Families of stable manifolds of invariant sets of systems of parabolic equations

Abstract

The wave pattern created by a pressure point moving with constant velocity over the free surface of an inviscid MHD plasma bordered by a magnetic field is investigated. If the depth of the plasma is taken to be Mnite, the results are the same as found in the case of an ordinary fluid provided that the velocity of the pressure point exceeds the Alfvbvelocity. In this case a new expression is given for the surface elevation in front of the pressure point. Moreover some new results are found in the cases of a fluid and a plasma of iinite depth.

Finite-dimensional dynamics on attractors of non-linear parabolic equations

“…It is important to cite very interesting papers by Kamaev [15], [16], in which an invariant C 1 -continuous family of smooth stable manifolds of finite codimension is constructed for the attractors of equations similar to (2) and the corresponding systems of equations. It would be interesting to elicit (in the general case) the relation between the existence of such a family of manifolds and the finite-dimensionality of the limiting dynamics of the evolution problems (1).…”

“…In this definition the B(u, v) are operators of scalar type (see [19]) for all u, v ∈ A. In the case when T 0 = 0 and u = v the representation B = S −1 HS in Definition 2.2 was actually used by Kamaev [15], [16] in his study of phase dynamics near the attractors of (scalar or vector) equations of a somewhat wider class than (2). In these papers either the conventional Liouville transformation or [16] a modification of it was applied to the right-hand side of the linearized equation.…”

“…Let ω > ln 2 ρ(u, v) + 1 on N . Then (16) implies that Σ ⊂ Ω(k, θ), θ = 1/2, k = 2(ω − 1) 1/2 . In fact Σ is contained in the domain |y| k(x − 1) θ , whence Σ ⊂ Ω(k 1 , 1) with k 1 = k 1 (ω) > 0.…”

“…. , u l ), with the Dirichlet boundary condition (the coefficient d(x) > 0 is smooth on [0, 1]) if we use the above-mentioned analogue of the Liouville transformation [16]. Apparently, it can be proved that semilinear parabolic equations of order greater than two on (0, 1) have finite-dimensional limiting dynamics if the boundary conditions are not too pathological and the non-linear part satisfies appropriate conditions.…”