Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping

Haitong Li; La-Su Mai; Shiyu Wang

doi:10.1515/anona-2025-0102

Article Open Access

Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping

Haitong Li , La-Su Mai and Shiyu Wang

Published/Copyright: August 29, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, we consider the well-posedness of the local smooth solutions to the physical vacuum free boundary problem of the cylindrical symmetric Euler equations with time-dependent damping − μ ( 1 + t ) λ ρ u , μ > 0 , and λ > 0 . In this case, the free boundary is moving in the radial direction and the radial velocity will affect the angular velocity, but the axial velocity can be explicitly expressed by ω = ω 0 ( 1 + t ) − μ when λ = 1 and ω = ω 0 e μ 1 − λ [ 1 − ( 1 + t ) 1 − λ ] when λ ≠ 1 . Thus, when 0 < λ < 1 , ω converges to 0 as t → + ∞ , and when λ > 1 , ω converges to ω 0 e μ 1 − λ as t → + ∞ . This is essentially different from the case of μ = 1 , λ = 0 in (Meng-Mai-Mei, JDE, 2022), whose axial velocity is only expressed by ω = ω 0 e − t . Moreover, at the moving boundary, the compressible Euler equations with time-dependent damping become a degenerate hyperbolic system. Thanks to the Hardy-type inequality and cut-off functions, we choose a suitable weighted Sobolev spaces to construct a priori estimates to overcome the degeneracy of the system in the Lagrangian coordinates, which confirms the existence of local smooth solution. Gronwall’s inequality guarantees the uniqueness of local smooth solution.

Keywords: compressible Euler equations; time-dependent damping; cylindrical symmetry; physical vacuum; free boundary; local smooth solutions

MSC 2010: 35R35; 35L60; 35Q35; 35Q31

1 Introduction

In this article, we will investigate the well-posedness of local smooth solutions to the free boundary problem of the cylindrical symmetric compressible Euler equations with time-dependent damping:

(1.1) ∂ t ρ + div ( ρ u ) = 0 , ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇ p ( ρ ) = − μ ( 1 + t ) λ ρ u ,

where ρ and u denote the density and the velocity for fluids, respectively, p is the pressure and satisfies the γ law: p ( ρ ) = ρ γ , γ > 1 is an adiabatic exponent. − μ ( 1 + t ) λ ρ u is the time-dependent damping term with the positive constants μ > 0 and λ > 0 . In order to describe our problem, we introduce the following cylindrical symmetric transformations

(1.2) u = u ˜ x 1 r − v ˜ x 2 r , u ˜ x 2 r + v ˜ x 1 r , w ˜ , u ˜ = u ˜ ( r , t ) , v ˜ = v ˜ ( r , t ) , w ˜ = w ˜ ( r , t ) ,

where r = x 1 2 + x 2 2 , t > 0 . u ˜ , v ˜ , w ˜ are the radial, angular, and axial components of u , respectively. Then, system (1.1) can be transformed into

(1.3) ∂ t ( r ρ ) + ∂ r ( r ρ u ˜ ) = 0 , ρ u ˜ t + u ˜ u ˜ r − v ˜ 2 r + μ ( 1 + t ) λ u ˜ + p r = 0 , v ˜ t + u ˜ v ˜ r + u ˜ v ˜ r + μ ( 1 + t ) λ v ˜ = 0 , w ˜ t + u ˜ w ˜ r + μ ( 1 + t ) λ w ˜ = 0 .

System (1.1) is subjected to the following initial data and free boundary conditions on ( 0 , R ( t ) ) × [ 0 , T ] :

(1.4) ρ > 0 , x ∈ [ 0 , R ( t ) ) , ρ ( R ( t ) , t ) = 0 , u ˜ ( 0 , t ) = 0 , d R ( t ) d t = u ˜ ( R ( t ) , t ) , R ( 0 ) = 1 , ( ρ , u ˜ , v ˜ , w ˜ ) ( x , 0 ) = ( ρ 0 , u 0 , v 0 , w 0 ) , ρ 0 ( x ) > 0 , x ∈ [ 0 , 1 ) , − ∞ < d ρ 0 γ − 1 d x < 0 , r = 1 ,

where condition (1.4)₅ is called the physical vacuum condition [4,16,22].

Main results. Under the Lagrangian transformation (2.1), we transform the free boundary problem (1.3) and (1.4) into the initial boundary-value problem (2.8) and (2.9). Then, we will mainly construct the a priori estimates in (2.18), which confirms the existence and the uniqueness of local smooth solutions. Namely, the well-posedness of local smooth solutions to problem (2.8) and (2.9) will be stated in Theorem 2.1.

Background of relevant research. In recent years, physical vacuum free boundary problem of compressible Euler equations with damping and without damping has been studied extensively by many authors:

(I) For the case of μ = 0 , system (1.1) becomes the standard Euler equations. Coutand et al. [3–5], Hao [11], and Jang and Masmoudi [16,17] investigated the local-in-time well-posedness for the one, and three-dimensional physical vacuum free boundary problem by different methods. Luo et al. [24] proved a new local-in-time well-posedness of the solutions for spherically symmetric motions without imposing the compatibility condition and obtained a general uniqueness theorem for three-dimensional motions with or without self-gravitation. In the literature [32,33], Sideris established global smooth solutions for the spherically symmetric case and global affine solutions for the three-dimensional case. Besides, Hadžić and Jang [9] constructed a global-in-time unique solution under the assumption that the initial data are sufficiently close to a class of expanding compactly supported affine fluid motions constructed by Sideris [33]. For other results related to the free boundary problem to the relativistic motions for the compressible Euler equations, we refer to these interesting works [10,15,26,27] and references therein.

(II) For the case of μ > 0 , λ = 0 , system (1.1) turns into the compressible Euler equations with constant-coefficient damping. In the pioneering work [21], Liu proved that the constructed explicit solutions time-asymptotically converge to the self-similar solutions of the porous media equations and obtained the convergence rate. He also showed that if the initial value is sufficiently smooth, there exists a waiting time for the solutions near the free boundary. After that time, the boundary will move due to the influence of pressure. Along this direction, Liu and Yang [22] first found that if the initial density function has compact support, the solutions will not be global for the three-dimensional case. They also studied the local existence of solutions for the one-dimensional case by using the energy method and the characteristics method. Subsequently, Liu and Yang [23] obtained the local existence of solutions by a coordinate transformation capturing the physical sigularity at the vacuum. Thanks to the transformation introduced in [23], Xu and Yang [34] proved a local existence theorem by using the Littlewood-Paley theory. Later on, Yang [35] summarized the relevant results on the free boundary of the one-dimensional standard Euler equations and the Euler equations with linear damping. On the basis of higher-order nonlinear energy estimates and elliptic estimates, Luo and Zeng [25] generalized the results of [21] to the case of general solutions and proved that the global solutions converge to the Barenblatt self-similar solutions for the porous media equation. Then, Zeng [36] generalized the results of [25] to the case of spherical symmetry. Furthermore, Zeng [37] proved the almost global existence of smooth solutions for the three-dimensional case, which are close to the open question in multiple dimensions proposed by Liu [21]. By establishing a suitable weighted Sobolev space and using Hardy’s inequality, Meng et al. [28] studied the local well-posedness of smooth solutions from the perspective of cylindrical symmetry and overcame the singularity at the center point and the vacuum appearing on the moving boundary successfully. Other results about week solutions for the compressible Euler equations with damping and vacuum refer to the interesting works [12–14] and references therein.

However, compared to cases (I) and (II), there are few results for the case of μ > 0 and λ ≠ 0 . In this case, system (1.1) transforms into the compressible Euler equations with time-dependent damping. In the spirits of ideas of the the compressible Euler equations with constant-coefficient damping [25,36], Pan [29–31] considered the one-dimensional and the spherically symmetric global existence of smooth solutions and convergence to the different types of Barenblatt solutions of the related porous media equation with time-dependent dissipation. But, these articles do not provide a detailed proof of the well-posedness of local smooth solutions, respectively. This article aims to fill in the gap of smooth local solutions by considering the cylindrical symmetry cases. Without vacuum, some results related to the large time behavior of compressible Euler equations with time-dependent damping, we refer to the interesting works [1,2,6,7,19,20] and references therein for more detail.

New observation. For the time-dependent case, the axial velocity can be explicitly expressed by ω = ω 0 ( 1 + t ) − μ when λ = 1 , and ω = ω 0 e μ 1 − λ [ 1 − ( 1 + t ) 1 − λ ] when λ ≠ 1 . Thus, when 0 < λ < 1 , ω converges to 0 as t → + ∞ , and when λ > 1 , ω converges to ω 0 e μ 1 − λ as t → + ∞ . This is quite different from the case of μ = 1 , λ = 0 in [28], whose axial velocity is only expressed by ω = ω 0 e − t .

Main difficulties and technical issues. To obtain the main result Theorem 2.1, one of the main difficulties is how to deal with the degenerate of system (1.3) near the free boundary, because standard methods of symmetric hyperbolic systems cannot be applied directly to prove the short time existence of classical solutions. The second difficulty is caused by the singularity at the center point 0. To overcome these two difficulties, we introduce σ ( x ) ≔ x ρ 0 γ − 1 which is equivalent to the distance function d ( x ) near the boundary point and the center point. Thus, σ ( x ) plays as the weight in the weighted Sobolev embedding inequality (3.1) and can be the basic weight in the coefficient of system (2.8). Meanwhile, we choose the C ∞ cut-off functions ξ ( t ) and η ( x ) which satisfy (2.10)–(2.11) to help us to obtain elliptic estimates. The third difficulty is brought by the time-dependent damping. Compared to the case of μ = 1 , λ = 0 in [28], the time-dependent damping leads more complicated calculations but can derive different mechanisms.

The rest of this article is organized as follows. In Section 2, we transform the physical vacuum free boundary value problem (1.3) and (1.4) into the initial boundary problem (2.8) and (2.9), introduce the cut-off functions to conquer the singularities caused by degeneracy, and construct the higher-order weighted energy functional. Finally, we state the main results of this article. In Section 3, we clarify some preliminaries. In Section 4, we devote ourselves to establishing the energy estimates for higher-order time derivatives under the a priori assumptions. In Section 5, elliptic estimates for the higher-order spatial derivatives near the center point x = 0 and the boundary x = 1 are obtained with the help of cut-off functions. In Section 6, we address the existence and uniqueness of the local smooth solutions.

2 Reformulation and main results

In this section, our main target is to transform the physical vacuum free boundary problem (1.3) and (1.4) into the initial boundary value problem (2.8) and (2.9) and present the main results of this article.

Reformulation. We define the Lagrangian variable as

(2.1) ∂ t r ( x , t ) = u ˜ ( r ( x , t ) , t ) , t > 0 and r ( x , 0 ) = x .

Set the Lagrangian density and velocity by

(2.2) f ( x , t ) = ρ ( r ( x , t ) , t ) , u ( x , t ) = u ˜ ( r ( x , t ) , t ) , v ( x , t ) = v ˜ ( r ( x , t ) , t ) , w ( x , t ) = w ˜ ( r ( x , t ) , t ) .

System (1.3) becomes

(2.3) ∂ t ( r f ) + r f u x r x = 0 , f ∂ t u − v 2 r + μ ( 1 + t ) λ u + ( f γ ) x r x = 0 , ∂ t v + u v r + μ ( 1 + t ) λ v = 0 , ∂ t w + μ ( 1 + t ) λ w = 0 .

From (2.3)₁, we have

(2.4) f = 1 r x x r ρ 0 .

Then, system (2.3) turns into

(2.5) x r ρ 0 ∂ t u − v 2 r + μ ( 1 + t ) λ u + ∂ x 1 r x x r ρ 0 γ = 0 , ∂ t v + u v r + μ ( 1 + t ) λ v = 0 , ∂ t w + μ ( 1 + t ) λ w = 0 .

Define

(2.6) σ ( x ) ≔ x ρ 0 γ − 1 .

