Time-Periodic Solutions of the Hydrodynamic Equations for a Reacting Mixture in n-Dimension ()
1. Introduction
Mathematical models for mixtures of the hydrodynamic equations in the space
have been studied for quite a few years. Notice that if the radiation effect is neglected, the existence, uniqueness and dynamic behavior of solutions were recognized by Chen [1] [2] and Li [3] under the initial value satisfies certain assumptions; if the radiation effect is considered, for (1.1) in one dimensional, the existence and uniqueness of the global solution to the Cauchy problem is obtained by Liao and Zhao [4] under the assumption of constant viscosity coefficient; the global existence and uniqueness of solutions for initial boundary value problems of viscous radiative reactive gases were achieved very well by Liao and Zhao [5], Ducomet [6], Jiang and Zheng [7] [8] and Umehara [9] [10]. Besides, for (1.1) in multidimensional, global existence and exponential stability of spherically symmetric solutions in a bounded annular domain for compressible viscous radiative reactive gases were well obtained by Qin, Zhang, and Su [11] and Liao, Wang and Zhao [12] for spherical solutions in an exterior domain. When the initial data is in the neighborhood of the trivial stable solution, the global existence and uniqueness of the strong solution of Cauchy problem are proved effectively by Wang and Wen [13]. In addition, there are lots of researches on the time-periodic solution of the Navier-Stokes system, what we want to say most is that the existence of a time-periodic solution for the Navier-Stokes equations with time-periodic external force under some assumptions when the space dimension
was well proved by Ma, Ukai and Yang [14]; The existence of time-periodic solutions for compressible Navier-Stokes equations under general external forces when space dimension
was defined by Jin [15]; For the time-periodic parallel flow problem in n-dimensional space, there is a time-periodic solution for the Navier-Stokes equation with a special time-periodic external force was controlled by Brezina and Kagei by [16]. In summary, it’s still open whether the time-periodic solution to (1.1) exists and is unique in n dimensions.
In this paper, we consider the Cauchy problem of a model for the combustion of the hydrodynamic equations with a time-periodic external force in
:
(1.1)
for
, where
,
,
,
denote the density, the velocity, the temperature and the mass fraction of the reactant, respectively. The last term on the right side of the energy Equation (1.1)3 is the reactant energy difference
, which means the difference between the rate of gained energy for the product and that of lost energy for the reactant. The constant
is the reaction rate. q is the difference of the stoichiometric coefficients for components appearing as reactant and product.
represents the density of the reactant.
is the reaction function which is assumed to satisfy the first-order Arrhenius law as follows (see [6] ):
where A is a positive constant and stands for the activation energy.
is the ignition temperature. Combustion will occur when the temperature of the given fluid particle rise above
. Then, the reactant is transformed to the product via an irreversible reaction governed by the function
.
The heat flux
satisfies the Fourier law
(1.2)
where
is the heat conductivity coefficient.
,
and b are positive constants.
The function F that we assumed denotes the diffusion velocity and satisfy Fick’s law
(1.3)
where
stands for the reactant flux diffusion coefficient and the positive constant
is the species diffusion in the reaction.
The total pressure P in the gas and the corresponding specific internal energy e have the following form
(1.4)
where constants
, a and
are positive.
The viscous stress tensor is given in system (1.1) by
(1.5)
where
is the heat viscosity coefficient,
with the bulk viscosity coefficient
. Thus
(1.6)
In what follows, we first make two assumptions:
A.1.
is periodic on time with period
.
A.2.
is a smooth function in a neighborhood of the constant state
with
,
. In addition,
satisfies
and
.
In this paper, our main purpose is to obtain a time-periodic solution of (1.1) around the constant state
which has the same period as the periodic function
. Our main idea is to combine the energy method with spectral analysis to get the optimal decay estimates of the linearized solution operator
, which we will introduce in Section 2 and obtain the decay rates of
in Section 4.
Let
, we define the solution space by
(1.7)
with some constant
. The corresponding norm
is defined as follows:
(1.8)
In this paper, we will study the existence and uniqueness of time-periodic solutions and associate optimal time-decay estimates for the time-periodic solution which we obtained in
. Now we are in a position to state our main results.
