Startseite Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
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Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions

  • Jinbo Geng , Ke Hu , Ning-An Lai EMAIL logo und Manwai Yuen
Veröffentlicht/Copyright: 17. Oktober 2024
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 13 Heft 1

Abstract

Compressible Euler equations with space-dependent damping in high dimensions R n ( n = 2 , 3 ) are considered in this article. Assuming that the small initial velocity and small perturbation of the initial density have compact support, we establish finite-time blow-up results for the Euler system, by combining energy estimate and new test functions constructed by the solutions of the following linear elliptic partial differential equations system:

− G 1 ( x ) + ∇ ⋅ G 2 → ( x ) = 0 , − G 2 → ( x ) + ∇ G 1 ( x ) = μ G 2 → ( x ) ( 1 + ∣ x ∣ ) λ .

This result generalizes the one in the literature from 1 − D to high dimension R n ( n = 2 , 3 ) .

Keywords: compressible Euler equations; damping; blow-up; lifespan; test function method
MSC 2010: 35Q31; 35L65; 35L67; 76N15

1 Introduction

In this article, we consider the compressible Euler equations with space-dependent damping in R n ( n = 2 , 3 ) :

(1) ρ t + ∇ ⋅ ( ρ u ) = 0 , ( ρ u ) t + ∇ ⋅ ( ρ u ⊗ u ) + ∇ P + μ ρ u ( 1 + ∣ x ∣ ) λ = 0 , t = 0 : u ( 0 , x ) = ε u 0 ( x ) , ρ ( 0 , x ) = 1 + ε ρ 0 ( x ) ,

where ρ = ρ ( t , x ) ∈ R and u = u ( t , x ) ∈ R n ( n = 2 , 3 ) are unknown functions, representing the density and velocity of the flow, respectively. The state equation for the gas is in the form

P = A ρ γ ,

with A > 0 , γ > 1 are constants. In addition, μ > 0 , λ > 1 denote the damping coefficient and decay rate, respectively, and ε > 0 is a parameter representing the smallness of the initial data that satisfy

(2) ( u 0 ( x ) , ρ 0 ( x ) ) ∈ C 0 ∞ ( R n ) , supp ( u 0 ( x ) , ρ 0 ( x ) ) ⊂ { x ∣ ∣ x ∣ ≤ 1 } .

Euler equations constitute a fundamental system in fluid dynamics, which is a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. There is extensive literature on the study of long-time behavior for Euler equations, it is really hard for us to survey all of them, and hence, we may focus on the closely related results. If μ = 0 , (1) stands for the classical isentropic compressible Euler equations. The study of the formation of singularity for a compressible Euler system in high dimensions was initiated by Sideris [27], in which several blow-up results were established to 3 − D Euler equations for a nonisentropic polytropic, ideal fluid with both large data and small initial perturbation. Rammaha [26] obtained similar results for the 2 − D case. We should mention that Liu [20] and Li et al. [19] studied finite blow-up and global existence to the first-order quasilinear hyperbolic system in 1 − D , and the results can be applied to isentropic Euler equations in the Lagrangian coordinates. Lei et al. [17] showed blow-up results for both 2 − D and 3 − D isentropic cases, under the assumptions that the large initial data are radial symmetrical and the initial density contains vacuum states. Lai et al. [13] studied the generalized Riemann problem governed by compressible isothermal ( P = ρ ) Euler equations in R 2 and R 3 and established blow-up results for the fan-shaped wave structure solution, which is a perturbation of plane waves. Jin and Zhou [11] studied the isentropic case in R n ( n = 1 , 2 , 3 ) with small initial data and established the upper bound of the lifespan estimate. We also refer to [1,4,23,35] (and the references therein) for shock formation by using the characteristic method.

If μ ≠ 0 and λ = 0 , (1) reduces to a compressible Euler system with constant damping, which has a resistant influence on the formation of singularities. In this case, the properties of solutions depend on the size of the initial data. In general, there exist global solutions if the initial data are sufficiently small in some norm. See [9,10,22,34] for the global existence and large-time behavior of solutions with small perturbations around a constant state. Sideris et al. [28] proved the global existence if the initial energy is small and finite time blow-up for large initial data.