Due to the physical vacuum condition (1.4)₅, it is easy to know that σ ( x ) is equivalent to the distance function d ( x ) near the center point x = 0 and the boundary point x = 1 . Thus, (2.5)₁ can be rewritten as

(2.7) σ ρ 0 γ − 2 ∂ t u − v 2 r + μ ( 1 + t ) λ u + ∂ x 1 ρ 0 γ − 2 σ 2 x x r γ − 1 1 r x γ − 1 ρ 0 γ − 2 σ 2 r 2 1 r x γ − 1 x r γ − 2 = 0 .

In this article, we focus on the case of γ = 2 . Then, system (2.5) can be written as

(2.8) σ ∂ t u − v 2 r + μ ( 1 + t ) λ u + ∂ x σ 2 r 1 r x 2 − σ 2 x 2 x r 2 1 r x = 0 , ∂ t v + u v r + μ ( 1 + t ) λ v = 0 , ∂ t w + μ ( 1 + t ) λ w = 0 .

When γ = 2 , the initial data and the free boundary conditions (1.4) become

(2.9) ρ 0 ( x ) > 0 , x ∈ [ 0 , 1 ) , ρ 0 ( 1 ) = 0 , − ∞ < d ρ 0 ( 1 ) d x < 0 , u ( 0 , t ) = 0 , { x = 0 } × ( 0 , T ] , ( u , v , w ) ( x , 0 ) = ( u 0 , v 0 , w 0 ) , x ∈ ( 0 , 1 ) .

Cut-off functions. In order to overcome the degeneracy at the boundary point x = 1 and the singularity at the center point x = 0 , we choose the C ∞ cut-off functions ξ ( t ) and η ( x ) satisfying

(2.10) ξ ( x ) = 1 , x ∈ [ 0 , δ ] ; ξ ( x ) = 0 , x ∈ [ 2 δ , 1 ] ; ∣ ξ ′ ( x ) ∣ ≤ C 0 δ , x ∈ [ 0 , 1 ] ;

(2.11) η ( x ) = 1 , x ∈ [ δ , 1 ] ; η ( x ) = 0 , x ∈ 0 , δ 2 ; ∣ η ′ ( x ) ∣ ≤ C 0 δ , x ∈ [ 0 , 1 ] ,

where C 0 and δ are both positive constants, and δ is given later.

Weighted energy functional. Define the higher-order weighted energy functional as

(2.12) E ( t ) ≔ E ( u ) + E ( v ) ,

where

(2.13) E ( u ) = ∥ σ ∂ t 5 u ( t ) ∥ 0 2 + σ x ∂ t 4 ∂ x u ( t ) 0 2 + σ x ∂ t 4 u x ( t ) 0 2 + ∥ u ( t ) ∥ 2 2 + u x ( t ) 1 2 + ∥ σ u ( t ) ∥ 3 2 + ∥ ∂ t 4 u ( t ) ∥ 1 2 + ∥ σ ∂ t 2 u ( t ) ∥ 3 2 + ∥ ∂ t 2 u ( t ) ∥ 1 2 + σ x ∂ t 2 u ( t ) 2 2 + ∂ t 2 ∂ x u x ( t ) 0 2 + ∂ t 2 u x x ( t ) 0 2 + ∑ s = 0 1 ∂ t 2 s + 1 u x 3 2 − s 2 + ∂ t 2 s + 1 u x 1 − s 2 + ∥ σ ∂ t 2 s + 1 ∂ x 3 − s u ∥ 0 2 + ∥ ∂ t 2 s + 1 ∂ x 2 − s u ∥ 0 2 + ∑ s = 1 2 ( ∥ ξ σ ∂ t 5 − 2 s u ( t ) ∥ s + 1 2 + ∥ ξ ∂ t 5 − 2 s u ( t ) ∥ s 2 ) ,

(2.14) E ( v ) = ∥ σ ∂ t 4 v ( t ) ∥ 0 2 + ∥ σ ∂ t 3 v ( t ) ∥ L 4 2 + σ x ∂ t 2 ∂ x 2 v ( t ) 0 2 + ∥ ∂ t 2 ∂ x v ( t ) ∥ 0 2 + ∥ ∂ t ∂ x v ( t ) ∥ 0 2 + v , v x , v x , ∂ t v , ∂ t v x , ∂ t 2 v , ∂ t 2 v x , σ x ∂ t ∂ x v , σ x ∂ t 3 v ( t ) L ∞ 2 .

In order to obtain the existence result, for 1 ≤ k ≤ 5 , the compatibility conditions should be satisfied

(2.15) ∂ t k u ( x , 0 ) ≔ ∂ t k − 1 v 0 2 x − u 0 − 2 σ x + σ x ,

(2.16) ∂ t k v ( x , 0 ) ≔ ∂ t k − 1 − u 0 x v 0 − v 0 .

Main results. Now, we state the main results of this article.

Theorem 2.1

Let the initial data ( ρ 0 , u 0 , v 0 ) ∈ C 2 [ 0 , 1 ] satisfy (2.9), (2.15), (2.16), and

(2.17) E ( 0 ) < + ∞ .

Then, there exists a positive constant T 0 such that system (2.8) and (2.9) has a unique smooth solution ( r , u , v ) in [ 0 , 1 ] × [ 0 , T 0 ] satisfying

(2.18) sup t ∈ [ 0 , T 0 ] E ( t ) ≤ 2 P 0 ,

where P 0 = P ( E ( 0 ) ) , and P denotes a general polynomial function.

Remark 1

(2.5)₃ yields

(2.19) ω = ω 0 e − ∫ 0 t μ ( 1 + τ ) λ d τ ,

this implies the definition of E ( t ) in (2.12) does not involve the higher order energy for the axial velocity ω . From (2.19), we can observe that λ = 1 yields ω = ω 0 ( 1 + t ) − μ , and λ ≠ 1 yields ω = ω 0 e μ 1 − λ [ 1 − ( 1 + t ) 1 − λ ] . Thus, when 0 < λ < 1 , ω converges to 0 as t → + ∞ , and when λ > 1 , ω converges to ω 0 e μ 1 − λ as t → + ∞ . This is essentially different from the case of μ = 1 , λ = 0 in [28], whose axial velocity only expressed by ω = ω 0 e − t .

Remark 2

When γ = 2 , system (2.5) can be rewritten as the relatively simple formation (2.8), we first present the main results for γ = 2 in this article. When γ ≠ 2 , it can cause quite different difficulties in analyses and calculations; therefore, we leave the case of γ ≠ 2 as our next work in future.

3 Preliminaries

In this section, we will clarify some notations used throughout this article, present some embedding estimates for weighted Sobolev spaces, and derive some bounds which follows directly from the definition of the high-order weighted energy functional and the a priori assumptions.

Notation. We employ the notation a ≲ b to denote a ≤ C b , where C > 0 is a generic constant which may change from one line to another. The notation ′ denotes ∂ x , and ‖ ⋅ ‖ s denotes the norm of the standard Sobolev space ‖ ⋅ ‖ H s ( I ) for s ≥ 0 . In particular,

‖ ⋅ ‖ 0 ≕ ‖ ⋅ ‖ L 2 ( I ) , ‖ ⋅ ‖ L ∞ ≕ ‖ ⋅ ‖ L ∞ ( I ) .

Embedding of weighted Sobolev spaces. Let I ≔ ( 0 , 1 ) and d ( x ) denote the distance function to the boundary ∂ I , with

d ( x ) ≔ dist ( x , ∂ I ) = min { x , 1 − x } , x ∈ I .

For any a > 0 and nonnegative integer b , the weighted Sobolev space H a , b ( I ) is given by

H a , b ( I ) ≔ d a 2 F ∈ L 2 ( I ) : ∫ I d a ∣ D x k F ∣ 2 d x < ∞ , 0 ≤ k ≤ b

with the norm

‖ F ‖ H a , b ( I ) 2 ≔ ∑ k = 0 b ∫ I d a ∣ D x k F ∣ 2 d x .

Then, for b ≥ a 2 , it holds the following embedding of weighted Sobolev spaces [18]:

H a , b ( I ) ↪ H b − a ⁄ 2 ( I )

with the estimate

‖ F ‖ H b − a ⁄ 2 ( I ) ≲ ‖ F ‖ H a , b ( I ) .

In particular, we have

(3.1) ‖ F ‖ H 1 − a ⁄ 2 ( I ) 2 ≲ ∫ I d ( x ) a ( ∣ F ( x ) ∣ 2 + ∣ D F ( x ) ∣ 2 ) d x , a = 1 or 2 .

A polynomial-type inequality [3]. For a constant M 0 ≥ 0 , suppose that f ( t ) ≥ 0 , t ↦ f ( t ) is continuous, and

(3.2) f ( t ) ≤ M 0 + C t P ( f ( t ) ) ,

where P denotes a general polynomial function as aforementioned, and C is a generic constant. Then, for t taken sufficiently small, we have the bound

f ( t ) ≤ 2 M 0 .

A priori assumptions. As [24], suppose that there exists a smooth solution ( r , u , v ) to system (2.8) and (2.9) on [ 0 , 1 ] × [ 0 , T ] satisfies the following a priori assumptions:

(3.3) sup t ∈ [ 0 , T ] u x , u x , v x , v x ( t ) L ∞ ≲ M 0 ,

where the constant M 0 > 0 will be determined later. A direct calculation gives that there exists a sufficiently small time T 1 satisfying 0 < T 1 < T such that for any ( x , t ) ∈ [ 0 , 1 ] × ( 0 , T 1 ] holds

(3.4) 1 2 ≤ r ( x , t ) x ≤ 3 2 , 1 2 ≤ r ′ ( x , t ) ≤ 3 2 .

Lemma 3.1

[8] Let s ≥ 1 be a given integer and suppose that u ∈ H s ( I ) ∩ H 0 1 ( I ) , and d is the distance function to ∂ I , then we have that u d ∈ H s − 1 ( I ) with

(3.5) u d H s − 1 ≲ ∥ u ∥ H s .

Lemma 3.2

Let T > 0 and ( r , u , v ) be a smooth solution to system (2.8) and (2.9) on [ 0 , 1 ] × [ 0 , T ] , then for any 1 < p < ∞ , there are

(3.6) u x , u x , σ u x x , ∂ t u , ∂ t u x , ∂ t 2 u , σ x ∂ t 2 u x , σ x ∂ t 2 ∂ x u ( t ) L ∞ + ξ ∂ t ∂ x u , ξ ∂ t u x x , ξ σ ∂ t ∂ x 2 u , ξ ∂ t 3 u , ξ σ ∂ t 3 ∂ x u ( t ) L ∞ + σ x ∂ t 3 ∂ x u , σ x ∂ t 3 u x , ∂ t ∂ x u x , ∂ t u x x ( t ) L p ≲ E ( u ) ,

and

(3.7) σ x ∂ t 3 u ( t ) 3 2 2 + σ x ∂ t u ( t ) 5 2 2 ≲ E ( t ) .

The proof of Lemma 3.2 is on the basis of weighted Sobolev inequality, Minkowski’s inequality and the embedding inequalities ‖ ⋅ ‖ L ∞ ≲ ‖ ⋅ ‖ 1 , ‖ ⋅ ‖ L p ≲ ‖ ⋅ ‖ 1 ⁄ 2 , 1 < p < ∞ . We omit it.

In addition, throughout this article, another type of estimate is also used. That is, for any norm

(3.8) ∥ ∂ t i u ( t ) ∥ = ∂ t i u ( 0 ) + ∫ 0 t ∂ t i + 1 u ( τ ) d τ ≤ ∥ ∂ t i u ( 0 ) ∥ + ∫ 0 t ∥ ∂ t i + 1 u ( τ ) ∥ d τ ≤ ∥ ∂ t i u ( 0 ) ∥ + t ⋅ sup τ ∈ [ 0 , t ] ∥ ∂ t i + 1 u ( τ ) ∥ , i = 0 , 1 , 2 , 3 .

