On définit des invariants entiers mesurant l’irrégularité d’un point singulier d’un système différentiel. Les propriétés de ces invariants, l’étude de la variation de l’ordre de la singularité par perturbation linéaire ainsi qu’une généralisation d’un théorème de W. Jurkat et D.A. Lutz permettent de donner une méthode de calcul de cet ordre.
Inertial manifold with delay (IMD) for dissipative systems of second order in time is constructed. This result is applied to the study of different asymptotic properties of solutions. Using IMD, we construct approximate inertial manifolds containing all the stationary solutions and give a new characterization of the K-invariant manifold.
We study the corrector matrix to the conductivity equations. We show that if converges weakly to the identity, then for any laminate at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal.158 (2001) 155-171]. We use this...
We study the corrector matrix to the conductivity equations. We show that if converges weakly to the identity, then for any laminate at almost every point. This simple property is shown to be false for generic microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear]. In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal. 158 (2001) 155-171]. We use this...
We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution , the graph of its blow-up points and the set of all characteristic points and show that is locally finite. Finally, given , we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that forms a...
Given a family of Lévy measures ν={ν(x, ⋅)}x∈ℝd, the present work deals with the regularity of harmonic functions and the Feller property of corresponding jump processes. The main aim is to establish continuity estimates for harmonic functions under weak assumptions on the family ν. Different from previous contributions the method covers cases where lower bounds on the probability of hitting small sets degenerate.
We consider the linearized elasticity system in a multidomain of . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and tend to zero simultaneously, with , we identify the limit problem. This limit problem involves six junction conditions.
The long-time asymptotics of certain nonlinear , nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [4] guaranteeing eventual relaxation to equilibrium velocities in a spatially homogencous model of granular flow is extended and quantified by computing explicit relaxation rates. Our arguments rely on establishing generalizations of logarithmic Sobolev inequalities and mass transportation inequalities,...