Recently, a compressible Euler system with time-dependent damping

(3) ρ t + ∇ ⋅ ( ρ u ) = 0 , ( ρ u ) t + ∇ ⋅ ( ρ u ⊗ u ) + ∇ P + μ ρ u ( 1 + t ) λ = 0

attracts more and more attention. Hou and Yin [7] and Hou et al. [8] studied the long-time behavior for (3) in R n ( n = 2 , 3 ) with small initial perturbations and showed global existence for

  • 0 ≤ λ < 1 , μ > 0 , ∇ × u 0 ≠ 0 ;

  • λ = 1 , μ > 2 , ∇ × u 0 ≠ 0 and λ = 1 , 3 − n < μ ≤ 2 , ∇ × u 0 = 0

and finite-time blow-up for

  • λ > 1 , μ > 0 ;

  • λ = 1 , 0 < μ ≤ 3 − n and n = 2 .

Pan [24,25] obtained similar results for the 1 − D case

  • if λ = 1 , μ > 2 or 0 ≤ λ < 1 , μ > 0 , there exists global solution;

  • if λ = 1 , 0 ≤ μ ≤ 2 or λ > 1 , μ ≥ 0 , the C 1 solutions will blow up in finite time.

Lai and Schiavone [15] introduced the Orlicz spaces technique and obtained improved lifespan estimates for μ ≥ 0 , λ > 1 , γ > 1 , n = 2 , 3 and 0 ≤ μ ≤ 3 − n , λ = 1 , γ > 1 , n = 2 .

We should also refer to some recent results to show the formation of singularity for the Cauchy problem (3) in 1 − D by using a characteristic method, see [2,31–33] for time-dependent damping and [29,30] for space-time-dependent damping.

In [5], the finite-time blow-up for the Cauchy problem (1) in 1 − D with μ > 0 , λ > 1 and γ ≥ 2 has been established by using a special test function and energy equality. In this article, we consider the Cauchy problem (1) of isentropic compressible Euler equations with space-dependent damping and small perturbation and wave strength in R n ( n = 2 , 3 ) . To the best of our knowledge, there are few known results for compressible Euler equations with space-dependent damping, especially in high-dimensional case. By combining energy estimate and test function method inspired by [11,13], a Riccati-type differential inequality is established for a solution-related functional, and hence, blow-up result is obtained. In addition, the upper bound of lifespan will be established, which we conjecture is optimal.

Remark 1.1

Compared to the 1 − D case, in order to construct the appropriate test functions, we have to solve a first-order differential system (15) (essentially a second-order elliptic equation) in a high-dimensional case instead of a second-order ordinary differential equation, which is much more complicated. We overcome such kind of difficulty by combining elliptic equation theory in a bounded domain and ordinary differential equation theory in its complementary domain.

Remark 1.2

The analysis of asymptotic behavior of the test functions used in this article is motivated by the works [14,16] for nonlinear wave equations with space-dependent damping.

Our main result states as follows.

Theorem 1.3

Assume λ > 1 , P = 1 γ ρ γ , γ ≥ 2 . We further assume that the initial perturbations satisfy (2) and

C 0 ≜ ∫ R 3 G 1 ( x ) ρ 0 ( x ) d x + ∫ R 3 G → 2 ⋅ u 0 d x > 0 ,

where G 1 ( x ) , G → 2 ( x ) are the solutions to the ordinary differential system (15) in Lemma  2.4. Then, system (1) has no global solutions and the lifespan satisfies

(4) T ( ε ) ≤ C ε − 2 , n = 2 , exp ( C ε − 1 ) , n = 3 .

Hereinafter, C denotes a generic positive constant independent of ε .

Remark 1.4

For convenience, we demonstrate the detailed proof of Theorem 1.3 for R 3 with γ = 2 .

2 Preliminaries

In this section, we are going to give some important preliminary lemmas. The first one is related to the local existence and finite speed of propagation.

Let γ > 1 and

σ ( ρ ) = P ′ ( ρ ) , θ ≜ 2 γ − 1 ( σ ( ρ ) − 1 ) = 2 γ − 1 ρ γ − 1 2 − 1 ,

then the Euler system (1) can be transformed into

(5) θ t + ∇ ⋅ u = − u ⋅ ∇ θ − γ − 1 2 θ ∇ ⋅ u , u t + ∇ θ = − μ u ( 1 + ∣ x ∣ ) λ − u ⋅ ∇ u − γ − 1 2 θ ∇ θ , t = 0 : u ( 0 , x ) = ε u 0 ( x ) , θ ( 0 , x ) = θ 0 ( x )

with

θ 0 ( x ) = 2 γ − 1 ( σ ( ρ 0 ( x ) ) − 1 ) .

