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1.
We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u T (x,t)=u(T x,T t) is pre-compact as t→∞ in L loc 1(? d +1 +, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L 1 loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l ?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l ?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.  相似文献   

2.
The Navier-Stokes system for a steady-state barotropic nonlinear compressible viscous flow, with an inflow boundary condition, is studied on a polygon D. A unique existence for the solution of the system is established. It is shown that the lowest order corner singularity of the nonlinear system is the same as that of the Laplacian in suitable L q spaces. Let ω be the interior angle of a vertex P of D. If \(\) and \(\), then the velocity u is split into singular and regular parts near the vertex P. If α < 2 and \(\) or if α > 2 and 2 < q < ∞&;, it is shown that u∈ (H 2, q (D))2.  相似文献   

3.
Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component u j of the velocity field u is determined by the scalar θ through \({u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}\) , where \({\mathcal{R}}\) is a Riesz transform and Λ = (?Δ)1/2. The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. We develop tools to bound \({\|\nabla u||_{L^\infty}}\) for a general class of operators P and establish the global regularity for the Loglog-Euler equation for which P(Λ) = (log(I + log(I ? Δ))) γ with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ β with 0 ≦ β ≦ 1 is also obtained.  相似文献   

4.
This article is concerned with interface problems for Lipschitz mappings f +:? n +→? n and f ?:? n ?→? n in the half spaces, which agree on the common boundary ? n ? 1=?? n +=?? n ?. These naturally occur in mathematical models for material microstructures and crystals. The task is to determine the relationship between the sets of values of the differentials Df + and Df ?. For some time it has been thought that the polyconvex hulls [Df +] pc and [Df ?] pc satisfy Hadamard's jump condition or are at least rank-one connected. Our examples here refute this idea.The estimates of the Jacobians we obtain in the course of solving the so-called Monge-Ampère inequalities seem also to be of independent interest. As an application, we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar Cauchy-Riemann equations, for which the unique continuation property fails.  相似文献   

5.
6.
The long-time asymptotics is analyzed for all finite energy solutions to a model\(\mathbf{U}(1)\)-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e?iω t. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time spectrum in the spectral gap [ ? m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single harmonic \(\omega\in[-m,m]\).The research is inspired by Bohr’s postulate on quantum transitions and Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled\(\mathbf{U}(1)\)-invariant Maxwell–Schrödinger and Maxwell–Dirac equations.  相似文献   

7.
The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β T and a solutal mobility ratio β C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β T is seen to enhance the instability for fixed β C , Le and λ. For fixed β C and β T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β C , but β T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β C only.  相似文献   

8.
In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v 0, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ 0 + S ψ*0 + x(C ψ 1 ? S ψ 2) + y(C ψ 2 + S ψ 1), where C(v) and S(v) are the operator functions described in [1] and ψ 0(u), …, ψ 2(u) are the desired functions. The boundary conditions for the function F posed for v = ±v 0 are equivalent to two operator equations for ψ 1(u) and ψ 2(u) and to two ordinary differential equations of first order for ψ 0(u) and ψ*0(u) [2]. By finding the functions ψ j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v 0 = 1) loaded on the boundary v = v 0 by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X and to the transverse and pure bending of the strip due to the transverse force Y and the constant moment M , respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.  相似文献   

9.
The influences of fuel Lewis number Le F on localised forced ignition of inhomogeneous mixtures are analysed using three-dimensional compressible Direct Numerical Simulations (DNS) of turbulent mixing layers for Le F  = 0.8, 1.0 and 1.2 and a range of different root-mean-square turbulent velocity fluctuation u′ values. For all Le F cases a tribrachial flame has been observed in case of successful ignition. However, the lean premixed branch tends to merge with the diffusion flame on the stoichiometric mixture fraction isosurface at later stages of the flame evolution. It has been observed that the maximum values of temperature and reaction rate increase with decreasing Le F during the period of external energy addition. Moreover, Le F is found to have a significant effect on the behaviours of mean temperature and fuel reaction rate magnitude conditional on mixture fraction values. It is also found that reaction rate and mixture fraction gradient magnitude \(\vert \nabla \xi \vert \) are negatively correlated at the most reactive region for all values of Le F explored. The probability of finding high values of \(\vert \nabla \xi \vert \) increases with increasing Le F . For a given value of u′, the extent of burning decreases with increasing Le F . A moderate increase in u′ gives rise to an increase in the extent of burning for Le F  = 0.8 and 1.0, which starts to decrease with further increases in u′. For Le F  = 1.2, the extent of burning decreases monotonically with increasing u′. The extent of edge flame propagation on the stoichiometric mixture fraction ξ = ξ st isosurface is characterised by the probability of finding burned gas on this isosurface, which decreases with increasing u′ and Le F . It has been found that it is easier to obtain self-sustained combustion following localised forced ignition in case of inhomogeneous mixtures than that in the case of homogeneous mixtures with the same energy input, energy deposition duration when the ignition centre is placed at the stoichiometric mixture. The difficultly to sustain combustion unaided by external energy addition in homogeneous mixture is particularly prevalent in the case of Le F  = 1.2.  相似文献   

