Strong Solution Of The Viscosity Coefficient And Density Of The Navier-stokes-poisson Equation, Existence And Uniqueness

In this paper, we consider the following initial-boundary problem for the nonhomogeneousincompressible Navier-Stokes-Poisson equations with a densitydependentviscosity.whereΩis an open bounded subset of R³ with a smooth boundary. The unknown functions u = u(x, t),ρ=ρ(x, t),p = p(x, t) are the velocity, density and pressure of the flow, respectively. f = f(x,t) is the known external potential andΦdenotes Newtonian gravitational potential, d = 1/2(▽u +▽^Tu) is the deformation tensor, where▽u denotes the gradient matrix ((?)u^j/(?)x_i) of u and▽^Tu its transpose. Moreover, u₀ =u₀(x) andρ₀ =ρ₀(x)≥0 are the initial velocity and density respectively. The viscosity coefficientμ=μ(ρ) is in general a function of the densityρand is assumed to satisfy thatμ∈C¹[0,∞) andμ> 0 on [0,∞).In this paper, we mainly study the local existence of unique strong solutions for all initial data satisfying a natural compatibility codition. Moreover, we provide a blow-up criterion for the regularity of the strong solution. The contents of the paper include two parts:First, we consider the existence and uniqueness of the strong solution of the linearized 3-D non-homogeneous incompressible Navier-Stokes-Poisson equations.To do so, we employ the "semi-Galerkin method" to construct the sequence of approximate solutions, and further to get the unifirm estimates of the approx imate solutions and theirderivatives. Then we employ the compactness arguments to obtain the existence and uniquness of the strong solutions of the linearized problem.Second, we consider the local existence and uniqueness of strong solutions. We construct the approximate solutions by solving iteratively linearized problems, and derive (local in time) uniform bounds of approximate solutions and then show that they converge to a strong solution of the original nonlinear problem. Moreover, we prove a blow-up criterion of the strong solution.