global Solution the Family of Global Attractors Hausdorff Dimension and Fractal Dimension of Global Attractor

A Family of Global Attractors for the Generalized Kirchhoff-Beam Equations

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Journal of Applied Mathematics and Physics 2023年第7期11卷 1945-1963页

作者： guoguang lin Boshi Chen Department of Mathematics Yunnan University Kunming China

In this paper, we discuss the existence and uniqueness of global solutions, the existence of the family of global attractors and its dimension estimation for generalized Beam-Kirchhoff equation under initial conditions and boundary conditions, using the previous research results for reference. Firstly, the existence of bounded absorption set is proved by using a prior estimation, then the existence and uniqueness of the global solution of the problem is proved by using the classical Galerkin’s method. Finally, Housdorff dimension and fractal dimension of the family of global attractors are estimated by linear variational method and generalized Sobolev-Lieb-Thirring inequality.

关键词： Beam-Kirchhoff Equation Galerkin’s Method Family of Global Attractors Housdorff Dimension Fractal Dimension

A Family of the Exponential Attractors and the Inertial Manifolds for a Class of Generalized Kirchhoff Equations

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Journal of Applied Mathematics and Physics 2021年第10期9卷 2399-2413页

作者： guoguang lin Lujiao Yang Department of Mathematics Yunnan University Kunming China

In this paper, we studied a family of the exponential attractors and the inertial manifolds for a class of generalized Kirchhoff-type equations with strong dissipation term. After making appropriate assumptions for Kirchhoff stress term and nonlinear term, the existence of exponential attractor is obtained by proving the discrete squeezing property of the equation, then according to Hadamard’s graph transformation method, the spectral interval condition is proved to be true, therefore, the existence of a family of the inertial manifolds for the equation is obtained.

关键词： Kirchhoff-Type Equation Spectral Interval Condition A Family of the Exponential Attractors A Family of the Inertial Manifolds

A Family of the Global Attractor for Higher Order Nonlinear Kirchhoff Equation

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Open Journal of Applied Sciences 2021年第6期11卷 750-765页

作者： guoguang lin Yingguo Wang Department of Mathematics Yunnan University Kunming China

In this paper,we study the wellness and long time dynamic behavior of the solution of the initial boundary value problem for a class of higher order Kirchhoff equations with strong damping terms. We will properly assume the stress term M(s) and nonlinear term g(ut). First, we can prove the existence and uniqueness of the solution of the equation via a prior estimate and Galerkin’s method, then the existence of the family of global attractor is obtained. At last, we can obtain that the Hausdorff dimension and Fractal dimension of the family of global attractor are finite.

关键词： Kirchhoff-Type Equations Prior Estimation Galerkin’s Method The Family of Global Attractor Hausdorff Dimension Fractal Dimension