Hopf's conjecture for a class of chemical kinetics equations

Attractors of partial differential evolution equations and estimates of their dimension

“…The Hausdorff dimension of an attractor for a two-dimensional Navier-Stokes system with zero boundary conditions was found to be bounded by an exponential in l/v (see [15] and [35]). We obtain in §8 a power estimate for this case (see [46]) (4) dim 21 < Cv-* = С (Re) 4 , Re = 1/v…”

Families of stable manifolds for one-dimensional parabolic equations

“…(~o) ,-,,,~g'), By virtue of this inequality, the mapping K: n is well defined for sufficiently large n. |n According to the standard pattern of the graph transformation technique, to prove the differentiability and Cl-continuity of the family (21), it suffices to prove that the mapping K: n has a unique attracting fixed In point. Let us use the contraction mapping principle for bundle mappings [8, w By this principle, in our case it suffices to prove that for sufficiently large n the Lipschitz constant of the mapping p2/Ctn : fl(en) --' fl(e,) is less than unity for any 7/E s162 But this can be readily done similarly to the derivation of the estimates for the mapping ~'~ [3]. This proves item 1 of Theorem 2.…”

“…In particular, it can be studied by constructing invariant families of manifolds. In the present paper, we prove the existence of a family of stable manifolds associated with an invariant subset of the dynamical system generated by the one-dimensional parabolic equation The existence of invariant manifolds for infinite-dimensional dynamical systems generated by equations of the form (1) was studied in [2][3][4] under the assumption that f does not depend on uz. This restriction is removed in the present paper.…”

Inertial manifold for a reaction diffusion equation model of competition in a chemostat

“…There are a number of existence theories for inertial manifolds (e.g. Kamaev [4], Mora [6], Foias, Sell and Teman [2], Mallet-Paret and Sell [5], Chow and Lu [1] and Teman [9]). In this section we recall one that is immediately applicable to (1.8) and (1.9).…”