EXISTENCE TIME OF SOLUTION OF THE (1+2)D KNOBLOCH EQUATION WITH INITIAL-BOUNDARY VALUE PROBLEM

Abstract

The equation of pattern formation induced by buoyancy or by surface-tension gradient in finite systems confined between horizontal poor heat conductors is introduced by Knobloch[1990] ∂u∂t=αu-μ∇2u-∇4u+k∇⋅(|∇u|2∇u+β∇2u∇u-γu∇u+δ∇|∇u|2)where u is the planform function, µ is the scaled Rayleigh number, K = 1 and α represents the effects of a heat transfer finite Biot number. The cofficients β, δ and γ do not vanish when the boundary conditions at top and bottom are not identical ( β≠0,δ≠0) or non-Boussinesq effects are taken into account ( γ≠0). In this paper, the Knobloch equation with α > 0 is considered, the global existence in L2-space and the finite existence time of solution in V2-space have been obtained respectively.