10.
In technological processes of rod bending, the critical time is determined [1] by the criterion of unbounded increase A → ∞ in the bent axis amplitude, which is equivalent to the requirement A ? A 0, where A 0 is value of the amplitude at the initial time t = 0. In this case, the mathematical models of the process of buckling of rods and plates [2] are constructed in the framework of the theory of small displacements. This contradiction can be removed by the assumption that the critical state is realized for deflections A of the order of several A 0, i.e., at the time instant corresponding to a sharp increase in displacements. Naturally, this assumption is of local character, because the instant of the transition to the accelerated increase in deflections depends on specific conditions such as, for example, the support conditions, the creep coefficient, the type of the system imperfectness, the value of A 0, and the eccentricity of the load application.In what follows, we show that, in the case of longitudinal bending (buckling), the time instant directly preceding the beginning of the catastrophic increase in deflections can be determined by the variation in the phase volume of the system.  相似文献   

11.
An experimental study was carried out to investigate the effect of periodic blowing and suction on a turbulent boundary layer. Particle image velocimetry (PIV) was used to probe the characteristics of the flow. Local forcing was introduced to the boundary layer via a sinusoidally-oscillating jet issuing from a thin spanwise slot. Three forcing frequencies (f+=0.44, 0.66 and 0.88) with a fixed forcing amplitude (A+=0.6) were employed at Re θ =690. The effect of three different forcing angles (α=60°, 90° and l20°) was investigated under a fixed forcing frequency (f+=0.088). The PIV results showed that the wall-region velocity decreases on imposition of the local forcing. Inspection of the phase-averaged velocity profiles revealed that spanwise large-scale vortices are generated downstream of the slot and persist farther downstream. The highest reduction in skin friction was achieved at the highest forcing frequency (f+=0.088) and a forcing angle of α=120°. The spatial fraction of the vortices was examined to analyze the skin friction reduction.  相似文献   

12.
We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x t , x t ). The state space is a closed subset in a manifold of C 2-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).  相似文献   

13.
A scale-similarity model of a two-point two-time Lagrangian velocity correlation(LVC) was originally developed for the relative dispersion of tracer particles in isotropic turbulent flows(HE, G. W., JIN, G. D., and ZHAO, X. Scale-similarity model for Lagrangian velocity correlations in isotropic and stationary turbulence. Physical Review E, 80, 066313(2009)). The model can be expressed as a two-point Eulerian space correlation and the dispersion velocity V. The dispersion velocity denotes the rate at which one moving particle departs from another fixed particle. This paper numerically validates the robustness of the scale-similarity model at high Taylor micro-scale Reynolds numbers up to 373, which are much higher than the original values(R_λ = 66, 102). The effect of the Reynolds number on the dispersion velocity in the scale-similarity model is carefully investigated. The results show that the scale-similarity model is more accurate at higher Reynolds numbers because the two-point Lagrangian velocity correlations with different initial spatial separations collapse into a universal form compared with a combination of the initial separation and the temporal separation via the dispersion velocity.Moreover, the dispersion velocity V normalized by the Kolmogorov velocity V_η≡η/τ_η in which η and τ_η are the Kolmogorov space and time scales, respectively, scales with the Reynolds number R_λ as V/V_η∝ R_λ~(1.39) obtained from the numerical data.  相似文献   

