Time-Periodic Solutions of the Hydrodynamic Equations for a Reacting Mixture in n -Dimension in multidimensional, existence and

In this paper, we investigate the existence of time-periodic solutions to the n-dimension hydrodynamic model for a reacting mixture with a time-periodic external force when the dimension 5 ≥ n is under some smallness assumption. The energy method combined with the spectral analysis is used to obtain the optimal decay estimates on the linearized solution operator. We study the existence and uniqueness of the time-periodic solution in some suitable function space by using a fixed point method and the decay estimates. Further-more, we obtain the time asymptotic stability of the time-periodic solution.


Introduction
Mathematical models for mixtures of the hydrodynamic equations in the space  n 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 5 > n 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 4 = n 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 ( ) ϕ ϕ θ = 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. 0 θ ≥ I is the ignition temperature. Combustion will occur when the temperature of the given fluid particle rise above θ I . Then, the reactant is transformed to the product via an irreversible reaction governed by the function θ I .
where µ is the heat viscosity coefficient, In what follows, we first make two assumptions: 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 ( ) , f t x . Our main idea is to combine the energy method with spectral analysis to get the optimal decay estimates of the linearized solution operator ( ) , u S t s , which we will introduce in Section 2 and obtain the decay rates of ( ) , u S t s in Section 4.
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  n . Now we are in a position to state our main results.
and it holds that , .
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.

Reformations
From the system (1.1), we have In terms of the definition of ( )   1  2  3   2  1  2  3   , , ,  , div  ,  ,   ,  : , , , Taking a change of variables again by   2  3  4  1  2  3  4   ,  1  1  , , ,  ,  ,  ,  ,  ,  , . , Notice that 1 2 3 4 , , , G G G G have the following properties: , , , div div : : , Referred to the way of [14], we are using A and B ω to denote the ( ) ( ) 2 2 + × + n n matrix differential operators, we can write as ( ) To obtain the periodic solution of the above problem, we consider the following linear system for any given That's to say (2.10) can be written by From (2.11), we use the Duhamels principle to determine the solution of the system (2.10) Open Journal of Applied Sciences S t s is the corresponding linearized solution operator.

Energy Estimates
In this section, we assume that ( ) . We list the following inequalities for later use; cf. [17].
where m is defined in Lemma 3.1.

The Energy Estimates on the Higher Order Derivatives
σ ω v r be the solution of (2.10), it holds that x v respectively, and using Young's inequality and integration by parts over  n , one can get if n is even. For each multi-index k with 2 ≤ ≤ k N , using Lemma 3.1, Lemma 3.2 and the fact that for 5 ≥ n and 2 ≥ + N n , Now, we need to prove that the term can be estimated by Notice that for any ′′ , .
ε ε ε ε ε ρ θ ρ ρ ρ θ θ ρ θ ρ θ θ θ Now we turn to 4 I , for any ′ ≤ k k with For the term 6 I , we can get Now, we estimate the unknown function Z and ∇Z . Multiplying ∂ k x (2.10) 4 by ∂ k x r and integrating with respect to x over  n , we can get ( ) Similar to the estimation on 3 By using Hölder's inequality, it holds that And by using Youngs inequality, we obtain ( )

The Energy Estimates on the Lower Order Derivatives
In this subsection, the usual energy method does not work here for the system (2.10) σ ω V r be the solution of (2.10), it holds that c is suitably large and ε is small enough.
For the term 1 J , we obtain

Conclusion
In this section, we will present the proof of two Theorems in Section 1.   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.

U t H U T t S T t s G U s F s s S T t T s G U T s F T s s S t s G U s F s s
Since the period of And choosing a small constant 2 0 ε > such that 0 2 , from (4.11) , it can be written as

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.