On a Bound for Amplitudes of Navier�CStokes Flow with almost Periodic Initial Data

For any bounded (real) initial data it is known that there is a unique global solution to the two-dimensional Navier–Stokes equations. This paper is concerned with a bound for the sum of the modulus of amplitudes when initial velocity is spatially almost periodic in 2D. In the case of general dimension, it is bounded on local time of existence shown by Giga et al. (Methods Appl Anal 12:381–393,2005). A class of initial data is given such that the sum of the modulus of amplitudes of a solution is bounded on any finite time interval. It is shown by an explicit example that such a bound may diverge to infinity as the time goes to infinity at least for complex initial data.     

Relationship Between Singularities in Classical Mechanics for Complex Initial Conditions and for Complex Time

A relationship is established between the functional forms of two kinds of singularities in dynamical variables that arise in complexified versions classical mechanics: singularities that are treated as a functions of complex initial conditions for real time and those that are treated as a functions of complex time for real initial conditions. The analysis is verified by numerical calculations. The results imply that Kowaleskaya–Painlevé condition for integrability can be phrased in terms of singularities with respect to initial conditions.     

Experimental Research on Failure Waves in Soda-Lime Glass

A series of plate impact experiment with soda-lime glass specimens in different thicknesses are conducted on a 57 mm diameter one-stage gas gun in order to further investigate the so-called failure wave phenomena under dynamic compressive loads. With the aid of the VISAR technique, the failure wave trajectory is explored, which shows that, apart from a constant failure wave velocity, an initial delay time for the failure wave to initiate at the impact surface of the specimen should be taken into consideration. Comparing our experimental results with the available data presented in the previous open literature shows that, with the increasing magnitude of the impact loads, the failure wave velocity increases and the initial delay time decreases. Moreover, the derived initial delay time τ = 0.694 μs for the soda-lime glass specimens under the impact stress of 4.7 GPa is the same order of magnitude as that of the incubation time proposed by Morozov and Petrov (2000), which shows that the incubation time plays a dominant role in the total initial delay time, and it also provides an reasonable explanation to the fundamental question pointed out by Clifton (Appl Mech Rev 46: 540-546, 1993).     

Bounds on Coarsening Rates for the Lifschitz–Slyozov–Wagner Equation

This paper is concerned with the large time behavior of solutions to the Lifschitz–Slyozov–Wagner (LSW) system of equations. Point-wise in time upper and lower bounds on the rate of coarsening are obtained for solutions with fairly general initial data. These bounds complement the time averaged upper bounds obtained by Dai and Pego, and the point-wise in time upper and lower bounds obtained by Niethammer and Velasquez for solutions with initial data close to a self-similar solution.     

On the Two-Dimensional Euler Equations with Spatially Almost Periodic Initial Data

We consider the nonstationary Euler equations in \mathbbR²{\mathbb{R}}^2 with almost periodic unbounded vorticity. We show that a unique solution is always spatially almost periodic at any time when the almost periodic initial data belongs to some function space. In order to prove this, we demonstrate the continuity with respect to initial data which do not decay at spatial infinity. The proof of the continuity with respect to initial data is based on that of Vishik’s uniqueness theorem.     

Study of the analytical solution to the heat transfer problem and surface temperature in a semi-infinite body with a constant heat flux at the surface and an initial temperature distribution

In many practical cases, one heats a semi-infinite solid with a constant heat flux source. For such an unsteady heat transfer problem, if the body has a uniform initial temperature, the analytical solution has been given by Carslaw and Jaeger. The surface temperature of the semi-infinite body follows the $\sqrt t $ -rule, that is, the surface temperature changes in proportion to square root of heating time. But if, instead of the uniform initial temperature, the body has a temperature distribution at the beginning of heating, the analytical solution has not yet been developed. Analytical solutions to the same problem with an exponential or a linear initial temperature distribution are obtained in this paper. It is shown, that in the case of a linear initial temperature distribution the surface temperature also changes according to $\sqrt t $ -rule Approximating the initial temperature distribution near the surface by its tangent at the surface, it is found that the surface temperature within a short time after the start of heating should also satisfy the $\sqrt t $ -rule, in spite of an arbitrary initial temperature distribution. The experimental data support this argument. Furthermore, the constant heat flux can be calculated after relationship between the surface temperature and heating time according to the equation derived in this paper, if the initial temperature distribution or its first-order derivative at the surface is known.     

