On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted L ^∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L ^∞ norm of the perturbation decays as time goes to infinity.     

Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on $${\mathbb{R}^3}$$

The compressible Navier–Stokes–Korteweg system is considered on \({\mathbb{R}^3}\) when the external force is periodic in the time variable. The existence of a time periodic solution is proved for a sufficiently small external force by using the time-T-map related to the linearized problem around the motionless state with constant density and absolute temperature. The spectral properties of the time-T-map is investigated by a potential theoretic method and an energy method in some weighted spaces. The stability of the time periodic solution is proved for sufficiently small initial perturbations. It is also shown that the \({L^\infty}\) norm of the perturbation decays as time goes to infinity.     

Exponential Stability to a Contact Problem of Partially Viscoelastic Materials

In this paper we study models for contact problems of materials consisting of an elastic part (without memory) and a viscoelastic part, where the dissipation given by the memory is effective. We show that the solution of the corresponding viscoelastic equation decays exponentially to zero as time goes to infinity, provided the relaxation function also decays exponentially, no matter how small is the dissipative part of the material.     

Pointwise Green Function Bounds for Shock Profiles of Systems with Real Viscosity

Following the pointwise semigroup approach of [ZH,MZ.1], we establish sharp pointwise Green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic-parabolic systems of conservation laws of dissipative type, under the necessary assumptions ([Z.1,Z.3,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave; transversality of the connecting profile; and hyperbolic stability of the corresponding ideal shock of the associated inviscid system, with no additional assumptions on the structure or strength of the shock. These bounds are used in a companion paper [MZ.2] to establish nonlinear stability of small-amplitude Lax shocks of symmetrizable hyperbolic-parabolic systems.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> Stability of the Boltzmann Equation for the Hard-Sphere Model

We study the L¹ stability of classical solutions to the Boltzmann equation for a hard-sphere model, when initial datum is a small perturbation of a vacuum, and tends to zero exponentially fast at infinity in the phase space. For this, we introduce nonlinear functionals measuring potential interactions between particles with different velocities and L¹ distance between classical solutions. We use pointwise estimates for a solution and the gain term of a collision operator to control the time-evolution of nonlinear functionals.Dedicated to Marshall Slemrod on the occasion of his 60th birthday     

The attractor of a Navier-Stokes system in an unbounded channel-like domain

The Navier-Stokes system describes a flow of a fluid in an unbounded planar channel-like domain. It is proved that in the case when an external force decays at infinity, the semigroup generated by this system has a global attractor and its Hausdorff dimension is finite. Estimates in weighted Sobolev spaces are used as a main tool. Asymptotics, as the distance from the origin in the plane tends to infinity, of functions on the attactor is found. This asymptotics show that all dynamics on the attractor decays at infinity and the turbulence generated by the force does not propagate to infinity.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Finite Dimensional Attractor for a One-Phase Stefan Problem with Kinetics

For a one-phase free-boundary problem with kinetics, which is known to generate a rich dynamics, we study evolution of the infinitesimal volume along the trajectories in the attractor. We demonstrate that for sufficiently large m that is defined solely by the properties of the kinetics function the m-dimensional volume decays exponentially. This property combined with the uniform differentiability of the semigroup leads to the conclusion that the Hausdorff dimension of the attractor is finite.     

被均匀流缓慢调制的有限水深毛细重力波

非线性的存在会产生高次谐波,这些谐波又反作用于原来的低次谐波,使波幅发生缓慢变化,从而产生缓慢调制现象．这里从考虑均匀流作用下的毛细重力水波基本方程出发,在不可压缩、无旋、无黏条件假设下,使用多重尺度分析方法推导出了在均匀流影响下有限深水毛细重力波振幅所满足的非线性Schr?dinger方程(NLSE)．分析了NLSE解的调制不稳定性．给出了毛细重力波调制不稳定的条件和钟型孤立波产生的条件．分析了无量纲最大不稳定增长率随无量纲水深和表面张力的变化趋势．同时给出了无量纲不稳定增长率随无量纲微扰动波数变化的曲线,呈现出了先增大后减小的趋势．最后指出均匀顺流减小了无量纲不稳定增长率及最大增长率,逆流增大了它们．由表面张力作用产生的毛细波及重力与表面张力共同作用产生的毛细重力波,与流的相互作用可以改变海表粗糙度和海洋上层流场结构,进而影响海气界面动量、热量及水汽的交换．了解海表这些短波动力机制,对卫星遥感的精确测量、海气相互作用的研究及海气耦合模式的改进等有重要意义．      

Linearization and correction method for nonlinear problems

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Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.     