14.
Turbulent flows in channels with intense distributed injection are modeled using the large eddy method and the two-equation k-? turbulence model. The calculations are carried out for different velocities of injection from the channel walls. For a channel with one-sided injection the results of large eddy simulation are in good agreement with the measured data, whereas the calculations in accordance with the k-? model give a less convex cross-sectional velocity profile and an appreciable error in determining the surface friction coefficient on the impermeable wall and also have certain other shortcomings. In the case of two-sided injection, the results of the calculations by the large eddy method and the k-? model are in good agreement with one another and the data of physical experiments.  相似文献   

15.
We determine all the \({\mathcal{C}^1}\) planar vector fields with a given set of orbits of the form y ? y(x) = 0 satisfying convenient assumptions. The case when these orbits are branches of an algebraic curve is also study. We show that if a quadratic vector field admits a unique irreducible invariant algebraic curve \({g(x, y) = \sum_{j=0}^S a_j(x) y^{S-j}= 0}\) with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S.  相似文献   

16.
On the basis of an asymptotic analysis of the Navier-Stokes system of equations for large Reynolds numbers (Re → ∞), the plane incompressible fluid flow near a surface having a convex corner with a small angle 2θ* is investigated. It is shown that for θ* = O(Re?1/4), in addition to the known solution that describes a separated flow completely localized in a thin “viscous” sublayer of the interaction region near the corner point, another solution corresponding to a flow with a developed separation zone is possible. For θ 0 = Re1/4 θ* = O(1), the longitudinal dimension of this zone varies from finite values up to values of the order of Re?3/8. The nonuniqueness of the solution is established on a certain range of variation of the parameter θ 0. The dependence of the drag coefficient on the angle θ* is found.  相似文献   

17.
Dynamic-stress concentrations due to elastic waves are analyzed by high-speed photoelasticity, and some differences between dynamic- and static-stress distributions are clarified. Specimens such as long struts with shoulders or notches are loaded dynamically by an elastic wave behind the front of a rectangular pulse with a plateau of constant stress ρ cv (ρ=density,c=elasticwave velocity,v=tensile velocity) and of long duration. Dynamic isochromatic patterns caused in polyurethanerubber specimens are recorded with a “Himac 16H” high-speed camera (framing speed: 10,000 pictures/ sec). It is found that dynamic-stress-concentration factors, that is, maximum fringe order in passage of the wave front and the initial part of the plateau, divided by the fringe order corresponding to the height of the plateau, are about 0.5–0.6 times the static-stress-concentration factors, for the specimen dimensions considered. An approximate theory of dynamic-stress concentration factor is derived by considering notches as the discontinuities in the cross-sectional area of strut. This theoretical consideration correlates somewhat with the values of dynamic-stress-concentration factors obtained experimentally.  相似文献   

18.
In a bounded domain \({\Omega \subset \mathbb R^2}\) with smooth boundary we consider the problem
$\Delta u = 0 \quad {\rm{in }}\, \Omega, \qquad \frac{\partial u}{\partial \nu} = \frac1\varepsilon f(u) \quad {\rm{on }}\,\partial\Omega,$
where ν is the unit normal exterior vector, ε > 0 is a small parameter and f is a bistable nonlinearity such as f(u) = sin(π u) or f(u) = (1 ? u 2)u. We construct solutions that develop multiple transitions from ?1 to 1 and vice-versa along a connected component of the boundary ?Ω. We also construct an explicit solution when Ω is a disk and f(u) = sin(π u).
  相似文献   

19.
We study the values e σ(f) of the best approximation of integrals of functions from the spaces L p (A, dμ) by integrals of rank σ. We determine the orders of the least upper bounds of these values as σ → ∞ in the case where the function ? is the product of two nonnegative functions one of which is fixed and the other varies on the unit ball U p (A) of the space L p (A, dμ). We consider applications of the obtained results to approximation problems in the spaces S p ? .  相似文献   

20.
In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let \(\hat n\) be the outward unit normal to S, ▽ S the surface gradient on S, I S the metric tensor on S, gij the four covariant components of I S (i,j = 1, 2), h ij the four covariant components of -\(\hat n\)xI S , and D i covariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ? and ψ on S, unique to within additive constants, such that \(u = \nabla _s \varphi - \hat n \times \nabla _s \psi \); the covariant components of u are \(u_i = D_i \varphi + h_i^j D_j \psi \). This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are
$$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$  相似文献   

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