Therefore, it can be obtained

(3.9) ∑ s = 0 3 σ x ∂ t s u ( t ) 5 − s 2 2 + ∂ t s u x ( t ) 3 − s 2 2 + ∂ t 2 u x ( t ) 0 2 + ∂ t u x x ( t ) 0 2 + u x ( t ) 1 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Since we consider the local solutions in this article, it usually assume the time t ≤ 1 .

4 Energy estimates

The aim of this section is to establish the higher-order energy estimates of local solutions to (2.8) and (2.9) on [ 0 , 1 ] × [ 0 , T ] under the assumptions (3.3).

Lemma 4.1

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] . Then, there exists a small time T 2 satisfying 0 < T 2 < T , such that for any t ∈ ( 0 , T 2 ] , it holds that

(4.1) ∥ σ ∂ t 5 u ∥ 0 2 + σ x ∂ t 4 ∂ x u 0 2 + σ x ∂ t 4 u x 0 2 + ∫ 0 t ∫ 0 1 σ μ ( 1 + τ ) λ ( ∂ t 5 u ) 2 d τ d x ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Proof

Taking the ( k + 1 ) th time derivative of (2.8)₁, we obtain

(4.2) σ ∂ t k + 2 u − σ ∂ t k + 1 v 2 r + σ ∂ t k + 1 μ ( 1 + t ) λ u − σ 2 x x 2 r 2 r x 2 ∂ t k u x + 2 x r r x 3 ∂ t k ∂ x u x + σ 2 x 2 2 x 3 r 3 r x ∂ t k u x + x 2 r 2 r x 2 ∂ t k ∂ x u + ∑ l = 1 2 A l k = 0 ,

where

(4.3) A 1 k = − ∑ i = 0 k − 1 C k i σ 2 x ∂ t k − i x 2 r 2 r x 2 ∂ t i u x + 2 σ 2 x ∂ t k − i x r r x 3 ∂ t i ∂ x u x ,

(4.4) A 2 k = ∑ i = 0 k − 1 C k i 2 σ 2 x 2 ∂ t k − i x 3 r 3 r x ∂ t i u x + σ 2 x 2 ∂ t k − i x 2 r 2 r x 2 ∂ t i ∂ x u ,

with C k i = k ! i ! ( k − i ) ! .

When k = 4 , we multiply (4.2) by ∂ t 5 u and integrate the resultant equation over ( 0 , 1 ) × ( 0 , t ) , then a straightforward calculation gives

(4.5) ∫ 0 1 σ 2 ( ∂ t 5 u ) 2 + σ 2 x x r r x 1 r x 2 ( ∂ t 4 ∂ x u ) 2 + x r r x ∂ t 4 ∂ x u ∂ t 4 u x + x 2 r 2 ∂ t 4 u x 2 d x 0 t − ∫ 0 t ∫ 0 1 σ ∂ t 5 v 2 r ∂ t 5 u d x d τ + ∫ 0 t ∫ 0 1 σ ∂ t 5 μ ( 1 + τ ) λ u ∂ t 5 u d x d τ = ∫ 0 t ∫ 0 1 σ 2 x ∂ t x r r x 3 ( ∂ t 4 ∂ x u ) 2 + ∂ t x 2 r 2 r x 2 ∂ t 4 ∂ x u ∂ t 4 u x + ∂ t x 3 r 3 r x ∂ t 4 u x 2 d x d τ − ∑ l = 1 2 ∫ 0 t ∫ 0 1 A l 4 ∂ t 5 u d x d τ ≕ B 1 + B 2 .

In order to estimate the terms on the right-hand side of (4.5), it follows from (3.4) that for any nonnegative integers m and n , we have

(4.6) ∂ t k + 1 x m r m r x n ≲ A k , k = 0 , 1 , 2 , 3 , 4 ,

where

(4.7) A 0 = u x + ∣ u x ∣ , A k = ∂ t k u x + ∣ ∂ t k ∂ x u ∣ + A k − 1 A 0 + A k − 2 A 1 , k = 1 , 2 , 3 , 4 ,

with A i = 0 ( i < 0 ) . Analogous to [24,28], and by (2.13), (3.6), the H o ¨ lder inequality and ‖ ( σ , ρ 0 ) ‖ L ∞ ≤ C , we can conclude that for any 1 < p < ∞ ,

(4.8) ∥ A 0 ∥ L p + ∥ A 0 ∥ L ∞ + ∥ A 1 ∥ L p + ∥ A 1 ∥ L ∞ + ∥ A 2 ∥ 0 + ∥ σ A 2 ∥ L p + ∥ σ A 2 ∥ L ∞ + ∥ σ A 3 ∥ L p + ∥ σ A 4 ∥ 0 ≲ P ( E ) .

Furthermore, with the help of (3.8), it holds that

(4.9) ∥ A 0 ∥ L p 2 + ∥ A 1 ∥ 0 2 + ∥ σ A 2 ∥ L p 2 + ∥ σ A 3 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Now, we estimate the terms on the right-hand side of (4.5). For the first term, it is easy to obtain

(4.10) B 1 ≲ ∫ 0 t ∥ A 0 ∥ L ∞ σ x ∂ t 4 ∂ x u 0 2 d τ + ∫ 0 t ∥ A 0 ∥ L ∞ σ x ∂ t 4 u x 0 2 d τ ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

The second term B 2 in (4.5) can be written as

(4.11) B 2 = − ∫ 0 t ∫ 0 1 A 1 4 ∂ t 5 u d x d τ − ∫ 0 t ∫ 0 1 A 2 4 ∂ t 5 u d x d τ .

The estimate of the first term of (4.11) is

(4.12) − ∫ 0 t ∫ 0 1 A 1 4 ∂ t 5 u d x d τ = − ∫ 0 1 σ x ( I 11 + 2 I 12 ) σ x ∂ t 4 ∂ x u d x 0 t + ∫ 0 t ∫ 0 1 σ x ( ∂ t I 11 + 2 ∂ t I 12 ) σ x ∂ t 4 ∂ x u d x d τ ≕ B 21 ∣ 0 t + B 22 ,

where

(4.13) I 11 = ∑ i = 0 3 C 4 i ∂ t 4 − i x 2 r 2 r x 2 ∂ t i u x , I 12 = ∑ i = 0 3 C 4 i ∂ t 4 − i x r r x 3 ∂ t i ∂ x u .

For B 21 in (4.12), Young’s inequality implies that for any positive constant ε ,

(4.14) ∣ B 21 ∣ ≲ ∫ 0 1 ε σ x ∂ t 4 ∂ x u 2 + C ( ε ) σ x I 11 2 + σ x I 12 2 d x ≲ ε σ x ∂ t 4 ∂ x u 0 2 + C ( ε ) ∑ i = 0 3 σ x A 3 − i ∂ t i u x 0 2 + σ x A 3 − i ∂ t i ∂ x u 0 2 .

For ∑ i = 0 3 σ x A 3 − i ∂ t i u x 0 2 + σ x A 3 − i ∂ t i ∂ x u 0 2 , when i = 3 , from (3.8) and Hölder’s inequality, we have

(4.15) σ x A 0 ( t ) ∂ t 3 u x ( t ) 0 2 + σ x A 0 ( t ) ∂ t 3 ∂ x u ( t ) 0 2 ≤ 2 ∥ A 0 ( t ) ∥ L 4 2 σ x ∂ t 3 u x ( 0 ) L 4 2 + σ x ∂ t 3 ∂ x u ( 0 ) L 4 2 + 2 ∥ A 0 ( t ) ∥ L ∞ 2 ∫ 0 t σ x ∂ t 4 u x ( τ ) 0 2 + σ x ∂ t 4 ∂ x u ( τ ) 0 2 d τ ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

In addition, the remaining items are as follows:

(4.16) ∑ i = 0 2 σ x A 3 − i ∂ t i u x ( t ) 0 2 + σ x A 3 − i ∂ t i ∂ x u ( t ) 0 2 ≲ ∥ A 1 ( t ) ∥ 0 2 σ x ∂ t 2 u x ( 0 ) L ∞ 2 + σ x ∂ t 2 ∂ x u ( 0 ) L ∞ 2 + ∥ A 1 ( t ) ∥ L 4 2 ∫ 0 t σ x ∂ t 3 u x ( τ ) L 4 2 + σ x ∂ t 3 ∂ x u ( τ ) L 4 2 d τ + ∥ σ A 2 ( t ) ∥ L 4 2 ∂ t u x x ( 0 ) L 4 2 + ∂ t ∂ x u x ( 0 ) L 4 2 + ∥ σ A 2 ( t ) ∥ L ∞ 2 ∫ 0 t ∂ t 2 u x x ( τ ) 0 2 + ∂ t 2 ∂ x u ( τ ) x 0 2 d τ + ∥ σ A 3 ( t ) ∥ 0 2 u x 2 ( 0 ) L ∞ 2 + u x x ( 0 ) L ∞ 2 + ∥ σ A 3 ( t ) ∥ L 4 2 ∫ 0 t ∂ t u x x ( τ ) L 4 2 + ∂ t ∂ x u x ( τ ) L 4 2 d τ ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

From (4.14) to (4.16), it follows that

(4.17) ∣ B 21 ∣ ≲ ε σ x ∂ t 4 ∂ x u 0 2 + C ( ε ) [ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ] .

Similarly, by virtue of (3.6), (4.6), (4.8), and (4.13), we obtain

(4.18) ∣ B 22 ∣ ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

The second term of B 2 in (4.11) can be similarly estimated as (4.12). Thus, it follows that

(4.19) B 2 ≲ ε σ x ∂ t 4 ∂ x u 0 2 + σ x ∂ t 4 u x 0 2 + C ( ε ) [ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ] .

Taking (4.10) and (4.19) into (4.5) yields

(4.20) ∫ 0 1 σ 2 ( ∂ t 5 u ) 2 + σ 2 x x r r x 1 r x 2 ( ∂ t 4 ∂ x u ) 2 + x r r x ∂ t 4 ∂ x u ∂ t 4 u x + x 2 r 2 ∂ t 4 u x 2 d x 0 t − ∫ 0 t ∫ 0 1 σ ∂ t 5 v 2 r ∂ t 5 u d x d τ + ∫ 0 t ∫ 0 1 σ ∂ t 5 μ ( 1 + τ ) λ u ∂ t 5 u d x d τ ≲ ε σ x ∂ t 4 ∂ x u 0 2 + σ x ∂ t 4 u x 0 2 + P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

On the contrary, due to

(4.21) σ 2 x x r r x 1 r x 2 ( ∂ t 4 ∂ x u ) 2 + x r r x ∂ t 4 ∂ x u ∂ t 4 u x + x 2 r 2 ∂ t 4 u x 2 ≥ 1 32 σ 2 x ( ∂ t 4 ∂ x u ) 2 + ∂ t 4 u x 2 ,

and μ , λ > 0 , for a small time T 2 , 0 < T 2 < T , for any t ∈ ( 0 , T 2 ] ,

(4.22) ∂ t i μ ( 1 + t ) λ ≤ C , i = 0 , 1 , 2 , 3 , 4 , 5 ,

and

(4.23) ∂ t 5 μ ( 1 + t ) λ u = ∑ k = 0 4 C 5 k ∂ t 5 − k μ ( 1 + t ) λ ∂ t k u + μ ( 1 + t ) λ ∂ t 5 u ,

then (4.20) becomes

(4.24) 1 2 ∥ σ ∂ t 5 u ∥ 0 2 + 1 32 σ x ∂ t 4 ∂ x u 0 2 + σ x ∂ t 4 u x 0 2 + ∫ 0 t ∫ 0 1 σ μ ( 1 + τ ) λ ( ∂ t 5 u ) 2 d x d τ ≤ ∑ k = 0 4 ∫ 0 t ∫ 0 1 σ ∂ t k u ∂ t 5 u d x d τ + ∫ 0 t ∫ 0 1 σ ∂ t 5 v 2 r ∂ t 5 u d x d τ + P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ≤ ε ∫ 0 t ∥ σ ∂ t 5 u ∥ 0 2 d τ + ∫ 0 t ∫ 0 1 σ ∂ t 5 v 2 r ∂ t 5 u d x d τ + P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

where we have used

(4.25) ∑ k = 0 4 ∫ 0 t ∫ 0 1 σ ∂ t k u ∂ t 5 u d x d τ ≤ ε ∫ 0 t ∥ σ ∂ t 5 u ∥ 0 2 d τ + C ( ε ) ∑ k = 0 4 ∫ 0 t ∥ σ ∂ t k u ∥ 0 2 d τ ≤ ε ∫ 0 t ∥ σ ∂ t 5 u ∥ 0 2 d τ + C ( ε ) ∫ 0 t ∥ u ∥ 2 2 + ∂ t u x 1 2 + σ x ∂ t 2 u 2 2 + ∂ t 3 u x 0 2 + σ x ∂ t 4 u x 0 2 d τ ≤ ε ∫ 0 t ∥ σ ∂ t 5 u ∥ 0 2 d τ + P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Now, we handle the term of v in (4.24). Taking the k th time derivative of (2.8)₂ gives

(4.26) ∂ t k + 1 v + μ ( 1 + t ) λ + x r u x ∂ t k v + v x r ∂ t k u x = − v ∑ i = 0 k − 1 C k i ∂ t k − i x r ∂ t i u x − ∑ i = 1 k − 1 C k i ∂ t k − i v ∂ t i x r u x − ∑ i = 1 k C k i ∂ t i μ ( 1 + t ) λ ∂ t k − i v .

