Existence and stability of contact discontinuities to piston problems

Xiaomin Zhang; Huimin Yu

doi:10.1515/anona-2025-0131

Article Open Access

Existence and stability of contact discontinuities to piston problems

Xiaomin Zhang and Huimin Yu

Published/Copyright: November 21, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

This study investigates the existence and stability of the contact discontinuity to a piston problem, which is governed by one-dimensional compressible Euler equations under the condition of zero relative velocity between the piston and tube gas. We prove that a contact discontinuity is stable under the assumptions that the perturbations of the initial data and the piston velocity are both small enough in some suitable function spaces. Moreover, using the method developed in this study, similar stability result can be obtained when the contact discontinuity is sealed by two pistons on both sides.

Keywords: compressible Euler equations; piston problem; contact discontinuity; stability

MSC 2020: 35B20; 76L99; 76N10; 76N15

1 Introduction

The piston problem constitutes a particular initial-boundary value problem in compressible flow dynamics (cf. [3]). In a slender tube with one end sealed by a piston and the other end open, gas dynamics become fully coupled to the piston kinematics through wave-mediated interactions. To be more specific, when the piston is pushed forward relative to the gas, a shock wave emerges and propagates ahead of the piston. Conversely, when the piston is pulled backward relative to the gas, a rarefaction wave comes into existence. Moreover, if the piston remains stationary relative to the gas within the tube, a contact discontinuity is formed. In this study, we present a mathematical analysis on the piston problem with the piston being stationary relative to the gas.

In recent years, numerous studies have been conducted on the contact discontinuity solutions of compressible flows. For example, in 2008, Huang et al. [11] studied the large time asymptotic stability of contact discontinuities with a convergence rate for the Navier-Stokes equations and Boltzmann equation under general initial perturbations. For the steady system with multiple space dimensions, Wang and Yu [14] explored the structural stability of supersonic contact discontinuities for three-dimensional (3D) steady compressible isentropic Euler equations in 2015. Later, in 2016, Ruan et al. [12] studied the linear stability of constant and curved vortex sheet solutions for the two-dimensional (2D) inviscid liquid-gas two-phase flow. Moreover, the vortex sheet corresponds to a contact discontinuity in the terminology of hyperbolic conservation law systems. In 2019, Bae and Park [1,2] analyzed the existence of a subsonic contact discontinuity for the 2D and 3D axisymmetric compressible Euler equations in an infinitely long nozzle. At the same time, Huang et al. [9] investigated the stability of supersonic contact discontinuity in two-dimensional steady compressible Euler equations within a finitely long nozzle with varying cross sections. Subsequently, in 2021, they extended this finding to the transonic contact discontinuity in [10]. Recently, in 2025, Weng and Zhang [15] and Gao et al. [8] explored the structural stability of supersonic contact discontinuity for 2D steady compressible rotating and combustion Euler equations, respectively. Concerning contact discontinuities to the piston problem, in 2018, Ding [4] studied the stability of contact discontinuity to the 1D piston problem for the compressible Euler equations in the B V space. Subsequently, in 2021, she extended this result to the relativistic full Euler equations and investigated the non-relativistic limits in [5]. For other extended discussions related to the contact discontinuity, we refer to [6,13].

In this study, we reconsider the stability of the contact discontinuity in the piston problem and assume the gas in the tubes is controlled by the following 1D compressible Euler equations:

(1.1) ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 + p ) x = 0 , ( ρ E ) t + ( ρ E u + p u ) x = 0 ,

where ρ , u , and p stand for the density, velocity, and pressure, respectively, and the total energy

(1.2) E = u 2 2 + p ( γ − 1 ) ρ ,

with the adiabatic exponent γ > 1 . Moreover, the pressure p and density ρ satisfy the constitutive relation:

(1.3) p = p ( ρ , S ) = ρ γ φ ( S )

with S denoting the entropy of the gas. Then, by (1.2) and (1.3), we rewrite (1.1) as

(1.4) p t + u p x + γ p u x = 0 , u t + u u x + p − 1 γ φ 1 γ ( S ) p x = 0 , S t + u S x = 0 .

For any given T 0 > 0 , we study the state of the gas in the domain D = { ( x , t ) ∣ x ≥ B ( t ) , t ∈ [ 0 , T 0 ] } , where B ( t ) denotes the movement curve of the piston with the speed B ′ ( t ) . Assume the initial data of equations (1.4) are

(1.5) ( p , u , S ) ( x , 0 ) = ( p 0 , u 0 , S 0 ) ( x ) = ( p 0 , − , u 0 , − , S 0 , − ) ( x ) , B ( 0 ) ≤ x < x 0 , ( p 0 , + , u 0 , + , S 0 , + ) ( x ) , x > x 0 ,

which are divided by x = x 0 and satisfy

(1.6) u 0 , − ( x 0 ) = u 0 , + ( x 0 ) , p 0 , − ( x 0 ) = p 0 , + ( x 0 ) .

The velocity of the gas at the boundary x = B ( t ) is the same as the piston velocity, that is,

(1.7) u ( B ( t ) , t ) = B ′ ( t ) .

Let the location of the contact discontinuity be x = ϒ ( t ) , which divides the domain D into two subdomains

D − = { ( x , t ) ∣ B ( t ) ≤ x < ϒ ( t ) , t ∈ [ 0 , T 0 ] } , D + = { ( x , t ) ∣ x > ϒ ( t ) , t ∈ [ 0 , T 0 ] } .

Along the contact discontinuity x = ϒ ( t ) , with the aid of the Rankine-Hugoniot conditions, we have

(1.8) u ( ϒ ( t ) + , t ) = u ( ϒ ( t ) − , t ) = ϒ ′ ( t ) , p ( ϒ ( t ) + , t ) = p ( ϒ ( t ) − , t ) .

We consider a special kind of contact discontinuity, which is considered as the background solution in this study and described as follows:

(1.9) ( p ̲ , u ̲ , S ̲ ) = ( p ̲ − , u ̲ − , S ̲ − ) , ( x , t ) ∈ [ b ( t ) , υ ( t ) ) × [ 0 , T 0 ] , ( p ̲ + , u ̲ + , S ̲ + ) , ( x , t ) ∈ ( υ ( t ) , + ∞ ) × [ 0 , T 0 ] ,

where p ± , u ± , S ± are some nonzero constants satisfying

u ̲ − = u ̲ + = u ̲ , p ̲ − = p ̲ + = p ̲ , S ̲ + = S ̲ − + σ

with the constant σ ≠ 0 and b ( t ) < υ ( t ) < + ∞ . Moreover, we assume both the piston speed b ′ ( t ) and the contact discontinuity velocity υ ′ ( t ) are equal to u ̲ .

Different from the former works [4,5], which use a modified wave front tracking method in B V space, we reconsider this problem by a well-designed iteration scheme in C 1 space. Specifically, we first demonstrate the problem by two free boundary problems in the x − t plane. Then, employing the Lagrangian coordinates transformation, we straighten the contact discontinuity and change the free boundary into a fixed one. Subsequently, by constructing an ingenious linear iteration scheme, we rigorously demonstrate the existence and stability of contact discontinuity. Finally, based on the inverse Lagrangian coordinates transformation, we show that the original free boundary admits two layers of smooth flows separated by a smooth contact discontinuity. Here is the main theorem of this study.

Theorem 1.1

(Existence and stability of the contact discontinuity) For any given T 0 > 0 , there exist some positive constants ε 0 , C S , such that for any given ε ∈ ( 0 , ε 0 ) , if the initial data ( p 0 , u 0 , S 0 ) ( x ) and piston velocity B ′ ( t ) satisfy

(1.10) ‖ ( p 0 , − , u 0 , − ) ( x ) − ( p ̲ − , u ̲ − ) ‖ C 1 ( [ B ( 0 ) , x 0 ) ) + ‖ ( p 0 , + , u 0 , + ) ( x ) − ( p ̲ + , u ̲ + ) ‖ C 1 ( ( x 0 , + ∞ ) ) + ‖ S 0 , − ( x ) − S ̲ − ‖ C 2 ( [ B ( 0 ) , x 0 ) ) + ‖ S 0 , + ( x ) − S ̲ + ‖ C 2 ( ( x 0 , + ∞ ) ) + ∣ x 0 − υ ( 0 ) ∣ + ‖ B ′ ( t ) − u ̲ ‖ C 1 ( [ 0 , T 0 ] ) ≤ ε ,

then the initial and boundary value problem (IBVP) (1.4)–(1.8) admit a unique piecewise smooth solution ( p , u , S ) ( x , t ) with the contact discontinuity x = ϒ ( t ) , which has the following form:

( p , u , S ) ( x , t ) = ( p − , u − , S − ) ( x , t ) , ( x , t ) ∈ D − = { ( x , t ) ∣ B ( t ) ≤ x < ϒ ( t ) , t ∈ [ 0 , T 0 ] } , ( p + , u + , S + ) ( x , t ) , ( x , t ) ∈ D + = { ( x , t ) ∣ x > ϒ ( t ) , t ∈ [ 0 , T 0 ] } ,

and satisfies

(1.11) ‖ ( p − , u − ) ( x , t ) − ( p ̲ − , u ̲ − ) ‖ C 1 ( D − ) + ‖ ( p + , u + ) ( x , t ) − ( p ̲ + , u ̲ + ) ‖ C 1 ( D + ) + ‖ S − ( x , t ) − S ̲ − ‖ C 2 ( D − ) + ‖ S + ( x , t ) − S ̲ + ‖ C 2 ( D + ) ≤ C S ε .

