Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects

1 Introduction

This article investigates the qusi-one-dimensional non-isentropic compressible Euler equations:

for ( t , x ) ∈ R + × [ 0 , L ] . (1.1) can describe the motion of flows in a nozzle with generally varied cross-sectional area, where L > 0 is a constant denoting the considered nozzle length.

The unknown functions u ( t , x ) , n ( t , x ) , and S ( t , x ) denote velocity, density, and entropy of the considered fluids, respectively. The pressure function is p ( n , S ) = e S n γ , with the adiabatic gas index γ ∈ ( 1 , 3 ) . And the function a ( x ) = A ′ ( x ) A ( x ) , where A ( x ) ∈ C 2 ( [ 0 , L ] ) represents the cross-section of the duct. When A ( x ) are 1 , x , x 2 , system (1.1) denotes the classical 1 D , 2 D rotationally, and 3 D spherically symmetric Euler equations, respectively. We further assume there exist positive constants D 0 , D 1 , and D 2 such that

and

for any x ∈ [ 0 , L ] .

There are many literature about the compressible Euler flows moving through nozzles. For example, by a numerical scheme, Liu [14] constructed the global solutions for general quasilinear hyperbolic systems, which include the model of flows in variable area pipelines. The stability and instability of transonic flows through a nozzle were discussed in [15]. Later on, an efficient shock-capturing scheme for solving the Euler equations with geometric structure was proposed in [4,5]. We can refer to [2,6,7,11,12,16,17,24,27,28] for some other existing results of global solutions. Meanwhile, many works have been done on the multidimensional non-isentropic steady Euler flows. In [3], Chen considered the non-isentropic inviscid subsonic steady flows with large vorticity in a two-dimensional infinitely long nozzle and built the existence and uniqueness of the subsonic problem. Duan and Luo[9] investigated the problem in three-dimensional axisymmetric nozzles and removed the constraint on the smallness of the vorticity in [8]. Then [20] gained the optimal convergence rates of the subsonic non-isentropic steady Euler equations through a three-dimensional axisymmetric infinitely long nozzle.

At the same time, time-periodic problems have attracted much attention for its applicability in industry. For the temporal periodic solutions excited by the time-periodic external force, we can see [1,18,19,23]. However, the time-periodic solutions motivated by boundary conditions are significant in the feedback boundary control. In [32], Yuan first gained the existence and stability of isentropic supersonic time-periodic Euler flows which are motivated by periodic boundary conditions. Inspired by this result, [31] and [21] further studied the time-periodic supersonic solutions with the source term β ρ ∣ u ∣ α u for isentropic and non-isentropic cases. It is worth noting that the supersonic assumption, which means all characteristics propagate forward in both space and time, plays an important role in [21,31,32]. Therefore, after a specific start-up time T * > 0 , the boundary condition can control the entire region completely, thus we gain the existence of time-periodic solutions. However, it becomes more complicated for subsonic case since the characteristics propagate both forward and backward, which leads to wave interactions between different families. For one-dimensional subsonic isentropic compressible Euler system, [26,33] investigated the time-periodic solutions for linear and nonlinear damping, respectively. There are also some important researches on the existence of space and time periodic problem for non-isentropic compressible Euler systems, see [29,30]. As for more general quasi-linear hyperbolic systems, we can refer to [10,25] for details.

To our knowledge, in general ducts, few studies have been made on the subsonic temporal periodic flows driven by periodic boundary conditions so far. However, considering this problem is one of the basic steps in studying time-periodic transonic shock solutions. In this article, motivated by [10,25,26,33], we study the non-isentropic subsonic temporal periodic solutions to compressible Euler system with geometric effects. By using an iterative method, we prove that there exists an initial data such that the periodic perturbation on the dissipative boundaries can trigger a time-periodic subsonic solution. Owing to the appearance of the geometric source term, it is important to make full use of the dissipative effects brought by boundaries and construct appropriate iterative formats. Unfortunately, a new challenge, derivative loss, emerges from the iterative format after diagonalization. We overcome this difficulty by noticing a significant property of entropy, which enables us to switch the derivative of entropy with respect to x into a derivative along the direction of the first or third characteristic.

The remaining parts of this article are organized as follows. In Section 2, we presented some elementary properties of the C 1 subsonic and supersonic steady solutions. Section 3 was devoted to reformulate (1.1) and the steady states system by Riemann invariants and present main results. Then in Section 4–7, we gave proofs of Theorems 3.1–3.4, respectively.

2 Steady solutions

In this section, we study the solution Q ˜ ( x ) = ( n ˜ ( x ) , u ˜ ( x ) , S ˜ ( x ) ) ⊤ of the time-independent system

or equivalently

We give the inflow data at x = 0 :

with u − > 0 and n − > 0 . By (2.2)₁, (2.2)₃, and (2.3), we know that the cross-sectional flux and entropy are two constants, i.e.,

One critical parameter in fluid dynamics is the Mach number, which can be defined as follows:

where the sound speed c ˜ ( n ˜ , S ˜ ) = ∂ p ∂ n ˜ = γ e S ˜ 2 n ˜ γ − 1 2 . The fluid is supersonic for M > 1 and subsonic while M < 1 . From (2.2), we could solve the continuous flows with Mach number M ≠ 1 , i.e.,

Then by the definition of M , it can be deduced that

Therefore, if we assume the fluid velocity is supersonic (i.e., c − < u − ) or subsonic (i.e., c − > u − ) at the entrance x = 0 , then problem (2.2) admits a C 1 supersonic or subsonic solution on [ 0 , L * ) , where L * is the life span of the corresponding supersonic or subsonic smooth solution. In this article, we expect the steady flow to keep its entrance supersonic or subsonic situation in the whole duct. Therefore, the length of the duct L should be smaller than L * . While the maximum duct length L * can be determined by (2.6), which denotes L * only depends on γ , the initial data ( n − , u − , S − ) and the given function a ( x ) . We will consider some other cases, such as the transonic smooth solutions and transonic shocks in the following article.

