### Abstract

We consider nonstationary 3-D flow of a compressible viscous heat-conducting micropolar
fluid in the domain to be the subset of

##### Keywords:

micropolar fluid; generalized solution; spherical symmetry; weak and strong convergence### 1 Introduction

The theory of micropolar fluids is introduced in [1] by Eringen. Various problems with different initial and boundary conditions for incompressible micropolar fluid are presented in [2], but the theory of compressible micropolar fluid is still in the beginning. N. Mujakovic in [3] developed the model for one-dimensional isotropic, viscous, compressible micropolar fluid which is in thermodynamical sense perfect and polytropic. In the same work, the local existence of the solution for homogeneous boundary conditions is proved. N. Mujakovic in [4] and in the references cited therein proved the local and global existence of inhomogeneous boundary conditions for velocity and microrotation as well as stabilization and regularity. In [5] the Cauchy problem for the described problem was also considered. In the last years we find some interesting works with different kind of problems concerning micropolar fluid, e.g., [6,7], but till now the described model of compressible micropolar fluid in three-dimensional case has not been considered.

In this work we consider the three-dimensional model with spherical symmetry. The first article in which the problem of spherical symmetry was described is [8], but for classical fluid. The spherical symmetry for classical fluid is also considered in articles [9-12].

In the setting of the field equations we use the Eulerian description.

In what follows we use the notation:

*ρ* - mass density

**v** - velocity

*p* - pressure

**T** - stress tensor

^{a}

** ω** - microrotation velocity

**C** - couple stress tensor

*θ* - absolute temperature

*E* - internal energy density

**q** - heat flux density vector

**f** - body force density

**g** - body couple density

*δ* - body heat density

The problem we consider here is based on local forms of the conservation laws for mass, momentum, momentum moment and energy, which are stated respectively as follows:

where
**a**:

The scalar product of tensors **A** and **B** is defined by

The linear constitutive equations for stress tensor, couple stress tensor and heat flux density vector are respectively in the forms:

where

*λ*, *μ* - coefficients of viscosity,

*k* - heat conduction coefficient

are constants with the properties

Assuming that the fluid is perfect and polytropic, for pressure and internal energy we have the equations

where *R* and

Let
*a* and *b*. The boundary of the described domain is

for

for
** ν** is an exterior unit normal vector.

For simplicity we also assume that

The initial boundary problems for the system (1)-(13) so far have not been considered in three-dimensional case. The same and similar models in one-dimensional case were considered in [3,5,13] and [4]. In [2] the three-dimensional model was considered but for an incompressible micropolar fluid.

In this paper we prove the local existence of generalized spherically symmetric solution to the problem (1)-(13) in the domain Ω, assuming that the initial functions are also spherically symmetric. In the proof we use the Faedo-Galerkin method. We follow some ideas of [14] where this method was applied to a classical fluid (where microrotation is equal to zero) in one-dimensional case as well as the ideas from [3] and [13] where the same result as here was provided for one-dimensional case.

The paper is organized as follows. In the second section, we derive a spherically
symmetric form of (1)-(4), introduce Lagrangian description, and present the main
result. In the third section, we consider an approximate problem and get an approximate
solution for each

### 2 Spherically symmetric form and the main result

We first derive the spherically symmetric form of (1)-(7) and (10)-(11). A spherically symmetric solution of (1)-(7) has the form:

where

where

with the following initial and boundary conditions

To investigate the local existence, it is convenient to transform the system (16)-(19)
to that in Lagrangian coordinates. The Eulerian coordinates

where

We introduce the new function *η* by

Note that if
*L* be defined as

From (16) we can easily get the equation

i.e.,

It is useful to introduce the next coordinate

and the following functions

Similarly as in [15], for a new coordinate

Taking into account (26) and (24), we obtain that the functions

in

for

for

From

putting

where

We assume the inequalities

where

Before stating the main result, we introduce the following definition.

**Definition 2.1** A generalized solution of the problem (30)-(38) in the domain

where

that satisfies Equations (30)-(33) a.e. in

**Remark 2.1** From the embedding and interpolation theorems (e.g., [16] and [17]) one can conclude that from (43) and (44) it follows:

It is easy to check that the solution (42) with properties (43)-(44) satisfies the condition for a strong solution of the described problem.

The aim of this paper is to prove the following statements.