Proposition 2.1

If the initial data ( u 0 ( x ) , θ 0 ( x ) ) ∈ H 3 ( R 3 ) , then there exists a unique local solution ( u ( t , x ) , θ ( t , x ) ) ∈ C ( [ 0 , T ) × H 3 ( R 3 ) ) ∩ C 1 ( [ 0 , T ) × H 2 ( R 3 ) ) for system (5) and

supp ( u ( t , x ) , θ ( t , x ) ) ⊂ { x ∣ ∣ x ∣ ≤ t + 1 } , f o r 0 ≤ t < T .

We refer to [12,21] for the proof of the local existence of the hyperbolic system (5), and the finite speed of propagation can be obtained by using similar argument in [28] with minor modification.

Lemma 2.2

The following energy equality holds for the Euler system (1) in R 3 with γ = 2 :

(6) ∫ R 3 1 2 ( ρ − 1 ) 2 + 1 2 ρ u 2 d x + ∫ 0 t ∫ R 3 μ ρ u 2 ( 1 + ∣ x ∣ ) λ d x d τ = ε 2 2 ∫ R 3 ( ( 1 + ε ρ 0 ) u 0 2 + ρ 0 2 ) d x .

Proof

We multiply (1)2 by u . For the first term, according to (1)1, we can substitute − ∇ ⋅ ( ρ u ) for ρ t twice and obtain

(7) ∂ t ( ρ u ) ⋅ u = − ∇ ⋅ ( ρ u ) ∣ u ∣ 2 + 1 2 ( ρ u 2 ) t + 1 2 ∇ ⋅ ( ρ u ) u 2 .

At the same time, we have

(8) ∇ ⋅ ( ρ u ⊗ u ) ⋅ u = ∇ ⋅ ( ρ u ) ∣ u ∣ 2 + 1 2 ∇ ⋅ ( ρ u ∣ u ∣ 2 ) − 1 2 ∇ ⋅ ( ρ u ) u 2 .

For the third term, we substitute − ∇ ⋅ ( ρ u ) for ( ρ − 1 ) t and obtain

(9) ∇ P ⋅ u = ρ ∇ ( ρ − 1 ) ⋅ u = ∇ ⋅ [ ρ ( ρ − 1 ) u ] − ( ρ − 1 ) ∇ ⋅ ( ρ u ) = ∇ ⋅ [ ρ ( ρ − 1 ) u ] + 1 2 [ ( ρ − 1 ) 2 ] t .

Adding (7)–(9) together, we obtain

1 2 ( ρ u 2 ) t + 1 2 [ ( ρ − 1 ) 2 ] t + 1 2 ∇ ⋅ ( ρ u ∣ u ∣ 2 ) + ∇ ⋅ [ ρ ( ρ − 1 ) u ] + μ ρ u 2 ( 1 + ∣ x ∣ ) λ = 0 ,

integrating which over [ 0 , t ] × R 3 we come to

(10)□ ∫ 0 t ∫ R 3 1 2 ( ρ − 1 ) 2 + 1 2 ρ u 2 τ d x d τ + ∫ 0 t ∫ R 3 μ ρ u 2 ( 1 + ∣ x ∣ ) λ d x d τ = 0 .

Hence, the energy equality (6) follows.

Remark 2.3

For general γ > 1 , the energy equality (6) will be changed. Actually, if we define the pressure potential Q ( ρ ) as the solution to the ordinary differential equation

(11) ρ Q ′ ( ρ ) − Q ( ρ ) = P ( ρ ) , Q ( 0 ) = 0 ,

then the potential energy balance for (1) can be obtained

(12) ∂ t Q ( ρ ) + ∇ ⋅ ( Q ( ρ ) u ) + P ∇ ⋅ u = 0 .

Hence, by combining the corresponding kinetic energy balance

(13) ∂ t 1 2 ρ u 2 + ∇ ⋅ 1 2 ρ u u 2 + ∇ P ⋅ u + μ ρ u 2 ( 1 + ∣ x ∣ ) λ = 0 ,

the energy inequality for general γ > 1 follows as

(14) ∫ R 3 Q ( ρ ) − Q ′ ( 1 ) ( ρ − 1 ) − Q ( 1 ) + 1 2 ρ u 2 + ∫ 0 t ∫ R 3 μ ρ u 2 ( 1 + ∣ x ∣ ) λ d x d τ = ∫ R 3 1 γ ( γ − 1 ) ( ρ γ − 1 − γ ( ρ − 1 ) ) + 1 2 ρ u 2 d x + ∫ 0 t ∫ R 3 μ ρ u 2 ( 1 + ∣ x ∣ ) λ d x d τ ≜ ∫ R 3 N ( ρ ) γ ( γ − 1 ) + 1 2 ρ u 2 d x + ∫ 0 t ∫ R 3 μ ρ u 2 ( 1 + ∣ x ∣ ) λ d x d τ = ∫ R 3 1 γ ( γ − 1 ) ( ( 1 + ε ρ 0 ) γ − 1 − γ ε ρ 0 ) + 1 2 ( 1 + ε ρ 0 ) ( ε u 0 ) 2 d x ≤ C ( γ , ρ 0 , u 0 ) ε 2 ,