Spectral Properties of the Linearization at the Burgers Vortex in the High Rotation Limit

We study a linearized operator of the equation for the axisymmetric Burgers vortex which gives a stationary solution to the three dimensional Navier–Stokes equations with an axisymmetric background straining flow. It is numerically known that the Burgers vortex obtains better stabilities as the circulation number (or the vortex Reynolds number) is increasing. Although the global stability of the axisymmetric Burgers vortex is already proved rigorously, mathematical understanding of this numerical observation has been lacking. In this paper we study a linearized operator that includes the circulation number as a parameter, and prove that if the operator is restricted on a suitable invariant subspace, then its spectrum moves to the left as the circulation number goes to infinity. As an application, we show that the axisymmetric Burgers vortex with a high rotation has a better stability, in the sense that the non-radially symmetric part of solutions to the associated evolution equation decays faster in time if the circulation number is sufficiently large.     

Stability of a Steady Viscous Incompressible Flow Past an Obstacle

We derive a sufficient condition for stability of a steady solution of the Navier–Stokes equation in a 3D exterior domain Ω. The condition is formulated as a requirement on integrability on the time interval (0, +∞) of a semigroup generated by the linearized problem for perturbations, applied to a finite family of certain functions. The norm of the semigroup is measured in a bounded sub-domain of Ω. We do not use any condition on “smallness” of the basic steady solution.      

Asymptotic behaviour of solution for fourth order wave equation with dispersive and dissipative terms

This paper studies the initial boundary value problem of fourth order wave equation with dispersive and dissipative terms. By using multiplier method, it is proven that the global strong solution of the problem decays to zero exponentially as the time approaches infinite, under a very simple and mild assumption regarding the nonlinear term.     

Stability Balls in the Rotating Bénard System

Nonlinear stability in the standard rotating Bénard system with free boundaries is investigated. The perturbations are assumed to be three-dimensional and to be periodic in the horizontal directions. Below the critical value of the Rayleigh numberRa there is conditional stability, i.e. there are nonvanishing stability balls such that perturbations with initial values (measured in a suitable norm) in these balls decay exponentially in time. We give here explicit bounds to these stability balls from below in terms of the parameters of the system, i.e.Ra, the Taylor numberT and the Prandtl numberPr as well as the size of the periodicity cell of the perturbation. The bounds are valid in the entire parameter space; in particular, forPr<1 and for arbitrarily large values ofT. They provide a qualitative explanation for the experimental observation of subcritical instabilities in the rangePr<1. The method is based on a mode expansion of the perturbation equations and explicit estimates of the semigroup operator as well as of the nonlinearity.     

Stability of Viscous Profiles: Proofs Via Dichotomies

In this paper we give a self-contained approach to a nonlinear stability result, as t → ∞, for a viscous profile corresponding to a strong shock of a system of conservation laws. The initial perturbation is assumed to be small and to have zero mass. As t→ ∞, the solution with perturbed initial data is shown to approach the viscous profile in maximum norm.A complete proof of the stability result is given under slightly weaker assumptions than those in [Comm. Pure Appl. Math. LI (1998) 1397]; our assumptions, techniques, and results also differ from those in [Indiana Univ. Math. J. 47 (1998) 741]. To derive resolvent estimates for a linearized problem, we use the theory of exponential dichotomies for ODEs extensively. A main tool provided by this theory is a quantitative L ₁ perturbation theorem for dichotomies, which yields the delicate resolvent estimates for s near zero.When showing that the resolvent estimates imply nonlinear stability, we essentially follow the arguments in [Comm. Pure Appl. Math. LI (1998) 1397; SIAM J. Math. Anal. 20 (1999) 401], but note some simplifications.     

Screening in Interacting Particle Systems

We consider the Green's function of the Laplace operator in domains with spherical holes (particles). Under natural assumptions on the distribution of particles we show that the Green's function decays exponentially over distances larger than the screening length. This result is fundamental for example when deriving effective equations for coarsening systems in unbounded domains.     

Controlled synchronization of discrete-time chaotic systems under communication constraints

This paper investigates the controlled synchronization problem for a class of nonlinear discrete-time chaotic systems subject to limited communication capacity. A general chaotic master system and its slave system with a controller are connected via a limited capacity channel. In this case, the effect of quantization errors is considered. A practical quantized scheme is proposed so that the synchronization error is input-to-state stable with respect to the transmission error. Meanwhile, the transmission error decays to zero exponentially. This implies that the synchronization error converges to zero under a limited communication channel. A?simulation example for the Fold chaotic system is presented to illustrate the effectiveness of the proposed method.     

Controlling the inverted pendulum by means of a nested saturation function

A nonlinear controller is presented for the stabilization of the underactuated inverted pendulum mounted on a cart. The fact that this system can be expressed as a chain of integrators, with an additionally nonlinear perturbation, allows us to use a nested saturation control technique to bring the pendulum to the top position, with zero displacement of the cart. The obtained closed-loop system is semiglobal, asymptotically stable, and locally exponentially stable, under the assumption that the position of the angle is initialized above the upper half plane.     

A uniformly convergent finite difference method for a singularly perturbed initial value problem

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The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh–Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.