**Theorem 2.1***Let the functions*

*satisfy conditions* (41). *Then there exists*
*such that the problem* (30)-(38) *has a generalized solution in*
*having the property*

*For the function**r*, *it holds*

**Remark 2.2** Notice that the function

From (40) and (41) we get

where

The proof of Theorem 2.1 is essentially based on a careful examination of a priori
estimates and a limit procedure. We first study, for each
*n* by utilizing a technique of Kazhikov [14,18] and Mujakovic [3,13] for one-dimensional case. Using the obtained a priori estimates and results of weak
compactness, we extract the subsequence of approximate solutions, which, when *n* tends to infinity, has limit in the same weak sense on

### 3 Approximate solutions

We shall find a local generalized solution to the problem (30)-(38) as a limit of approximate solutions

obtained in what follows. First, we introduce the approximations
*v* and *r* by

where

Then, we can write the solution

in the similar way as in [3] and [13] in the form

Since

We also introduce the approximations
*ω* and *θ* respectively by

where

Evidently, the boundary conditions

for

According to the Faedo-Galerkin method, we take the following approximate conditions:

for

Let

Let

We take the initial conditions for

Let

then we have

where

Here we have

Notice that the functions on the right-hand side of (79)-(84) satisfy the conditions of the Cauchy-Picard theorem [19,20] and we can easily conclude that the following statements are valid.

**Lemma 3.1***For each*
*there exists such*
*that the Cauchy problem* (79)-(86) *has a unique solution defined on*
*The functions*
*and*
*defined by the formulas* (57), (62) *and* (63) *belong to the class*
*and satisfy conditions* (71)-(73).

From (77) and (78) we can also easily conclude that

**Lemma 3.2***There exists*
*such that the functions*
*and*
*satisfy the conditions*

*on*
*The constants**m*, *a*,
*M**and*
*are introduced by* (40), (41), (53) *and* (55).

*Proof* The statements follow from (90)-(91), (41), (53) and (55). □

### 4 A priori estimates

Our purpose is to find

In what follows we denote by

We also use the notation

Some of our considerations are very similar or identical to that of [3] or [13]. In these cases we omit proofs or details of proofs making references to corresponding pages of the articles [3] or [13].

**Lemma 4.1***For*
*it holds*

*Proof* From (58) follows

and using Remark 2.2 we get (95) immediately. □

**Lemma 4.2***For*
*the following inequality holds*:

*Proof* Multiplying (66) by

Integrating over

Using (92), we get (96). □

In what follows, we use the inequalities

(for a function *f* vanishing at

**Lemma 4.3***For*
*the following inequality holds*:

*Proof* Multiplying (65) by

Integrating over

Taking into account (96), (71), (73), and (97) we obtain (98). □

**Lemma 4.4** ([3], Lemma 5.3)

*For*
*the following inequality holds*:

**Lemma 4.5***For*
*the following inequality holds*:

*Proof* Taking the derivative of the function
*x* and using the estimates (92)-(94), we obtain

With the help of (97) applied to the function

**Lemma 4.6***For*
*it holds*

*where*

*Proof* As in [3] Lemma 5.5, [14] pp.63-66 and in [13] Lemma 5.6, multiplying (65), (66) and (67) respectively by

where

Taking into account (92)-(94) and (95)-(100), we estimate the terms on the right-hand side of (102). For instance,

Applying the Young inequality, we get

where

Using these inequalities with sufficiently small *ε* and estimates (92)-(94), from (102) we get (101). □

**Lemma 4.7***There exists*
*such that for each*
*the Cauchy problem* (79)-(86) *has a unique solution defined on*
*Moreover*, *the functions*
*and*
*satisfy the inequalities*

(*a*,
*m**and**M**are defined by* (41) *and* (53)-(55)).

*Proof* To get the estimate (103) we use an approach similar to that in [3] (Lemma 5.6) and [14] (pp.64-67). First, we introduce the function

Using Lemma 4.6, we find that the function

There exists a constant

and we can conclude that

We compare the solution of the problem (109)-(110) with the solution of the Cauchy problem

Let

Let

and from (101) it follows

Integrating (101) over

Now, using the inequalities (97) for the function *v*, we easily get

Using (116), we derive the following estimates:

where *C* and *B* are from (103). Assuming that

and using (117) and (118) from (58) and (60), we get (104)-(105).

Because of (57), (62) and (63), from (103) and (98), we easily get that for

and we can conclude that the solution of the problem (79)-(86) is defined on

From (119) and (120), we can easily conclude that

and from (95), (100) and (99) it follows

**Lemma 4.8***Let*
*be defined by Lemma *4.7. *Then for each*
*the inequalities*

*hold true*.

*Proof* Multiplying (65) by