Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model ()
1. Introduction
In this paper, we consider the one-dimensional viscous micropolar fluid model in Lagrangian coordinates:
(1)
Here
is the Lagrangian space variable,
the time variable and the primary dependent variable are the specific volume
, the velocity
, the absolute temperature
and the microrotation velocity
. The positive constants
,
and A denote the viscosity coefficient, heat conduction coefficient and microviscosity coefficient, respectively. The pressure p and the internal energy e are given by the state equations
where
is the adiabatic exponent, s is the entropy of fluid, R and B are the positive constants. We impose the following initial and far field conditions:
(2)
where
,
,
are given constants, and we assume
,
,
as compatibility conditions.
It is known that the large-time behavior of solutions of the Cauchy problem (1) and (2) is closely related to the Riemann problem of the compressible Euler system:
(3)
with the Riemann initial data
(4)
It is well-known that the above system has three eigenvalues:
which implies that the first and third characteristic fields are genuinely nonlinear and the second field is linearly degenerate. Then it is known that the basic Riemann solutions of the problem (3)-(4) are dilation invariant solutions: shock waves, rarefaction waves, contact discontinuities, and the linear combinations of these basic waves. In particular, the contact discontinuity solution of the Riemann problems (3) and (4) takes the form [1] :
(5)
provided that
(6)
The viscous contact wave
with
corresponding to the contact discontinuity
with
defined in (5) and
for the compressible micropolar fluid model (1) becomes smooth and behaviors as a diffusion waves due to the effect of heat conductivity. As in [2] , we can define the viscous contact wave
as follows.
Since the pressure of the profile
is expected to be almost constant, we set
(7)
Then the leading part of the energy Equation (1)3 is
(8)
Using the Equation (7),
and (8), we get a nonlinear diffusion equation
(9)
which has a unique self-similar solution
,
because
of [3] [4] . Furthermore,
is a monotone function, increasing if
and decreasing if
. On the other hand, there exists some positive constant
, such that for
,
satisfies
(10)
where
and
are two positive constants depending only on
and
. Once
is determined, the viscous contact wave profile
is defined as follows:
(11)
It is straightforward to check that
satisfies
which means the nonlinear diffusion wave
approximates the contact discontinuity
to the Euler system (3) in
norm,
on any finite time interval as the heat conductivity coefficient
tends to zero. And it is easy to check that the viscous contact wave
satisfies the system
(12)
where
(13)
and
(14)
We are now in a position to state our main results. Let
For interval
, we define a function space
as
We can get main result from [5] and it is stated as the following theorem.
Theorem 1.1. For any given
, suppose that
satisfies (6), Let
be the viscous contact wave defined in (11) with strength
. Then there exist positive constants
and
, such that if
and the initial data
satisfies
then the Cauchy problem (1)-(2) admits a unique global solution
satisfying
and
(15)
When the relation (6) fails, the basic theory of hyperbolic systems of conservation laws (for example, see [6] ) implies that for any given constant state
with
and
, there exists a suitable neighborhood
of
such that for any
, the Riemann problem of the Euler system (3)-(4) has a unique solution. In this paper, we only consider the stability of the superposition of the viscous contact wave and rarefaction waves. In this situation, we assume that
where
with
and
It is well-known [6] that there exists some suitably small
such that for
(16)
there exists a positive constant
and a unique pair of points
and
in
satisfying
and
Moreover, the points
and
belong to the 1-rarefaction wave curve
and the 3-rarefaction wave curve
, respectively, where
The points
and
may coincide with
and
, respectively. The 1-rarefaction wave
(respectively the 3-rarefaction wave
) connecting
and
(respectively
and
) is the weak solution of the Riemann problem of the Euler system (3) with the following initial Riemann data
(17)
Since the rarefaction waves
are not smooth enough solutions, it is convenient to construct smooth approximate ones. Motivated by [7] , the smooth solutions of Euler system (3),
, which approximate
are given by
(18)
where
(respectively
) is the solution of the initial problem for the typical Burgers equation:
(19)
with
and
(respectively
and
).
