TIME PERIODIC SOLUTION TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS IN A PERIODIC DOMAIN

This article is concerned with the time periodic solution to the isentropic compressible Navier-Stokes equations in a periodic domain. Using an approach of parabolic regularization, we first obtain the existence of the time periodic solution to a regularized problem under some smallness and symmetry assumptions on the external force. The result for the original compressible Navier-Stokes equations is then obtained by a limiting process.The uniqueness of the periodic solution is also given.     

On the Existence of Time-periodic Solution to the Compressible Heat-conducting Navier-Stokes Equations

We study in this article the compressible heat-conducting Navier-Stokes equations in periodic domain driven by a time-periodic external force. The existence of the strong time-periodic solution is established by a new approach. First, we reformulate the system and consider some decay estimates of the linearized system.Under some smallness and symmetry assumptions on the external force, the existence of the time-periodic solution of the linearized system is then identi?ed as the ?xed point of a Poincare′ map which is obtained by the Tychonoff ?xed point theorem.Although the Tychonoff ?xed point theorem cannot directly ensure the uniqueness,but we could construct a set-valued function, the ?xed point of which is the timeperiodic solution of the original system. At last, the existence of the ?xed point is obtained by the Kakutani ?xed point theorem. In addition, the uniqueness of timeperiodic solution is also studied.     

Stability of Navier–Stokes System with Singular External Force in Fourier–Herz Space

We prove the asymptotic properties of the solutions to the 3D Navier–Stokes system with singular external force, by making use of Fourier localization method, the Littlewood–Paley theory and some subtle estimates in Fourier–Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339(2022)]. We deal either with the nonstationary problem or with the stationary problem where solution may be singular due to singular external force. In this p...     

Asymptotic stability of periodic solution for compressible viscous van der Waals fluids

This paper is concerned with the asymptotic stability of the periodic solution to a one-dimensional model system for the compressible viscous van der Waals fluid in Eulerian coordinates. If the initial density and initial momentum are suitably close to the average density and average momentum, then the solution is proved to tend toward a stationary solution as t -→∞.     

TIME PERIODIC SOLUTIONS OF NON-ISENTROPIC COMPRESSIBLE MAGNETOHYDRODYNAMIC SYSTEM

In this paper, we study the non-isentropic compressible magnetohydrodynamic system with a time periodic external force in Rn. Under the condition that the optimal time decay rates are obtained by spectral analysis, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension n ≥ 5. Our proof is based on a combination of the energy method and the contraction mapping theorem.     

WEAK TIME-PERIODIC SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

The compressible Navier-Stokes equations driven by a time-periodic external force are considered in this article. We establish the existence of weak time-periodic solutions and improve the result from [3] in the following sense: we extend the class of pressure functions, that is, we consider lower exponent γ.     

Existence of weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in R3 for any specific heat ratio γ 1.The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density,and the method of weak convergence.According to the author's knowledge,it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with γ 1.     

EXISTENCE OF INFINITE ENERGY SOLUTION TO THE INELASTIC BOLTZMANN EQUATION WITH EXTERNAL FORCE

In this paper,the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy.More precisely,under the assumptions on the bicharacteristic generated by external force,we prove the global existence of solution for small initial data compared to the local Maxwellian exp{ p|x v|2},which has infinite mass and energy.     

Periodic solutions in unbounded domains for the Boussinesq system

Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of Rn. Our analysis is based on the framework of weak-Lp spaces.     

Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum

We study the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in a bounded annulus Ω of R3. And a result on the existence and uniqueness of global spherically symmetric classical solutions is obtained. Here the initial data could be large and initial vacuum is allowed.     

非自治Ginzburg-Landau方程的周期解和全局周期吸引子

研究受周期外力影响的非自治Ginzburg-Landau方程的解的长时间行为.首先证明系统在空间H上存在周期解,而且周期解包含在空间V中的一个有界吸收集内.然后证明了当耗散系数λ满足一定条件时,该系统在空间H上具有唯一的周期解,该周期解指数吸引H中的任意有界集.     

有阻尼受迫sine-Gordon方程的全局周期吸引子

本文证明了当阻尼与扩散系数在一定的参数范围内时,有阻尼的受迫sineGordon方程的狄氏问题对于任意非自治时间周期受迫力均具有唯一的指数吸引有界集的周期解．并且,如果受迫力是自治的,则全局吸引子恰是系统唯一的指数吸引有界集的平衡解．     

具有粘弹性和热粘弹性方程组的全局周期吸引子

证明了具有粘弹性和热粘弹性方程组在Dirichlet边界条件下,对于任意的非自治时间周期受迫力,均具有唯一的指数吸引任何有界集的周期解,即全局周期吸引子.并且如果受迫力是自治的,则全局周期吸引子恰是系统唯一的指数吸引有界集的平衡解.     

Time‐periodic solution to the compressible nematic liquid crystal flows in periodic domain

In this paper, we consider the time‐periodic solution to a simplified version of Ericksen‐Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals with a time‐periodic external force in a periodic domain in . By using an approach of parabolic regularization and combining with the topology degree theory, we establish the existence of the time‐periodic solution to the model under some smallness and symmetry assumptions on the external force. Then, we give the uniqueness of the periodic solution of this model.     

Forced oscillations of nonlinear damped equation of suspended string

We shall study the existence of time-periodic solutions of nonlinear damped equation of suspended string to which a periodic nonlinear force works. We shall be conterned with weak, strong and classical time-periodic solutions and also the regularity of the solutions. To formulate our results, we shall take suitable weighted Sobolev-type spaces introduced by [M. Yamaguchi, Almost periodic oscillations of suspended string under quasiperiodic linear force, J. Math. Anal. Appl. 303 (2) (2005) 643-660; M. Yamaguchi, Infinitely many time-periodic solutions of nonlinear equation of suspended string, Funkcial. Ekvac., in press]. We shall study properties of the function spaces and show inequalities on the function spaces. To show our results we shall apply the Schauder fixed point theorem and the fixed point continuation theorem in the function spaces.     

Dynamics of a coupled atmosphere–ocean model

We consider a coupled atmosphere–ocean model, which involves hydrodynamics, thermodynamics and nonautonomous interaction at the air–sea interface. First, we show that the coupled atmosphere–ocean system is stable under the external fluctuation in the atmospheric energy balance relation. Then, we estimate the atmospheric temperature feedback in terms of the freshwater flux, heat flux and the external fluctuation at the air–sea interface, as well as the earth's longwave radiation coefficient and the shortwave solar radiation profile. Finally, we prove that the coupled atmosphere–ocean system has time-periodic, quasiperiodic and almost periodic motions, whenever the external fluctuation in the atmospheric energy balance relation is time-periodic, quasiperiodic and almost periodic, respectively.     

The Navier-Stokes equations in the weak- LnL^n space with time-dependent external force

Abstract. We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension . We give conditions on the external force sufficient for the unique existence of small solutions in the weak- space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak- space with time-independent external force. Received: 30 March 1999 / Accepted: 21 September 1999 / Published online: 28 June 2000     

Time-Periodic Fitzhugh-Nagumo Equation and the Optimal Control Problems

In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function.     

Time periodic solutions of the diffusive Nicholson blowflies equation with delay

This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.