A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations

We give simple proofs that a weak solution u of the Navier–Stokes equations with H ¹ initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L ^s (0, T;L ^r,∞(Ω)) or if its L ^s,∞(0, T;L ^r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.     

Minimizing Sequences Selected via Singular Perturbations, and their Pattern Formation

We study the Cahn-Hilliard energy E _ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E ₀(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E _ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E _ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property. Accepted January 21, 2000?Published online November 24, 2000     

Liquid–crystalline nematic polymers revisited

A new model for nematic polymers is proposed, based on the probability ψ(u,u,t) for a macromolecule to be oriented along direction u while embedded in a u environment created by its neighbours. The potential of the internal forces is written Φ(u,u) accordingly. The free energy contains a contribution ν Φ + k_BT ln ψ where the brackets mean an average over the probability distribution, while ν is the (uniform) polymer number density. An equation is derived for the time-evolution of the order parameter S = uu − I/3, together with an expression for the stress tensor. These two results offer a generalization of the Doi Model in so far as they include a distortional energy, analogue to the Frank elastic energy for low molecular mass nematics. Extending the Maier–Saupe variational procedure, we specify the way that the internal potential Φ(u,u) must be written for it to favour non-zero values of the order parameter, while giving a penalty to situations with gradients of the order parameter. The result is quite different from the potential proposed a decade ago by Marrucci and Greco (their Φ depends on u only), while it has a clear connection with the so-called Landau-de Gennes (LdG) tensor models, which are based on a free-energy depending on the order parameter and its gradients.     

Local energy decay for the wave equation on the exterior of two balls or convex bodies

An algebraic rate of decay of local energy, nonuniform with respect to the initial data, is established for solutions of the Dirichlet and Neumann problems for the scalar wave equation defined on the exterior V³ of two balls or of two convex bodies. That is, for given initial data f(x)=u(x), 0 and g(x)= u _t (x, 0), if u solves u _tt in V with either u(x, t)=0 or u _n (x,t)+(x) u(x,t,)-0 ((x)0) on V, then there exists a constant T ₀, depending upon (f, g), such that the local energy (the energy in any compact set) of u at t=T is bounded from above by QE(0)T ^–1 for TT ₀, where E(0) is the total initial energy of u and Q is a positive constant, independent of u, that depends upon V.     

Existence of Time-Periodic Solutions for a Magnetoelastic System in Bounded Domains

We investigate the existence of periodic solutions for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domains with boundaries of class C ². The mathematical model includes a nonlinear mechanical dissipation like ρ(u′)=|u′| ^p u′ and a periodic forcing function of period T. We prove the existence of T-periodic weak solutions when p∈[3,4] (p=0 being a simpler case). In the corresponding two-dimensional case, the existence result holds under the assumption that p≥2.     

Entire Solutions with Merging Fronts to Reaction–Diffusion Equations

We deal with a reaction–diffusion equation u _t = u _xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ₁(x + c ₁ t) (c ₁ < 0) and ψ₂(x + c ₂ t) (c ₂ > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all . We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ₁(x + c ₁ t) and ψ₂(x + c ₂ t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c ₁, we show the existence of an entire solution which behaves as ψ₁( − x + c ₁ t) in and φ(x + ct) in for t≈ − ∞.     

The stress conjugate to logarithmic strain

A stress S is said to be conjugate to a strain measure E if the inner product S · E̊ is the power per unit volume. The logarithmic strain In U, with U the right stretch tensor, has been considered an interesting strain measure because of the relationship of its material time derivative (In U) with the stretching tensor D. In a previous article (Int. J. Solids Structures 22, 1019–1032 (1986)) a formula for (In U) was obtained in direct notation for the cases where the principal stretches are repeated, as well as for the case where they are all distinct. Here the formula for (In U) and the definition of conjugate stress are used to derive an explicit, properly invariant expression for the stress conjugate to the logarithmic strain.     

Backward Uniqueness for Parabolic Equations

It is shown that a function u satisfying |_t+u|M(|u|+|u|), |u(x, t)|Me^M|x|2 in (ⁿ \ (B_R) × [0, T] and u(x, 0) = 0 for xⁿ \ B_R must vanish identically in ⁿ \ B_R×[0, T].     

