Steady Boussinesq Equation With Damping Term Weak Solution Posedness

This paper is concerned with the well-possedness of the very weak solution to the stationary Boussinesq system with damping |u|^α-1u.We prove that whenα=1 or 2≤α＜3 and when v large enough,there exists a very weak solution if the boundary function is a L² function.Furthermore,we prove the uniqueness and continuous dependence of the very weak solution.In our approach,we first prove the existence and uniqueness of the linear equations with respect toθ(x,t) and then prove that whenθ(x,t) is a known function the existence and uniqueness of the linear equations with respect to u(x,t).Finally,by using the Leray-Shauder fixed point theorem,we prove the well-possedness of the solution of the original nonlinear problem.