Moreover, the contact discontinuity x = ϒ ( t ) is a stream line, which satisfies

(1.12) ‖ ϒ ( t ) − υ ( t ) ‖ C 2 ( [ 0 , T 0 ] ) ≤ C S ε ,

and ϒ ( 0 ) = x 0 .

Remark 1.1

This Theorem only considers the tube that is closed at one end by a piston and open at the other end. In fact, from the proof of Theorem 1.1, similar results can be obtained for the tube that is closed by two pistons, which move statically relatively to the internal gas.

The rest of the study is organized as follows. In Section 2, we introduce the Lagrangian coordinates transformation to reformulate this problem as an initial-boundary problem with fixed boundary. Then, we make a series of transformations to turn this system into equations of diagonalized form. In Section 3, we establish a linearized iteration scheme and employ the fixed point theorem to prove Theorem 1.1.

2 Mathematical formulation

2.1 Mathematical problem in the Lagrangian coordinates

Given that the gradient of the contact discontinuity x = ϒ ( t ) coincides with the flow velocity on either side of x = ϒ ( t ) , and the velocity B ′ ( t ) of the piston is equal to the flow velocity at the piston curve x = B ( t ) , it is expedient to employ the Lagrangian coordinates transformation to transform the free boundary x = ϒ ( t ) into a fixed one.

By the conservation of mass, i.e., (1.1)₁, for t ∈ [ 0 , T 0 ] , we have

(2.1) ∫ B ( t ) ϒ ( t ) ρ ( x , t ) d x = F − ,

where

F − = ∫ B ( 0 ) x 0 ρ ( x , 0 ) d x = ∫ B ( 0 ) x 0 p 0 , − 1 γ φ − 1 γ ( S 0 , − ) ( x ) d x .

Let

y ( x , t ) = ∫ B ( t ) x ρ ( s , t ) d s − F − ,

then it follows from (1.1)₁

(2.2) ∂ y ∂ x = ρ , ∂ y ∂ t = − ρ u .

Now, we introduce the Lagrangian coordinates transformation as

(2.3) y = y ( x , t ) , τ = t .

By (2.2), we obtain

∂ ( y , τ ) ∂ ( x , t ) = ρ − ρ u 0 1 ,

thus the Lagrangian coordinates transformation is invertible if and only if ρ ≠ 0 .

Under the coordinates transformation (2.3), the domain D becomes

D ˆ = { ( y , τ ) ∣ y ≥ F − , τ ∈ [ 0 , T 0 ] } .

By (2.1), we obtain

y ( ϒ ( t ) , t ) = ∫ B ( t ) ϒ ( t ) ρ ( x , t ) d x − F − = 0 ,

so the free boundary x = ϒ ( t ) is turned into the fixed straight line y = 0 . Furthermore, the domain D ˆ is divided into

D ˆ − = { ( y , τ ) ∣ − F − ≤ y < 0 , τ ∈ [ 0 , T 0 ] } , D ˆ + = { ( y , τ ) ∣ y > 0 , τ ∈ [ 0 , T 0 ] } .

Denote

( p ˆ , u ˆ , S ˆ ) ( y , τ ) = ( p ˆ − , u ˆ − , S ˆ − ) ( y , τ ) , ( y , τ ) ∈ D ˆ − , ( p ˆ + , u ˆ + , S ˆ + ) ( y , τ ) , ( y , τ ) ∈ D ˆ + ,

is the piecewise smooth solution in the domain D ˆ .

By (2.2) and (2.3), we obtain

∂ x = ρ ∂ y , ∂ t = ∂ τ − ρ u ∂ y ,

then equation (1.4) in the Lagrangian coordinates become

(2.4) ∂ τ p ˆ ι + γ p ˆ ι 1 γ + 1 φ − 1 γ ( S ˆ ι ) ∂ y u ˆ ι = 0 , ∂ τ u ˆ ι + ∂ y p ˆ ι = 0 , ∂ τ S ˆ ι = 0 ,

with ι = + , − . The initial data (1.5) and the boundary conditions (1.7) and (1.8) become

(2.5) τ = 0 : ( p ˆ , u ˆ , S ˆ ) ( y , 0 ) = ( p ˆ 0 , u ˆ 0 , S ˆ 0 ) ( y ) = ( p ˆ 0 , − , u ˆ 0 , − , S ˆ 0 , − ) ( y ) , − F − ≤ y < 0 , ( p ˆ 0 , + , u ˆ 0 , + , S ˆ 0 , + ) ( y ) , y > 0 ,

(2.6) y = − F − : u ˆ − ( − F − , τ ) = B ′ ( τ ) ,

(2.7) y = 0 : u ˆ − ( 0 , τ ) = u ˆ + ( 0 , τ ) = ϒ ′ ( τ ) , p ˆ − ( 0 , τ ) = p ˆ + ( 0 , τ ) .

In the same way, the background solution in the new coordinates corresponding to (1.9) is

(2.8) ( p ˆ ̲ , u ˆ ̲ , S ˆ ̲ ) = ( p ̲ − , u ̲ − , S ̲ − ) , ( y , τ ) ∈ [ − F ̲ − , 0 ) × [ 0 , T 0 ] , ( p ̲ + , u ̲ + , S ̲ + ) , ( y , τ ) ∈ ( 0 , + ∞ ) × [ 0 , T 0 ] ,

where F ̲ − = ρ ̲ − .

Therefore, the initial-boundary problems (1.4)–(1.8) with the free boundary in the Eulerian coordinates is reformulated as the initial-boundary problems (2.4)–(2.7) with the fixed boundary in the Lagrangian coordinates. Theorem 1.1 can be rephrased as follows.

Theorem 2.1

For any given T 0 > 0 , there exist positive constants ε ˆ 0 , C ˆ S , such that for any given ε ∈ ( 0 , ε ˆ 0 ) , if

(2.9) ‖ ( p ˆ 0 , − , u ˆ 0 , − ) ( y ) − ( p ̲ − , u ̲ − ) ‖ C 1 ( [ − F − , 0 ) ) + ‖ ( p ˆ 0 , + , u ˆ 0 , + ) ( y ) − ( p ̲ + , u ̲ + ) ‖ C 1 ( ( 0 , + ∞ ) ) + ‖ S ˆ 0 , − ( y ) − S ̲ − ‖ C 2 ( [ − F − , 0 ) ) + ‖ S ˆ 0 , + ( y ) − S ̲ + ‖ C 2 ( ( 0 , + ∞ ) ) + ‖ B ′ ( τ ) − u ̲ ‖ C 1 ( [ 0 , T 0 ] ) ≤ ε ,

then the IBVPs (2.4)–(2.7) admit a unique piecewise smooth solution ( p ˆ , u ˆ , S ˆ ) ( y , τ ) with the contact discontinuity y = 0 , which satisfies

(2.10) ‖ ( p ˆ − , u ˆ − ) ( y , τ ) − ( p ̲ − , u ̲ − ) ‖ C 1 ( D ˆ − ) + ‖ ( p ˆ + , u ˆ + ) ( y , τ ) − ( p ̲ + , u ̲ + ) ‖ C 1 ( D ˆ + ) + ‖ S ˆ − ( y , τ ) − S ̲ − ‖ C 2 ( D ˆ − ) + ‖ S ˆ + ( y , τ ) − S ̲ + ‖ C 2 ( D ˆ + ) ≤ C ˆ S ε .

Remark 2.1

Under the Lagrangian coordinates transformation (2.3), Theorems 1.1 and 2.1 are equivalent. Since for small enough ε , one has

det ∂ ( y , τ ) ∂ ( x , t ) = ρ ≠ 0 ,

the inverse Lagrangian coordinates transformation exists and can be expressed as follows:

x = B ( τ ) + ∫ − F − y 1 ρ ( s , τ ) d s , t = τ .

Then, if Theorem 2.1 holds, we define ( p , u , S ) ( x , t ) = ( p ˆ , u ˆ , S ˆ ) ( y ( x , t ) , τ ( t ) ) and

ϒ ( t ) = B ( t ) + ∫ − F − 0 1 ρ ( s , t ) d s .

It follows from (2.4) that

ϒ ′ ( t ) = u ( ϒ ( t ) , t )

and then by (1.10) and (1.11), we obtain ϒ ( t ) ∈ C 2 [ 0 , T 0 ] . Thus, we deduce that Theorem 1.1 also holds.

2.2 Derivation of equivalent system

In this subsection, we make some transformations to diagonalize system (2.4). To be specific, we first write (2.4) as the equations about perturbed variables, then utilize the eigen decomposition method to transform it into the diagonal form. By (2.4)₃, we deduce S ˆ − = S ˆ 0 , − and S ˆ + = S ˆ 0 , + , which means we only require solving equations (2.4)₁ and (2.4)₂.

Let

p ¯ ι = p ˆ ι − p ̲ , u ¯ ι = u ˆ ι − u ̲ , ϕ ¯ ι = ( p ¯ ι , u ¯ ι ) , ι = + , − ,

then equations (2.4)₁ and (2.4)₂ are rewritten as

(2.11) ∂ τ ϕ ¯ ι + G ¯ ( y , ϕ ¯ ι ) ∂ y ϕ ¯ ι = 0 ,

where

G ¯ ( y , ϕ ¯ ι ) = 0 γ ( p ¯ ι + p ̲ ) 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) 1 0 .

