Asymptotic Analysis

We continue the study of the operator of generalized Maxwell equations and completely discover the behavior of the solutions of the time-harmonic equations as the frequency tends to zero. Thereby we identify degenerate operators in terms of special ‘polynomially growing’ solutions of a corresponding static problem, which must be added to the ‘usual’ Neumann series in order to describe the low frequency asymptotic adequately.

We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we use the Kirchgässner reduction to obtain a spatial dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. In particular we describe the complete set ℬ of all small bounded solutions. In the classical Kolmogorov case, ℬ consists of periodic profiles, homoclinic pulses and a heteroclinic front–back pair. This is a consequence of the symmetry of the external force.

We study the asymptotic behavior of least energy solutions to the equation Δ²u=c₀K(x)u^p_ε with the Navier boundary condition as ε→+0, where Ω is a smooth bounded domain in R^N (N≥5), c₀=(N−4)(N−2)N(N+2) and p_ε=(N+4)/(N−4)−ε, ε>0. Under some assumptions on the coefficient function K, we obtain fairly precise asymptotics of the L^∞-norm of least energy solutions.

We consider the Cauchy problem for a doubly nonlinear parabolic equation, obtaining optimal estimates both for the sup norm of nonnegative solutions, and for their support for large times.