for small enough ε , and C ( γ , ρ 0 , u 0 ) is a positive constant depending on γ , ρ 0 , u 0 . For our choice of pressure P = 1 γ ρ γ , the pressure potential would be

Q ( ρ ) = 1 γ ( γ − 1 ) ρ γ ,

which is a convex function. Therefore, it is easy to obtain

N ( ρ ) = ρ γ − 1 − γ ( ρ − 1 ) = γ ( γ − 1 ) [ Q ( ρ ) − Q ′ ( 1 ) ( ρ − 1 ) − Q ( 1 ) ] ≥ 0 .

The next lemma is concerned to the existence and asymptotic behavior of the solution to an elliptic system, which will be used as the test functions.

Lemma 2.4

Consider the first-order differential system:

(15) − G 1 ( x ) + ∇ ⋅ G 2 → ( x ) = 0 , − G 2 → ( x ) + ∇ G 1 ( x ) = μ G 2 → ( x ) ( 1 + ∣ x ∣ ) λ ,

where μ > 0 , λ > 1 , G 1 ( x ) , and G 2 → ( x ) represent a scalar function and a vector-valued function, respectively. Let

(16) G 1 ( x ) = h ( x ) F ( x ) , G 2 → ( x ) = ∇ F ( x ) + F ( x ) g → ( x ) ,

where F ( x ) = ∫ S 2 e x ⋅ w d w . Then, the differential system (15) admits C 2 solutions in the form of (16) satisfying

(17) lim ∣ x ∣ → + ∞ h ( x ) = 1 , lim ∣ x ∣ → + ∞ ∂ x i h ( x ) = 0 , lim ∣ x ∣ → + ∞ g → ( x ) = 0 , lim ∣ x ∣ → + ∞ ∂ x i g j → ( x ) = 0 .

Proof

Denoting Q ( x ) = μ ( 1 + ∣ x ∣ ) λ , then according to (15) 2 , we can express g → ( x ) with h ( x ) as

(18) g → ( x ) = ∇ h + h ∇ ln F 1 + Q − ∇ ln F .

Substituting (18) into (15) 1 , we obtain an elliptic equation for h ( x )

(19) Δ h + 2 ∇ ln F + λ ( 1 + r ) r Q 1 + Q x ⋅ ∇ h + − Q + λ ( 1 + r ) r Q 1 + Q x ⋅ ∇ ln F h = 0 .

Noting the exact formula for F in (39), and the fact that all the coefficients are radially symmetrical, there exists radial solution h ( r ) for (19), and it can be rewritten as

(20) h ″ ( r ) + 2 + 4 e − r e r − e − r + λ Q ( 1 + Q ) ( 1 + r ) h ′ ( r ) + − Q + λ Q ( 1 + Q ) ( 1 + r ) e r + e − r e r − e − r − 1 r h ( r ) = 0 .

Let

Y ( r ) = h ( r ) h ′ ( r ) ,

we then obtain the equation in matrix form for Y ( r ) from (20)

(21) Y ′ ( r ) = ( A + B ( r ) ) h ( r ) h ′ ( r ) ,

where

A = 0 1 0 − 2 , B ( r ) = 0 0 m ( r ) n ( r )

with

m ( r ) = − − Q + λ Q ( 1 + Q ) ( 1 + r ) e r + e − r e r − e − r − 1 r , n ( r ) = − 4 e − r e r − e − r + λ Q ( 1 + Q ) ( 1 + r ) .

It is easy to verify that B ( r ) ∈ L 1 [ 1 , ∞ ) , we then can apply the Levinson theorem (see, e.g., [[3], Chapter 3, Theorem 8.1]) to the differential system (21) to conclude that there exist two linearly independent solutions

(22) Y 1 ( r ) = 1 + o ( 1 ) o ( 1 ) , Y 2 ( r ) = 1 + o ( 1 ) − 2 + o ( 1 ) e − 2 r .