Moreover, there exists an invertible matrix

Q ( y ) = 1 − γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) 1 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) ,

such that Q ( y ) G ¯ ( y , 0 ) Q − 1 ( y ) is a diagonal matrix. If we let

ϕ ι = ( ϕ 1 , ι , ϕ 2 , ι ) ⊤ = Q ( y ) ϕ ¯ ι , ι = + , − ,

then (2.11) becomes

(2.12) ∂ τ ϕ ι + G ( y , ϕ ι ) ∂ y ϕ ι = f ( y , ϕ ι ) ,

where

G ( y , ϕ ι ) = Q ( y ) G ¯ ( y , Q − 1 ( y ) ϕ ι ) Q − 1 ( y ) , f ( y , ϕ ι ) = G ( y , ϕ ι ) Q ′ ( y ) Q − 1 ( y ) ϕ ι .

Furthermore, the eigenvalues of the matrix G ( y , ϕ ι ) are

λ 1 = λ 1 ( y , ϕ ι ) = − γ ϕ 1 , ι + ϕ 2 , ι 2 + p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) , λ 2 = λ 2 ( y , ϕ ι ) = γ ϕ 1 , ι + ϕ 2 , ι 2 + p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) ,

and the left eigenvectors are

l 1 ( y , ϕ ι ) = ( 1 , h ( y , ϕ ι ) ) , l 2 ( y , ϕ ι ) = ( h ( y , ϕ ι ) , 1 ) ,

where

h = h ( y , ϕ ι ) = γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) − γ ϕ 1 , ι + ϕ 2 , ι 2 + p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) + γ ϕ 1 , ι + ϕ 2 , ι 2 + p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) ,

with

(2.13) h ( y , 0 ) = 0 .

Then, by pre-multiplying both sides of system (2.12) with the eigenvector l ( y , ϕ ι ) = ( l 1 , l 2 ) ⊤ ( y , ϕ ι ) , we obtain

(2.14) ∂ ϕ 1 , ι ∂ τ + λ 1 ( y , ϕ ι ) ∂ ϕ 1 , ι ∂ y = − h ( y , ϕ ι ) ∂ ϕ 2 , ι ∂ τ + λ 1 ( y , ϕ ι ) ∂ ϕ 2 , ι ∂ y + g ( y , ϕ ι ) ,

(2.15) ∂ ϕ 2 , ι ∂ τ + λ 2 ( y , ϕ ι ) ∂ ϕ 2 , ι ∂ y = − h ( y , ϕ ι ) ∂ ϕ 1 , ι ∂ τ + λ 2 ( y , ϕ ι ) ∂ ϕ 1 , ι ∂ y + g ( y , ϕ ι ) ,

where

(2.16) g = g ( y , ϕ ι ) = ( 1 − h ( y , ϕ ι ) ) γ ϕ 1 , ι + ϕ 2 , ι 2 + p ̲ 1 γ + 1 φ − 1 + 2 γ γ ( S ˆ 0 , ι ( y ) ) φ ′ ( S ˆ 0 , ι ( y ) ) S ˆ ′ 0 , ι ( y ) 4 γ ( ϕ 1 , ι − ϕ 2 , ι ) .

Using (2.5)–(2.7), we obtain that the initial data and the boundary conditions of systems (2.14) and (2.15) are

(2.17) τ = 0 : ϕ 1 , ι ( y , 0 ) = ϕ 1 0 , ι ( y ) = p ˆ 0 , ι − p ̲ − γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) ( u ˆ 0 , ι ( y ) − u ̲ ) , ϕ 2 , ι ( y , 0 ) = ϕ 2 0 , ι ( y ) = p ˆ 0 , ι − p ̲ + γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , ι ( y ) ) ( u ˆ 0 , ι ( y ) − u ̲ ) ,

(2.18) y = − F − : ( ϕ 2 , − − ϕ 1 , − ) ( − F − , τ ) = 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) ( B ′ ( τ ) − u ̲ ) ,

(2.19) y = 0 : φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) ( ϕ 2 , − − ϕ 1 , − ) ( 0 , τ ) = φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) ( ϕ 2 , + − ϕ 1 , + ) ( 0 , τ ) , ( ϕ 2 , − + ϕ 1 , − ) ( 0 , τ ) = ( ϕ 2 , + + ϕ 1 , + ) ( 0 , τ ) .

In the following proofs, we focus on solving IBVPs (2.14)–(2.19) and prove the following theorem.

Theorem 2.2

For any given T 0 > 0 , there exist constants ε 1 ∈ ( 0 , ε ˆ 0 ) , C 0 > 0 , C s > 0 , such that for any given ε ∈ ( 0 , ε 1 ) , if

(2.20) ‖ ( ϕ 1 0 , − , ϕ 2 0 , − ) ( y ) ‖ C 1 ( [ − F − , 0 ) ) + ‖ ( ϕ 1 0 , + , ϕ 2 0 , + ) ( y ) ‖ C 1 ( ( 0 , + ∞ ) ) + ‖ S ˆ 0 , − ( y ) − S ̲ − ‖ C 2 ( [ − F − , 0 ) ) + ‖ S ˆ 0 , + ( y ) − S ̲ + ‖ C 2 ( ( 0 , + ∞ ) ) + ‖ B ′ ( τ ) − u ̲ ‖ C 1 ( [ 0 , T 0 ] ) ≤ C 0 ε ,

then IBVPs (2.14)–(2.19) admit a unique piecewise smooth solution

ϕ ( y , τ ) = ϕ − ( y , τ ) = ( ϕ 1 , − , ϕ 2 , − ) ⊤ ( y , τ ) , ( y , τ ) ∈ D ˆ − , ϕ + ( y , τ ) = ( ϕ 1 , + , ϕ 2 , + ) ⊤ ( y , τ ) , ( y , τ ) ∈ D ˆ + ,

with the contact discontinuity y = 0 , which satisfies

(2.21) ‖ ϕ − ( y , τ ) ‖ C 1 ( D ˆ − ) + ‖ ϕ + ( y , τ ) ‖ C 1 ( D ˆ + ) ≤ C s ε .

3 Existence and stability of the contact discontinuity

In this section, we mainly prove the existence and stability of the contact discontinuity to IBVPs (2.14)–(2.19) (i.e., Theorem 2.2). After proving Theorem 2.2, Theorem 2.1 follows from the definition of ϕ ( y , τ ) , and then with the aid of the inverse Lagrangian coordinates transformation, we obtain Theorem 1.1.

First, we establish the linearized iteration scheme of (2.14), (2.15), and (2.17)–(2.19). For n ∈ N + , ι = + , − ,

(3.1) ∂ ϕ 1 , ι ( n ) ∂ τ + λ 1 ( y , ϕ ι ( n − 1 ) ) ∂ ϕ 1 , ι ( n ) ∂ y = − h ( y , ϕ ι ( n − 1 ) ) ∂ ϕ 2 , ι ( n − 1 ) ∂ τ + λ 1 ( y , ϕ ι ( n − 1 ) ) ∂ ϕ 2 , ι ( n − 1 ) ∂ y + g ( y , ϕ ι ( n − 1 ) ) ,

(3.2) ∂ ϕ 2 , ι ( n ) ∂ τ + λ 2 ( y , ϕ ι ( n − 1 ) ) ∂ ϕ 2 , ι ( n ) ∂ y = − h ( y , ϕ ι ( n − 1 ) ) ∂ ϕ 1 , ι ( n − 1 ) ∂ τ + λ 2 ( y , ϕ ι ( n − 1 ) ) ∂ ϕ 1 , ι ( n − 1 ) ∂ y + g ( y , ϕ ι ( n − 1 ) ) ,

(3.3) τ = 0 : ( ϕ 1 , ι ( n ) , ϕ 2 , ι ( n ) ) ( y , 0 ) = ( ϕ 1 0 , ι , ϕ 2 0 , ι ) ( y ) ,

(3.4) y = − F − : ( ϕ 2 , − ( n ) − ϕ 1 , − ( n ) ) ( − F − , τ ) = 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) ( B ′ ( τ ) − u ̲ ) ,

(3.5) y = 0 : φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) ( ϕ 2 , − ( n ) − ϕ 1 , − ( n ) ) ( 0 , τ ) = φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) ( ϕ 2 , + ( n ) − ϕ 1 , + ( n ) ) ( 0 , τ ) , ( ϕ 2 , − ( n ) + ϕ 1 , − ( n ) ) ( 0 , τ ) = ( ϕ 2 , + ( n ) + ϕ 1 , + ( n ) ) ( 0 , τ ) .

We start to iterate from

(3.6) ϕ ι ( 0 ) = ( ϕ 1 , ι ( 0 ) , ϕ 2 , ι ( 0 ) ) = ( 0 , 0 ) , ι = + , − .

Remark 3.1

In this section, the proof of Theorem 2.2 is established through the application of the Banach fixed point theorem, which requires the imposition of smallness conditions in the C 2 norm on both the initial data and the piston velocity. However, if we employ the Cauchy sequence and Arzelà-Ascoli theorem (cf. [7,16]), we only need the smallness of the C 1 norm of the initial data and piston velocity to prove Theorem 2.2. We will not go into detail here.

