Solvability of a three-dimensional stationary flowing problem

Abstract

where v is the velocity field of the fluid, H ≡ P +v2/2 is the Bernoulli function, and P is the pressure. We suppose that the flow domain Ω ⊂ R3 is bounded and fixed. Denote the boundary of Ω by Γ. The normal component γ of v on Γ gives rise to the partition of Γ into three disjoint sets: the rigid wall Γ0, the inlet part Γ+ through which the fluid flows into the domain, and the outlet part Γ− through which the fluid flows out the domain: Γ0 = {x ∈ Γ : γ(x) = 0}, Γ+ = {x ∈ Γ : γ(x) 0}. The set Γ+ is referred to as the inlet. If γ 6≡ 0 then we have the flowing problem for Ω. Without additional boundary conditions, a solution to the problem is nonunique and depends on arbitrary functions. In the present article, we study the stationary flowing problem (SFP) that comprises the Euler equations (1), the circulation equations ∮