Correctors for the Homogenization of Monotone Parabolic Operators

Abstract

We prove the existence of non-decaying real solutions of the Johnson equation, vanishing as x → +∞. We obtain asymptotic formulas as t → ∞ for the solutions in the form of an infinite series of asymptotic solitons with curved lines of constant phase and varying amplitude and width.

Cylindrical Kadomtsev-Petviashvili equation: Old and new results

“…There exists a remarkable link, discovered in 1986 [6], between the solutions of the widely known KP equation (8) and the solutions of the CKP equation: the map…”

“…The results in[5],[6] were recently used to prove the existence of nondecaying solutions of the CKP equation vanishing as x → ∞[8]. Some additional results concerning physical applications of the CKP equation can be found in[9].…”

On homogenization of space-time dependent and degenerate random flows II

“…However, the uniformly elliptic setting provides us with exact problems (or PDEs) to compute A, C, U . The reader is referred to [1], Chapter 2, Section 1, or [16,20] for the periodic case and to [4] for the random case.…”

“…To get an idea of the scaling effect, let us focus on the 1-dimensional periodic case. In this setting, the matrix A is given by (see [20], Proposition 2.2)…”

“…Briefly, the time dependence of the process brings about a strong time degeneracy of the underlying Dirichlet form, which satisfies no sector condition, even weak. To face such a degeneracy, analytical methods, more precisely compactness methods, carried through the homogenization procedure for a uniformly elliptic diffusion matrix a in periodic media [1,20] or in random media [4]. In [8], critical scaling is considered (α = 2 and β = 1) and an annealed invariance principle is proved for the diffusion (1) by means of probabilistic tools under the assumptions V = 0 and a = Id.…”