Define the iteration set

D δ = { ( ϕ − ( n ) , ϕ + ( n ) ) : ‖ ϕ − ( n ) ‖ C 2 ( D ˆ − ) + ‖ ϕ + ( n ) ‖ C 2 ( D ˆ + ) ≤ δ } ,

with some δ > 0 . Then, we introduce the map

ℳ : D C * ε → D C * ε .

For given ( ϕ − ( n − 1 ) , ϕ + ( n − 1 ) ) ∈ D C * ε , we solve the linearized IBVPs (3.1)–(3.5) to obtain ( ϕ − ( n ) , ϕ + ( n ) ) ∈ D C * ε . Then, the map ℳ is defined such that

(3.7) ( ϕ − ( n ) , ϕ + ( n ) ) = ℳ ( ϕ − ( n − 1 ) , ϕ + ( n − 1 ) ) .

Next we shall prove that for the map ℳ , the following facts hold:

ℳ is well-defined, i.e., ℳ exists and maps from D C * ε to itself;
ℳ is a contraction map.

Once the above facts are proved, the convergence follows from the Banach fixed point theorem, and one can obtain that the limit is the solution of IBVPs (2.14)–(2.19). The first fact is proved in the following Proposition.

Proposition 3.1

For any given T 0 > 0 , there exist positive constants C * and ε 1 , such that for any given ε ∈ ( 0 , ε 1 ) , if ( ϕ − ( n − 1 ) , ϕ + ( n − 1 ) ) ∈ D C * ε and

(3.8) ‖ ( ϕ 1 0 , − , ϕ 2 0 , − ) ( y ) ‖ C 2 ( [ − F − , 0 ) ) + ‖ ( ϕ 1 0 , + , ϕ 2 0 , + ) ( y ) ‖ C 2 ( ( 0 , + ∞ ) ) + ‖ S ˆ 0 , − ( y ) − S ̲ − ‖ C 3 ( [ − F − , 0 ) ) + ‖ S ˆ 0 , + ( y ) − S ̲ + ‖ C 3 ( ( 0 , + ∞ ) ) + ‖ B ′ ( τ ) − u ̲ ‖ C 2 ( [ 0 , T 0 ] ) ≤ C 0 ε ,

then the solution ( ϕ − ( n ) , ϕ + ( n ) ) to the linearized IBVPs (3.1)–(3.5) satisfies

(3.9) ‖ ϕ − ( n ) ‖ C 2 ( D ˆ − ) + ‖ ϕ + ( n ) ‖ C 2 ( D ˆ + ) ≤ C * ε .

Remark 3.2

In Proposition 3.1, we demand the initial entropy satisfying a small condition in the C 3 norm. This is because, after performing a transformation on the equations, the non-homogeneous terms in the equations contain the first-order derivatives of the initial entropy. Consequently, when establishing C 2 estimates of the solution, we need to control the C 3 norm of the initial entropy.

To prove the Proposition 3.1, first, we divide the domains D ˆ − and D ˆ + into several different subdomains. As shown in Figure 1, let D 1 be bounded by the initial τ = 0 , the characteristics of (3.1) with ι = − corresponding to λ 1 starting from ( 0 , 0 ) (denoted by ξ 1 , − ) and the characteristics of (3.2) with ι = − corresponding to λ 2 starting from ( − F − , 0 ) (denoted by ξ 2 , − ). Assume ξ 1 , − intersects with y = − F − at the point ( − F − , τ 1 , − * ) and ξ 2 , − intersects with y = 0 at the point ( 0 , τ 1,0 * ) , and denote τ 1 * = min { τ 1 , − * , τ 1,0 * } . Let D 2 be bounded by the initial τ = 0 , the straight line τ = τ 1 * and the characteristics of (3.2) with ι = + corresponding to λ 2 starting from ( 0 , 0 ) (denoted by ξ 2 , + ), D 3 be bounded by the boundary y = − F − , ξ 1 , − and ξ 2 , − , D 4 be bounded by the boundary y = 0 , ξ 1 , − and ξ 2 , − , D 5 be bounded by ξ 1 , − , ξ 2 , − and the straight line τ = τ 1 * , and D 6 be bounded by the boundary y = 0 , ξ 2 , + and the straight line τ = τ 1 * . Let ( y 1 * , τ 1 * ) be the intersection point of ξ 2 , + and the straight line τ = τ 1 * with y 1 * ∈ [ 0 , + ∞ ) .

$Figure 1 Division of regions D ˆ − {\hat{D}}_{-} and D ˆ + {\hat{D}}_{+} into several sub-regions by boundaries and characteristics.$

Figure 1

Division of regions D ˆ − and D ˆ + into several sub-regions by boundaries and characteristics.

Next we will proceed with the proof in three steps.

Step 1: The estimates of ϕ 1 , − ( n ) in D 1 ∪ D 3 , ϕ 2 , − ( n ) in D 1 ∪ D 4 , ϕ 1 , + ( n ) in D 2 ∪ D 6 and ϕ 2 , + ( n ) in D 2 .

For y 0 ∈ [ − F − , + ∞ ) , define y = Y 1 , ι ( n ) ( τ , y 0 ) and y = Y 2 , ι ( n ) ( τ , y 0 ) as the characteristic curves of ϕ 1 , ι ( n ) and ϕ 2 , ι ( n ) , respectively, i.e.,

(3.10) d Y 1 , ι ( n ) d τ ( τ , y 0 ) = λ 1 ( Y 1 , ι ( n ) , ϕ ι ( n − 1 ) ( Y 1 , ι ( n ) , τ ) ) , Y 1 , ι ( n ) ( 0 , y 0 ) = y 0 , ι = + , − ,

(3.11) d Y 2 , ι ( n ) d τ ( τ , y 0 ) = λ 2 ( Y 2 , ι ( n ) , ϕ ι ( n − 1 ) ( Y 2 , ι ( n ) , τ ) ) , Y 2 , ι ( n ) ( 0 , y 0 ) = y 0 . ι = + , − ,

Then, along the characteristic curves (3.10) and (3.11), one has

(3.12) y = y 0 + ∫ 0 τ λ i ( Y i , ι ( n ) ( s , y 0 ) , ϕ ι ( n − 1 ) ( Y i , ι ( n ) ( s , y 0 ) , s ) ) d s , i = 1 , 2 , ι = + , − .

By (3.12), for any given point ( y , τ ) ∈ D ˆ − ∪ D ˆ + , there exists a unique y 0 i such that the characteristic curve corresponding to λ i issuing from ( 0 , y 0 i ) respectively passes through ( y , τ ) . Thus, we can regard y 0 i as a function of ( y , τ ) in D ˆ − ∪ D ˆ + . For simplicity of notation, we write y 0 i as y 0 . Moreover, for the sake of further proof, we provide the estimates of y 0 .

Lemma 3.1

For any ( ϕ − ( n − 1 ) , ϕ + ( n − 1 ) ) ∈ D C * ε , there exists a constant M 0 > 0 , such that

(3.13) ‖ ( ∂ y y 0 , ∂ τ y 0 ) ‖ C 0 ( D ˆ − ∪ D ˆ + ) + ‖ ( ∂ y y 2 y 0 , ∂ y τ 2 y 0 , ∂ τ τ 2 y 0 ) ‖ C 0 ( D ˆ − ∪ D ˆ + ) ≤ M 0 .

Proof

Without loss of generality, we only consider the estimates of (3.13) in D 1 , since otherwise we may need to consider the reflection of characteristics by the boundary, which can be uniformly bounded by a constant. The discussion of the reflection of the characteristics by the boundary is deferred to Lemma 3.2.

Let

λ i ( Y i , ι ( n ) ( s , y 0 ) , ϕ ι ( n − 1 ) ( Y i , ι ( n ) ( s , y 0 ) , s ) ) = λ i ( s , y 0 ) ,

then taking derivatives on (3.12) with respect to y , τ , we obtain the formulas of the first- and second-order partial derivatives of y 0 with respect to y , τ , for example,

∂ y 0 ∂ y = 1 1 + ∫ 0 τ ∂ λ i ∂ y 0 ( s , y 0 ) d s ,

∂ 2 y 0 ∂ y 2 = − 1 1 + ∫ 0 τ ∂ λ i ∂ y 0 ( s , y 0 ) d s 2 ∫ 0 τ ∂ 2 λ i ∂ y 0 2 ( s , y 0 ) d s ∂ y 0 ∂ y .

Other derivatives can also be written similarly. From these formulas, we can derive the estimates for ‖ ( ∂ y y 0 , ∂ τ y 0 ) ‖ C 0 ( D ˆ − ∪ D ˆ + ) and ‖ ( ∂ y y 2 y 0 , ∂ y τ 2 y 0 , ∂ τ τ 2 y 0 ) ‖ C 0 ( D ˆ − ∪ D ˆ + ) . Thus, we complete the proof of Lemma 3.1.□

For any given point ( y 0 , 0 ) with y 0 ∈ [ − F − , 0 ] , we integrate (3.1) along the characteristics y = Y 1 , − ( n ) ( τ , y 0 ) to obtain

ϕ 1 , − ( n ) ( y , τ ) = ϕ 1 0 , − ( y 0 ) + ∫ 0 τ − h ( Y 1 , − ( n ) ( s , y 0 ) , ϕ − ( n − 1 ) ) ∂ ϕ 2 , − ( n − 1 ) ∂ τ + λ 1 ( Y 1 , − ( n ) ( s , y 0 ) , ϕ − ( n − 1 ) ) ∂ ϕ 2 , − ( n − 1 ) ∂ y + g ( Y 1 , − ( n ) ( s , y 0 ) , ϕ − ( n − 1 ) ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) d s .