Theorem 1.1. Let
,
. Suppose
. Under assumptions A.1-A.2, there exists a fixed constant
given in the proof, such that if
for some sufficiently small constant
, then the system (1.1) admits a unique time-periodic solution
with period
, which satisfies
.
We consider the Cauchy problem of the system (1.1) with the initial data
(1.9)
for some fixed initial time
. When the initial data is a small perturbation of the periodic solution
stated in Theorem 1.1, we obtain the stability of the solution around this time-periodic solution.
Theorem 1.2. Assume that
is stated in Theorem 1.1. For each
, let
is sufficiently small under the same conditions of Theorem 1.1, then the Cauchy problem (1.1) and (1.9) has a unique global solution
which satisfies
(1.10)
(1.11)
(1.12)
and it holds that
(1.13)
Moreover, if
, then exists a constant
such that
(1.14)
Notations For a multi-index
, we denote
and
.
,
, for
,
. Here,
for
in the Sobolev Space
.
The rest of the paper is organized as follows. In the second section, we introduce appropriate variable transformations to linearize the transformed equations. In Section 3, we first show the energy estimates of the solution to (1.1). Then we study the periodicity of the solution of the linearized system with respect to time. Finally, we obtain the time-periodic solution of the nonlinear equation according to the expression of the solution of the equations. In Section 4, the proof of Theorem 1.1 is given. In the last section, we study the stability of the time-periodic solution.
2. Reformations
From the system (1.1), we have
(2.1)
In terms of the definition of
in (1.3), we can see that
(2.2)
By using (2.1)1, we have
(2.3)
Taking a change of variables by
,
,
,
, and using (2.3), then the problem (1.1) can be reformulated as
(2.4)
where
with
Taking a change of variables again by
(2.5)
and
Then, the regularized problem (2.4) can be reformulated as
(2.6)
where
(2.7)
Notice that
have the following properties:
Let
,
,
.
Then, we can get
(2.8)
Referred to the way of [14], we are using A and Bω to denote the
matrix differential operators, we can write as
From (2.6), we can write as
(2.9)
To obtain the periodic solution of the above problem, we consider the following linear system
(2.10)
for any given
satisfying
That’s to say (2.10) can be written by
(2.11)
From (2.11), we use the Duhamels principle to determine the solution of the system (2.10)
(2.12)
where
is the corresponding linearized solution operator.
3. Energy Estimates
In this section, we assume that
for all
. We list the following inequalities for later use; cf. [17].
Lemma 3.1. Let
, we have
(3.1)
Lemma 3.2. Assume that
. If
, we have
(3.2)
(3.3)
where m is defined in Lemma 3.1.
3.1. The Energy Estimates on the Higher Order Derivatives
Lemma 3.3. Assume that
,
. Let
be the solution of (2.10), it holds that
(3.4)
where constants
and
with
is large suitably.
Proof. For each multi-index k with
. Applying
to (2.10)1 - (2.10)3, multiplying them by
,
,
respectively, and using Young’s inequality and integration by parts over
, one can get
(3.5)
For each multi-index k with
, using Hölder’s inequality and Lemma 3.1, we have
where
if n is odd,
if n is even. For each multi-index k with
, using Lemma 3.1, Lemma 3.2 and the fact that for
and
,
, we have
Thus for
and
,
and
, we can get
(3.6)
For the term
, let
with
, we have
(3.7)
Now, we need to prove that the term
(3.8)
can be estimated by
, where we show the estimate on the term of
which can be written as
(3.9)
Notice that for any
with
,
; for any
with
,
. Then, Lemma 3.1 implies that
which is the desired estimate. Note that the other terms can be estimated similarly. Hence, we have
(3.10)
Similar to
and
, it follows from Lemma 3.1 and Lemma 3.2 that
(3.11)
and
Now we turn to
, for any
with
, we have
(3.12)
For the term
, we can get
(3.13)
Now, we estimate the unknown function Z and
. Multiplying
(2.10)4 by
and integrating with respect to x over
, we can get
(3.14)
Similar to the estimation on
, let
with
, we have
(3.15)
Then, we have
(3.16)
Meanwhile, to estimate
for
, by applying
to (2.10)2, multiplying them by
and integrating them on
, we obtain
(3.17)
By applying
to (2.10)1, multiplying them by
, integrating them on
, we obtain
(3.18)
Meanwhile, using integration by parts and putting (3.18) into (3.17), we have
(3.19)
By using Hölder’s inequality, it holds that
(3.20)
And by using Youngs inequality, we obtain
(3.21)
Similar to the estimation on
, we have
(3.22)
Hence, from (3.19) to (3.22), we can get
(3.23)
Combining
, (3.17) and (3.23) yields the inequality (3.4). This completes the proof of the lemma.