From (2.5)₁ and (2.5)₂, we directly derive

Then, when the incoming flow is subsonic, namely, c − > u − > 0 , it follows from (2.5)₁, (2.5)₂, and (2.6) that

Similarly, when the incoming flow is supersonic, i.e., u − > c − > 0 , one has

Formulas (2.8)–(2.11) imply that if the duct is divergent, the flow always keeps its state of entrance for any duct length L > 0 , regardless subsonic or supersonic. While, if the duct is convergent, to keep the upstream subsonic or supersonic state, there must exist a unique positive constant L * < + ∞ , such that the flow is subsonic or supersonic for x ∈ [ 0 , L * ) . In addition, after some straightforward computations, we obtain that:

1. If a ( x ) > 0 , according to (1.3) and (2.4), for x ∈ [ 0 , + ∞ ) , we have

   (2.12) n − < n ˜ ( x ) < n − A ( x ) A ( 0 ) u − 2 c − 2 − u − 2 , u − A ( 0 ) A ( x ) c − 2 c − 2 − u − 2 < u ˜ ( x ) < u − , for c − > u − > 0 ,

   and

   (2.13) n − A ( 0 ) A ( x ) u − 2 u − 2 − c − 2 < n ˜ ( x ) < n − , u − < u ˜ ( x ) < u − A ( x ) A ( 0 ) c − 2 u − 2 − c − 2 , for u − > c − > 0 .

2. If a ( x ) < 0 , for x ∈ [ 0 , L ] and L < L * , we have

   (2.14) n − e ∫ 0 L a ( ξ ) M 2 ( ξ ) 1 − M 2 ( ξ ) d ξ < n ˜ ( x ) < n − , u − < u ˜ ( x ) < u − e ∫ 0 L a ( ξ ) 1 − M 2 ( ξ ) d ξ , for c − > u − > 0 ,

   and

   (2.15) n − < n ˜ ( x ) < n − e ∫ 0 L a ( ξ ) M 2 ( ξ ) M 2 ( ξ ) − 1 d ξ , u − e ∫ 0 L a ( ξ ) M 2 ( ξ ) − 1 d ξ < u ˜ ( x ) < u − , for u − > c − > 0 .

3 Reformulation and main results

In this section, we rewrite systems (1.1) by Riemann invariants. Besides, we state our main results of this article. First, by simple calculations, the Riemann invariants of system (1.1) are as follows:

Then system (1.1) changes into

where

Similarly, the steady system (2.2) can be rewritten as follows:

where

and

As noted in Section 1, the supersonic case is much elementary as it can be directly solved by wave decomposition method. More details can be seen in [21,22], here we omit the supersonic case.

To formulate the subsonic problem, we now specify the initial data and boundary conditions. Assuming that at t = 0 :

and at x = L :

while at x = 0 :

where the boundary conditions (3.6)–(3.8) are dissipative and time-periodic. That is, max { ∣ s 1 ∣ + ∣ s 1 * ∣ , ∣ s 2 ∣ , ∣ s 3 ∣ } = s < 1 , and ς p ( t ) , η p ( t ) , υ p ( t ) are temporal periodic functions with a constant time period T . The solid boundaries (3.6)–(3.8) possess the dissipative structure in the sense of Li [13], who pointed out that without this boundary dissipation, the initial-boundary value problem of hyperbolic equations may blow up in finite time, even when the initial data is small.

Taking the steady solution Q ˜ ( x ) = ( n ˜ ( x ) , u ˜ ( x ) , S ˜ ( x ) ) ⊤ as a background solution, now we define the perturbation variables

and rename the steady solution as follows:

Then combining (3.2) and (3.4) yields

Correspondingly, the initial boundary conditions are as follows:

where Γ 1 p ( t ) = ς p ( t ) − ς ˜ ( L ) , Γ 2 p ( t ) = υ p ( t ) − υ ˜ ( 0 ) , Γ 3 p ( t ) = η p ( t ) − η ˜ ( 0 ) , and Γ i p ( t ) , ( i = 1 , 2 , 3 ) satisfy

To gain the global existence of time-periodic solutions, we require that the incoming sound speed is a small quantity, namely, there is a small positive constant ε 0 , such that

In fact, we can acquire the upper bound for the small constant ε 0 , which depends on the relative slope a ( x ) , the duct length L , the dissipation coefficient s , and the inflow data (2.3). However, to specify this upper bound would cause a tedious primary calculation which deviates from our main purpose in this article. Concerning the slope of nozzles, it is worth pointing out that we only require the cross-section A ( x ) be C 2 continuous, without any other restrictions.

Now we are ready to state our main results in this article.

Take t → + ∞ in (3.24), so we obtain the uniqueness result.

4 Existence of the temporal periodic solution

In this section, we will prove the existence of subsonic time-periodic solutions to the non-isentropic compressible Euler equations by constructing iterative formats in details. Now we focus on the time-dependent system (3.11)–(3.17) and consider its periodic solutions driven by boundary conditions.

In this process, we will encounter a difficulty, which raises from the appearance of ∂ x Γ 2 in (3.11) and (3.13). This term can lead to the derivative loss in C 1 prior estimates. Fortunately, we notice Γ 2 is conserved along the second characteristic curve and overcome this difficulty by transforming ∂ x Γ 2 into the first and third characteristic directions. That is, by using (3.12), we can transform (3.11) and (3.13) into