Hence, the following second-order ordinary differential equation:

(23) h ″ ( r ) + ( 2 − n ( r ) ) h ′ ( r ) − m ( r ) h = 0 , r ≥ 1 , h ( 1 ) = 1 , h ′ ( 1 ) = δ > 0 ,

admits a unique solution in C 2 ( [ 1 , ∞ ) ) , in addition, there exist two constants C 1 , C 2 such that the solution of (23) must have the form

(24) h ( r ) h ′ ( r ) = C 1 1 + o ( 1 ) o ( 1 ) + C 2 1 + o ( 1 ) − 2 + o ( 1 ) e − 2 r .

To meet the initial conditions in (23), it is easy to conclude that the constants C 1 , C 2 should satisfy

C 1 > 0 , C 2 < 0 ,

which in turn implies that

(25)□ h ( r ) → C 1 , h ′ ( r ) → 0 , as r → + ∞ .

Actually, we can give some convincing figures for specific examples by numerical modeling, which are listed below and show the asymptotic behavior for h ( r ) as r → ∞ .

The above picture represents the behavior for h ( r ) with μ = 1 , λ = 2 , while the below one corresponds to the case μ = 0.5 , λ = 1.2 , by assuming that h ( 0.01 ) = h ′ ( 0.01 ) = 1 .

Set

h ˜ ( r ) = h ( r ) C 1 ,

then h ˜ ( r ) still solves (19) and satisfies

(26) h ˜ ( r ) → 1 , h ˜ ′ ( r ) → 0 , as r → + ∞ .

In the following, we still denote h ˜ ( r ) by h ( r ) for simplicity. Combining the asymptotic behavior (26) and equation (20), it is easy to see

(27) h ″ ( r ) → 0 , as r → + ∞ ,

which implies

(28) lim ∣ x ∣ → ∞ ∂ i h = lim ∣ x ∣ → ∞ h ′ ( r ) x i r = 0 , lim ∣ x ∣ → ∞ ∂ i ∂ j h = lim ∣ x ∣ → ∞ h ″ ( r ) x i x j r 2 + h ′ ( r ) δ i j r − h ′ ( r ) x i x j r 3 = 0 .

Noting formula (18) and the fact

lim ∣ x ∣ → ∞ Q = 0 , lim ∣ x ∣ → ∞ h ( r ) = 1 ,

we have

(29) lim ∣ x ∣ → ∞ g → ( x ) = lim ∣ x ∣ → ∞ ∇ h + h ∇ ln F 1 + Q − ∇ ln F = 0 .

Also, taking the gradient to (18), we come to

(30) ∇ g → = ∇ 2 h + ∇ h ⊗ ∇ ln F + h ∇ 2 ln F 1 + Q + Q ( 1 + Q ) 2 λ r ( 1 + r ) x ⊗ ( ∇ h + h ∇ ln F ) − ∇ 2 ln F .

Noting the fact that ∣ ∇ F ∣ ≤ K (see (38) in Lemma 2.6), it is not difficult to verify that

(31) lim ∣ x ∣ → + ∞ ∂ i g j ( x ) = 0 .

In the region ∣ x ∣ ≤ 1 , we will consider the elliptic system (15) directly. It is easy to see that G 1 ( x ) satisfies

(32) Δ G 1 − ∂ x i 1 + μ ( 1 + ∣ x ∣ ) λ ⋅ ∂ x i G 1 1 + μ ( 1 + ∣ x ∣ ) λ − 1 + μ ( 1 + ∣ x ∣ ) λ G 1 = 0 ,

we then may consider the Dirichlet problem for G 1 ( x ) in ∣ x ∣ ≤ 1 with a boundary condition

(33) G 1 ( x ) = h ( x ) F ( x ) , ∣ x ∣ = 1 .

By using proposition (2.5) from elliptic theory stated below, the Dirichlet problem (32) and (33) admits a unique solution in C 2 ( B 1 ¯ ) . Then, Lemma 2.4 follows from uniqueness of solution.

Proposition 2.5

(Corollary 6.9 in [6]) Denote the elliptic operator

L = a i j ( x ) D i j + b i ( x ) D i + c ( x ) .

Let Ω be a ball in R n , and let the operator L be strictly elliptic in Ω with coefficients in C α ( Ω ¯ ) and with c ( x ) ≤ 0 . Then, if f ∈ C α ( Ω ¯ ) and φ ∈ C 2 , α ( Ω ¯ ) , the Dirichlet problem:

L u ( x ) = f ( x ) i n Ω , u ( x ) = φ ( x ) o n ∂ Ω

has a unique solution u ∈ C 2 , α ( Ω ¯ ) .

