Narrow Results

We study the magnetic Laplacian and the Ginzburg–Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some nonlinear regime, we determine the structure of the order parameter and compute the super-current along the boundary of the sample. Our results are in agreement with what was observed in the Little–Parks experiment, for a thin cylindrical sample.

We study spiral waves in a mathematical model of a nonlinear optical system with a feedback loop. Starting from a delayed scalar diffusion equation in a thin annulus with oblique derivative boundary conditions, we shrink the annulus and derive the limiting equation on a circle. Based on the explicitly constructed normal form of the Hopf bifurcation for the one-dimensional delayed scalar diffusion equation, we make predictions about the existence and stability of two-dimensional spirals that we verify in direct numerical simulations, observing pulsating and rotating spiral waves.

We study the asymptotic behavior of the inductance coefficient for a thin toroidal inductor whose thickness depends on a small parameter ε > 0. We give an explicit form of the singular part of the corresponding potential u^ε which allows to construct the limit potential u (as ε → 0) and an approximation of the inductance coefficient L^ε. We establish some estimates of the deviation u^ε - u and of the error of approximation of the inductance. We show that L^ε behaves asymptotically as ln ε, when ε → 0.

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.

We investigate the limiting behavior of dynamics of non-autonomous stochastic FitzHugh–Nagumo equations driven by a nonlinear multiplicative colored noise on unbounded thin domains. We first establish the existence and uniqueness of random attractors for the equations on the thin domains and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.

This paper deals with the limiting behavior of invariant measures of stochastic reaction–diffusion equations on thin domains. We first show the existence of invariant measures of the stochastic equations in a bounded domain in ${ℝ}^{n+1}$ which can be viewed as a perturbation of a bounded domain in ${ℝ}^n$. We then prove that the set of all invariant measures of the perturbed equations is tight and any limit of invariant measures of the perturbed systems must be an invariant measure of the limiting system when the $(n+1)$-dimensional thin domains collapses onto an $n$-dimensional domain.