Some new classes of inverse coefficient problems in non-linear mechanics and computational material science

Three classes of inverse coefficient problems arising in engineering mechanics and computational material science are considered. Mathematical models of all considered problems are proposed within the J₂-deformation theory of plasticity. The first class is related to the determination of unknown elastoplastic properties of a beam from a limited number of torsional experiments. The inverse problem here consists of identifying the unknown coefficient g(ξ²) (plasticity function) in the non-linear differential equation of torsional creep −(g(|∇u|²)u_x1)_x1−(g(|∇u|²)u_x2)_x2=2?, x∈Ω⊂R², from the torque (or torsional rigidity) T(?), given experimentally. The second class of inverse problems is related to the identification of elastoplastic properties of a 3D body from spherical indentation tests. In this case one needs to determine unknown Lame coefficients in the system of PDEs of non-linear elasticity, from the measured spherical indentation loading curve P=P(α), obtained during the quasi-static indentation test. In the third model an inverse problem of identifying the unknown coefficient g(ξ²(u)) in the non-linear bending equation is analyzed. The boundary measured data here is assumed to be the deflections w_i[τ_k]?w(λ_i;τ_k), measured during the quasi-static bending process, given by the parameter τ_k, , at some points , of a plate. An existence of weak solutions of all direct problems are derived in appropriate Sobolev spaces, by using monotone potential operator theory. Then monotone iteration schemes for all the linearized direct problems are proposed. Strong convergence of solutions of the linearized problems, as well as rates of convergence is proved. Based on obtained continuity property of the direct problem solution with respect to coefficients, and compactness of the set of admissible coefficients, an existence of quasi-solutions of all considered inverse problems is proved. Some numerical results, useful from the points of view of engineering mechanics and computational material science, are demonstrated.     

Darboux problem for a differential equation with fractional derivative

We find conditions for the unique solvability of the problem u _xy (x, y) = f(x, y, u(x, y), (D ₀ ^r u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D ₀ ^r u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r ₁, r ₂), 0 < r ₁, r ₂ < 1, in the class of functions that have the continuous derivatives u _xy (x, y) and (D ₀ ^r u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005.     

Heat Flow of Harmonic Maps Whose Gradients Belong to L^{n}_{x}L^{\infty}_{t}

For any compact n-dimensional Riemannian manifold (M, g) without boundary, a compact Riemannian manifold without boundary, and 0 < T ≦ +∞, we prove that for n ≧ 4, if u : M × (0, T] → N is a weak solution to the heat flow of harmonic maps such that , then u ∈C ^∞(M × (0, T], N). As a consequence, we show that for n ≧3, if 0 < T < +∞ is the maximal time interval for the unique smooth solution u ∈C ^∞(M × [0, T), N) of (1.1), then blows up as t ↑ T.     

A symmetry analysis of some classes of evolutionary nonlinear (2+1)-diffusion equations with variable diffusivity

In this paper the (2+1)-nonlinear diffusion equation u _t ?div(f(u)grad u)=0 with variable diffusivity is considered. Using the Lie method, a complete symmetry classification of the equation is presented. Reductions, via two-dimensional Lie subalgebras of the equation, to first- or second-order ordinary differential equations are given. In a few interesting cases exact solutions are presented.     

Large bending behavior of creased paperboard. I. Experimental investigations

The large bending behavior of a creased paperboard is studied in the range of rotation θ ? [0°, 180°] – new results, apparently not reported previously in literature – with the aim to point out some crucial aspect involved in an adaptive robotic manipulation of the industrial cartons.The loading tests show a great variability of the mechanical behavior, depending dramatically on the crease indentation depth (also for the specimens obtained from the same carton): (a) when the damage induced during the crease formation is relatively small, the bending response is unusually complex: the moment constitutive function, m_L(θ), presents (up to) two peaks followed by unstable branches; (b) for greater indentation, the m_L(θ) is monotone.In the unloading case the response m_U(θ) is always monotone and is practically independent of the formation conditions of the crease. These behaviors can be easily described analytically using (piecewise) third degree splines.In a companion paper, the erection of a typical carton corner with unstable constitutive behavior is fully analyzed to detect the possible criticalities.     

A Locally Quadratic Glimm Functional and Sharp Convergence Rate of the Glimm Scheme for Nonlinear Hyperbolic Systems

Consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one space dimension

Determination of source parameter in parabolic equations

The authors consider the problem of finding u=u(x, t) and p=p(t) which satisfy u = Lu + p(t) + F(x, t, u, _x, p(t)) in Q _T=Ω×(0, T], u(x, 0)=ø(x), x∈Ω, u(x, t)=g(x, t) on ?Ω×(0, T] and either ∫_G(t) Φ(x,t)u(x,t)dx = E(t), 0 ? t ? T or u(x₀, t)=E(t), 0≤t≤T, where Ω?R ⁿ is a bounded domain with smooth boundary ?Ω, x ₀∈Ω, L is a linear elliptic operator, G(t)?Ω, and F, ø, g, and E are known functions. For each of the two problems stated above, we demonstrate the existence, unicity and continuous dependence upon the data. Some considerations on the numerical solution for these two inverse problems are presented with examples.     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result

