Another Proof of Stability for Global Weak Solutions of 2D Degenerated Shallow Water Models

We consider non-linear viscous shallow water models with varying topography, extra friction terms and capillary effects, in a two-dimensional framework. Water-depth dependent laminar and turbulent friction coefficients issued from an asymptotic analysis of the three-dimensional free-surface Navier–Stokes equations are considered here. A new proof of stability for global weak solutions is given in periodic domain Ω = T², adapting the method introduced by J. Simon in [15] for the non-homogeneous Navier–Stokes equations. Existence results for such solutions can be obtained from this stability analysis.     

Analysis on non-oscillatory singularity behaviors of mode Ⅱ interface crack tip in orthotropic bimaterial

The fracture behaviors near the mode Ⅱ interface crack tip for orthotropic bimaterial are studied. The non-oscillatory field, where the stress singularity exponent is a real number, is discussed by the complex function method and the undetermined coefficient method. From the research fracture problems, the stress functions with ten undetermined coefficients and an unknown singularity exponent are introduced when?_1 0 and ?_2 0. By the existence theorem of non-trival solutions for the system of eight homogeneous linear equations, the characteristic equation, the stress singularity exponent, and the discriminating condition of the non-oscillatory singularity are found.By the uniqueness theorem of the solutions for the system of twelve non-homogeneous linear equations with ten unknowns, the ten undermined coefficients in the stress functions are uniquely determined. The definitions of the stress intensity factors are given with the help of one-sided limit, and their theoretical formulae are deduced. The analytic solutions of the stresses near the mode Ⅱ interface crack tip are derived. The classical results for orthotropic material are obtained.     

On the solution of the model of two co-affected species

ＯＮＴＨＥＳＯＬＵＴＩＯＮＯＦＴＨＥＭＯＤＥＬＯＦＴＷＯＣＯ－ＡＦＦＥＣＴＥＤＳＰＥＣＩＥＳＺｈａｎｇＷｅｉ－ｆｕ（张为付）,ＬｕＲｏｎｇ－ｑｉｎｇ（吕荣庆）（ＴｈｅＡｉｒＦｏｒｃｅＩｎｓｔｉｔｕｔｅｏｆＭｅｔｅｏｒｏｌｏｇｙ,Ｎａｎｊｉｎｇ００３Ｂ...     

Asymptotic Solutions of Boundary Value Problems for Third-Order Ordinary Differential Equations with Turning Points

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｓｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈｎｏｔｕｒｎｉｎｇｐｏｉｎｔｈａｄｂｅｅｎｓｔｕｄｉｅｄｖｅｒｙｅａｒｌｙ ,ｂｕｔｔｈｅｓｔｕｄｙｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｗｉｔｈｔｕｒｎｉｎｇｐｏｉｎｔｐｒｏｃｅｅｄｅｄｒａｔｈｅｒｓｌｏｗｌｙｏｗｉｎｇｔｏｔｈｅｓｉｎｇｕｌａｒｐｏｉｎｔｏｆｔｈｅｄｅｇｅｎｅｒａｔｅｄｅｑｕａｔｉｏ…     

Fast Decays of Strong Global Solutions of the Navier–Stokes Equations

In the paper we study the asymptotic dynamics of strong global solutions of the Navier Stokes equations. We are concerned with the question whether or not a strong global solution w can pass through arbitrarily large fast decays. Avoiding results on higher regularity of w used in other papers we prove as the main result that for the case of homogeneous Navier–Stokes equations the answer is negative: If [0, 1/4) and δ₀ > 0, then the quotient remains bounded for all t ≥ 0 and δ∈[0, δ₀]. This result is not valid for the non-homogeneous case. We present an example of a strong global solution w of the non-homogeneous Navier–Stokes equations, where the exterior force f decreases very quickly to zero for while w passes infinitely often through stages of arbitrarily large fast decays. Nevertheless, we show that for the non-homogeneous case arbitrarily large fast decays (measured in the norm cannot occur at the time t in which the norm is greater than a given positive number.      

Further improvement on fundamental solutions of plane problems for orthotropic materials

On the basis of the ecisting fundamental solutions of displacements, further improvement is made, and then the general fundamental solutions of both plane elastic and plane plastic problems for orthotropic materials are obtained. Two parameters based on material constants α₁ = α₁ are used to derive the relevant expressions in a real variable form. Additionally an analytical method of solving the singular integral for the internal stresses is introduced, and the corresponding results are given. If α₁ = α₁ = 1, all the expressions obtained for orthotropy can be reduced to the corresponding ones for isotropy. Because all these expressions and results can be directly used for both isotropic problems and orthotropic problems, it is convenient to use them in engineering with the boundary element method (BEM).     

