Local Strong Solution Of The Density-dependent Boussinesq Equations, Existence And Uniqueness

In this paper ,We consider the following initial-boundary value problem of homogeneous incompressible Boussinesq equations.whereΩis an open bounded subset of R³ with a smooth boundary.The unknown functions are u = u(x, t),ρ=ρ(x, t),p =ρ(x, t) andθ=θ(x,t).which represent the velocity field, the density, the pressure and the temperature of the flow, respectively. f= f(x,t) is the known external potential.ρ₀u₀=ρ₀u₀(x) is the initial momentum andρ₀ =ρ₀(x) is the initial density.μ:=μ(ρ.θ),κ:=κ(ρ,θ), C_v:= C_v(ρ,θ) is the viscosity coefficient. heat conductivity coefficient and specific heat at constant volume of the flow, respectively. They are all positive functions ofρandθ.In this paper,we mainly study the local existence and uniqueness of the strong solutions of the problem (*). The contents of the paper include two parts:First , we consider a linearized problem of (*). It is proved that the strong solution of the linearized problem exists uniquely. And it is obtained that the strong solutions satisfy some uniform estimates, which is independent of the lower bound of the initial density.Second, we prove the existence and uniqueness of strong solutions of the problem (*) by applying the classical iteration argument , based on the above uniform estimates.