Let (M, g) be a n-dimensional ( ${n\geqq 2}Let (M, g) be a n-dimensional ( n\geqq 2{n\geqq 2}) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ^∞. This paper is concerned with the study of the wave equation on (M, g) with locally distributed damping, described by

l utt - Dgu+ a(x) g(ut)=0,   on M×] 0,￥[ ,u=0 on ?M ×] 0,￥[, \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right.      19. Deduction of Saint-Venant"s Principle for the Asymptotic Residue of Elastic Rods      H. Irago 《Journal of Elasticity》1999,57(1):55-83 Let u(ε) be a rescaled 3-dimensional displacement field solution of the linear elastic model for a free prismatic rod Ωε 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.      20. Crack size and speed interaction characteristics at micro-, meso- and macro-scale      G. C. Sih  R. Jones 《Theoretical and Applied Fracture Mechanics》2003,39(2):127 The motivation to examine physical events at even smaller size scale arises from the development of use-specific materials where information transfer from one micro- or macro-element to another could be pre-assigned. There is the growing belief that the cumulated macroscopic experiences could be related to those at the lower size scales. Otherwise, there serves little purpose to examine material behavior at the different scale levels. Size scale, however, is intimately associated with time, not to mention temperature. As the size and time scales are shifted, different physical events may be identified. Dislocations with the movements of atoms, shear and rotation of clusters of molecules with inhomogeneity of polycrystals; and yielding/fracture with bulk properties of continuum specimens. Piecemeal results at the different scale levels are vulnerable to the possibility that they may be incompatible. The attention should therefore be focused on a single formulation that has the characteristics of multiscaling in size and time. The fact that the task may be overwhelmingly difficult cannot be used as an excuse for ignoring the fundamental aspects of the problem.Local nonlinearity is smeared into a small zone ahead of the crack. A “restrain stress” is introduced to also account for cracking at the meso-scale.The major emphasis is placed on developing a model that could exhibit the evolution characteristics of change in cracking behavior due to size and speed. Material inhomogeneity is assumed to favor self-similar crack growth although this may not always be the case. For relatively high restrain stress, the possible nucleation of micro-, meso- and macro-crack can be distinguished near the crack tip region. This distinction quickly disappears after a small distance after which scaling is no longer possible. This character prevails for Mode I and II cracking at different speeds. Special efforts are made to confine discussions within the framework of assumed conditions. To be kept in mind are the words of Isaac Newton in the Fourth Regula Philosophandi: “Men are often led into error by the love of simplicity which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is … We may learn something of the way in which nature operates from fact and observation; but if we conclude that it operates in such a manner, only because to our understanding that operates to be the best and simplest manner, we shall always go wrong.”––Isaac Newton Article Outline 1. Introduction 2. Elastodynamic equations and moving coordinates 3. Moving crack with restrain stress zone 3.1. Mode I crack 3.2. Mode II crack 4. Strain energy density function 4.1. Mode I 4.2. Mode II 5. Conclusions Acknowledgements References 1. Introduction Even though experimental observations could reveal atomic scale events, in principle, analytical predictions of atomic movements fall short of expectation by a wide margin. Classical dislocation models have shown to be inadequate by large scale computational schemes such as embedded atoms and molecular dynamics. Lacking in particular is a connection between interatomic (10−8 cm) processes and behavior on mesoscopic scale (10−4 cm) [1]. Relating microstructure entities to macroscopic properties may represent too wide of a gap. A finer scale range may be needed to understand the underlying physics. Segmentation in terms of lineal dimensions of 10−6–10−5, 10−5–10−3 and 10−3–10−2 cm may be required. They are referred to, respectively, as the micro-, meso- and macro-scale. Even though the atomistic simulation approach has gained wide acceptance in recent times, continuum mechanics remains as a power tool for modeling material behavior. Validity of the discrete and continuum approach at the different length scales has been discussed in [2 and 3].Material microstructure inhomogeneities such as lattice configurations, phase topologies, grain sizes, etc. suggest an uneven distribution of stored energy per unit volume. The size of the unit volume could be selected arbitrarily such as micro-, meso- or macroscopic. When the localized energy concentration level overcomes the microstructure integrity, a change of microstructure morphology could take place. This can be accompanied by a corresponding redistribution of the energy in the system. A unique correspondence between the material microstructure and energy density function is thus assumed [4]. Effects of material structure can be reflected by continuum mechanics in the constitutive relations as in [5 and 6] for piezoelectric materials.In what follows, the energy density packed in a narrow region of prospective crack nucleation sites, the width of this region will be used as a characteristic length parameter for analyzing the behavior of moving cracks in materials at the atomic, micro-, meso- and macroscopic scale level. Nonlinearity is confined to a zone local to the crack tip. The degree of nonlinearity can be adjusted by using two parameters (σ0,ℓ) or (τ0,ℓ) where σ0 and τ0 are referred to, respectively, as the stresses of “restraint” owing to the normal and shear action over a local zone of length ℓ. The physical interpretation of σ0 and τ0 should be distinguished from the “cohesive stress” and “yield stress” initiated by Barenblatt and Dugdale although the mathematics may be similar. The former has been regarded as intrinsic to the material microstructure (or interatomic force) while the latter is triggered by macroscopic external loading. Strictly speaking, they are both affected by the material microstructure and loading. The difference is that their pre-dominance occurs at different scale levels. Henceforth, the term restrain stress will be adopted. For simplicity, the stresses σ0 and τ0 will be taken as constants over the segment ℓ and they apply to the meso-scale range as well. 2. Elastodynamic equations and moving coordinates Navier’s equation of motion is given by(1)in which u and f are displacement and body force vector, respectively. Let the body force equal to zero, and introduce dilatational displacement potential φ(x,y,t) and the distortional displacement potential ψ(x,y,t) such that(2)u=φ+×ψThis yields two wave equations as(3)where 2 is the Laplacian in x and y while dot represents time differentiation. The dilatational and shear wave speeds are denoted by cd and cs, respectively.For a system of coordinates moving with velocity v in the x-direction,(4)ξ=x−vt, η=ythe potential function φ(x,y,t) and ψ(x,y,t) can be simplified to(5)φ=φ(ξ,η), ψ=ψ(ξ,η)Eq. (3) can thus be rewritten as(6)in which(7)In view of Eqs. (7), φ and ψ would depend on (ξ,η) as(8)φ(ξ,η)=Re[F(ζd)], ψ(ξ,η)=Im[G(ζs)]The arguments ζj(j=d,s) are complex:(9)ζj=ξ+iαjη for j=d,sThe stress and displacement components in terms of φ and ψ are given as(10)uy(ξ,η)=−Im[αdF′(ζd)+G′(ζs)]The stresses are(11)σxy(ξ,η)=−μ Im[2αdF″(ζd)+(1+αs2)G″(ζs)]σxx(ξ,η)=μ Re[(1−αs2+2αd2)F″(ζd)+2αsG″(ζs)]σyy(ξ,η)=−μ Re[(1+αs2)F″(ζd)+2αsG″(ζs)]with μ being the shear modulus of elasticity. 3. Moving crack with restrain stress zone The local stress zone is introduced to represent nonlinearity; it can be normal or shear depending on whether the crack is under Mode I or Mode II loading. For Mode I, a uniform stress σ∞ is applied at infinity while τ∞ is for Mode II. The corresponding stress in the local zone of length ℓ are σ0 are τ0. They are shown in Fig. 1 for Mode I and Fig. 2 for Mode II. Assumed are the conditions in the Yoffé crack model. What occurs as positive at the leading crack edge, the negative is assumed to prevail at the trailing edge.

