In this paper, we establish the existence and uniqueness of strong solutions for the viscous incompressible chemically active fluid in an unbounded domain differing somewhat from those previously known.

The Equations of a Viscous Incompressible Chemically Active Fluid: Existence and Uniqueness of Strong Solutions in an Unbounded Domain

In this paper, we are concerned with the nonsteady Boussinesq system under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions, velocity, static (or total) pressure, rotation, and stress (or total stress) together, and the boundary conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. Relying on the relations among strain, rotation, normal derivative of velocity, and shape of the boundary surface, we get variational formulation. The formulations consist of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature. For the case of boundary conditions including the static pressure and stress, we prove that if the data of the problem are small enough and compatibility conditions at the initial instance are satisfied, then there exists a unique solution on the given interval. For the case of boundary conditions including the total pressure and total stress, we prove the existence of a solution without restriction on the data and parameters of the problem.

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The Nonsteady Boussinesq System with Mixed Boundary Conditions including Conditions of Friction Type

In this chapter we are concerned with the non-steady Boussinesq problem corresponding to the steady problem in Chap. 7. The formulations consist of a non-steady variational inequality for velocity and a non-steady variational equation for temperature. For the problem with boundary conditions involving the static pressure and stress, it is proved that if the data of problem are small enough and compatibility conditions at the initial time for velocity and temperature are satisfied, then there exists a unique solution on the given interval. For the problem with boundary conditions involving the total pressure and total stress, the existence of a solution is proved without restriction on the data of problem.

The Non-steady Boussinesq System

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Kazuo Ōeda

Papers

Initial Value Problems for the Heat Convection Equations in Exterior Domains