Taking k = 4 for (4.26), we have

(4.27) ∂ t 5 v + μ ( 1 + t ) λ + x r u x ∂ t 4 v + v x r ∂ t 4 u x = − v ∑ i = 0 3 C 4 i ∂ t 4 − i x r ∂ t i u x − ∑ i = 1 3 C 4 i ∂ t 4 − i v ∂ t i x r u x − ∑ i = 1 4 C 4 i ∂ t i μ ( 1 + t ) λ ∂ t 4 − i v .

Then, the term related to v in (4.24) has the form

(4.28) ∫ 0 t ∫ 0 1 σ ∂ t 5 v 2 r ∂ t 5 u d x d τ = ∫ 0 t ∫ 0 1 σ x ∂ t 5 u 2 x r v ∂ t 5 v + 10 x r ∂ t v ∂ t 4 v + 20 x r ∂ t 2 v ∂ t 3 v + ∑ i = 0 4 C 5 i ∂ t 5 − i x r ∂ t i ( v 2 ) d x d τ ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Combining (4.24) with (4.28), the desired estimate (4.1) in Lemma 4.1 is proved.□

5 Elliptic estimates

In order to further estimate the higher-order spatial derivatives in the Lagrangian coordinates, this section establishes elliptic estimates for the local smooth solutions to problems (2.8) and (2.9) on [ 0 , 1 ] × [ 0 , T ] under the a priori assumptions (3.3). Since the degeneracy of the system is of different orders near the center x = 0 and the boundary x = 1 , it is necessary to make interior estimates for each term at x = 0 and boundary estimates for each term at x = 1 , respectively. This section is divided into three subsections, which will give the interior estimates of all terms in the first subsection, the boundary estimates of all terms in the second subsection, and summarize the estimates of E ( t ) in the third subsection.

Multiplying (4.2) by x 2 σ yields

(5.1) σ ∂ t k ∂ x 2 u + σ ′ ∂ t k ∂ x u − σ ′ ∂ t k u x = x 2 ∂ t k + 1 μ ( 1 + t ) λ u − x ρ 0 x ∂ t k ∂ x u − 2 ρ 0 x ∂ t k u + B k ,

where

(5.2) B k = x 2 σ ∑ l = 1 4 A l k + A 5 k ,

with A l k ( l = 1 , 2 ) is given by (4.3), (4.4), and

(5.3) A 3 k = σ ∂ t k + 2 u − σ ∂ t k + 1 v 2 r ,

(5.4) A 4 k = − σ 2 x x 2 r 2 r x 2 − 1 ∂ t k u x + 2 x r r x 3 − 1 ∂ t k ∂ x u x ,

(5.5) A 5 k = σ 2 x 2 x 3 r 3 r x − 1 ∂ t k u x + x 2 r 2 r x 2 − 1 ∂ t k ∂ x u .

Next, we determine the constant δ in (2.10) and (2.11). Since

σ ′ ( x ) = ρ 0 ( x ) + x ρ 0 x ( x ) , σ ′ ( 0 ) = ρ 0 ( 0 ) > 0 , σ ″ ( x ) = 2 ρ 0 x ( x ) + x ρ 0 x x ( x ) ,

there exists a positive constant δ 0 (depending only on ρ 0 ( x ) ) such that for all x ∈ [ 0 , δ 0 ] ,

(5.6) ρ 0 ( 0 ) 2 ≤ ρ 0 ( x ) ≤ 3 ρ 0 ( 0 ) 2 ,

(5.7) ρ 0 ( 0 ) 2 ≤ σ ′ ( x ) ≤ 3 ρ 0 ( 0 ) 2 ,

and then

(5.8) ρ 0 ( 0 ) 2 x ≤ σ ( x ) ≤ 3 ρ 0 ( 0 ) 2 x .

Taking a general positive constant M 0 , which depends only on

max x ∈ [ 0 , 1 ] { ρ 0 ( x ) , ∣ ρ 0 x ( x ) ∣ , ∣ ρ 0 x x ( x ) ∣ , ∣ x ρ 0 x x x ( x ) ∣ } ,

then, for all x ∈ [ 0 , δ 0 ] ,

(5.9) σ ( x ) x − σ ′ ( x ) = ∣ x ρ 0 x ∣ ≲ M 0 , ∣ σ ″ ( x ) ∣ ≲ M 0 , ∣ σ ‴ ( x ) ∣ ≲ M 0 .

Now, we take δ as 0 < 2 δ ≤ δ 0 .

5.1 Interior estimates

Before proceeding, we first derive some estimates which will be used later. In addition to (3.6), (3.7), and (3.9), analogous to [24], we have the following interior bounds:

(5.10) ∥ ( ξ ∂ t ∂ x u , ξ σ ∂ t ∂ x 2 u , ξ ∂ t 3 u , ξ σ ∂ t 3 ∂ x u ) ( t ) ∥ L ∞ ≲ E ( t ) ,

(5.11) ∥ ξ ∂ t 2 u ( t ) ∥ 1 2 + ∥ ( ξ σ ∂ t 2 u , ξ u ) ( t ) ∥ 2 2 + ∥ ξ σ u ( t ) ∥ 3 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

(5.12) ‖ ( ξ u x , ξ σ u x x , ξ ∂ t 2 u , ξ σ ∂ t 2 ∂ x u ) ( t ) ‖ L ∞ 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

(5.13) v , v x , v x , ∂ t v , σ ∂ t 2 v ( t ) L ∞ 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

this, together with (4.8) and (4.9), yields that for 1 < p < ∞ ,

(5.14) ∥ A 0 ∥ L ∞ + ∥ A 1 ∥ L p + ∥ ξ A 1 ∥ L ∞ + ∥ A 2 ∥ 0 ≲ P ( E ) ,

(5.15) ∥ A 0 ∥ L p 2 + ∥ ξ A 0 ∥ L ∞ 2 + ∥ A 1 ∥ 0 2 + ∥ ξ A 2 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

5.1.1 Interior estimates for ∂ t 3 u , ∂ t 2 u

Lemma 5.1

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] . Then, there exists a small time T 3 satisfying 0 < T 3 < T , such that for any t ∈ ( 0 , T 3 ] , it holds that

(5.16) ξ σ x ∂ t 3 ∂ x 2 u x 0 2 + ξ σ ∂ t 3 ∂ x u x 0 2 + ξ σ ′ ∂ t 3 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

(5.17) ξ σ x ∂ t 2 ∂ x 2 u x 0 2 + ξ σ ∂ t 2 ∂ x u x 0 2 + ξ σ ′ ∂ t 2 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Proof

Taking k = 3 for (5.1) yields

(5.18) σ ∂ t 3 ∂ x 2 u + σ ′ ∂ t 3 ∂ x u − σ ′ ∂ t 3 u x = x 2 ∂ t 4 μ ( 1 + t ) λ u − x ρ 0 x ∂ t 3 ∂ x u − 2 ρ 0 x ∂ t 3 u + B 3 .

Note that

(5.19) ∂ t k ∂ x j u = x ∂ t k ∂ x j u x + j ∂ t k ∂ x j − 1 u x , j = 1 , 2 , … ,

and x σ ′ = x 2 ρ 0 x + σ , then (5.18) can be changed into

(5.20) σ x ∂ t 3 ∂ x 2 u x + 3 σ ∂ t 3 ∂ x u x + 3 σ ′ ∂ t 3 u x = x 2 ∂ t 4 μ ( 1 + t ) λ u − 2 x 2 ρ 0 x ∂ t 3 ∂ x u x + 3 σ x ∂ t 3 u x + B 3 .

Multiplying (5.20) by ξ and taking L 2 -norm, we have

(5.21) ξ σ x ∂ t 3 ∂ x 2 u x + 3 ξ σ ∂ t 3 ∂ x u x + 3 ξ σ ′ ∂ t 3 u x 0 2 ≲ 1 2 ξ x ∂ t 4 μ ( 1 + t ) λ u 0 2 + − 2 ξ x 2 ρ 0 x ∂ t 3 ∂ x u x + 3 ξ σ x ∂ t 3 u x 0 2 + ∥ ξ B 3 ∥ 0 2 .

Now, we estimate the right-hand side of (5.21). From (4.22) together with (3.9), (4.1), and (5.11), we obtain

(5.22) 1 2 ξ x ∂ t 4 μ ( 1 + t ) λ u 0 2 = 1 2 ξ x ∂ t 4 μ ( 1 + t ) λ u + 2 ξ x ∂ t 3 μ ( 1 + t ) λ ∂ t u + 3 ξ x ∂ t 2 μ ( 1 + t ) λ ∂ t 2 u + 2 ξ x ∂ t μ ( 1 + t ) λ ∂ t 3 u + 1 2 ξ x μ ( 1 + t ) λ ∂ t 4 u 0 2 ≲ ∥ ξ σ u ∥ 3 2 + σ x ∂ t u 2 2 + ∥ ξ σ ∂ t 2 u ∥ 2 2 + σ x ∂ t 3 u 1 2 + σ x ∂ t 4 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Besides, from (3.8) and (5.9), we have

(5.23) − 2 ξ x 2 ρ 0 x ∂ t 3 ∂ x u x + 3 ξ σ x ∂ t 3 u x 0 2 = − 2 ξ x ρ 0 x ∂ t 3 ∂ x u + 2 ξ σ ′ − σ x ∂ t 3 u x + 3 ξ σ x ∂ t 3 u x 0 2 ≲ ∥ ξ ∂ t 3 u ∥ 1 2 ≲ ∥ ξ ∂ t 3 u ( 0 ) ∥ 1 2 + t ⋅ sup τ ∈ [ 0 , t ] ∥ ∂ t 4 u ( τ ) ∥ 1 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

For the last term ∥ ξ B 3 ∥ 0 2 on the right-hand side of (5.21), the estimates for these terms are given below. Before which, a similar analysis as (4.6) shows that for any nonnegative integers m and n , it holds that

(5.24) ∂ t k + 1 x m r m r x n x ≲ ℑ k , k = 0 , 1 , 2 ,

where

(5.25) ℑ k = ∂ t k u x x + ∣ ∂ t k ∂ x 2 u ∣ + ℋ 0 A k + ℑ 0 A k − 1 + ℑ 1 A k − 2 , k = 1 , 2 ,

with

(5.26) ℋ 0 = x r x + ∣ r x x ∣ = r x − 1 x + ∣ r x x ∣ ≲ r x x + ∣ r x x ∣ .

