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Global solution to 3D problem of a compressible viscous micropolar fluid with spherical symmetry and a free boundary. (English) Zbl 1359.35155

Summary: We consider a nonstationary 3D flow of a compressible viscous heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic, in the domain bounded with two concentric spheres in \(\mathbf{\mathbb R}^3\). In this paper we establish the existence of a global solution to the free boundary problem defined with the homogeneous boundary conditions for velocity, microrotation, heat flux on the fixed border and homogeneous boundary conditions for strain, microrotation, and heat flux on the free boundary. We suppose that the initial data are spherically symmetric, with positive initial density and temperature, having the zero density on the free boundary. Because of the spherical symmetry, the starting three-dimensional problem is transformed to the one-dimensional problem in Lagrangian coordinates in the domain that is a segment. The solution to our problem is obtained as a limit of the sequence of approximate solutions derived from suitable semi-discrete finite difference approximate systems. By using the derived bounded estimates of the approximate solutions and the results of the weak and strong compactness, we establish the convergence to the generalized solution of our problem globally in time.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
80A20 Heat and mass transfer, heat flow (MSC2010)
35B06 Symmetries, invariants, etc. in context of PDEs
35R35 Free boundary problems for PDEs
76U05 General theory of rotating fluids
65N06 Finite difference methods for boundary value problems involving PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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