By (2.13), (2.16), and (3.8), we have

(3.14) ‖ ϕ 1 , − ( n ) ‖ C 0 ( D 1 ∪ D 3 ) ≤ C 0 ε + C ε 2 ,

where C > 0 is a generic constant, which is different in different places. Here we used ϕ − ( n − 1 ) ∈ D C * ε and (2.16), (3.8) to deduce

(3.15) ‖ g ( y , ϕ − ( n − 1 ) ) ‖ C 2 ( D ˆ − ) ≤ C ε 2 .

Differentiating equation (3.1) with respect to y and τ , we have

(3.16) ∂ ∂ τ ( ∂ y ϕ 1 , − ( n ) ) + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ y ϕ 1 , − ( n ) ) = − ∂ λ 1 ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y ϕ 1 , − ( n ) − ∂ h ∂ y + ∑ i = 1 2 ∂ h ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ ϕ 2 , − ( n − 1 ) ∂ τ + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ϕ 2 , − ( n − 1 ) ∂ y − h ( y , ϕ − ( n − 1 ) ) ∂ ∂ τ ( ∂ y ϕ 2 , − ( n − 1 ) ) + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ y ϕ 2 , − ( n − 1 ) ) − h ( y , ϕ − ( n − 1 ) ) ∂ λ 1 ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y ϕ 2 , − ( n − 1 ) + ∂ g ∂ y + ∑ i = 1 2 ∂ g ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y = △ − ∂ λ 1 ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y ϕ 1 , − ( n ) + Ξ 1 , − y , ( n − 1 ) ,

(3.17) ∂ ∂ τ ( ∂ τ ϕ 1 , − ( n ) ) + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ τ ϕ 1 , − ( n ) ) = − ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y ϕ 1 , − ( n ) − ∑ i = 1 2 ∂ h ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ ϕ 2 , − ( n − 1 ) ∂ τ + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ϕ 2 , − ( n − 1 ) ∂ y − h ( y , ϕ − ( n − 1 ) ) ∂ ∂ τ ( ∂ τ ϕ 2 , − ( n − 1 ) ) + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ τ ϕ 2 , − ( n − 1 ) ) − h ( y , ϕ − ( n − 1 ) ) ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y ϕ 2 , − ( n − 1 ) + ∑ i = 1 2 ∂ g ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ = △ − ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y ϕ 1 , − ( n ) + Ξ 1 , − τ , ( n − 1 ) .

Then, integrating (3.16) and (3.17) along the characteristics y = Y 1 , − ( n ) ( τ , y 0 ) , one has

(3.18) ∂ y ϕ 1 , − ( n ) ( y , τ ) = ∂ y ϕ 1 0 , − ( y 0 ) + ∫ 0 τ − ∂ λ 1 ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y ϕ 1 , − ( n ) + Ξ 1 , − y , ( n − 1 ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) d s = ∂ y 0 ∂ y ∂ y 0 ϕ 1 0 , − ( y 0 ) + ∫ 0 τ − ∂ λ 1 ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y ϕ 1 , − ( n ) + Ξ 1 , − y , ( n − 1 ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) d s ,

(3.19) ∂ τ ϕ 1 , − ( n ) ( y , τ ) = ∂ τ ϕ 1 0 , − ( y 0 ) + ∫ 0 τ − ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y ϕ 1 , − ( n ) + Ξ 1 , − τ , ( n − 1 ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) d s = ∂ y 0 ∂ τ ∂ y 0 ϕ 1 0 , − ( y 0 ) + ∫ 0 τ − ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y ϕ 1 , − ( n ) + Ξ 1 , − τ , ( n − 1 ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) d s .

By (2.13), (3.8), (3.13), (3.15), (3.18), and (3.19), we have

∣ ∂ y ϕ 1 , − ( n ) ( y , τ ) ∣ ≤ M 0 C 0 ε + C ε ∫ 0 τ ∣ ∂ y ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ d s + C ε 2 , ∣ ∂ τ ϕ 1 , − ( n ) ( y , τ ) ∣ ≤ M 0 C 0 ε + C ε ∫ 0 τ ∣ ∂ y ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ d s + C ε 2 .

Then, it follows from the Gronwall inequality that

(3.20) ‖ ∂ y ϕ 1 , − ( n ) ‖ C 0 ( D 1 ∪ D 3 ) + ‖ ∂ τ ϕ 1 , − ( n ) ‖ C 0 ( D 1 ∪ D 3 ) ≤ ( 2 M 0 C 0 ε + C ε 2 ) e C ε τ 1 * ≤ 2 ( 1 + C ε ) M 0 C 0 ε .

We differentiate equations (3.16) and (3.17) with respect to y and τ again to obtain

∂ ∂ τ ( ∂ y y 2 ϕ 1 , − ( n ) ) + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ y y 2 ϕ 1 , − ( n ) ) = − 2 ∂ λ 1 ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y y 2 ϕ 1 , − ( n ) − ∂ 2 λ 1 ∂ y 2 + 2 ∑ i = 1 2 ∂ 2 λ 1 ∂ y ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y + ∑ i , j = 1 2 ∂ 2 λ 1 ∂ ϕ i , − ∂ ϕ j , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ ϕ j , − ( n − 1 ) ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ 2 ϕ i , − ( n − 1 ) ∂ y 2 ∂ y ϕ 1 , − ( n ) + ∂ y Ξ 1 , − y , ( n − 1 ) ,

∂ ∂ τ ( ∂ y τ 2 ϕ 1 , − ( n ) ) + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ y τ 2 ϕ 1 , − ( n ) ) = − ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y y 2 ϕ 1 , − ( n ) − ∂ λ 1 ∂ y + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y τ 2 ϕ 1 , − ( n ) − ∑ i = 1 2 ∂ 2 λ 1 ∂ y ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ + ∑ i , j = 1 2 ∂ 2 λ 1 ∂ ϕ i , − ∂ ϕ j , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ ϕ j , − ( n − 1 ) ∂ τ + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ 2 ϕ i , − ( n − 1 ) ∂ y ∂ τ ∂ y ϕ 1 , − ( n ) + ∂ τ Ξ 1 , − y , ( n − 1 ) ,

∂ ∂ τ ( ∂ τ τ 2 ϕ 1 , − ( n ) ) + λ 1 ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ τ τ 2 ϕ 1 , − ( n ) ) = − 2 ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y τ 2 ϕ 1 , − ( n ) − ∑ i , j = 1 2 ∂ 2 λ 1 ∂ ϕ i , − ∂ ϕ j , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ ϕ j , − ( n − 1 ) ∂ τ + ∑ i = 1 2 ∂ λ 1 ∂ ϕ i , − ∂ 2 ϕ i , − ( n − 1 ) ∂ τ 2 ∂ y ϕ 1 , − ( n ) + ∂ τ Ξ 1 , − τ , ( n − 1 ) .

Then, integrating them along the characteristics y = Y 1 , − ( n ) ( τ , y 0 ) and by (2.13), (3.8), (3.13), (3.15), (3.20), we obtain

∣ ∂ y y 2 ϕ 1 , − ( n ) ( y , τ ) ∣ ≤ ∣ ∂ y y 2 ϕ 1 0 , − ( y 0 ) ∣ + C ε ∫ 0 τ ∣ ∂ y y 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ d s + C ε 2 ≤ ∂ y 0 ∂ y 2 ‖ ∂ y 0 y 0 2 ϕ 1 0 , − ‖ C 0 ( [ − F − , 0 ] ) + ∂ 2 y 0 ∂ y 2 ‖ ∂ y 0 ϕ 1 0 , − ‖ C 0 ( [ − F − , 0 ] ) + C ε ∫ 0 τ ∣ ∂ y y 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ d s + C ε 2 ≤ ( M 0 2 + M 0 ) C 0 ε + C ε ∫ 0 τ ∣ ∂ y y 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ d s + C ε 2 ,

∣ ∂ y τ 2 ϕ 1 , − ( n ) ( y , τ ) ∣ ≤ ∣ ∂ y τ 2 ϕ 1 0 , − ( y 0 ) ∣ + C ε ∫ 0 τ ( ∣ ∂ y y 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ + ∣ ∂ y τ 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ ) d s + C ε 2 ≤ ( M 0 2 + M 0 ) C 0 ε + C ε ∫ 0 τ ( ∣ ∂ y y 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ + ∣ ∂ y τ 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ ) d s + C ε 2 ,

∣ ∂ τ τ 2 ϕ 1 , − ( n ) ( y , τ ) ∣ ≤ ∣ ∂ τ τ 2 ϕ 1 0 , − ( n ) ( y 0 ) ∣ + C ε ∫ 0 τ ∣ ∂ y τ 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ d s + C ε 2 ≤ ( M 0 2 + M 0 ) C 0 ε + C ε ∫ 0 τ ∣ ∂ y τ 2 ϕ 1 , − ( n ) ( Y 1 , − ( n ) ( s , y 0 ) , s ) ∣ d s + C ε 2 .