Let
be the viscous contact wave constructed in (11) and (9) with
replaced by
, respectively. We define
(20)
and
Then our main result of this paper is as follows:
Theorem 1.2. For any given
, suppose that (16) holds for some small
. Let
be as in (20) with strength
. Then there exist positive constants
and
, such that if
and the initial data
satisfies
then the Cauchy problem (1)-(2) admits a unique global solution
satisfying
and
(21)
Now, we briefly recall some related work in this aspect and make some comments on the analysis in this paper. The nonlinear stability of some basic wave patterns has been studied by many authors. The stability toward contact waves for solutions of systems of viscous conservation laws was first studied by Xin [8] who proved the nonlinear stability of a weak contact discontinuity for the compressible Euler equations with uniform viscosity. Later, Liu and Xin [9] showed the stability of contact discontinuities for a class of general systems of nonlinear conservation laws with uniform viscosity. And this result was improved by Xin and Zeng in [10] . The large-time asymptotic nonlinear stability of the supposition of viscous shock waves and contact discontinuities for system of viscous conservation laws with artificial viscosity under small initial perturbations was proved by Zeng [11] . Some interesting results have been obtained for compressible Navier-Stokes system. The asymptotics toward the rarefaction waves for compressible Navier-Stokes system is established in [7] [12] [13] [14] . For a free-boundary value problem, the asymptotic stability of a viscous contact wave of the one-dimensional compressible Navier-Stokes system was first proved by an elementary energy method by Huang, Matsumura, and Shi [15] , where the initial perturbation and the strength of the contact discontinuity are suitably small. The asymptotic stability of the linear combination wave of viscous contact wave and the rarefaction waves for the Cauchy problem of the one-dimensional compressible Navier-Stokes system was obtained by Huang, Li and Matsumura in [16] , and provided the strength of the combination wave is suitably small. The viscous shock profiles and viscous rarefaction waves have been shown to be asymptotically stable for quite general perturbation for the compressible Navier-Stokes system and more general systems of viscous strictly hyperbolic conservation laws [4] [12] [14] [17] - [22] . There are many results have been obtained for the nonlinear stability of some basic wave patterns consisting of viscous shock waves, rarefaction waves, viscous contact discontinuities and their certain linear superpositions with small perturbation. We refer to [23] - [31] and the references therein for viscous shock waves, [13] [14] [20] for rarefaction waves, [2] [32] [33] [34] [35] for viscous contact discontinuities, [16] [36] [37] [38] for the composition of a viscous contact wave and rarefaction waves. For the corresponding results with large initial perturbation, see [39] [40] [41] and the references cited therein.
The compressible micropolar fluid model has become an important area of interest for mathematicians in the last several decades. The model for compressible flow of micropolar fluid in the one-dimensional case was first studied by N. Mujaković. She considered the local-in-time existence and uniqueness [42] , the global existence [43] and regularity of solutions [44] to an initial-boundary value problem with homogeneous boundary conditions of the compressible one-dimensional micropolar fluid system respectively. Other results were proved in [45] [46] [47] for the corresponding non-homogeneous boundary value problems. Besides, she also analyzed large time behavior of the solutions and the stabilization of solutions to the Cauchy problem [48] of the one-dimensional model. There are other authors in [5] [49] showed the nolinear stability of some basic waves (such as rarefaction waves and viscous contact wave etc.). The stability of composite wave for one-dimensional compressible micropolar fluid model without viscosity was studied by Zheng, Chen and Zhang [50] . For the three-dimensional compressible micropolar fluid model, I. Dražić and N. Mujaković in [51] [52] [53] [54] [55] studied the local existence, global existence, uniqueness, large time behavior and regularity of spherical symmetry solutions.
