Commutability of homogenization and linearization at identity in finite elasticity and applications

We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Müller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.     

Energy method for multi-dimensional balance laws with non-local dissipation

In this paper, we are concerned with a class of multi-dimensional balance laws with a non-local dissipative source which arise as simplified models for the hydrodynamics of radiating gases. At first we introduce the energy method in the setting of smooth perturbations and study the stability of constants states. Precisely, we use Fourier space analysis to quantify the energy dissipation rate and recover the optimal time-decay estimates for perturbed solutions via an interpolation inequality in Fourier space. As application, the developed energy method is used to prove stability of smooth planar waves in all dimensions n?2, and also to show existence and stability of time-periodic solutions in the presence of the time-periodic source. Optimal rates of convergence of solutions towards the planar waves or time-periodic states are also shown provided initially L¹-perturbations.     

Compactness for maps minimizing the n-energy under a free boundary constraint

Let X denote a compact subset of ℝ ⁿ and B the unit ball in ℝ ⁿ . In this paper we investigate analytical and topological compactness properties of minimizing sequences for the n-energy in the class of maps , the homotopy class . Received: 5 June 2000     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Dimensional reduction for energies with linear growth involving the bending moment

A Γ-convergence analysis is used to perform a 3D–2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of functions with bounded variation.     

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

Local solutions for a coupled system of Kirchhoff type

We investigate the existence of local solutions of the following coupled system of Kirchhoff equations subject to nonlinear dissipation on the boundary: (∗) Here {Γ₀,Γ₁} is an appropriate partition of the boundary Γ of Ω and ν(x), the outer unit normal vector at x∈Γ₁.By applying the Galerkin method with a special basis for the space where lie the approximations of the initial data, we obtain local solutions of the initial-boundary value problem for (∗).     

Kac–Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory Cw of mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that Cw is a Frobenius category and that its stable category is a Calabi–Yau category of dimension two. We show that Cw yields a cluster algebra structure on the coordinate ring C[N(w)] of the unipotent group N(w):=N∩(w−1N₋w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify C[N(w)] with a subalgebra of , the graded dual of the universal enveloping algebra U(n) of n. Let S^? be the dual of Lusztig?s semicanonical basis S of U(n). We show that all cluster monomials of C[N(w)] belong to S^?, and that S^?∩C[N(w)] is a C-basis of C[N(w)]. Moreover, we show that the cluster algebra obtained from C[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C[Nw] of regular functions on the unipotent cell Nw of the Kac–Moody group with Lie algebra g. We obtain a corresponding dual semicanonical basis of C[Nw]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.     

A Saint-Venant principle for shear band localization

A type of Saint-Venant principle is derived for a two-dimensional model of shear band formation in thermoviscoplastic solids. To establish that the thermal energy generated during the formation process remains highly localized, a spatially decaying upper bound on the temperature is derived. It is found that the temperature bound decays exponentially along the direction perpendicular to the band, with a rate that decreases in time. The result is established by using maximum principles for second-order nonlinear parabolic partial differential equations.Mathematics Subject Classification (2000). 74C20, 74G50.     

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka–Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c^? such that for each wave speed c?c^?, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c<c^? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c>c^?.     

Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production

This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non-cut-off collision kernels (γ>−n and s∈(0,1)) in the trilinear L²(Rn) energy 〈Q(g,f),f〉. These new estimates prove that, for a very general class of g(v), the global diffusive behavior (on f) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works (Gressman and Strain, 2010 [15], 2011 [16]). We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non-cut-off Boltzmann collision operator in the energy space L²(Rn).     

Global uniqueness and reconstruction for the multi-channel Gel?fand–Calderón inverse problem in two dimensions

We study the multi-channel Gel?fand–Calderón inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation −Δψ+v(x)ψ=0, x∈D, where v is a smooth matrix-valued potential defined on a bounded planar domain D. We give an exact global reconstruction method for finding v from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.     

Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval

In this paper, we study the existence of mild solutions for initial value problems for semilinear Volterra integrodifferential equations in a Banach space. The arguments are based on the concept of measure of noncompactness in Fréchet space and the Tikhonov fixed point theorem.     

Failure of the Hasse principle for Enriques surfaces

We construct an Enriques surface X over Q with empty étale-Brauer set (and hence no rational points) for which there is no algebraic Brauer–Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.     

The Baum–Connes conjecture for free orthogonal quantum groups

We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ=1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C^?-algebras do not contain nontrivial idempotents.Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle? sphere.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.     

Blow-up behavior outside the origin for a semilinear wave equation in the radial case

We consider the semilinear wave equation in the radial case with conformal subcritical power nonlinearity. If we consider a blow-up point different from the origin, then we exhibit a new Lyapunov functional which is a perturbation of the one-dimensional case and extend all our previous results known in the one-dimensional case. In particular, we show that the blow-up set near non-zero non-characteristic points is of class C¹, and that the set of characteristic points is made of concentric spheres in finite number in for any R>1.     

Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity

We show that the averaged response of random isotropic Cauchy elastic material can be described analytically. It leads to a higher gradient model with explicit expressions for the dependence on the second derivatives of the mean field. A subsequent penalty formulation coincides with a linear elastic micro-stretch model with specific choice of constitutive parameters, depending only on the average cut-off length (the internal characteristic length scale L_c > 0). Thus the microstretch displacement field can be interpreted as an approximated mean field response for these parameter ranges. The mean field free energy in this micro-stretch formulation is not uniformly pointwise positive, nevertheless, the model is well posed.      

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.