Furthermore, applying the Gronwall inequality, one has

(3.21) ‖ ∂ y y 2 ϕ 1 , − ( n ) ‖ C 0 ( D 1 ∪ D 3 ) + ‖ ∂ y τ 2 ϕ 1 , − ( n ) ‖ C 0 ( D 1 ∪ D 3 ) + ‖ ∂ τ τ 2 ϕ 1 , − ( n ) ‖ C 0 ( D 1 ∪ D 3 ) ≤ ( 3 ( M 0 2 + M 0 ) C 0 ε + C ε 2 ) e C ε τ 1 * ≤ 3 ( 1 + C ε ) ( M 0 2 + M 0 ) C 0 ε .

Thus, by (3.14), (3.20), and (3.21), we can deduce that there exists a constant C 1 > 0 , such that

(3.22) ‖ ϕ 1 , − ( n ) ‖ C 2 ( D 1 ∪ D 3 ) ≤ C 1 ε .

By imitating the proof process of estimate (3.22), we also obtain that there exist positive constants C 2 , C 3 , C 4 , such that

(3.23) ‖ ϕ 2 , − ( n ) ‖ C 2 ( D 1 ∪ D 4 ) ≤ C 2 ε ,

(3.24) ‖ ϕ 1 , + ( n ) ‖ C 2 ( D 2 ∪ D 6 ) ≤ C 3 ε ,

(3.25) ‖ ϕ 2 , + ( n ) ‖ C 2 ( D 2 ) ≤ C 4 ε .

Step 2: The estimate of ϕ 2 , − ( n ) in D 3 ∪ D 5 .

Let τ = T 2 , − ( n ) ( y , τ − ) be the characteristic curve of ϕ 2 , − ( n ) , i.e.,

(3.26) d T 2 , − ( n ) d y ( y , τ − ) = 1 λ 2 ( y , ϕ − ( n − 1 ) ( y , T 2 , − ( n ) ) ) , T 2 , − ( n ) ( − F − , τ − ) = τ − ,

where τ − ∈ [ 0 , τ 1 * ] . Then, by (3.26), we obtain

(3.27) τ = τ − + ∫ − F − y 1 λ 2 ( y , ϕ − ( n − 1 ) ( s , T 2 , − ( n ) ( s , τ − ) ) ) d s .

Furthermore, for any given point ( y , τ ) ∈ D 3 ∪ D 5 , there exists a unique τ − such that the characteristics corresponding to λ 2 passes through ( y , τ ) . Thus, we can regard τ − as a function of ( y , τ ) in D 3 ∪ D 5 and then obtain the following lemma.

Lemma 3.2

For any ( ϕ 1 , − ( n ) , ϕ 2 , − ( n ) ) ∈ D C * ε , there exists a constant M 1 > 0 , such that

(3.28) ‖ ( ∂ y τ − , ∂ τ τ − ) ‖ C 0 ( D 3 ∪ D 5 ) + ‖ ( ∂ y y 2 τ − , ∂ y τ 2 τ − , ∂ τ τ 2 τ − ) ‖ C 0 ( D 3 ∪ D 5 ) ≤ M 1 .

Proof

Similar to the proof of Lemma 3.1, taking derivatives on (3.27) with respect to y , τ , we obtain the formulas of the first- and second-order partial derivatives of τ − with respect to y , τ . Then, from these formulas, we deduce the estimates for ‖ ( ∂ y τ − , ∂ τ τ − ) ‖ C 0 ( D 3 ∪ D 5 ) and ‖ ( ∂ y y 2 τ − , ∂ y τ 2 τ − , ∂ τ τ 2 τ − ) ‖ C 0 ( D 3 ∪ D 5 ) . We complete the proof of Lemma 3.2.□

We multiply 1 λ 2 ( y , ϕ − ( n − 1 ) ) on both sides of equation (3.2) simultaneously to obtain

(3.29) ∂ ϕ 2 , − ( n ) ∂ y + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∂ ϕ 2 , − ( n ) ∂ τ = − h ( y , ϕ − ( n − 1 ) ) ∂ ϕ 1 , − ( n − 1 ) ∂ y + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∂ ϕ 1 , − ( n − 1 ) ∂ τ + 1 λ 2 ( y , ϕ − ( n − 1 ) ) g ( y , ϕ − ( n − 1 ) ) .

Then, integrating it along the characteristics τ = T 2 , − ( n ) ( y , τ − ) and by (2.13), (3.8), (3.15), and (3.22), one has

(3.30) ‖ ϕ 2 , − ( n ) ‖ C 0 ( D 3 ∪ D 5 ) ≤ C 1 ε + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) C 0 ε + C ε 2 .

We take the first derivative of (3.29) with respect to y and τ and integrate them along the characteristics τ = T 2 , − ( n ) ( y , τ − ) to obtain

∂ y ϕ 2 , − ( n ) ( y , τ ) = ∂ y ϕ 2 , − ( n ) ( − F − , τ − ) + ∫ − F − y − ∂ 1 λ 2 ∂ y + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ τ ϕ 2 , − ( n ) + Ξ 2 , − y , ( n − 1 ) ( s , T 2 , − ( n ) ( s , τ − ) ) d s ,

∂ τ ϕ 2 , − ( n ) ( y , τ ) = ∂ τ ϕ 2 , − ( n ) ( − F − , τ − ) + ∫ − F − y − ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ τ ϕ 2 , − ( n ) + Ξ 2 , − τ , ( n − 1 ) ( s , T 2 , − ( n ) ( s , τ − ) ) d s ,

where

Ξ 2 , − y , ( n − 1 ) = △ − ∂ h ∂ y + ∑ i = 1 2 ∂ h ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ ϕ 1 , − ( n − 1 ) ∂ y + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∂ ϕ 1 , − ( n − 1 ) ∂ τ − h ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ y ϕ 1 , − ( n − 1 ) ) + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∂ ∂ τ ( ∂ y ϕ 1 , − ( n − 1 ) ) − h ( y , ϕ − ( n − 1 ) ) ∂ 1 λ 2 ∂ y + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ τ ϕ 1 , − ( n − 1 ) + ∂ 1 λ 2 ∂ y + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y g ( y , ϕ − ( n − 1 ) ) + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∂ g ∂ y + ∑ i = 1 2 ∂ g ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ,

Ξ 2 , − τ , ( n − 1 ) = △ − ∑ i = 1 2 ∂ h ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ ϕ 1 , − ( n − 1 ) ∂ y + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∂ ϕ 1 , − ( n − 1 ) ∂ τ − h ( y , ϕ − ( n − 1 ) ) ∂ ∂ y ( ∂ τ ϕ 1 , − ( n − 1 ) ) + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∂ ∂ τ ( ∂ τ ϕ 1 , − ( n − 1 ) ) − h ( y , ϕ − ( n − 1 ) ) ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ τ ϕ 1 , − ( n − 1 ) + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ g ( y , ϕ − ( n − 1 ) ) + 1 λ 2 ( y , ϕ − ( n − 1 ) ) ∑ i = 1 2 ∂ g ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ .

At the boundary y = − F − , we have

∂ y ϕ 2 , − ( n ) ( − F − , τ − ) = ∂ τ − ∂ y ∂ τ − ϕ 2 , − ( n ) ( − F − , τ − ) = ∂ τ − ∂ y ∂ τ − ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ″ ( τ − ) , ∂ τ ϕ 2 , − ( n ) ( − F − , τ − ) = ∂ τ − ∂ τ ∂ τ − ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ″ ( τ − ) .

Then, by (2.13), (3.8), (3.15), (3.22), (3.28), and the Gronwall inequality, one has

(3.31) ‖ ∂ y ϕ 2 , − ( n ) ‖ C 0 ( D 3 ∪ D 5 ) + ‖ ∂ τ ϕ 2 , − ( n ) ‖ C 0 ( D 3 ∪ D 5 ) ≤ 2 M 1 ( C 1 ε + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) C 0 ε ) + C ε 2 e C ε F − ≤ 2 ( 1 + C ε ) M 1 C 1 ε + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) C 0 ε .