In this paper, we shall show the asymptotic stability of combination of contact discontinuity with rarefaction waves for the Cauchy problem of the one-dimensional viscous micropolar fluid model, provided the strength of the combination wave is suitably small. After stating some notations, we will reformulate the original problem and give some preliminary lemmas and a priori estimates of solutions to the Cauchy problem (22) in Section 2. Finally, section 3 completes the proof of Theorem 1.2.
Notation. Throughout this paper, several positive generic constants are denoted by C, c without confusion. For functional spaces,
denotes the lth order Sobolev space with its norm
2. Reformation of the Problem and Preliminaries
Noticing that
satisfies Euler system (3) and
satisfies (1)1 and (8). To make it more convenient to prove Theorem 1.2, in this section, we will reformulate the problem (1), then the system (1)-(2) be rewritten as
(22)
where
(23)
and
(24)
We derive an elementary inequality concerning the heat kernel which will play an essential role later. For
, we define
(25)
It is easy to check that
(26)
Then we have
Lemma 2.1 (see [16] ) For
, suppose that
satisfies
Then the following estimate holds:
(27)
For the proof Lemma 2.1, one refers to [16] . Next, we summarize some basic properties of the viscous contact wave
.
Lemma 2.2 (see [36] ) Assume that
for some positive constant
. Then there exists two positive constants
and C such that the viscous contact wave
satisfies the following estimates:
(28)
(29)
Lemma 2.2 can be proved directly from Equations (10) and (11), the details are omitted here. The solution
of the Cauchy problem (19) has the following properties.
Lemma 2.3 (see [7] ) For given
, and
, let
Then the problem (19) has a unique smooth global solution in time satisfying the following:
i)
ii) For any
, there exists some positive constant
such that for
and
,
iii) If
, for any
iv) If
, for any
,
v) For the Riemann solution
of the scalar Equation (19)1 with the Riemann initial data
we have
We use Lemma 2.3 to investigate the properties of the smooth rarefaction waves
constructed in (20) and the viscous contact wave
, we divided the domain
into three parts, that is,
with
and
Then, Lemma 2.3 and (10) easily give
Lemma 2.4. (see [16] ) For any given
, assume that
satisfies (16) with
Then the smooth rarefaction waves
constructed in (18) and the viscous contact discontinuity wave
satisfy the following:
i)
.
ii) For any
, there exists some positive constant
such that for
and
,
iii) There exists some positive constant
such that for
and
we have in
that
and in
,
iv) For the rarefaction waves
determined by (3) and (17), it holds
Finally, we give a Sobolev inequality without proof.
Lemma 2.5. (see [32] ) For any
, we have
(30)
Since the local existence of the solution is well known (for example, see [42] ), to prove the global existence part of Theorems 1.2, we only have to establish the following a priori estimates.
Proposition 2.1
(A priori estimate) There exist positive constants
,
and C, such that for
and
satisfying
(31)
it follows the estimate
(32)
Once Proposition 2.1 is proved, we can extend the unique local solution
which can be obtained as in [42] , to
. Estimate (32) together with the Equation (22) implies that
which as well as (32) and the Sobolev inequality easily leads to the asymptotic behavior of the solutions, that is, (21).
From now on until the end of this paper, we always assume that
. Proposition 2.1 is an easy consequence of the following lemmas.
3. Energy Estimates
In this section we will drive some a priori energy estimates for the solutions to the system (1). Since Theorem 1.1 has been proved by Liu and Yin that we can see the details in [5] , we will give here the proof of Theorem 1.2 for brevity. We first give the following key estimate.
Lemma 3.1. For
and
satisfying (31) with suitably small
, we have for
,
(33)
Proof. First, we use Lemma 2.4 to investigate some aspects of F and G.