It can be checked that

(5.27) ∥ ℋ 0 ∥ 0 2 + ∥ σ ℋ 0 ∥ L ∞ 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

(5.28) ∥ ℑ 0 ∥ 0 2 + ∥ σ ℑ 0 ∥ L ∞ 2 + ∥ σ ℑ 1 ∥ L p 2 + ∥ ξ σ ℑ 1 ∥ L ∞ 2 + ∥ σ ℑ 2 ∥ 0 2 ≤ C P ( E ( t ) ) + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

(5.29) ∥ ξ ℑ 0 ∥ 0 2 + ∥ σ ℑ 0 ∥ L p 2 + ∥ ξ σ ℑ 0 ∥ L ∞ 2 + ∥ σ ℑ 1 ∥ 0 2 + ∥ ξ σ ℑ 2 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

A simple calculation gives

(5.30) ξ x 2 σ A i 3 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) , i = 1 , 2 , 3 .

In addition, we have

(5.31) ξ x 2 σ A 4 3 0 2 ≲ ξ x 2 r 2 r x 2 − 1 ∂ t 3 u x 0 2 + ξ x r r x 3 − 1 ∂ t 3 ∂ x u 0 2 + ξ σ x 2 r 2 r x 2 x ∂ t 3 u x 0 2 + ξ σ x r r x 3 x ∂ t 3 ∂ x u 0 2 + ξ σ x 2 r 2 r x 2 − 1 ∂ t 3 u x x 0 2 + ξ σ x r r x 3 − 1 ∂ t 3 ∂ x 2 u 0 2 ≲ ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ ( ∥ ξ ∂ t 3 u ∥ 1 2 + ∥ ξ σ ∂ t 3 u ∥ 2 2 ) + ∥ σ ℋ 0 ∥ L ∞ 2 ∥ ξ ∂ t 3 u ∥ 1 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

where we have used

(5.32) σ ′ − σ 2 x ≤ 1 2 σ ′ + 1 2 σ ′ − σ x ≲ M 0 ,

(5.33) x r r x 3 x ≲ ∂ x x r + r x x ≲ ℋ 0 ,

(5.34) x 2 r 2 r x 2 x ≲ ∂ x x r + r x x ≲ ℋ 0 ,

and

(5.35) x 2 r 2 r x 2 − 1 L ∞ 2 ≲ x 4 r 4 1 − r 4 x 4 L ∞ 2 + 1 r x 4 ( 1 − r x 4 ) L ∞ 2 ≲ r 4 ( x , 0 ) x 4 − r 4 x 4 L ∞ 2 + ∥ r x 4 ( x , 0 ) − r x 4 ∥ L ∞ 2 ≲ ∫ 0 t u x L ∞ 2 d τ + ∫ 0 t ∥ u x ∥ L ∞ 2 d τ ,

and

(5.36) x r r x 3 − 1 L ∞ 2 ≲ ∫ 0 t u x L ∞ 2 d τ + ∫ 0 t ∥ u x ∥ L ∞ 2 d τ .

Similarly,

(5.37) ∥ ξ A 5 3 ∥ 0 2 ≲ ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ ∥ ξ ∂ t 3 u ∥ 1 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

By (5.30), (5.31), and (5.37), we have

(5.38) ∥ ξ B 3 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Substituting (5.22), (5.23), and (5.38) into (5.21) yields

(5.39) ξ σ x ∂ t 3 ∂ x 2 u x + 3 ξ σ ∂ t 3 ∂ x u x + 3 ξ σ ′ ∂ t 3 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

On the contrary, we estimate the left-hand side of (5.21).

(5.40) ξ σ x ∂ t 3 ∂ x 2 u x + 3 ξ σ ∂ t 3 ∂ x u x + 3 ξ σ ′ ∂ t 3 u x 0 2 = ξ σ x ∂ t 3 ∂ x 2 u x 0 2 + 9 ξ σ ∂ t 3 ∂ x u x 0 2 + 9 ξ σ ′ ∂ t 3 u x 0 2 + 6 ∫ 0 1 ξ σ x ∂ t 3 ∂ x 2 u x ξ σ ∂ t 3 ∂ x u x d x + 6 ∫ 0 1 ξ σ x ∂ t 3 ∂ x 2 u x ξ σ ′ ∂ t 3 u x d x + 18 ∫ 0 1 ξ σ ∂ t 3 ∂ x u x ξ σ ′ ∂ t 3 u x d x ,

where integration by parts gives

(5.41) 6 ∫ 0 1 ξ σ x ∂ t 3 ∂ x 2 u x ξ σ ∂ t 3 ∂ x u x d x = − 6 ∫ 0 1 ξ ξ ′ σ 2 x ∂ t 3 ∂ x u x 2 d x − 6 ∫ 0 1 ξ 2 σ σ ′ x ∂ t 3 ∂ x u x 2 d x − 3 ∫ 0 1 ξ 2 σ 2 ∂ t 3 ∂ x u x 2 d x

and

(5.42) 6 ∫ 0 1 ξ σ x ∂ t 3 ∂ x 2 u x ξ σ ′ ∂ t 3 u x d x + 18 ∫ 0 1 ξ σ ∂ t 3 ∂ x u x ξ σ ′ ∂ t 3 u x d x = − 12 ∫ 0 1 ξ ξ ′ σ σ ′ x ∂ t 3 ∂ x u x ∂ t 3 u x d x − 6 ∫ 0 1 ξ 2 ( σ ′ ) 2 x ∂ t 3 ∂ x u x ∂ t 3 u x d x − 6 ∫ 0 1 ξ 2 σ σ ″ x ∂ t 3 ∂ x u x ∂ t 3 u x d x − 6 ∫ 0 1 ξ 2 σ σ ′ x ∂ t 3 ∂ x u x 2 d x + 12 ∫ 0 1 ξ 2 σ σ ′ ∂ t 3 ∂ x u x ∂ t 3 u x d x .

For the last term in (5.42),

(5.43) 12 ∫ 0 1 ξ 2 σ σ ′ ∂ t 3 ∂ x u x ∂ t 3 u x d x = − 12 ∫ 0 1 ξ ξ ′ σ σ ′ ∂ t 3 u x 2 d x − 6 ∫ 0 1 ξ 2 ( σ ′ ) 2 ∂ t 3 u x 2 d x − 6 ∫ 0 1 ξ 2 σ σ ″ ∂ t 3 u x 2 d x .

Substituting (5.41)–(5.43) into (5.40) yields

(5.44) ξ σ x ∂ t 3 ∂ x 2 u x + 3 ξ σ ∂ t 3 ∂ x u x + 3 ξ σ ′ ∂ t 3 u x 0 2 ≥ ξ σ x ∂ t 3 ∂ x 2 u x 0 2 + ( 6 − 24 ε 1 ) ξ σ ∂ t 3 ∂ x u x 0 2 + ( 3 − 30 ε 2 ) ξ σ ′ ∂ t 3 u x 0 2 − P 0 − C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) − 6 ∫ 0 1 ξ 2 σ σ ″ ∂ t 3 u x 2 d x .

Taking ε 1 = 1 12 , ε 2 = 1 15 , and by

(5.45) 6 ∫ 0 1 ξ 2 σ σ ″ ∂ t 3 u x 2 d x ≲ ξ σ ∂ t 3 u x 0 2 + ξ σ ″ ∂ t 3 u x 0 2 ≲ ∥ ξ ∂ t 3 u ( 0 ) ∥ 1 2 + t ⋅ sup τ ∈ [ 0 , t ] ∥ ∂ t 4 u ( τ ) ∥ 1 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Thus, in view of (5.44), we obtain

(5.46) ξ σ x ∂ t 3 ∂ x 2 u x 0 2 + ξ σ ∂ t 3 ∂ x u x 0 2 + ξ σ ′ ∂ t 3 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

Therefore, the interior estimate for ∂ t 3 u is proved.□

Moreover, from (5.11), we can see that

(5.47) ξ σ x ∂ t 2 ∂ x 2 u x 0 2 + ξ σ ∂ t 2 ∂ x u x 0 2 + ξ σ ′ ∂ t 2 u x 0 2 ≲ ∥ ξ σ ∂ t 2 ∂ x 2 u ∥ 0 2 + ∥ ξ ∂ t 2 ∂ x u ∥ 0 2 + ξ ∂ t 2 u x 0 2 + ξ ∂ t 2 u x 0 2 ≲ ∥ ξ σ ∂ t 2 u ∥ 2 2 + ∥ ξ ∂ t 2 u ∥ 1 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Thus, we address the interior estimate for ∂ t 2 u .

5.1.2 Interior estimates for ∂ t u , u

Lemma 5.2

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] . Then, there exists a small time T 4 , 0 < T 4 < T , such that for any t ∈ ( 0 , T 4 ] , it holds that

(5.48) ξ x σ ∂ t ∂ x 3 u x 0 2 + ξ σ ∂ t ∂ x 2 u x 0 2 + ξ σ ′ ∂ t ∂ x u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

(5.49) ξ x σ ∂ x 3 u x 0 2 + ξ σ ∂ x 2 u x 0 2 + ξ σ ′ ∂ x u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Proof

Taking k = 1 for (5.1), we have

(5.50) σ ∂ t ∂ x 2 u + σ ′ ∂ t ∂ x u − σ ′ ∂ t u x = x 2 ∂ t 2 μ ( 1 + t ) λ u − x ρ 0 x ∂ t ∂ x u − 2 ρ 0 x ∂ t u + B 1 .

With the aid of (5.19), and differentiating (5.50) with respect to x has

(5.51) σ x ∂ t ∂ x 3 u x + 3 σ ∂ t ∂ x 2 u x + 3 σ ′ ∂ t ∂ x u x = 1 2 ∂ t 2 μ ( 1 + t ) λ u + x 2 ∂ t 2 ∂ x μ ( 1 + t ) λ u − 2 σ ′ x ∂ t ∂ x 2 u x − σ ″ x ∂ t ∂ x u x − ∂ x ( x ρ 0 x ∂ t ∂ x u ) − ∂ x ( 2 ρ 0 x ∂ t u ) + B x 1 .

Multiplying (5.51) by ξ and taking L 2 -norm, we obtain

(5.52) ξ σ x ∂ t ∂ x 3 u x + 3 σ ∂ t ∂ x 2 u x + 3 σ ′ ∂ t ∂ x u x 0 2 ≲ ξ ∂ t 2 μ ( 1 + t ) λ u + x ∂ t 2 ∂ x μ ( 1 + t ) λ u 0 2 + ξ 2 σ ′ x ∂ t ∂ x 2 u x + σ ″ x ∂ t ∂ x u x 0 2 + ∥ ξ [ ∂ x ( x ρ 0 x ∂ t ∂ x u ) + ∂ x ( 2 ρ 0 x ∂ t u ) ] ∥ 0 2 + ∥ ξ B x 1 ∥ 0 2 .

