An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

The estimation of solution of the boundary value problem of the systems for quasi-linear ordinary differential equations

This paper deals.with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equations where x,f, y, h, A, B and C all belong to Rm, and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.     

Periodic solutions of dynamical systems

Summary In this paper we look for T-periodic solutions of dynamical systems. Particularly we consider the system whereU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. We assume that the problem is asymptotically linear with a bounded nonlinearity. Under a resonance assumption, we find a multiplicity of T-periodic solutions for T large enough.

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Sommario In questo lavoro si cercano soluzioni periodiche di periodo T assegnato di sistemi dinamici. In particolare si considera un sistema di n equazioni differenziali del secondo ordine del tipo doveU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. Nel caso in cui il problema sia asintoticamente lineare, con termine nonlineare limitato e in condizioni di risonanza, troviamo che esiste tale che per il sistema ha una molteplicità di soluzioni.

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Presented at the VII A.I.M.E.T.A. and supported by M.P.I. (40% and 60%).     

Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations

We study the solutions of the nonstationary incompressible Navier–Stokes equations in , of self-similar form , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x).     

Extended approaches to covariant Maxwell electrodynamics

The field equations of covariant Maxwell electrodynamics are a set I of 8 equations for the determination of 6 variables, i.e., the independent components of the skew-symmetric Maxwell tensorF . Obviously 2 of these equations are not evolutive; however they cannot be eliminated without losing manifest covariance. This paper presents a new hyperbolic set S of 8 equations in the 8 variablesF ,x, y, wherex, y are new auxiliary quantities. The solutions of S withx =y = 0 are those of the set I. Moreover, S is expressed in covariant form and is equivalent to a symmetric hyperbolic system.     

Behavior of Solutions of Second-Order Differential Equations with Sublinear Damping

We investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation

The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｅｘｉｓｔｅｎｃｅｏｆ2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｔｏｔｈｅｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｏｆｆｉｒｓｔ＿ｏｒｄｅｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈａｄｅｖｉａｔｉｎｇａｒｇｕｍｅｎｔ ｘ(ｔ) =Ｂｘ(ｔ) Ｆ(ｘ(ｔ-τ) ) ｐ(ｔ) ,( 1 )ｗｈｅｒｅｘ(ｔ)∈Ｒ2 , ｘ(ｔ) =ｄｄｔｘ(ｔ) ,τ∈Ｒ ,Ｂ∈Ｒ2×2 ,Ｆ :Ｒ2 →Ｒ2 ｉｓｂｏｕｎｄｅｄａｎｄｐ∈Ｃ(…     

Self-similarity of nonstationary boundary-layer motions

In this paper we show that problems concerning the development of a boundary layer on a semi-infinite plate when the outer flow speed is of the form U = (1 + ct)^b a, and on a cylinder when the outer flow speed has the forms U = ctx^m and U = (1 + ct)^b ax^m, are self-similar. We present the results of numerical calculations for various values of , b, and m. We consider the problem of a stepwise nonstationary heating of a plate, impulsively set into motion in an incompressible fluid; we show that this problem is self-similar and obtain its solution numerically.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 122–125, July–August, 1975.     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

The existence and uniqueness of a solution to the nonstationary Navier–Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x₃ and find the solution (u, p) having the following form

BASIC EQUATIONS, THEORY AND PRINCIPLE OF COMPUTATIONAL STOCK MARKET (Ⅲ)-BASIC THEORIES

By basic equations, two basic theories are presented: 1. Theory of stock’s value v^*(t)=v^*(0) exp (ar₂^*t);2. Theory of conservation of stock’s energy. Let stock’s energy Φbe defined as a quadratic function of stock’s price v and its derivative v, Φ=Av²+Bv+Cv²+Dv, under the constraint of basic equation, the problem was reduced to a problem of constrained optimization along optimal path. Using Lagrange multiplier and Euler equation of variation method, it can be proved that Φkeeps conservation for any v,v. The application of these equations and theories on judgement and analysis of tendency of stock market are given, and the judgement is checked to be correct by the recorded tendency of Shenzhen and Shanghai stock markets.     

On the motion of a material point on a moving surface

Summary Sufficient conditions are given for the stability and instability of the equilibrium position x=y=z=0 in the mechanical system consisting of a material point constrained to move on the moving surface z=−λ(t)(x²+y²) (λ(t)>0) in a constant field of gravity (the axis 0z is directed vertically upward) under the action of viscous friction of total dissipation.

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Sommario Si danno condizioni sufficienti per la stabilità e la instabilità della posizione di equilibrio x=y=z=0 nel sistema meccanico che consiste di un punto materiale vincolato a muoversi sulla superficie mobile z=−λ(t)(x²+y²) (λ(t)>0) in un campo di gravità costante (l'asse 0z è diretto verticalmente e orientato verso l'alto) sotto l'azione di attriti viscosi con dissipazione completa.

     

Ellipsoidal Domains: Piecewise Nonuniform and Impotent Eigenstrain Fields

In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment. Consider the set of nested similar ellipsoidal domains {Ω₁, Ω₂,⋯,Ω _N+1}, which are embedded in an infinite isotropic medium. Suppose that

Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains

In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R³ and β ∈[1/2, 1) , then there exist C₀ > 1 and δ₀ ∈ (0, 1) such that

Deduction of Saint-Venant"s Principle for the Asymptotic Residue of Elastic Rods

Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ω^ε having cross section with diameter of order ε, and let u ⁽⁰⁾ –Bernoulli–Navier displacement – and u ⁽²⁾ be the two first terms derived from the asymptotic method. We analyze the residue r(ε) = u(ε) − (u ⁽⁰⁾ + ε² u ⁽²⁾) and if the cross section is star-shaped, we prove such residue presents a Saint-Venant"s phenomenon near the ends of the rod. This revised version was published online in August 2006 with corrections to the Cover Date.     

Boundary control of the generalized Korteweg–de Vries–Burgers equation

This paper considers the boundary control problem of the generalized Korteweg–de Vries–Burgers (GKdVB) equation on the interval [0, 1]. We derive a control law of the form and α is a positive integer, and prove that it guarantees L ²-global exponential stability, H ¹-global asymptotic stability, and H ¹-semiglobal exponential stability. Numerical results supporting the analytical ones for both the controlled and uncontrolled equations are presented using a finite element method.     

Uniform Exponential Dichotomy and Continuity of Attractors for Singularly Perturbed Damped Wave Equations

For , we consider a family of damped wave equations , where − Λ denotes the Laplacian with zero Dirichlet boundary condition in L ²(Ω). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space semigroups which have global attractors A _η, . We show that the family , behaves upper and lower semicontinuously as the parameter η tends to 0⁺.     

Incompressible Fluids in Thin Domains with Navier Friction Boundary Conditions (I)

This study is motivated by problems arising in oceanic dynamics. Our focus is the Navier–Stokes equations in a three-dimensional domain Ω_ɛ, whose thickness is of order O(ɛ) as ɛ → 0, having non-trivial topography. The velocity field is subject to the Navier friction boundary conditions on the bottom and top boundaries of Ω_ɛ, and to the periodicity condition on its sides. Assume that the friction coefficients are of order O(ɛ^3/4) as ɛ → 0. It is shown that if the initial data, respectively, the body force, belongs to a large set of H¹(Ω_ɛ), respectively, L²(Ω_ɛ), then the strong solution of the Navier–Stokes equations exists for all time. Our proofs rely on the study of the dependence of the Stokes operator on ɛ, and the non-linear estimate in which the contributions of the boundary integrals are non-trivial.     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.