Adjoint optimization of natural convection problems: differentially heated cavity

Optimization of natural convection-driven flows may provide significant improvements to the performance of cooling devices, but a theoretical investigation of such flows has been rarely done. The present paper illustrates an efficient gradient-based optimization method for analyzing such systems. We consider numerically the natural convection-driven flow in a differentially heated cavity with three Prandtl numbers (\(Pr=0.15{-}7\)) at super-critical conditions. All results and implementations were done with the spectral element code Nek5000. The flow is analyzed using linear direct and adjoint computations about a nonlinear base flow, extracting in particular optimal initial conditions using power iteration and the solution of the full adjoint direct eigenproblem. The cost function for both temperature and velocity is based on the kinetic energy and the concept of entransy, which yields a quadratic functional. Results are presented as a function of Prandtl number, time horizons and weights between kinetic energy and entransy. In particular, it is shown that the maximum transient growth is achieved at time horizons on the order of 5 time units for all cases, whereas for larger time horizons the adjoint mode is recovered as optimal initial condition. For smaller time horizons, the influence of the weights leads either to a concentric temperature distribution or to an initial condition pattern that opposes the mean shear and grows according to the Orr mechanism. For specific cases, it could also been shown that the computation of optimal initial conditions leads to a degenerate problem, with a potential loss of symmetry. In these situations, it turns out that any initial condition lying in a specific span of the eigenfunctions will yield exactly the same transient amplification. As a consequence, the power iteration converges very slowly and fails to extract all possible optimal initial conditions. According to the authors’ knowledge, this behavior is illustrated here for the first time.     

Large time asymptotics for solutions to a nonhomogeneous Burgers equation

In this article, the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed. In an interesting study, Kloosterziel （Journal of Engineering Mathematics 24, 213-236 （1990）） represented a solution to an initial value problem （IVP） for the heat equation, with an initial data in a class of rapidly decaying functions, as a series of self-similar solutions to the heat equation. This approach quickly revealed the large time behaviour for the solution to the IVP. Inspired by Kloosterziel＇s approach, the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation. The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained.     

Les solutions periodiques et les points fixes de l'application translation dans une equation differentielle ordinaire

We consider an ordinary differential equation E, x(t) = F(t, x(t)), with time periodic right hand side, with period T. The translation mapping Θ is the one which transforms an initial point Y at time t₀ into the value at time t₀ + T of the solution of E with initial conditions (t₀, Y). It is known that the solution with initial conditions (t₀, Z) is periodic with period T if and only if Z is a fixed point of Θ. In this paper the Newton's method is applied to locate the fixed points of the translation mapping Θ.     

Natural convection produced by unsteady heating of a viscoplastic liquid in a plane vertical layer

A study is made of the unsteady heating of a viscoplastic liquid in the space between two infinite vertical plates when one is thermally insulated and a constant, uniformly distributed heat flux is supplied to the other after a certain initial time. It is assumed that at the initial time the liquid is uniformly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 148–150, July–August, 1982.     

Life-span of classical solutions for one-dimensional hydromagnetic flow

The paper concerns Cauchy, problem for one-dimensional hydromagnetic dynamics with dissipative terms. When the dissipation coefficient is equal to zero it is shown that the smooth solutions develop shocks in the finite time if the initial amounts of entropy and magnetic field are smaller than those of sound waves; when it is larger than zero, and the initial amounts of entropy, this dissipation coefficient and the magnetic field in each period are smaller than those of sound waves, then the smooth solutions blow up in the finite time. Moreover, the life-span of the smooth solution is given.     

Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations

We study the initial boundary value problem of two-dimensional viscous Boussinesq equations over a bounded domain with smooth boundary. We show that the equations have a unique classical solution for H ³ initial data and the no-slip boundary condition. In addition, we show that the kinetic energy is uniformly bounded in time.     

用丁二酮激光诱导磷光测量流场可能性的研究

对混合在氮气中丁二酮三重态~3Au的激光诱导磷光性质的实验结果表明,磷光寿命是温度的函数,与密度及浓度无关;而初始磷光强度是密度的线性函数,对温度变化不敏感,用观察氮气中少量丁二酮磷光寿命与磷光初始强度的方法可以测量流场的温度与密度分布,磷光寿命长的特点可以用来测量速度分布,它在很大程度上克服了多普勒测速中因固体微粒质量大而滞后于气流的问题,因而可以在具有高加速度的流场中使用,本文还对磷光衰减机理进行了分析,结果与实验一致,上述同时测温度与密度的方法尚未见诸文献报道。     

Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method.     

Differentiability of Solutions with Respect to the Initial Data in Differential Equations with State-dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is strictly monotone increasing.     

Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

We study the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier–Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above, independent of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.     

Optimal disturbances above and upstream of a flat plate with an elliptic-type leading edge

Adjoint-based iterative methods are employed to compute linear optimal disturbances in a spatially growing boundary layer around an elliptic leading edge. The Lagrangian approach is used where an objective function is chosen and constraints are assigned. The optimisation problem is solved using power iterations combined with a matrix-free formulation, where the state is marched forward in time with a standard direct numerical simulation solver and backward with the adjoint solver until a chosen convergence criterion is fulfilled. We consider the global and, more relevant to receptivity studies, the upstream localised optimal initial condition leading to the largest possible energy amplification at time T. We find that the two-dimensional initial condition with the largest potential for growth is a Tollmien–Schlichting-like wave packet that includes the Orr mechanism and is located inside the boundary layer downstream of the leading edge. Three-dimensional optimal disturbances induce streaks by the lift-up mechanism. Requiring the optimal initial condition to be localised upstream of the plate enables us to better study the effects of the leading edge on the boundary layer receptivity mechanisms. Two-dimensional upstream disturbances are inefficient at triggering unstable eigenmodes, whereas three-dimensional disturbances induce streamwise streaks with significant growth.     

Finite-time generalized synchronization of chaotic systems with different order

In this paper, the generalized synchronization of chaotic systems with different order is studied. The definition of finite-time generalized synchronization is put forward for the first time. Based on the finite-time stability theory, two control strategies are proposed to realize the generalized synchronization of chaotic systems with different order in finite time. Besides the relation between the parameter β, the initial states of systems and the convergent time were obtained. The corresponding numerical simulations are presented to demonstrate the effectiveness of proposed schemes.     

Vortex pairing in an axisymmetric jet using two-frequency acoustic forcing at low to moderate strouhal numbers

 Hot-wire measurement and multi-smoke wire flow visualization method are employed to study vortex pairing in the jet column mode under two-frequency forcing with controlled initial phase differences. For the range of 0.3<St _D <0.6, vortex pairing can be easily controlled by means of the fundamental and its subharmonic forcing with varying initial phase differences. As stable vortex pairing dominates, the variation of the subharmonic component with the initial phase difference changes from a sine shape to a cusp-like shape. The harmonics of the subharmonic also show similar trends. The detuning induces the amplitude and phase modulations of the u-signal in the time trace and the sideband growth in the spectra. The u-signal reflects the subharmonic variation with the initial phase difference in its envelope. For 0.6<St _D <0.9, non-pairing advection of vortices due to improper phase difference is sometimes observed under single-frequency forcing. In this case, vortex pairing can be made to occur by the addition of a subharmonic with very small amplitude. As the initial level of this subharmonic is increased, the onset position of vortex pairing moves upstream. In this range, the initial phase difference is not an effective parameter in controlling vortex pairing. Received: 22 May 1997 / Accepted: 16 October 1997