In the following, the estimates of the right-hand side of (5.52) are given on [ 0 , 1 ] × [ 0 , T ] , respectively. For the first term,

(5.53) ξ ∂ t 2 μ ( 1 + t ) λ u + x ∂ t 2 ∂ x μ ( 1 + t ) λ u 0 2 = ξ ∂ t 2 μ ( 1 + t ) λ u + 2 ξ ∂ t μ ( 1 + t ) λ ∂ t u + ξ μ ( 1 + t ) λ ∂ t 2 u + x ξ ∂ t 2 μ ( 1 + t ) λ u x + 2 x ξ ∂ t μ ( 1 + t ) λ ∂ t ∂ x u + x ξ μ ( 1 + t ) λ ∂ t 2 ∂ x u 0 2 ≲ ∥ ξ u ∥ 0 2 + ∥ ξ ∂ t u ∥ 0 2 + ∥ ξ ∂ t 2 u ∥ 0 2 + ∥ ξ σ u x ∥ 0 2 + ∥ ξ σ ∂ t ∂ x u ∥ 0 2 + ∥ ξ ∂ t 2 ∂ x u ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

At the meanwhile,

(5.54) ξ 2 σ ′ x ∂ t ∂ x 2 u x + σ ″ x ∂ t ∂ x u x 0 2 + ∥ ξ [ ∂ x ( x ρ 0 x ∂ t ∂ x u ) + ∂ x ( 2 ρ 0 x ∂ t u ) ] ∥ 0 2 ≲ ξ σ ′ x ∂ t ∂ x 2 u x 0 2 + ξ σ ″ x ∂ t ∂ x u x 0 2 + ∥ ξ ρ 0 x ∂ t ∂ x u ∥ 0 2 + ∥ ξ x ρ 0 x x ∂ t ∂ x u ∥ 0 2 + ∥ ξ x ρ 0 x ∂ t ∂ x 2 u ∥ 0 2 + ∥ ξ ρ 0 x x ∂ t u ∥ 0 2 ≲ ∥ ξ σ ∂ t u ( t ) ∥ 3 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Next, we estimate the last term ∥ ξ B x 1 ∥ 0 2 on the right-hand side of (5.52). In a similar way as the derivation of (5.24), we have that for any nonnegative integers m and n ,

(5.55) ∂ t x m r m r x n x x ≲ ℒ ,

where

(5.56) ℒ = u x x x + ∣ u x x x ∣ + ℋ 0 ℑ 0 + ( ℋ 1 + ℋ 0 2 ) A 0 ,

with

(5.57) ℋ 1 = x r x x + ∣ r x x x ∣ = r x − 1 x x + ∣ r x x x ∣ ≲ r x x 2 + r x x x + ∣ r x x x ∣ .

It is easy to obtain

(5.58) ∥ σ ℋ 1 ∥ 0 2 ≲ σ ∫ 0 t u x d τ x 2 0 2 + σ ∫ 0 t u x d τ x x 0 2 + σ ∫ 0 t u d τ x x x 0 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

(5.59) ∥ σ ℒ ∥ 0 2 ≲ σ u x x x 0 2 + ∥ σ u x x x ∥ 0 2 + ∥ σ ℋ 0 ∥ L ∞ 2 ∥ ℑ 0 ∥ 0 2 + ( ∥ σ ℋ 1 ∥ 0 2 + ∥ σ ℋ 0 ∥ L ∞ ∥ ℋ 0 ∥ 0 2 ) ∥ A 0 ∥ L ∞ 2 ≤ C E ( t ) + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

and

(5.60) ∥ ξ σ ℒ ( t ) ∥ 0 2 ≲ ξ σ u x x x ( t ) 0 2 + ( ∥ σ ℋ 1 ( t ) ∥ 0 2 + ∥ σ ℋ 0 ( t ) ∥ L ∞ 2 ∥ ℋ 0 ( t ) ∥ 0 2 ) ∥ A 0 ( t ) ∥ L ∞ 2 + ∥ ξ σ u x x x ( t ) ∥ 0 2 + ∥ σ ℋ 0 ( t ) ∥ L ∞ 2 ∥ ξ ℑ 0 ( t ) ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Now, we estimate each term of ∥ ξ B x 1 ∥ 0 2 separately. It is easy to calculate that

(5.61) ξ ∂ x x 2 σ A i 1 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) , i = 1 , 2 , 3 .

In addition,

(5.62) ξ ∂ x x 2 σ A 4 1 0 2 ≲ ( ∥ ξ ∂ t u ∥ 2 2 + ∥ ξ σ ∂ t u ∥ 3 2 ) ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ + ∥ ℋ 0 ∥ 0 2 ξ ∂ t u x L ∞ 2 + ∥ ξ ∂ t ∂ x u ∥ L ∞ 2 + ξ ∂ t u x x L ∞ 2 + ∥ σ ξ ∂ t ∂ x 2 u ∥ L ∞ 2 + ( ∥ σ ℋ 0 ∥ L ∞ 2 ∥ ℋ 0 ∥ 0 2 + ∥ σ ℋ 1 ∥ 0 2 ) ξ ∂ t u x L ∞ 2 + ∥ ξ ∂ t ∂ x u ∥ L ∞ 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

where we have used the following inequalities:

(5.63) σ ″ − σ ′ 2 x + σ 2 x 2 ≤ ∣ σ ″ ∣ + 1 2 x σ ′ − σ x ≤ M 0 + 1 2 x x ρ 0 x ≲ M 0 ,

(5.64) 3 2 σ ′ − σ 2 x = σ ′ + 1 2 σ ′ − σ x ≤ ∣ σ ′ ∣ + 1 2 σ ′ − σ x ≲ M 0 ,

(5.65) σ x r r x 3 x x ≲ σ ℋ 0 2 + σ ℋ 1 ,

(5.66) σ x 2 r 2 r x 2 x x ≲ σ ℋ 0 2 + σ ℋ 1 .

A similar calculation leads to

(5.67) ∥ ξ ∂ x ( A 5 1 ) ∥ 0 2 ≲ ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ ∥ ξ ∂ t u ∥ 2 2 + ∥ ℋ 0 ∥ 0 2 ∂ t u x L ∞ 2 + ∥ ξ ∂ t ∂ x u ∥ L ∞ 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

where we have used

(5.68) x 3 r 3 r x x ≲ ∂ x x r + r x x ≲ ℋ 0 ,

(5.69) x 3 r 3 r x − 1 L ∞ 2 ≲ ∫ 0 t u x L ∞ 2 d τ + ∫ 0 t ∥ u x ∥ L ∞ 2 d τ .

Combining (5.61), (5.62), and (5.67), we have

(5.70) ∥ ξ B x 1 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Thus, substituting (5.53), (5.54), and (5.70) into (5.52) yields

(5.71) ξ σ x ∂ t ∂ x 3 u x + 3 σ ∂ t ∂ x 2 u x + 3 σ ′ ∂ t ∂ x u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

The following calculation is analogous to (5.40)–(5.45), therefore, we obtain the desired estimate (5.48). By a similar calculation as (5.47), we obtain the expected estimate (5.49) immediately.□

5.2 Boundary estimates

Applying x ρ 0 x = σ ′ − σ x to (5.1), and differentiating the resultant equation, we have

(5.72) σ ∂ t k ∂ x 3 u + 3 σ ′ ∂ t k ∂ x 2 u = 1 2 ∂ t k + 1 μ ( 1 + t ) λ u + x 2 ∂ t k + 1 μ ( 1 + t ) λ u x − 2 σ ″ ∂ t k ∂ x u + σ x 2 ∂ t k ∂ x u + σ x ∂ t k ∂ x 2 u − σ ″ ∂ t k u x + 3 σ ′ x ∂ t k u x − 4 σ x 2 ∂ t k u x + B x k .

To simplify the boundary estimates, we first introduce the following lemma.

Lemma 5.3

For any function f = f ( x , t ) and integer i ≥ 2 , it holds that

(5.73) ∥ η σ f ′ ∥ 0 2 + ∥ η σ ′ f ∥ 0 2 ≤ ∥ η ( σ f ′ + i σ ′ f ) ∥ 0 2 + C σ 1 2 f 0 2 ,

(5.74) η σ 3 2 f ′ 0 2 + η σ 1 2 σ ′ f 0 2 ≤ 5 η σ 1 2 ( σ f ′ + i σ ′ f ) 0 2 + C ∥ σ f ∥ 0 2 .

The proof of Lemma 5.3 is so simple because it is only based on the integration by parts and Minkowski’s inequality that we omit it [24].

By means of Lemma 5.3, taking f = ∂ t 3 ∂ x u , i = 2 for (5.74), we have

(5.75) η σ 3 2 ∂ t 3 ∂ x 2 u 0 2 + ∥ η σ σ ′ ∂ t 3 ∂ x u ∥ 0 2 ≤ 5 ∥ η σ ( σ ∂ t 3 ∂ x 2 u + 2 σ ′ ∂ t 3 ∂ x u ) ∥ 0 2 + C ∥ σ ∂ t 3 ∂ x u ∥ 0 2 ,

and taking f = ∂ t 2 ∂ x u , i = 2 for (5.73), we obtain

(5.76) ∥ η σ ∂ t 2 ∂ x 2 u ∥ 0 2 + ∥ η σ ′ ∂ t 2 ∂ x u ∥ 0 2 ≤ ∥ η ( σ ∂ t 2 ∂ x 2 u + 2 σ ′ ∂ t 2 ∂ x u ) ∥ 0 2 + C ∥ σ ∂ t 2 ∂ x u ∥ 0 2 ,

and taking f = ∂ t ∂ x 2 u , i = 3 for (5.74), we obtain

(5.77) η σ 3 2 ∂ t ∂ x 3 u 0 2 + ∥ η σ σ ′ ∂ t ∂ x 2 u ∥ 0 2 ≤ 5 ∥ η σ ( σ ∂ t ∂ x 3 u + 3 σ ′ ∂ t ∂ x 2 u ) ∥ 0 2 + C ∥ σ ∂ t ∂ x 2 u ∥ 0 2 ,

and taking f = u x x , i = 3 for (5.73), we can see that

(5.78) ∥ η σ ∂ x 3 u ∥ 0 2 + ∥ η σ ′ u x x ∥ 0 2 ≤ ∥ η ( σ ∂ x 3 u + 3 σ ′ u x x ) ∥ 0 2 + C ∥ σ u x x ∥ 0 2 .

5.2.1 Boundary estimates for u

Lemma 5.4

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] . Then, there exists a small time T 5 , 0 < T 5 < T , such that for any t ∈ ( 0 , T 5 ] , it holds that

(5.79) ∥ η σ ∂ x 3 u ∥ 0 2 + ∥ η σ ′ u x x ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Proof

Taking k = 0 for (5.72) reads

(5.80) σ ∂ x 3 u + 3 σ ′ u x x = 1 2 ∂ t μ ( 1 + t ) λ u + 1 2 μ ( 1 + t ) λ ∂ t u + x 2 ∂ t μ ( 1 + t ) λ u x + 1 2 μ ( 1 + t ) λ ∂ t ∂ x u − 2 σ ″ u x + σ x 2 u x + σ x u x x − σ ″ u x + 3 σ ′ u x 2 − 4 σ x 2 u x + B x 0 .

In a similar way to (5.10)–(5.13), the following estimate is obtained:

(5.81) ∥ u ( t ) ∥ 2 2 + ∥ σ u ( t ) ∥ 3 2 + σ x ∂ t 2 u 1 2 + ∂ t 2 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Multiplying (5.81) by η and taking L 2 -norm yields

(5.82) ∥ η ( σ ∂ x 3 u + 3 σ ′ u x x ) ∥ 0 2 ≤ η 1 2 ∂ t μ ( 1 + t ) λ u + 1 2 μ ( 1 + t ) λ ∂ t u + x 2 ∂ t μ ( 1 + t ) λ u x + 1 2 μ ( 1 + t ) λ ∂ t ∂ x u 0 2 + η − 2 σ ″ u x + σ x 2 u x + σ x u x x − σ ″ u x + 3 σ ′ u x 2 − 4 σ x 2 u x 0 2 + ∥ η B x 0 ∥ 0 2 .