3.2. The Energy Estimates on the Lower Order Derivatives
In this subsection, the usual energy method does not work here for the system (2.10) because the zero order derivative term
cannot be controlled. In
order to overcome this difficulty, we would like to rewrite the system (2.10) and then obtain a new system about some modes
(3.24)
Then the system (2.10) could be rewritten as
(3.25)
Lemma 3.4. Assume that
,
. Let
be the solution of (2.10), it holds that
(3.26)
where constants
,
and
with
is suitably large and
is small enough.
Proof. Multiplying (3.25)1, (3.25)2 and (3.25)3 by
,
and V, integration by part over
respectively, one obtains
(3.27)
For the term
, we obtain
For
, by using Lemma 3.2, we have
For
and
, Lemma 3.1 and Lemma 3.2 give
For
, one can get
For
, we obtain
For
, we have
Due to
,
, we get that
and
. Now we estimate the unknown function Z and
. Multiplying (3.25)4 by r and integrating with respect to x over
, we can get
(3.28)
Next, we need to estimate
, similar to the estimation on
, we obtain
(3.29)
Combining
, (3.28) and (3.29) yields the inequality (3.26). This completes the proof of the lemma.
4. Conclusion
In this section, we will present the proof of two Theorems in Section 1.
4.1. The Proof of Theorem 1.1
Let
be the solution operator for the case when
, by spectral analysis, we have the following time decay properties for
, cf. [13].
Proposition 4.1. Assume
,
, for any integer
, if
, then for
, we have
(4.1)
By Proposition 4.1 and the estimates obtained in Section 3, the solution operator
has the decay estimates as follows.
Proposition 4.2. Let
be a smooth solution to the system (2.10), integers
and
, there exist three positive constant
and
with
being suitably large and
, such that for
,
and
if
Then it holds that
1)
(4.2)
2)
(4.3)
From the assumptions of time-periodic solution and global solution given above, we will prove Theorem 1.1 as follows and the proof is divided into two steps.
Proof. (Theorem 1.1). 1) Choosing the time
for
, we can suppose that there exists a time-periodic solution
,
with period
for the system (2.10) with the initial data
at any given time
. Then (2.12) can be written
(4.4)
where
,
and f are periodic on time with period
in the system (2.10). Due to
, from proposition 4.2, if
, we obtain
(4.5)
In addition, since
is dense in
, from (4.5), we have
Therefore, since
when
, the fact that
for
, which means that the convergence of the integral can be guaranteed in (4.4). Then, it holds that
(4.6)
For any given perturbed solution
, we can define a map
:
(4.7)
And by (4.6), there exists a fixed point which is also the mild solution of (2.10) in map
; on the contrary, suppose that there exist a unique fixed point in map H, denoted by
. Because F and f have the same time period T, we set
, then
since
. We have
(4.8)
Thus from the uniqueness,
which is the desired periodic solution to the system (2.10).
2) For the periodic function f, we assume that
and
is small enough, then for some appropriate constant
, H has a unique fixed point in the space
. Firstly, Since the period of the time-periodic solution
is also T, for
and
, from lemma 3.3 and lemma 3.4, we can choose a small constant
, a large enough
and a constant
such that if
(4.9)
Since the period of
is T, and the inequality (4.9) is integrated with respect to t in
, it can be got
(4.10)
Then, from the definition of
, one can get
(4.11)
And choosing a small constant
such that
. Then if
, from (4.11) , it can be written as
(4.12)
Secondly, from (4.6), one gets
(4.13)
From (2.10), we have
(4.14)
Therefore, due to
for
, by (4.13), we can obtain
(4.15)
From (4.12) and (4.15), there exist two positive constants
,
independent of W, such that
(4.16)
Finally, set
,
, let
,
, if we set
. By (2.8), we have
(4.17)
where there is no relationship between
and W. Assume that
and
, we choose
and sufficiently small
, such that
Then, we can get through simple calculation
(4.18)
and
(4.19)
Hence if
for a positive constant
, the inequalities (4.18) and (4.19) are both allowed to make sense. When
satisfies (4.18) and (4.19), for
, H is a contraction in the complete space
. Then H has a unique fixed point in
. We have proved the Theorem 1.1.
