3-D flow of a compressible viscous micropolar fluid with spherical symmetry: a local existence theorem

Abstract

We consider nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of R 3 bounded with two concentric spheres that present solid thermoinsulated walls. In thermodynamical sense fluid is perfect and polytropic. Assuming that the initial density and temperature are strictly positive we will prove that for smooth enough spherically symmetric initial data there exists a spherically symmetric generalized solution locally in time.

3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution

“…R, L, c v , j I are positive constants. As in [4] (formulas (8) and (9)) we have that the coefficients of viscosity λ, μ, coefficients of microviscosity μ r , c 0 , c d and heat conduction coefficient k have the properties:…”

“…In this paper we study the large time behavior of a generalized solution which is defined in [4] as follows.…”

“…Here we analyze the large time behavior for the spherically symmetric solution of the homogeneous initial-boundary problem which describes the flow of the isotropic, viscous and heat-conducting micropolar fluid introduced in [4]. Similar problems were studied for the classical fluid with spherical symmetry for example by Yanagi in [22], Zheng and Qin in [23], Jiang in [6].…”

“…We assume the domain to be the subset of R 3 bounded with two concentric spheres with radii a and b (0 < a < b) that present solid thermo-insulated walls. The model for this type of flow was developed in [4] where the local existence theorem for homogeneous boundary data was proved. We also proved that this solution is unique in [17].…”

Regularity to the spherically symmetric compressible Navier–Stokes equations with density-dependent viscosity

“…For the 3-D flow of compressible fluid with spherical symmetry, there are some interesting results, [18] for the global well-posedness of classical solutions with large oscillations and vacuum; [33] for the global existence and uniqueness of the weak solution without a solid core; [14] for the structure of the solution; [21] for the global existence of the exterior problem and the initial boundary value problem. Besides, we would like to refer to [6,8] as regards the existence and regularity of solutions for micropolar fluid with spherical symmetry in the three-dimensional case.…”