On the unsteady rotational flow of a generalized Maxwell fluid through a circular cylinder

In this paper, the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized Maxwell fluid within an infinite circular cylinder are determined by means of the Laplace and finite Hankel transforms. Initially, the fluid is at rest, and the motion is produced by the rotation of the cylinder about its axis with a unsteady angular velocity. The solutions that have been obtained are presented under series form in terms of the generalized G _a,b,c (, t)-functions. The similar solutions for the ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases, when β → 1, respectively β → 1 and λ → 0, from general solutions. Finally, the solutions that have been obtained are compared by graphical illustrations, and the influence of the pertinent parameters on the fluid motion is also underlined by graphical illustrations.     

Generalized flow analysis of non-Newtonian visco-elastic fluid flow through fractal reservoir

Ｆｒａｃｔａｌｇｅｏｍｅｔｒｙｉｓａｐｏｗｅｒｆｕｌｔｏｏｌｔｏｄｅｓｃｒｉｂｅｃｏｍｐｌｅｘｐｈｅｎｏｍｅｎｏｎ．Ｅｓｐｅｃｉａｌｌｙｉｔｉｓａｐｐｒｏｐｒｉａｔｅｔｏｓｃａｌｅｔｈｅｎｏｎｕｎｉｆｏｒｍｉｔｙａｎｄｎｏｎｓｅｑｕｅｎｃｅｏｆｐｏｒｏｕｓｍｅｄｉａ．Ｉｆｔｈｅｍｅｃｈａｎｉｃｓｏｆｆｌｕｉｄｆｌｏｗｔｈｒｏｕｇｈｐｏｒｏｕｓｍｅｄｉａｉｓｓｔｕｄｉｅｄｂｙｕｓｉｎｇｆｒａｃｔａｌ,ｔｈｅｄｉｓｃｅｒｎｉｂｌｅａｎｄｃｏｇｎｉｔｉｖｅａｂｉｌｉｔｙｆｏｒｐｏｒｏｕｓｍｅｄｉａａｎ…     

Intermediate Asymptotics Beyond Homogeneity and Self-Similarity: Long Time Behavior for u t = Δϕ(u)

We investigate the long time asymptotics in L¹₊(R) for solutions of general nonlinear diffusion equations u_t = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d₂. In the particular case of ϕ(u) ~ u^m at first order when u ~ 0, we also obtain an optimal rate of convergence in L¹ towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation of the porous medium equation.     

Structure Functions in Complex Flows

The turbulence in the ocean and atmosphere is most of the time non-homogeneous in nature. These spatial changes could affect the structure of the turbulence. In this work a classification is proposed to determine the intermittency and mixing ability. The variation of the structure functions and the scaling exponent in decaying non-homogeneous turbulence produced by a grid and by a jet is measured with a sonic velocimeter SONTEK3-D. We use Extended Self Similarity (ESS) to obtain better estimates of the scaling exponents of the structure functions of order up to the 6th. We study the variation of the absolute scaling exponents _p and relative scaling exponents ¯_p as a function of distance from the source of turbulence. In most cases, the absolute scaling exponent ₃ is shown to vary as function of the separation distance l. On the other hand the relative scaling exponents ¯_p depend on the location of the flow and in most cases the deviations from the Kolmogorov 1941 scaling are related to the intermittency.     

Exact Solution of the Wave‐Resistance Problem for a Submerged Cylinder. I. Linearized Theory

. We consider the problem of finding a holomorphic function in a strip with a cut ${\cal A}= \{(x,y) : \, x\in\RE,\,\,0 satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for $|x|\to\infty. We consider the problem of finding a holomorphic function in a strip with a cut satisfying some prescribed linear conditions on the boundary. The problem has a one‐parameter family of solutions in the class of sectionally holomorphic functions in ?, vanishing for . A special solution can be selected by fixing the value of the circulation around the cut. The problem is obtained by linearization of the equations of the wave‐resistance problem for a “slender” cylinder submerged in a heavy fluid and moving at uniform speed in the direction orthogonal to its generators. The results obtained, besides their own interest, are a crucial step for the resolution of the non‐linear problem. (Accepted October 14, 1998)     

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 + , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.     

