We consider the parabolic Anderson problem ∂tu = ∆u + ξ(x)u on R+ × Z with localized initial condition u(0, x) = δ0(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) lies in the vicinity of, or beyond, the double-exponential distribution, we prove the following geometric characterisation of intermittency: with probability one, as t → ∞, the overwhelming contri...

Assume that Vh is a space of piecewise polynomials of degree less than r ≥ 1 on a family of quasi-uniform triangulation of size h. Then the following well-known upper bound holds for a sufficiently smooth function u and p ∈ [1,∞] inf vh∈Vh ‖u− vh‖j,p,Ω,h ≤ Ch |u|r,p,Ω, 0 ≤ j ≤ r. In this paper, we prove that, roughly speaking, if u 6∈ Vh, the above estimate is sharp. Namely, inf vh∈Vh ‖u− vh‖j,...

We study the weak boundary layer phenomenon of the Navier-Stokes equations with generalized Navier friction boundary conditions, u ·n = 0, [S(u)n] tan +Au = 0, in a bounded domain in R when the viscosity, ε > 0, is small. Here, S(u) is the symmetric gradient of the velocity, u, and A is a type (1, 1) tensor on the boundary. When A = αI we obtain Navier boundary conditions, and when A is the sha...

Sensation seeking is a core personality trait that declines with age in both men and women, as do also both density and availability of the dopamine D(2/3) receptors in striatum and cortical regions. In contrast, novelty seeking at a given age relates inversely to dopamine receptor availability. The simplest explanation of these findings is an inverted-U-shaped correlation between ratings of se...