3‐D flow of a compressible viscous micropolar fluid with cylindrical symmetry: uniqueness of a generalized solution

Abstract

In this paper, we consider a nonstationary 3‐D flow of a compressible viscous and heat‐conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. The fluid domain is a subset of R3 bounded with two coaxial cylinders that present solid thermoinsulated walls. The mathematical model is set up in Lagrangian description. If we assume that the initial mass density, temperature, as well as the velocity and microrotation vectors are smooth enough cylindrically symmetric functions, then our problem has a generalized cylindrically symmetric solution for a sufficiently small time interval. Here, we prove the uniqueness of this solution. Copyright © 2016 John Wiley & Sons, Ltd.

Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry

“…Later, Ye [42] obtained the global existence results for classical 3-D GMHD ( 1 2 ≤ α ≤ 1). As mentioned above, the regularity and exponential stability of generalized (global) solutions in H 2 (Ω) has never been studied for system ( 14)-( 21) with boundary conditions (12) and initial conditions (13). Therefore, we shall continue the work by Huang and Dražić [22] and establish the regularity and exponential stability of solutions with small initial data.…”

“…Recently, for the spherical symmetric model of described micropolar fluid in an exterior unbounded domain, we proved the large time behavior for spherically symmetric flow of viscous polytropic gas with large initial data in [26]. In the case of cylinder symmetry, which model described micropolar fluid in a bounded domain with two coxial cylinders that present the solid thermoinsulated walls, Dražić and Mujaković [11] established the local existence of generalized solutions, then they proved global existence [12] and the uniqueness [13], Huang and Dražić [21,22] studied the large time behavior of the cylindrically symmetric with small initial data, but the regularity is open. Besides, we would like to mention the work on the global wellposedness of the three-dimensional magnetohydrodynamic equations, Wang and Wang [41] obtained the global existence results for classical 3-D MHD (α = 1).…”

“…In this paper, we consider the three dimensional case of (1)-( 9) with the assumption of cylindrical symmetry, and we study the problem with homogeneous boundary conditions as in [13]:…”

Strong solutions to the three-dimensional barotropic compressible magneto-micropolar fluid equations with vacuum

“…The theory of micropolar fluids introduced by Eringen in the 1960s (see [10,11]) is a significant step toward the generalization of the classical Navier-Stokes model. Due to the profound physical background and important mathematical significance, the compressible micropolar fluid equations have been extensively studied, such as large time behavior of solutions [14,15,22], blow-up criterion of solutions [4,5], qualitative theory of symmetric solutions [7][8][9]17], low Mach number limit of solutions [18,19], and so on. By constructing global weak solutions as limits of smooth solutions, Chen et al [6] proved global existence of weak solutions to the three-dimensional compressible micropolar fluid system with initial data which may be discontinuous and may contain vacuum states.…”

Three-dimensional flow of a compressible viscous micropolar fluid with cylindrical symmetry: a global existence theorem

“…Using the Faedo-Galerkin method in [22], it is proved that the corresponding problem with homogeneous boundary conditions for velocity, microrotation, and heat flux has a generalized solution locally in time, that is, on the domain 0, LOE 0, T 0 OE, where T 0 > 0 is sufficiently small. In [23], the uniqueness of the generalized solution for the same problem is obtained. Here, we prove that the problem has a generalized solution globally in time, that is, on the domain 0, LOE 0, TOE, for any finite T > 0.…”