条形荷载下梯度非均匀非饱和土的动力响应分析

基于多孔介质混合物理论, 建立了梯度非均匀非饱和土地基模型, 研究了条形荷载作用下梯度非均匀非饱和土地基的动力响应问题. 通过傅里叶积分变换和Helmholtz矢量分解原理, 获得频域内非饱和土地基动力响应问题的通解, 结合回传射线矩阵法和边界条件, 求解获得了非均匀非饱和土层中位移、应力以及孔隙压力的计算列式. 假设沿深度方向梯度非均匀非饱和土的物理力学性质按幂函数连续变化, 通过数值傅里叶逆变换得到了非均匀非饱和土地基中的应力、位移以及孔隙压力等物理量的数值解, 分析讨论了土体非均匀性对非饱和土介质动力响应的影响规律. 结果表明: 土体非均匀性显著改变了非饱和土中竖向位移、正应力和孔隙压力在其深度方向上的振动模态, 其中孔隙气压在其深度方向的振动频率随着梯度因子的增加而不断增大, 波峰值不断靠近地表处附近; 竖向位移随着梯度因子的增大不断减小; 正应力和孔隙水压随着梯度因子的增大先增大后减小, 并且土体非均匀程度越高, 正应力与孔隙水压的幅值越大.      

Quasiconvexity at the Boundary and a Simple Variational Formulation of Agmon's Condition

We study the question of positivity of quadratic funtionals which typically arise as the second variation at a critical point u of a functional. For interior points x₁∈ Ω rank-one convexity of C₀(x₁) is a necessary condition for u to be a local minimizer. For boundary points x₂∈ ∂ Ω where ϕ is allowed to vary freely the stronger condition of quasiconvexity at the boundary is necessary. For quadratic functionals this condition is roughly equivalent to rank-one convexity and Agmon's condition. We derive an equivalent condition on C₀(x₂) which is purely algebraic; and, moreover, it is variational in the sense that it can be formulated in terms of positive semidefiniteness of Hermitian matrices. A connection to the solvability of matrix-valued Riccati equations is established. Several applications in elasticity theory are treated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Periodic solutions of a class of higher order neutral type equations

In this paper, by ssing the theory of Fourier series, some necessary and sufficient conditions of existence and uniqueness of periodic solutions of a class of higher order neutral type equations are obtained. The main results by Shi Jianguo in “Discussion on the periodic solutions for linear equation of neutral type with constant coefficients” are improved, i. e., the condition |b₀|≠1 instead of the condtion |b₀|<1/2 of Theorem 1 by Shi Jianguo is given. Other theorems by Shi are rebuilt and improved according to the new assumption. Foundation item: the Natural Science Foundation of Yunnan Province, China (97A012G)     

A Particular Class of Partially Invariant Solutions of the Navier—Stokes Equations

One class of partially invariant solutions of the Navier—Stokes equations is studied here. This class of solutions is constructed on the basis of the four-dimensional algebra L ₄ with the generators Systematic investigation of the case, where the Monge—Ampere equation (10) is hyperbolic (Lf _z + k + l ≥ 0) is given. It is shown that this class of solutions is a particular case of the solutions with linear velocity profile with respect to one or two space variables.     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（Ｉ）ＱｉｎＳｈｅｎｇ－Ｉｉ（秦圣立）ＨｕａｎｇＪｉａ－ｙｉｎ（黄家寅）（Ｑｕｆ...     

Mixed Convection Boundary-Layer Flow in a Porous Medium Filled with Water Close to its Maximum Density

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water close to its maximum density is considered for uniform wall temperature and outer flow. The problem can be reduced to similarity form and the resulting equations are examined in terms of a mixed convection parameter λ and a parameter δ which measures the difference between the ambient temperature and the temperature at the maximum density. Both assisting (λ > 0) and opposing flows (λ < 0) are considered. A value δ₀ is found for which there are dual solutions for a range λ_c < λ < 0 of λ (the value of λ_c dependent on δ) and single solutions for all λ ≥ 0. Another value of δ₁ of δ, with δ₁ > δ₀, is found for which there are dual solutions for a range 0 < λ < λ_c of positive values of λ, with solutions for all λ≤ 0. There is also a range δ₀ <  δ < δ₁ where there are solutions only for a finite range of λ, with critical points at both positive and negative values of λ, thus putting a finite limit on the range of existence of solutions.     

A class of incompletely exponentially fitted difference schemes for a singular perturbation problem

Some sufficient conditions are considered, under which the solutions of a class of incompletely exponentially fitted difference schemes converge uniformly in e, with orders one and two, to the solution of the singular perturbation problem: eu"+a(x)u’-b(x)u=f(x), for 0a>0, b(x)≥0. From these conditions.an incompletely exponentially fitted second-order scheme is derived. Finally, the results of some numerical experiments are given.