In this paper, we study the long-time behavior of solutions of the single-layer quasi-geostrophic model arising from geophysical fluid dynamics. We obtain the lower bound of the decay estimate of the solution. Utilizi...In this paper, we study the long-time behavior of solutions of the single-layer quasi-geostrophic model arising from geophysical fluid dynamics. We obtain the lower bound of the decay estimate of the solution. Utilizing the Fourier splitting method, under suitable assumptions on the initial data, for any multi-index α, we show that the solution Ψ satisfies .展开更多
In this paper, upper bounds of the L2-decay rate for the Boussinesq equations are considered. Using the L2 decay rate of solutions for the heat equation, and assuming that the solutions of the Boussinesq equations are...In this paper, upper bounds of the L2-decay rate for the Boussinesq equations are considered. Using the L2 decay rate of solutions for the heat equation, and assuming that the solutions of the Boussinesq equations are smooth, we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data. The decay results may then be obtained by passing to the limit of approximating sequences of solutions. The main tool is the Fourier splitting method.展开更多
Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp ra...Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple togeth- er the elementary uniform energy estimates of the global weak solutions and a well known Gronwall＇s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980＇s to study the op- timal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay esti- mates with sharp rates of the global weak solutions of the Cauchy problems for n-dimensional incompressible Navier-Stokes equations, for the n-dimensional magnetohydrodynamics equations and for many other very interesting nonlin- ear evolution equations with dissipations can be established.展开更多
In this paper, we investigate a system of the incompressible Navier-Stokes equations coupled with Landau-Lifshitz equations in three spatial dimensions. Under the assumption of small initial data, we establish the glo...In this paper, we investigate a system of the incompressible Navier-Stokes equations coupled with Landau-Lifshitz equations in three spatial dimensions. Under the assumption of small initial data, we establish the global solutions with the help of an energy method. Furthermore, we obtain the time decay rates of the higher-order spatial derivatives of the solutions by applying a Fourier splitting method introduced by Schonbek(SCHONBEK, M. E. L2decay for weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 88, 209–222(1985)) under an additional assumption that the initial perturbation is bounded in L1(R3).展开更多
In this article, we study the electromagnetic fluid system in three-dimensional whole space R^3. Under assumption of small initial data, we establish the unique global solution by energy method. Moreover, we obtain th...In this article, we study the electromagnetic fluid system in three-dimensional whole space R^3. Under assumption of small initial data, we establish the unique global solution by energy method. Moreover, we obtain the time decay rates of the higher-order spatial derivatives of the solution by combining the L^p-L^q estimates for the linearized equations and an elaborate energy method when the L^1-norm of the perturbation is bounded.展开更多
文摘In this paper, we study the long-time behavior of solutions of the single-layer quasi-geostrophic model arising from geophysical fluid dynamics. We obtain the lower bound of the decay estimate of the solution. Utilizing the Fourier splitting method, under suitable assumptions on the initial data, for any multi-index α, we show that the solution Ψ satisfies .
文摘In this paper, upper bounds of the L2-decay rate for the Boussinesq equations are considered. Using the L2 decay rate of solutions for the heat equation, and assuming that the solutions of the Boussinesq equations are smooth, we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data. The decay results may then be obtained by passing to the limit of approximating sequences of solutions. The main tool is the Fourier splitting method.
文摘Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple togeth- er the elementary uniform energy estimates of the global weak solutions and a well known Gronwall＇s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980＇s to study the op- timal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay esti- mates with sharp rates of the global weak solutions of the Cauchy problems for n-dimensional incompressible Navier-Stokes equations, for the n-dimensional magnetohydrodynamics equations and for many other very interesting nonlin- ear evolution equations with dissipations can be established.
基金supported by the National Natural Science Foundation of China(Nos.11501373,11701380,and 11271381)the Natural Science Foundation of Guangdong Province(Nos.2017A030307022,2016A030310019,and 2016A030307042)+2 种基金the Guangdong Provincial Culture of Seedling of China(No.2013LYM0081)the Education Research Platform Project of Guangdong Province(No.2014KQNCX208)the Education Reform Project of Guangdong Province(No.2015558)
文摘In this paper, we investigate a system of the incompressible Navier-Stokes equations coupled with Landau-Lifshitz equations in three spatial dimensions. Under the assumption of small initial data, we establish the global solutions with the help of an energy method. Furthermore, we obtain the time decay rates of the higher-order spatial derivatives of the solutions by applying a Fourier splitting method introduced by Schonbek(SCHONBEK, M. E. L2decay for weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 88, 209–222(1985)) under an additional assumption that the initial perturbation is bounded in L1(R3).
基金partially supported by the National Natural Science Foundation of China(11501373,11701380,11271381)Guangdong Provincial Culture of Seedling of China(2013LYM0081)+2 种基金the Natural Science Foundation of Guangdong Province(2017A030307022,2016A0300310019,2016A030307042)the Education Research Platform Project of Guangdong Province(2014KQNCX208)the Education Reform Project of Guangdong Province(2015558)
文摘In this article, we study the electromagnetic fluid system in three-dimensional whole space R^3. Under assumption of small initial data, we establish the unique global solution by energy method. Moreover, we obtain the time decay rates of the higher-order spatial derivatives of the solution by combining the L^p-L^q estimates for the linearized equations and an elaborate energy method when the L^1-norm of the perturbation is bounded.
