Asymptotic Analysis

We consider the one-dimensional Stark–Wannier type operators $\[H=-\dfrac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-Fx-q(x)+v(x),\quad F>0,\]$ where q is a smooth function slowly growing at infinity and v is periodic, $\[$v\in L_{1}(\mathbb{T})$$ , with the Fourier coefficients of the form (ln |n|)^−β, 0<β<1/2, as n→∞. We show that for suitable q and F the spectrum of the corresponding operator is purely singular continuous. This proves the sharpness of the a.c. spectrum stability result obtained in Comm. Math. Phys. 234 (2003), 359–381.

We propose a definition for the resonances of Schrödinger operators with slowly decaying 𝒞^∞ potentials without any analyticity assumption. Our definition is based on almost analytic extensions for these potentials, and we describe a systematic way to build such an extension that coincide with the function itself whenever it is analytic. That way, if the potential is dilation analytic, our resonances are the usual ones. We show that our resonances with negative real part are exactly the eigenvalues of the operator. We also prove that our definition coincides with the usual ones in the case of smooth exponentially decaying potentials.

Then we consider semiclassical results. We show that, if the trapped set for some energy E is empty, there is no resonance in any complex vicinity of E of size O(hlog (1/h)). Finally, we investigate the semiclassical shape resonances and generalize some results of Helffer and Sjöstrand.

The present paper deals with the homogenization of a lubrication problem, via two-scale convergence and periodic unfolding techniques. We study in particular the Elrod–Adams problem with highly oscillating roughness effects.

In this paper we prove the existence of a compact global attractor for a reaction–diffusion equation on $\[$\mathbb{R}^{N}$$ . We do not assume that the nonlinear term is differentiable (just continuous) and, also, we do not guarantee the uniqueness of solutions of the Cauchy problem. Besides, the growth and dissipative conditions are different from the ones used in previous papers on the topic. An application is given to the Fitz–Hugh–Nagumo system, which models the transmission of signals across axons.

The aim of this paper is the mathematical study of the time evolution of a stressed pore channel in an axisymmetric configuration. Under some conditions, morphological instabilities may appear at the material–vacuum interface. Assuming some formal asymptotic assumptions, we derive a nonlinear parabolic PDE (19) governing the cylindrical surface evolution. Local existence and unity of the solution of this PDE are shown and we also perform some numerical computations (with different parameters and initial condition), using a pseudo-spectral Galerkin method, yielding different behaviours for the solution to (19). In particular, we numerically observe what appears to be a finite time pinch-off.

The behaviour of Newtonian and non-Newtonian flows through a thin three-dimensional domain are widely studied in the literature. Usually, authors deal with special models related to particular concrete fluids. In this work, our aim is to present a general model, governing the behaviour of a large class of Newtonian and non-Newtonian fluids. Moreover, we deal with mixed boundary conditions, which are not often studied in the literature related to flows in thin domains. We consider a nonlinear model of a flow in a thin three-dimensional domain, and we study its behaviour when the thickness in one direction tends to zero. At the limit, we obtain a quasilinear two-dimensional problem for the pressure, a nonlinear Reynolds's law for the velocity and a nonlinear Darcy's law for the averaged velocity. Finally, we check that our results hold for a large class of non-Newtonian fluids by producing concrete examples.