From (4.22) and (5.81), the first term on the right-hand side of (5.82) becomes

(5.83) η 1 2 ∂ t μ ( 1 + t ) λ u + 1 2 μ ( 1 + t ) λ ∂ t u + x 2 ∂ t μ ( 1 + t ) λ u x + 1 2 μ ( 1 + t ) λ ∂ t ∂ x u 0 2 ≲ ∥ u ∥ 2 2 + ∂ t u x 1 2 + ∥ σ u ∥ 3 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

For the second term on the right-hand side of (5.82), one has

(5.84) η − 2 σ ″ u x + σ x 2 u x + σ x u x x − σ ″ u x + 3 σ ′ u x 2 − 4 σ x 2 u x 0 2 ≲ ∥ σ u ∥ 3 2 + ∥ u ∥ 2 2 + u x 1 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Estimating each term of ∥ η B x 0 ∥ 0 2 on the right-hand side of (5.82) as follows:

(5.85) η ∂ x x 2 σ A 3 0 0 2 ≲ ∂ t 2 u x 0 2 + σ x ∂ t 2 u 1 2 + v x L ∞ 2 ∥ ∂ t v ∥ L ∞ 2 + v x L ∞ 4 ∥ u ∥ 2 2 + ∥ v x ∥ L ∞ 2 ∥ ∂ t v ∥ L ∞ 2 + v x L ∞ 2 ∥ ∂ t ∂ x v ∥ 0 2 + v x L ∞ 2 ∥ v x ∥ L ∞ 2 ∥ u ∥ 2 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) )

and

(5.86) η ∂ x x 2 σ A 4 0 0 2 ≲ ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ u x 1 2 + ∥ u ∥ 2 2 + ∥ σ u ∥ 3 2 + ∥ ℋ 0 ∥ 0 2 u x L ∞ 2 + ∥ u x ∥ L ∞ 2 + σ u x x L ∞ 2 + ∥ σ u x x ∥ L ∞ 2 + ( ∥ σ ℋ 0 ∥ L ∞ 2 ∥ ℋ 0 ∥ 0 2 + ∥ σ ℋ 1 ∥ 0 2 ) u x L ∞ 2 + ∥ u x ∥ L ∞ 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

where we have used (5.33)–(5.36), (5.63), and (5.64). Moreover,

(5.87) ∥ η ∂ x ( A 5 0 ) ∥ 0 2 = η ∂ x σ x x 3 r 3 r x − 1 u x + σ 2 x x 2 r 2 r x 2 − 1 u x 0 2 ≲ ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ u x 1 2 + ∥ u ∥ 2 2 + ∥ ℋ 0 ∥ 0 2 u x L ∞ 2 + ∥ u x ∥ L ∞ 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

where we have used (5.33)–(5.36), (5.65), (5.66), (5.68), and (5.69). Thus, together with (5.85)–(5.87), we have

(5.88) ∥ η B x 0 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

It follows from (5.82)–(5.84) and (5.88) that

(5.89) ∥ η ( σ ∂ x 3 u + 3 σ ′ u x x ) ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Finally, the expected estimate (5.79) of Lemma 5.4 holds by feat of (5.78), (5.81), and (5.89).□

5.2.2 Boundary estimates for ∂ t u

Lemma 5.5

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] , there exists a small time T 6 , 0 < T 6 < T , such that for any t ∈ ( 0 , T 6 ] , it holds that

(5.90) η σ 3 2 ∂ t ∂ x 3 u 0 2 + ∥ η σ σ ′ ∂ t ∂ x 2 u ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Proof

Taking k = 1 in (5.72), multiplying by η σ and taking L 2 -norm gives

(5.91) ∥ η σ ( σ ∂ t ∂ x 3 u + 3 σ ′ ∂ t ∂ x 2 u ) ∥ 0 2 ≲ η σ 1 2 ∂ t 2 μ ( 1 + t ) λ u + ∂ t μ ( 1 + t ) λ ∂ t u + 1 2 μ ( 1 + t ) λ ∂ t 2 u + x 2 ∂ t 2 μ ( 1 + t ) λ u x + x ∂ t μ ( 1 + t ) λ ∂ t ∂ x u + x 2 μ ( 1 + t ) λ ∂ t 2 ∂ x u 0 2 + η σ − 2 σ ″ ∂ t ∂ x u + σ x 2 ∂ t ∂ x u + σ x ∂ t ∂ x 2 u − σ ″ ∂ t u x + 3 σ ′ x ∂ t u x − 4 σ x 2 ∂ t u x 0 2 + ∥ η σ B x 1 ∥ 0 2 .

From (3.9), (4.22), and (5.81), the first term on the right-hand side of (5.91) becomes

(5.92) η σ 1 2 ∂ t 2 μ ( 1 + t ) λ u + ∂ t μ ( 1 + t ) λ ∂ t u + 1 2 μ ( 1 + t ) λ ∂ t 2 u + x 2 ∂ t 2 μ ( 1 + t ) λ u x + x ∂ t μ ( 1 + t ) λ ∂ t ∂ x u + x 2 μ ( 1 + t ) λ ∂ t 2 ∂ x u 0 2 ≲ ∥ u ∥ 2 2 + σ x ∂ t u 2 2 + σ x ∂ t 2 u 1 2 + ∥ σ u ∥ 3 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Due to (3.9), the second term on the right-hand side of (5.91) has

(5.93) η σ − 2 σ ″ ∂ t ∂ x u + σ x 2 ∂ t ∂ x u + σ x ∂ t ∂ x 2 u − σ ″ ∂ t u x + 3 σ ′ x ∂ t u x − 4 σ x 2 ∂ t u x 0 2 ≲ σ x ∂ t u 2 2 + ∂ t u x 1 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Analogous to (5.61), it is easy to conclude that

(5.94) η σ ∂ x x 2 σ A i 1 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) , i = 1 , 2 , 3 .

Similarly to (5.62), we have

(5.95) η σ ∂ x x 2 σ A 4 1 0 2 ≲ ∂ t u x 1 2 + σ x ∂ t u 2 2 + ∥ σ ∂ t ∂ x 3 u ∥ 0 2 ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ + ∥ ℋ 0 ∥ 0 2 ∂ t u x L ∞ 2 + ∥ ∂ t ∂ x u ∥ L ∞ 2 + ∂ t u x x L ∞ 2 + ∥ σ ∂ t ∂ x 2 u ∥ L ∞ 2 + ∥ σ ℋ 1 ∥ 0 2 ∂ t u x L ∞ 2 + ∥ ∂ t ∂ x u ∥ L ∞ 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

As (5.67), we obtain

(5.96) ∥ η σ ∂ x ( A 5 1 ) ∥ 0 2 = η σ ∂ x σ x x 3 r 3 r x − 1 ∂ t u x + 1 2 σ x x 2 r 2 r x 2 − 1 ∂ t ∂ x u 0 2 ≲ ∫ 0 t u x L ∞ 2 + ∥ u x ∥ L ∞ 2 d τ ∂ t u x 1 2 + ∥ ∂ t ∂ x 2 u ∥ 0 2 + ∥ ℋ 0 ∥ 0 2 ∂ t u x L ∞ 2 + ∥ ∂ t ∂ x u ∥ L ∞ 2 ≤ C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

In view of (5.94)–(5.96), we obtain

(5.97) ∥ η σ B x 1 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Thus, together with (5.92), (5.93), and (5.97), we can see that (5.91) becomes

(5.98) ∥ η σ ( σ ∂ t ∂ x 3 u + 3 σ ′ ∂ t ∂ x 2 u ) ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

In conclusion, it follows from (5.90) that Lemma 5.5 holds.□

5.2.3 Boundary estimates for ∂ t 2 u

Lemma 5.6

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] , there exists a small time T 7 , 0 < T 7 < T , such that for any t ∈ ( 0 , T 7 ] , it holds that

(5.99) ∥ η σ ∂ t 2 ∂ x 2 u ∥ 0 2 + ∥ η σ ′ ∂ t 2 ∂ x u ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Proof

Taking k = 2 for (5.72), we have

(5.100) σ ∂ t 2 ∂ x 2 u + 2 σ ′ ∂ t 2 ∂ x u = x 2 ∂ t 3 μ ( 1 + t ) λ u + σ x ∂ t 2 ∂ x u − σ ′ ∂ t 2 u x + 2 σ x ∂ t 2 u x + B 2 .

Multiplying (5.100) by η and taking L 2 -norm leads to

(5.101) ∥ η ( σ ∂ t 2 ∂ x 2 u + 2 σ ′ ∂ t 2 ∂ x u ) ∥ 0 2 ≲ η x 2 ∂ t 3 μ ( 1 + t ) λ u + 3 x 2 ∂ t 2 μ ( 1 + t ) λ ∂ t u + 3 x 2 ∂ t μ ( 1 + t ) λ ∂ t 2 u + x 2 μ ( 1 + t ) λ ∂ t 3 u 0 2 + η σ x ∂ t 2 ∂ x u − σ ′ ∂ t 2 u x + 2 σ x ∂ t 2 u x 0 2 + ∥ η B 2 ∥ 0 2 .

By virtue of (4.22) and (5.81), the first term on the right-hand side of (5.101) becomes

(5.102) η x 2 ∂ t 3 μ ( 1 + t ) λ u + 3 x 2 ∂ t 2 μ ( 1 + t ) λ ∂ t u + 3 x 2 ∂ t μ ( 1 + t ) λ ∂ t 2 u + x 2 μ ( 1 + t ) λ ∂ t 3 u 0 2 ≲ ∥ σ u ∥ 3 2 + ∂ t u x 1 2 + ∂ t 2 u x 0 2 + ∂ t 3 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Estimating the second term on the right-hand side of (5.101) has

(5.103) η σ x ∂ t 2 ∂ x u − σ ′ ∂ t 2 u x + 2 σ x ∂ t 2 u x 0 2 ≲ σ x ∂ t 2 u 1 2 + ∂ t 2 u x 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

For the third term ∥ η B 2 ∥ 0 2 on the right-hand side of (5.101), similar calculation as (5.94)–(5.96) can be arrived at

(5.104) ∥ η B 2 ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Summarizing from (5.101) to (5.104) that

(5.105) ∥ η ( σ ∂ t 2 ∂ x 2 u + 2 σ ′ ∂ t 2 ∂ x u ) ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Hence, by (5.76), Lemma 5.6 follows.□

5.2.4 Boundary estimates for ∂ t 3 u

Lemma 5.7

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] , there exists a small time T 8 , 0 < T 8 < T , such that for any t ∈ ( 0 , T 8 ] , it holds that

(5.106) η σ 3 2 ∂ t 3 ∂ x 2 u 0 2 + ∥ η σ σ ′ ∂ t 3 ∂ x u ∥ 0 2 ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

Proof

Taking k = 3 for (5.72), multiplying the resultant equation by η σ , and calculating similar to (5.82)–(5.89), we can derive Lemma 5.7.□

5.3 Estimates for E ( t )

Lemma 5.8

Suppose that ( r , u , v ) is a smooth solution to system (2.8) and (2.9) satisfying the a priori assumptions (3.3) on [ 0 , 1 ] × [ 0 , T ] , there exists a small time T 0 satisfying 0 < T 0 < T , for any t ∈ ( 0 , T 0 ] , (2.18) holds.

Proof

Concluding from (5.16), (5.17), (5.48), (5.49), (5.79), (5.90), (5.99), and (5.106) that

(5.107) E ( u ) ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) .

The estimate of v can be derived by the same method as for u has

(5.108) E ( v ) ≤ P 0 + C t P ( sup τ ∈ [ 0 , t ] E ( τ ) ) ,

Taking T 0 = min 1 ≤ i ≤ 8 { T i } and using the polynomial-type inequality (3.2) introduced in Section 3 and the literature [3], it follows that Lemma 5.8 holds.□

6 Existence and uniqueness results

This section investigates the existence and uniqueness of the smooth solutions ( r , u , v ) to problems (2.8) and (2.9). The existence results can be summarized from the previous sections, and the main purpose of this chapter is to give the steps for proving the uniqueness results in detail.

Under the condition that the a priori assumptions (3.3) holds, this section summarizes the previous sections to establish the existence of smooth solutions ( r , u , v ) to problems (2.8) and (2.9). First, we can establish the corresponding degenerate parabolic approximation system by a similar approach to [24,27,28]. Thus, we can prove the existence of solutions by using the fixed point theorem [4,8,27], and we omit the details here.

We state the uniqueness of the local smooth solutions to system (2.8) and (2.9) as follows.

