## On the equation of nonstationary stratifid fluid motion: Uniqueness and existence of the solutions.(English)Zbl 0596.76119

Let $$\Omega$$ be a bounded domain in $${\mathbb{R}}^ 2$$ or $${\mathbb{R}}^ 3$$ with a smooth boundary. We consider an inhomogeneous viscous incompressible fluid occupying $$\Omega$$. Let $$\rho$$,u,p be the mass density, the velocity vector and the pressure of the fluid, respectively. Then these quantities obey the following system of equations $\partial \rho /\partial t+u\cdot \nabla \rho =0\quad (0<t,\quad x\in \Omega),$
$\rho \{\partial u/\partial t+(u\cdot \nabla)u\}=\Delta u-\nabla p\quad (0<t,\quad x\in \Omega),$
$div u=0\quad (0<t,\quad x\in \Omega).$ This system is a generalization of the Navier-Stokes system.
We employ the $$L^ 2$$-theory and prove the unique existence (local in time) of the solution. The main tool is the theory of linear evolution equations. This local existence theorem is used to prove the following global existence theorem:
I) In the two-dimensional problem the solution always exists globally in time.
II) In the three-dimensional problem the global solution is obtained, if the initial values are sufficiently small.
The important fact is that in the two-dimensional problem we do not need to assume smallness of the initial values.
This paper consists of seven sections. In section 2 we give various function spaces and we formulate an initial boundary value problem in the framework of the theory of evolution equations. Main theorems are also stated in this section. Sections 3, 4 and 5 deal with the proof of the local existence theorem. The global existence of the solution of the two- dimensional problem is proved in section 6. The three-dimensional case is considered in section 7.

### MSC:

 76V05 Reaction effects in flows 46N99 Miscellaneous applications of functional analysis 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)