Taking the second derivatives with respect to y and τ on (3.29) and then integrating them along the characteristics τ = T 2 , − ( n ) ( y , τ − ) , we have

∂ y y 2 ϕ 2 , − ( n ) ( y , τ ) = ∂ y y 2 ϕ 2 , − ( n ) ( − F − , τ − ) + ∫ − F − y − 2 ∂ 1 λ 2 ∂ y + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ y τ 2 ϕ 2 , − ( n ) − ∂ 2 1 λ 2 ∂ y 2 + 2 ∑ i = 1 2 ∂ 2 1 λ 2 ∂ y ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y + ∑ i , j = 1 2 ∂ 2 1 λ 2 ∂ ϕ i , − ∂ ϕ j , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ ϕ j , − ( n − 1 ) ∂ y + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ 2 ϕ i , − ( n − 1 ) ∂ y 2 ∂ τ ϕ 2 , − ( n ) + ∂ y Ξ 2 , − y , ( n − 1 ) ( s , T 2 , − ( n ) ( s , τ − ) ) d s , ∂ y τ 2 ϕ 2 , − ( n ) ( y , τ ) = ∂ y τ 2 ϕ 2 , − ( n ) ( − F − , τ − ) + ∫ − F − y − ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ y τ 2 ϕ 2 , − ( n ) − ∂ 1 λ 2 ∂ y + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ τ τ 2 ϕ 2 , − ( n ) − ∑ i = 1 2 ∂ 2 1 λ 2 ∂ y ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ + ∑ i , j = 1 2 ∂ 2 1 λ 2 ∂ ϕ i , − ∂ ϕ j , − ∂ ϕ i , − ( n − 1 ) ∂ y ∂ ϕ j , − ( n − 1 ) ∂ τ + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ 2 ϕ i , − ( n − 1 ) ∂ y ∂ τ ∂ τ ϕ 2 , − ( n ) + ∂ τ Ξ 2 , − y , ( n − 1 ) ( s , T 2 , − ( n ) ( s , τ − ) ) d s ,

∂ τ τ 2 ϕ 2 , − ( n ) ( y , τ ) = ∂ τ τ 2 ϕ 2 , − ( n ) ( − F − , τ − ) + ∫ − F − y − 2 ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ τ τ 2 ϕ 2 , − ( n ) − ∑ i , j = 1 2 ∂ 2 1 λ 2 ∂ ϕ i , − ∂ ϕ j , − ∂ ϕ i , − ( n − 1 ) ∂ τ ∂ ϕ j , − ( n − 1 ) ∂ τ + ∑ i = 1 2 ∂ 1 λ 2 ∂ ϕ i , − ∂ 2 ϕ i , − ( n − 1 ) ∂ τ 2 ∂ τ ϕ 2 , − ( n ) + ∂ τ Ξ 2 , − τ , ( n − 1 ) ) ( s , T 2 , − ( n ) ( s , τ − ) ) d s .

By boundary condition (3.4), one has

∂ y y 2 ϕ 2 , − ( n ) ( − F − , τ − ) = ∂ 2 τ − ∂ y 2 ∂ τ − ϕ 2 , − ( n ) ( − F − , τ − ) + ∂ τ − ∂ y 2 ∂ τ − τ − 2 ϕ 2 , − ( n ) ( − F − , τ − ) = ∂ 2 τ − ∂ y 2 ∂ τ − ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ″ ( τ − ) + ∂ τ − ∂ y 2 ∂ τ − τ − 2 ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ‴ ( τ − ) ,

∂ y τ 2 ϕ 2 , − ( n ) ( − F − , τ − ) = ∂ 2 τ − ∂ y ∂ τ ∂ τ − ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ″ ( τ − ) + ∂ τ − ∂ y ∂ τ − ∂ τ ∂ τ − τ − 2 ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ‴ ( τ − ) , ∂ τ τ 2 ϕ 2 , − ( n ) ( − F − , τ − ) = ∂ 2 τ − ∂ τ 2 ∂ τ − ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ″ ( τ − ) + ∂ τ − ∂ τ 2 ∂ τ − τ − 2 ϕ 1 , − ( n ) ( − F − , τ − ) + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) B ‴ ( τ − ) .

Thus, it follows from (2.13), (3.8), (3.15), (3.22), (3.28), (3.31), and the Gronwall inequality

(3.32) ‖ ∂ y y 2 ϕ 2 , − ( n ) ‖ C 0 ( D 3 ∪ D 5 ) + ‖ ∂ y τ 2 ϕ 2 , − ( n ) ‖ C 0 ( D 3 ∪ D 5 ) + ‖ ∂ τ τ 2 ϕ 2 , − ( n ) ‖ C 0 ( D 3 ∪ D 5 ) ≤ 3 ( M 1 2 + M 1 ) C 1 ε + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) C 0 ε + C ε 2 e C ε F − ≤ 3 ( 1 + C ε ) ( M 1 2 + M 1 ) C 1 ε + 2 γ p ̲ 1 γ + 1 φ − 1 γ ( S ˆ 0 , − ( − F − ) ) C 0 ε .

Therefore, by (3.30), (3.31), and (3.32), we derive that there exist constants C 5 > 0 , such that

(3.33) ‖ ϕ 2 , − ( n ) ‖ C 2 ( D 3 ∪ D 5 ) ≤ C 5 ε .

Step 3: The estimates of ϕ 1 , − ( n ) in D 4 ∪ D 5 and ϕ 2 , + ( n ) in D 6 .

In the same way, we first define the characteristic curves issuing from y = 0 . Let τ = T i , − ( n ) ( y , τ 0 , − ) ( i = 1 , 2 ) be the characteristics of ϕ i , − ( n ) , which is expressed as follows:

(3.34) d T i , − ( n ) d y ( y , τ 0 , − ) = 1 λ 1 ( y , ϕ − ( n − 1 ) ( y , T 1 , − ( n ) ) ) , T i , − ( n ) ( 0 , τ 0 , − ) = τ 0 , − , i = 1 , 2 ,

where τ 0 , − , τ 0 , + ∈ [ 0 , τ 1 * ] . Similarly, we regard τ 0 , − and τ 0 , + as the function of ( y , τ ) . Moreover, similar to Lemma 3.2, we also have the following lemma.

Lemma 3.3

For any ( ϕ − ( n − 1 ) , ϕ + ( n − 1 ) ) ∈ D C * ε , there exist constants M 2 , M 3 > 0 , such that

(3.35) ‖ ( ∂ y τ 0 , − , ∂ τ τ 0 , − ) ‖ C 0 ( D 4 ∪ D 5 ) + ‖ ( ∂ y y 2 τ 0 , − , ∂ y τ 2 τ 0 , − , ∂ τ τ 2 τ 0 , − ) ‖ C 0 ( D 4 ∪ D 5 ) ≤ M 2 ,

(3.36) ‖ ( ∂ y τ 0 , + , ∂ τ τ 0 , + ) ‖ C 0 ( D 6 ) + ‖ ( ∂ y y 2 τ 0 , + , ∂ y τ 2 τ 0 , + , ∂ τ τ 2 τ 0 , + ) ‖ C 0 ( D 6 ) ≤ M 3 .

By boundary condition (3.5) on y = 0 , we have

(3.37) ϕ 2 , + ( n ) ( 0 , τ ) = α 1 ϕ 1 , + ( n ) ( 0 , τ ) + α 2 ϕ 2 , − ( n ) ( 0 , τ ) ,

(3.38) ϕ 1 , − ( n ) ( 0 , τ ) = α 3 ϕ 1 , + ( n ) ( 0 , τ ) − α 1 ϕ 2 , − ( n ) ( 0 , τ ) ,

where

α 1 = φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) − φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) + φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) , α 2 = 2 φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) + φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) , α 3 = 2 φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) + φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) .

Subsequently, following the proof procedure in step 2 and employing (3.37) and (3.38), we derive that there exists a constant C 6 , C 7 > 0 , such that

(3.39) ‖ ϕ 1 , − ( n ) ‖ C 2 ( D 4 ∪ D 5 ) ≤ C 6 ε ,

(3.40) ‖ ϕ 2 , + ( n ) ‖ C 2 ( D 6 ) ≤ C 7 ε .

Let D ˆ 1 , + = D ˆ + ∩ { τ ≤ τ 1 * } , D ˆ 1 , − = D ˆ − ∩ { τ ≤ τ 1 * } . Then, there exists a constant C 1 * > 0 , such that

(3.41) ‖ ϕ − ( n ) ‖ C 2 ( D ˆ 1 , − ) + ‖ ϕ + ( n ) ‖ C 2 ( D ˆ 1 , + ) ≤ C 1 * ε .

Let ( 0 , τ 2,0 * ) be the intersection point of the characteristic curve corresponding to λ 2 starting from ( − F − , τ 1 * ) and the boundary y = 0 , and ( − F − , τ 2 , − * ) be the intersection point of the characteristic curve corresponding to λ 1 starting from ( 0 , τ 1 * ) and the boundary y = − F − . Denote τ 2 * = min { τ 2,0 * , τ 2 , − * } and D ˆ 2 , + = D ˆ + ∩ { τ 1 * ≤ τ ≤ τ 2 * } , D ˆ 2 , − = D ˆ − ∩ { τ 1 * ≤ τ ≤ τ 2 * } . Regarding the straight line τ = τ 1 * as the initial line τ = 0 and repeating the proof process in Steps 1–3 again, we obtain that there exists a constant C 2 * > 0 , such that

(3.42) ‖ ϕ − ( n ) ‖ C 2 ( D ˆ 2 , − ) + ‖ ϕ + ( n ) ‖ C 2 ( D ˆ 2 , + ) ≤ C 2 * ε .

Then, we repeat this procedure for ℓ times with ℓ = [ T 0 τ 1 * ] + 1 . Define τ ℓ * , D ˆ ℓ , + and D ˆ ℓ , − , then D ˆ − = ⋃ 1 ≤ i ≤ ℓ D ˆ ℓ , − and D ˆ + = ⋃ 1 ≤ i ≤ ℓ D ˆ ℓ , + . Thus summing all estimates (3.41) and (3.42) together for i = 1 , 2 , … , ℓ , we finally obtain (3.9) and complete the proof of Proposition 3.1.

The second fact, i.e., the mapping ℳ is a contraction map, is proved in the following Proposition.

