On a Nonlinear System of PDE’s Arising in Free Convection

Existence of Strong Solutions for Nonlinear Systems of PDEs Arising in Convective Flow

“…(1) e main advantage of Proposition 1 (particularly, Fredholm's alternative) is the generalization of the results ([8], eorem 1 and eorem 4, and [1], eorem 4), and more precisely, it leads us to show that there exist strong solutions to problem (15)- (18) without assuming that ‖Ψ‖ H 2 is bounded by certain constants and without assuming that ‖K • ∇Θ‖ 2 is quite small.…”

“…at is, (Λ(H), H) is the strong solution of problem ( 15)- (18). Furthermore, K • ∇(H + Θ) ∈ L 4 (Ω).…”

“…Let us assume that (Ψ, H) ∈ (H 2 (Ω)) 2 is a solution of problem (15)- (18). Multiplying equation ( 15) by a function u ∈ H 1 0 (Ω, Γ 1 ) and equation ( 16) by a function v ∈ H 1 0 (Ω), integrating on Ω, and using Green's formula, we yield…”

“…For numerical method, we refer the reader to [7]. e second natural way of dealing with this problem, which is the framework of this paper, is straightly related to the coupled partial differential equations (see [1,8,9]). e aim of this paper is to generalize the existence of strong solutions of the problem introduced in [1,8,9].…”

“…e second natural way of dealing with this problem, which is the framework of this paper, is straightly related to the coupled partial differential equations (see [1,8,9]). e aim of this paper is to generalize the existence of strong solutions of the problem introduced in [1,8,9]. More precisely, we will give new results about the existence of strong solutions, in particular, without assuming that ‖Ψ‖ H 2 is small (as in [8], eorem 4).…”