Existence And Uniqueness Of The Steady Navier-stokes Equations With Damping Term Weak Solution

We consider the following boundary problem to the steady incompressible Navier-Stokes equations with damping :HereΩ, is a bounded domain in Rⁿ, u = u(x) = (u¹(x), ...uⁿ(x))and p(x, t)stand for the velocity field and the pressure of the flow, respectively. f = f(x) = (f¹(x), ...fⁿ(x)) is a given external force. In damping term,α> 0 andβ≥1 are two constants.φis the boundary value of u satisfying (?)?φ= 0;γ> 0 represents the viscosity coefficient of the flow.In this paper, we are mainly concerned with the existence and the uniqueness of the weak solutions of the problem (_*1) in W^1,2(Ω). The contents of the paper include two parts:1.We consider the existence and uniqueness with homogeneous boundary data (i.e.u = 0 on(?)Ω) in space (?)₀^1,2(Ω). To do so, we use the classical Galerkin Method to construct the"approximating solution"and apply a priori estimates to obtain the existence of the weak solutions.The uniqueness of weak solution is proved by the regularity theorem.2.We consider the nonhomogeneous boundary problem of (_*1) in W^1,2(Ω). In order to obtain the existence and uniqueness of weak solution,we transform it to a homogeneous boundary problem. Based on this ,we obtain the existence of the nonhomogeneous boundary problem under certain conditions and the uniqueness of the solutions.