4.2. The Proof of Theorem 1.2
In this section, the proving of Theorem 1.2 can be arranged as follows. Firstly, under the same conditions of Theorem 1.1, in order to prove the global existence of the solution to the Cauchy problem (1.1) and (1.9), we can suppose that
without loss of generality. Let
be the time-periodic solution constructed in Theorem 1.1 and
be a solution to (1.1) and (1.9). As in Section 2, denote
Then the difference
is a solution to the Cauchy problem
(4.20)
(4.21)
where
,
,
. Then our problem is turned to prove the global existence and decay estimates on the solution to the Cauchy problem (4.20) and (4.21). Define the function space of Cauchy problem (4.20) and (4.21) by
, where for
. Thus we have
(4.22)
with the norm
is given by
(4.23)
Notice that
. As usual, the local existence of the Cauchy problem (4.20) and (4.21) can be given by the standard argument of the contracting map theorem. Hence we omit the details of this proof.
Proposition 4.3. (The local existence). Assume that
such that
and
. Then there exists a constant
depending on
such that the Cauchy problem (4.20) and (4.21) has a unique solution
, which satisfies
(4.24)
where
is independent of
.
Proposition 4.4. (A priori estimates). Suppose that
, let
for some positive constant
be a solution of the Cauchy problem (4.20) and (4.21). Thus if the solution for sufficiently small positive constants
and
satisfies
(4.25)
It holds the following estimate that
(4.26)
where
.
Remark 4.1. Here
is independent of
and
, set
, such that
(4.27)
Then, the global solution to the Cauchy problem (4.20) and (4.21) will be obtained by combining Propositions 4.3 and 4.4 by the standard continuity argument. Finally, we have the following decay property of the solution
.
Proposition 4.5. [14] Assume that
be such that
is small enough and
is bounded. Then the solution
to (4.20) and (4.21) has the decay property
(4.28)
Appendix
In this section, we will give the proof of propositions used by Theorem 1.1 and 1.2.
A.1. The Proof of Proposition 4.2
Proof. Let us rewrite the problem (2.11) when
,
as follow:
(A.1)
In addition, by the proof of Lemma 3.3 in Section 3, when
,
, and if the condition
is tenable, we can easily get
(A.2)
We can construct an energy function
on the higher order derivatives by
(A.3)
Because of the sufficient large constant
, we obtain
(A.4)
Notice that if there exist a constant
, it holds that
(A.5)
We solve above inequality by solving general solutions of ordinary differential equations, then
(A.6)
Thus, we calculate the time decay estimate (i) by using (6.1) and Proposition 4.1, then when taking
,
:
(A.7)
when taking
,
:
(A.8)
Notice that
Since
, from Lemma 3.1, we obtain
,
.
Thus, we can easily get
(A.9)
and
(A.10)
According to the definition of the energy equation
, the time decay estimation of
can be further obtained, assume that for the time decay estimation (i),
(A.11)
for the time decay estimation (ii),
(A.12)
We use the definition of
, (A.4) and (A.10), taking (A.11) and (A.12) into (A.6) to get
(A.13)
(A.14)
Similar to (A.13) and (A.14), by (A.12), we have
(A.15)
(A.16)
We can obtain that
Then, we complete the proof of the proposition 4.2.
A.2. The Proof of Proposition 4.4
Proof. Before proving this proposition, we need to recall the third section of energy estimation. By (4.22) and the smallness condition on
, then there exist some positive constants
and
with
being suitably large and
being sufficiently small, we can get by direct calculation of the energy estimates for
(A.17)
and
(A.18)
From (A.17) and (A.18), by noticing that
, we can choose a constant
suitably large such that
(A.19)
which implies (4.26) is hold. This completes the proof of proposition 4.4.
A.3. Some Useful Formulas
Here, we list some known formulas.
Lemma A.1. (Duhamels principle). Assume that the function
is the solution of the Cauchy problem
(A.20)
then the function
is the solution of the Cauchy problem
(A.21)
Lemma A.2. (Hölder inequality). Assume that
,
. If
,
, we have
(A.22)