One-dimensional flow of a compressible viscous micropolar fluid: a local existence theorem

An initial-boundary value problem for one-dimensional flow of a compressible viscous heat-conducting micropolar fluid is considered. It is assumed that the fluid is thermodinamicaly perfect and politropic. A global-in-time existence theorem is proved. The proof is based on a local existence theorem, obtained in the previous paper [4]. 1. Statement of the problem and the main result In this paper we consider an initial-boundary value problem for one-dimensional flow of a compressible viscous heat-conducting micropolar fluid, being in thermodinamical sense perfect and politropic (see [4] and references therein). Let p, v, 01 and e denotes respectively the mass density, velocity, microrotation velocity and temperature in the Lagrangean description. Then the problem that we consider has the formulation as follows: op 20V _ 0 ot + p ox , ov 0 (OV) 0 t = ox P ox K ox(pe), pow =A[P~(pOW) _ 01], ot ox ox oe 2 OV 2(OV)2 2(001)2 2 0 ( oe) PEii = -Kp e ox + p ox + p ox + 01 + Dp ox P x (1.4) in ]0, l[xR+, v(O, t) = v(l, t) = 0, 01(0, t) = 01(1, t) = 0, oe oe ox (0, t) = ox (1, t) = 0, for t E R+, p(x, 0) = Po(x), v(x, 0) = vo(x), (1.5) (1.6) (1. 7)