Theorem 6.1

Assume that ( r 1 , u 1 , v 1 ) and ( r 2 , u 2 , v 2 ) are local smooth solutions to system (2.8) and (2.9) on [ 0 , 1 ] × [ 0 , T ] satisfying the a priori assumptions (3.3), and

(6.1) r i = x + ∫ 0 t u i ( x , τ ) d τ , i = 1 , 2 ,

then ( r 1 , u 1 , v 1 ) ( x , 0 ) = ( r 2 , u 2 , v 2 ) ( x , 0 ) , x ∈ [ 0 , 1 ] it holds

(6.2) ( r 1 , u 1 , v 1 ) ( x , t ) = ( r 2 , u 2 , v 2 ) ( x , t ) , ( x , t ) ∈ [ 0 , 1 ] × [ 0 , T ] .

Proof

Set

(6.3) R ≔ r 2 − r 1 , R t = U ≔ u 2 − u 1 , V ≔ v 1 − v 2 .

A simple calculation about (2.8)₁ gives

(6.4) σ ∂ t U + σ μ ( 1 + t ) λ U − σ 2 x L 1 R x + L 2 R x x + σ 2 x 2 L 3 R x + L 4 R x + H 1 = 0 ,

where

(6.5) L 1 = x r 2 1 r 1 x 1 r 2 x 2 + x r 2 1 r 1 x 2 1 r 2 x , L 2 = x r 1 x r 2 1 r 1 x 2 , L 3 = x r 2 2 1 r 1 x 1 r 2 x , L 4 = x r 1 x r 2 2 1 r 1 x + x r 1 2 x r 2 1 r 1 x , H 1 = σ r 1 ( V + 2 v 2 ) V + R v 1 2 r 1 r 2 = σ v 1 2 r 1 r 2 R + 1 r 2 V 2 + 2 r 2 v 2 V .

Multiplying (6.4) by U has

(6.6) 1 2 σ U 2 t − σ 2 x L 1 R x U + L 2 R x U x + σ 2 x 1 2 L 1 R x 2 + L 2 R x R x + 1 2 L 4 R x 2 t + U H 1 = − σ μ ( 1 + t ) − λ U 2 + σ 2 x 1 2 L 1 t R x 2 + L 2 t R x R x + 1 2 L 4 t R x 2 + σ 2 x L 2 U x R x − L 3 R x U x ,

where

(6.7) U H 1 = σ 1 2 v 1 2 r 1 r 2 ( R 2 ) t + 1 r 2 V 2 U + 2 r 2 v 2 V R t .

From (2.8)₂, we have

(6.8) ( r v ) t + μ ( 1 + t ) λ v r = 0 ,

then

(6.9) R t v 2 = u 1 V + r 1 V t − R v 2 t − μ ( 1 + t ) λ v 2 R + μ ( 1 + t ) λ r 1 V .

Substituting (6.9) into (6.7) yields

(6.10) U H 1 = 1 2 σ v 1 2 r 1 r 2 R 2 t + σ 1 r 2 r 1 V 2 t − H 2 − H 3 − H 4 ,

where

(6.11) H 2 = σ v 1 v 1 t r 1 r 2 R 2 − 1 2 σ v 1 2 r 1 2 r 2 u 1 R 2 − 1 2 σ v 1 2 r 1 r 2 2 u 2 R 2 , H 3 = − σ 1 r 2 V 2 U − σ 2 r 2 u 1 V 2 − σ r 1 r 2 2 u 2 V 2 + σ 1 r 2 u 1 V 2 − 2 μ σ ( 1 + t ) − λ r 1 r 2 V 2 , H 4 = 2 σ 1 r 2 v 2 t V R + 2 μ σ ( 1 + t ) − λ 1 r 2 v 2 R V .

Thus, integrating (6.6) over [ 0 , 1 ] with respect to x gives

(6.12) 1 2 d d t ∫ 0 1 σ U 2 + σ v 1 2 r 1 r 2 R 2 + 2 σ r 1 r 2 V 2 + σ 2 x L 1 R x 2 + 2 L 2 R x R x + L 4 R x 2 d x = − ∫ 0 1 σ μ ( 1 + t ) − λ U 2 d x + ∫ 0 1 H 2 d x + ∫ 0 1 H 3 d x + ∫ 0 1 H 4 d x + ∫ 0 1 1 2 σ 2 x L 1 t R x 2 + 2 L 2 t R x R x + L 4 t R x 2 d x + ∫ 0 1 σ 2 x ( L 2 − L 3 ) R x U x d x .

Next, we estimate the right-hand side of (6.12). Since λ > 0 , we have

(6.13) 0 < ( 1 + t ) − λ ≤ 1 ,

then

(6.14) ∫ 0 1 σ μ ( 1 + t ) − λ U 2 d x ≲ ∫ 0 1 σ U 2 d x .

It follows from (3.4) that there exists a positive constant C ( P 0 ) , depending only on P 0 such that

(6.15) ∣ ∂ t L i ∣ + ∣ L i ∣ ≤ C ( P 0 ) , i = 1 , 2 , 3 , 4 ,

and

(6.16) ∣ ∂ t v ∣ ≤ C ( P 0 ) .

Obviously, from (3.3), one has

(6.17) U x = u 2 − u 1 x ≤ u 2 x + u 1 x ≲ M 0 ,

(6.18) ∣ v x ∣ ≤ ‖ v x ‖ L ∞ ≲ M 0 .

A direct calculation leads to

(6.19) ∣ v ∣ ≲ M 0 .

On the basis of (3.3), (3.4), (6.16), and (6.19), the second term on the right-hand side of (6.12) can be estimated as

(6.20) ∫ 0 1 H 2 d x ≤ ∫ 0 1 σ x 2 r 1 r 2 v 1 v 1 t R x 2 d x + ∫ 0 1 1 2 σ x 3 r 1 2 r 2 u 1 x R x 2 d x + ∫ 0 1 1 2 σ x 3 r 1 r 2 2 v 1 2 u 2 x R x 2 d x ≤ C ( P 0 , M 0 ) ∫ 0 1 σ 2 x R x 2 d x .

Note from (3.3), (3.4), (6.13), and (6.17) that

(6.21) ∫ 0 1 H 3 d x ≤ ∫ 0 1 σ x r 2 V 2 U x d x + 2 ∫ 0 1 σ x r 2 u 1 x V 2 d x + ∫ 0 1 σ x 2 r 2 2 r 1 x u 2 x V 2 d x + ∫ 0 1 σ x r 2 u 1 x V 2 d x + ∫ 0 1 2 σ μ ( 1 + t ) − λ x r 2 r 1 x V 2 d x ≤ C ( M 0 ) ∫ 0 1 σ V 2 d x .

Similarly, we can derive

(6.22) ∫ 0 1 H 4 d x ≤ ∫ 0 1 σ x r 2 v 2 t V 2 + R x 2 d x + ∫ 0 1 μ σ ( 1 + t ) − λ x r 2 v 2 V 2 + R x 2 d x ≤ C ( P 0 , M 0 ) ∫ 0 1 σ 2 x V 2 + R x 2 d x

and

(6.23) ∫ 0 1 1 2 σ 2 x L 1 t R x 2 + 2 L 2 t R x R x + L 4 t R x 2 d x ≤ C ( P 0 ) ∫ 0 1 σ 2 x R x 2 + R x 2 d x .

In addition,

(6.24) ∫ 0 1 σ 2 x ( L 2 − L 3 ) R x U x d x ≤ C ( M 0 ) ∫ 0 1 σ 2 x R x 2 + R x 2 d x ,

where

(6.25) ∣ L 2 − L 3 ∣ ≤ x 3 r 1 r 2 2 r 1 x 2 R x + x 2 r 2 2 r 1 x 2 r 2 x R x ≲ R x + ∣ R x ∣ .

Therefore, (6.12) becomes

(6.26) 1 2 d d t ∫ 0 1 σ U 2 + σ v 1 2 r 1 r 2 R 2 + 2 σ r 1 r 2 V 2 + σ 2 x L 1 R x 2 + 2 L 2 R x R x + L 4 R x 2 d x ≤ C ( P 0 , M 0 ) ∫ 0 1 σ U 2 + σ V 2 + σ 2 x R x 2 + R x 2 d x .

On the contrary, (3.4) gives

(6.27) x 2 r 1 r 2 ≥ 4 9 , x r 2 r 1 x ≥ 1 3 , L 1 = x r 2 1 r 1 x 1 r 2 x 2 + x r 2 1 r 1 x 2 1 r 2 x ≥ 32 81 , L 2 = x r 1 x r 2 1 r 1 x 2 ≥ 16 81 , L 4 = x r 1 x r 2 2 1 r 1 x + x r 1 2 x r 2 1 r 1 x ≥ 32 81 .

Then, the left-hand side of (6.12) becomes

(6.28) 1 2 d d t ∫ 0 1 σ U 2 + σ v 1 2 r 1 r 2 R 2 + 2 σ r 1 r 2 V 2 + σ 2 x L 1 R x 2 + 2 L 2 R x R x + L 4 R x 2 d x ≥ 1 2 d d t ∫ 0 1 σ U 2 + 2 3 σ V 2 + 16 81 σ 2 x R x 2 + R x 2 d x .

Combining (6.26) with (6.28), we have

(6.29) 1 2 d d t ∫ 0 1 σ U 2 + 2 3 σ V 2 + 16 81 σ 2 x R x 2 + R x 2 d x ≤ C ( P 0 , M 0 ) ∫ 0 1 σ U 2 + σ V 2 + σ 2 x R x 2 + R x 2 d x .

Integrating (6.29) with respect to t on [ 0 , T ] , we obtain that

(6.30) 1 2 ∫ 0 1 σ U 2 + 2 3 σ V 2 + 16 81 σ 2 x R x 2 + R x 2 d x ≤ 1 2 ∫ 0 1 σ U 2 + 2 3 σ V 2 + 16 81 σ 2 x R x 2 + R x 2 d x t = 0 + C ( P 0 , M 0 ) ∫ 0 t ∫ 0 1 σ U 2 + σ V 2 + σ 2 x R x 2 + R x 2 d x d τ .

When ( r 1 , u 1 , v 1 ) ( x , 0 ) = ( r 2 , u 2 , v 2 ) ( x , 0 ) , x ∈ [ 0 , 1 ] , there are

(6.31) U ( x , 0 ) = 0 , V ( x , 0 ) = 0 , R ( x , 0 ) = 0 .

Thus, (6.30) implies

(6.32) 1 2 ∫ 0 1 σ U 2 + 2 3 σ V 2 + 16 81 σ 2 x R x 2 + R x 2 d x ≤ C ( P 0 , M 0 ) ∫ 0 t ∫ 0 1 σ U 2 + σ V 2 + σ 2 x R x 2 + R x 2 d x d τ .

Therefore, with the help of Gronwall’s inequality, it follows that

(6.33) U = u 2 − u 1 = 0 , V = v 1 − v 2 = 0 , R = r 2 − r 1 = 0 , ( x , t ) ∈ ( 0 , 1 ) × [ 0 , T ] .

So far, we complete the proof of Theorem 6.1.□

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments that help us to improve the article.

Funding information: The research of H. Li was partially supported by the Natural Science Foundation of Jilin Province, China (No. 20220101037JC), the National Natural Science Foundation of China (No. 12301267) and the Science and Technology Research Project of Education Department of Jilin Province, China (No. JJKH20230744KJ). The research of L. Mai was partially supported by the National Natural Science Foundation of China (No. 11601246, No.12461045, and No. 11971014), Young Science and Technology Talents Science and Technology Talents Cultivation Project of Inner Mongolia University (No. 21221505), Research and Educational Reform Project for Graduate Education of Inner Mongolia Autonomous Region in 2023 (No. JGCG2023007), and Outstanding Youth fund of Inner Mongolia Natural Science Foundation (No. 2023JQ13).
Author contributions: All authors contributed equally to the writing of this article. All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data were used for the research described in this article.

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Received: 2024-09-11

Revised: 2025-03-20

Accepted: 2025-07-11

Published Online: 2025-08-29

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

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https://doi.org/10.1515/anona-2025-0102

Keywords for this article

compressible Euler equations; time-dependent damping; cylindrical symmetry; physical vacuum; free boundary; local smooth solutions

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