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.     

Crack size and speed interaction characteristics at micro-, meso- and macro-scale

The motivation to examine physical events at even smaller size scale arises from the development of use-specific materials where information transfer from one micro- or macro-element to another could be pre-assigned. There is the growing belief that the cumulated macroscopic experiences could be related to those at the lower size scales. Otherwise, there serves little purpose to examine material behavior at the different scale levels. Size scale, however, is intimately associated with time, not to mention temperature. As the size and time scales are shifted, different physical events may be identified. Dislocations with the movements of atoms, shear and rotation of clusters of molecules with inhomogeneity of polycrystals; and yielding/fracture with bulk properties of continuum specimens. Piecemeal results at the different scale levels are vulnerable to the possibility that they may be incompatible. The attention should therefore be focused on a single formulation that has the characteristics of multiscaling in size and time. The fact that the task may be overwhelmingly difficult cannot be used as an excuse for ignoring the fundamental aspects of the problem.Local nonlinearity is smeared into a small zone ahead of the crack. A “restrain stress” is introduced to also account for cracking at the meso-scale.The major emphasis is placed on developing a model that could exhibit the evolution characteristics of change in cracking behavior due to size and speed. Material inhomogeneity is assumed to favor self-similar crack growth although this may not always be the case. For relatively high restrain stress, the possible nucleation of micro-, meso- and macro-crack can be distinguished near the crack tip region. This distinction quickly disappears after a small distance after which scaling is no longer possible. This character prevails for Mode I and II cracking at different speeds. Special efforts are made to confine discussions within the framework of assumed conditions. To be kept in mind are the words of Isaac Newton in the Fourth Regula Philosophandi:

“Men are often led into error by the love of simplicity which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is … We may learn something of the way in which nature operates from fact and observation; but if we conclude that it operates in such a manner, only because to our understanding that operates to be the best and simplest manner, we shall always go wrong.”––Isaac Newton