Proposition 3.2

For any given T 0 > 0 , there exists a constant ε 1 > 0 , such that for any given ε ∈ ( 0 , ε 1 ) , if

(3.43) ‖ ( ϕ 1 0 , − , ϕ 2 0 , − ) ( y ) ‖ C 2 ( [ − F − , 0 ) ) + ‖ ( ϕ 1 0 , + , ϕ 2 0 , + ) ( y ) ‖ C 2 ( ( 0 , + ∞ ) ) + ‖ S ˆ 0 , − ( y ) − S ̲ − ‖ C 3 ( [ − F − , 0 ) ) + ‖ S ˆ 0 , + ( y ) − S ̲ + ‖ C 3 ( ( 0 , + ∞ ) ) + ‖ B ′ ( τ ) − u ̲ ‖ C 2 ( [ 0 , T 0 ] ) ≤ C 0 ε ,

then the sequence { ϕ − ( n ) , ϕ + ( n ) } ( n ∈ Z + ) constructed by the iteration schemes (3.1)–(3.5) is well-defined in D C * ε and convergent in C 1 ( D ˆ − ∪ D ˆ + ) , and its limit function is the solution of the IBVPs (2.14)–(2.19).

Proof

First, by (3.9), we obtain that the iteration sequence { ϕ − ( n ) , ϕ + ( n ) } is well-defined in D C * ε , and then the map ℳ : D C * ε → D C * ε is well-defined. Using (3.7), we can let ( ϕ − ( n + 1 ) , ϕ + ( n + 1 ) ) = ℳ ( ϕ − ( n ) , ϕ + ( n ) ) .

Next we prove that ℳ is a contraction mapping. Let

Δ ϕ − ( n ) = ϕ − ( n ) − ϕ − ( n − 1 ) , Δ ϕ + ( n ) = ϕ + ( n ) − ϕ + ( n − 1 ) , n ∈ Z + ,

with the aid of (3.1)–(3.5), we obtain that Δ ϕ ι ( n ) = ( Δ ϕ 1 , ι ( n + 1 ) , Δ ϕ 2 , ι ( n + 1 ) ) ( ι = + , − ) satisfies the following IBVP:

(3.44) ∂ Δ ϕ 1 , ι ( n + 1 ) ∂ τ + λ 1 ( y , ϕ ι ( n ) ) ∂ Δ ϕ 1 , ι ( n + 1 ) ∂ y = − ( λ 1 ( y , ϕ ι ( n ) ) − λ 1 ( y , ϕ ι ( n − 1 ) ) ) ∂ ϕ 1 , ι ( n ) ∂ y − ( h ( y , ϕ ι ( n ) ) − h ( y , ϕ ι ( n − 1 ) ) ) ∂ ϕ 2 , ι ( n ) ∂ τ + λ 1 ( y , ϕ ι ( n ) ) ∂ ϕ 2 , ι ( n ) ∂ y − h ( y , ϕ ι ( n − 1 ) ) ∂ Δ ϕ 2 , ι ( n ) ∂ τ + λ 1 ( y , ϕ ι ( n ) ) ∂ Δ ϕ 2 , ι ( n ) ∂ y − h ( y , ϕ ι ( n − 1 ) ) ( λ 1 ( y , ϕ ι ( n ) ) − λ 1 ( y , ϕ ι ( n − 1 ) ) ) ∂ ϕ 2 , ι ( n − 1 ) ∂ y + g ( y , ϕ ι ( n ) ) − g ( y , ϕ ι ( n − 1 ) ) ,

(3.45) ∂ Δ ϕ 2 , ι ( n + 1 ) ∂ τ + λ 2 ( y , ϕ ι ( n ) ) ∂ Δ ϕ 2 , ι ( n + 1 ) ∂ y = − ( λ 2 ( y , ϕ ι ( n ) ) − λ 2 ( y , ϕ ι ( n − 1 ) ) ) ∂ ϕ 2 , ι ( n ) ∂ y − ( h ( y , ϕ ι ( n ) ) − h ( y , ϕ ι ( n − 1 ) ) ) ∂ ϕ 1 , ι ( n ) ∂ τ + λ 2 ( y , ϕ ι ( n ) ) ∂ ϕ 1 , ι ( n ) ∂ y − h ( y , ϕ ι ( n − 1 ) ) ∂ Δ ϕ 1 , ι ( n ) ∂ τ + λ 2 ( y , ϕ ι ( n ) ) ∂ Δ ϕ 1 , ι ( n ) ∂ y − h ( y , ϕ ι ( n − 1 ) ) ( λ 2 ( y , ϕ ι ( n ) ) − λ 2 ( y , ϕ ι ( n − 1 ) ) ) ∂ ϕ 1 , ι ( n − 1 ) ∂ y + g ( y , ϕ ι ( n ) ) − g ( y , ϕ ι ( n − 1 ) ) ,

(3.46) τ = 0 : Δ ϕ − ( n + 1 ) = 0 , Δ ϕ + ( n + 1 ) = 0 ,

(3.47) y = − F − : ( Δ ϕ 2 , − ( n + 1 ) − Δ ϕ 1 , − ( n + 1 ) ) ( − F − , 0 ) = 0 ,

(3.48) y = 0 : φ − 1 2 γ ( S ˆ 0 , + ( 0 ) ) ( Δ ϕ 2 , − ( n + 1 ) − Δ ϕ 1 , − ( n + 1 ) ) ( 0 , τ ) = φ − 1 2 γ ( S ˆ 0 , − ( 0 ) ) ( Δ ϕ 2 , + ( n + 1 ) − Δ ϕ 1 , + ( n + 1 ) ) ( 0 , τ ) , Δ ϕ 2 , − ( n + 1 ) + Δ ϕ 1 , − ( n + 1 ) ( 0 , τ ) = Δ ϕ 2 , + ( n + 1 ) + Δ ϕ 1 , + ( n + 1 ) ( 0 , τ ) .

Similar to the proof of Proposition 3.1, we deduce that there exists a constant C > 0 , such that

(3.49) ‖ Δ ϕ − ( n + 1 ) ‖ C 1 ( D ˆ − ) + ‖ Δ ϕ + ( n + 1 ) ‖ C 1 ( D ˆ + ) ≤ C ε ( ‖ Δ ϕ − ( n ) ‖ C 1 ( D ˆ − ) + ‖ Δ ϕ + ( n ) ‖ C 1 ( D ˆ + ) ) .

Then, taking ε > 0 small enough such that C ε ≤ 1 2 , we have

(3.50) ‖ Δ ϕ − ( n + 1 ) ‖ C 1 ( D ˆ − ) + ‖ Δ ϕ + ( n + 1 ) ‖ C 1 ( D ˆ + ) ≤ 1 2 ( ‖ Δ ϕ − ( n ) ‖ C 1 ( D ˆ − ) + ‖ Δ ϕ + ( n ) ‖ C 1 ( D ˆ + ) ) .

Furthermore, one has

(3.51) ‖ ℳ ( Δ ϕ − ( n ) , Δ ϕ + ( n ) ) ‖ C 1 ( D ˆ − ∪ D ˆ + ) ≤ 1 2 ‖ ( Δ ϕ − ( n ) , Δ ϕ + ( n ) ) ‖ C 1 ( D ˆ − ∪ D ˆ + ) ,

which indicates that ℳ is a contraction map. Thus, it follows from the Banach fixed point theorem that the sequence { ϕ − ( n ) , ϕ + ( n ) } is convergent. We denote its limit as ( ϕ − , ϕ + ) . Obviously, ( ϕ − , ϕ + ) is a piecewise smooth solution of IBVPs (2.14)–(2.19) and satisfies (2.21). Therefore, we complete the proof of Proposition 3.2.□

Proof of Theorem 2.2

Since we have obtained the existence of solutions to IBVPs (2.14)–(2.19) in Proposition 3.2, we only need to prove the uniqueness. We consider two solutions ϕ a = ( ϕ − a , ϕ + a ) and ϕ b = ( ϕ − b , ϕ + b ) , both of which satisfy estimate (2.21). Define

Δ ϕ − = ϕ − a − ϕ − b , Δ ϕ + = ϕ + a − ϕ + b .

Then, imitating the proof process of Proposition 3.2, we obtain that ( Δ ϕ − , Δ ϕ + ) satisfies (3.49) for C 0 norm, where Δ ϕ ι ( n + 1 ) and Δ ϕ ι ( n ) are replaced by Δ ϕ ι for ι = + , − . This indicates Δ ϕ ι = 0 and then one has ϕ − a = ϕ − b , ϕ + a = ϕ + b . We complete the proof of Theorem 2.2.□

Acknowledgments

The authors are deeply grateful to the anonymous reviewers and editors for their valuable comments on the manuscript.

Funding information: This work was supported in part by NSFC Grant No. 12271310 and Natural Science Foundation of Shandong Province ZR2022MA088.
Author contributions: Xiaomin Zhang performed formal mathematical analysis and wrote the original draft. Huimin Yu supervised the theoretical rigor and critically revised the manuscript. Both authors participated in interpreting findings, approved the final version, and ensured compliance with academic standards.
Conflict of interest: The authors declared that they have no conflict of interest.
Data availability statement: No data were used for the research described in the article.

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Received: 2025-06-28

Revised: 2025-10-08

Accepted: 2025-10-29

Published Online: 2025-11-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2025-0131

Keywords for this article

compressible Euler equations; piston problem; contact discontinuity; stability

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