Since
direct calculation yields that
(34)
It follows from (20) that
thus, it derives from Lemma 2.4 that
(35)
We can treat the other terms on the righthand side of (34) in the same way to obtain
(36)
Since
(37)
the estimate (10) and Lemma 2.4 imply
(38)
(39)
and
(40)
The estimates (36)-(40) give
(41)
Similar to (36), we have
(42)
Since
(43)
and
(44)
It follows from (10) and Lemma 2.4 that
(45)
(46)
and
(47)
Thus, one derives from (34), (42)-(47) that
(48)
Now, multiplying (22)1 by
, (22)2 by
and (22)3 by
, (22)4 by
, then adding the resulting equations together, and using that
(49)
(50)
then we have
(51)
where
(52)
Notice that
and
, there exists positive constants
and
such that
(53)
Noticing that
(54)
we have
(55)
where
(56)
and
(57)
satisfies
(58)
due to Lemma 2.4 and (10). It is easy to compute
(59)
After integrating (51) on
, we deduce from (30), (41), (48), (22), (56), (28) and (59) that
(60)
Lemma 3.1 thus follows directly from (60) and the following Lemma 3.2 by choosing
in (61) and
suitably small. ,
Lemma 3.2. For
and w defined in (25), there exists some positive constant C depending on
such that the following estimate holds
(61)
Proof. The proof of (61) is divided into the following two parts:
(62)
and for any
,
(63)
In fact, adding (63) to (62) and taking first
then
suitably small thus implies (61) easily.
Here we used the same method as in [16] and combined with [50] , then, we can complete the proof of Lemma 3.2. We omit the details for simplicity. ,
Lemma 3.3. Suppose that
satisfies (31) with suitably small
. Then it holds for
,
(64)
Proof. We rewrite Equation (22)2 as
(65)
where we have used the following simple fact
due to
. Multiplying (65) by
, using
, noticing that
we have
(66)
Using (36), we obtain by direct calculation
(67)
It follows from Lemma 2.4 that
(68)
so, we have
(69)
The Cauchy inequality leads to
(70)
The estimate (10) and Lemma 2.4 yield that
(71)
and
(72)
Lemma 3.3 thus follows directly from Lemma 3.1 and (61), (66)-(72) by first choosing
suitably small then
suitably small.
Lemma 3.4. Suppose that
satisfies (31) with suitably small
. Then it holds for
,
(73)
Proof. Multiplying (22)2 by
, we have
(74)
with
Now, we need to control the term
. Using (10), (37), Lemma 2.4 and Lemma 2.5, we obtain by direct calculation
(75)
It follows from (10), (36), (75) and Lemma 2.4 that
(76)
The estimate (76) yields that
(77)
The Cauchy inequality leads to
So, we have
(78)
where we have used
due to (77).
Integrating (74) over
, using (78), and first choosing
suitably small then
suitably small, we can obtain Equation (73). This completes the proof of Lemma 3.4. ,
Lemma 3.5. Suppose that
satisfies (31) with suitably small
. Then it holds for
,
(79)
Proof. Multiplying Equation (22)3 by
, we obtain
(80)
Now, we are devoted to controlling the term
. Using (10), (44) and Lemma 2.4, we obtain by direct calculation
(81)
it follows from (43), (46) and (81) that
(82)
Using Lemma 2.2, Lemma 2.4, (31), the Hölder inequality and the Young inequality, we have
(83)
The estimate (82) and (83) yields that
(84)
The Cauchy inequality and Lemma 2.5 lead to
(85)
and
(86)
Then, by using the same argument as the previous, and integrating (80) over
, using (84)-(86) and following Lemma 3.6, and first choosing
suitably small then
suitably small, one can obtain equation (79). This completes the proof of Lemma 3.5. ,
The next Lemma is devoted to controlling the term
.
Lemma 3.6. Suppose that
satisfies (31) with suitably small
. Then it holds for
,
(87)
Proof. Multiplying (22)4 by
, we have
(88)
We have by Lemma 2.5 (31), the Hölder inequality and the Young inequality that
(89)
and
(90)
Integrating (88) over
, using (89) and (90), and first choosing
suitably small then
suitably small, one can obtain Equation (87). This completes the proof of Lemma 3.6. ,
4. Conclusion
Thus, we finish the proof of Proposition 2.1, and so the proof of Theorem 1.2 is completed.