Lemma 2.6

Let F ( x ) = ∫ S 2 e x ⋅ w d w , it holds

(34) ( ∇ 2 F ) ⋅ ( u ⊗ u ) ≥ ∂ r r F ( x ⋅ u ) 2 r 2 ,

(35) F ( x ) ∼ e r 1 + r ,

(36) ∂ r F ≲ F ( x ) ,

(37) ∂ r r F ∼ e r 1 + r ∼ F ( x ) ,

(38) ∣ ∇ ln F ∣ ≤ K ,

where K > 0 is a constant.

Proof

In 3 − D , the radial function F ( x ) can be written as (see page 308 in [18])

(39) F ( r ) = C ∫ 0 1 e r w 1 d w 1 + ∫ 0 1 e − r w 1 d w 1 = C r ( e r − e − r ) ,

then (35) follows. Also, we have

(40) ∂ r F ( r ) = C r ( e r + e − r ) − ( e r − e − r ) r 2 , ∂ r r F ( r ) = C e r − e − r r − 2 ( e r + e − r ) r 2 + 2 ( e r − e − r ) r 3 ,

it is easy to obtain (36) and (37). Next, we prove (34). Since

∂ i F = ∂ r F x i r ,

∂ i ∂ j F = ∂ j ∂ r F x i r = ∂ r r F x i x j r 2 + ∂ r F δ i j r − x i x j r 3 ,

then we have

∑ i , j ( ∂ i ∂ j F ) u i u j = ∂ r r F ( x ⋅ u ) 2 r 2 + ∂ r F u 2 r − ( x ⋅ u ) 2 r 3 ≥ ∂ r r F ( x ⋅ u ) 2 r 2 ,

which implies (34). Finally, we prove (38). According to (36), it is easy to see there exists a positive constant K such that

∣ ∇ ln F ∣ = ∣ ∇ F ∣ F = ∣ ∂ r F x r ∣ F ≤ ∂ r F F ≤ K ,

then (38) follows.□

3 Proof of the Main Theorem for γ = 2

Now, we are going to prove Theorem 1.3.

Let ϕ ( t , x ) = e − t G 1 ( x ) , ψ → ( t , x ) = e − t G 2 → ( x ) , where G 1 ( x ) , G 2 → ( x ) are defined as in Lemma 2.4. Rewrite the first equation in (1) as

( ρ − 1 ) t + ∇ ⋅ ( ρ u ) = 0 .

Then, multiplying it with ϕ ( t , x ) , we have

(41) [ ϕ ( ρ − 1 ) ] t − ϕ t ( ρ − 1 ) + ∇ ⋅ ( ϕ ρ u ) − ∇ ϕ ⋅ ( ρ u ) = 0 .

Multiplying the second equation in (1) with ψ → ( t , x ) , since ∇ P = ∇ ( ρ 2 − 1 2 ) , we have

(42) [ ψ → ⋅ ( ρ u ) ] t − ψ → t ⋅ ( ρ u ) + ∇ ⋅ [ ψ → ⋅ ( ρ u ⊗ u ) ] − ( ∇ ψ → ) ⋅ ( ρ u ⊗ u ) + ∇ ⋅ ψ → ⋅ ρ 2 − 1 2 − ∇ ⋅ ψ → ( ρ − 1 ) 2 2 + ( ρ − 1 ) + ψ → ⋅ μ ρ u ( 1 + ∣ x ∣ ) λ = 0 .

Adding (41) and (42) together and then integrating over R 3 , we obtain

(43) d d t ∫ R 3 ϕ ( ρ − 1 ) d x + ∫ R 3 ψ → ⋅ ( ρ u ) d x − ∫ R 3 ( ϕ t + ∇ ⋅ ψ → ) ( ρ − 1 ) d x − ∫ R 3 ∇ ϕ + ψ → t − μ ψ → ( 1 + ∣ x ∣ ) λ ⋅ ( ρ u ) d x − ∫ R 3 ( ∇ ψ → ) ⋅ ( ρ u ⊗ u ) d x − ∫ R 3 ∇ ⋅ ψ → ( ρ − 1 ) 2 2 d x = 0 .

According to the differential system (15) in Lemma 2.4, we have

ϕ t + ∇ ⋅ ψ → = 0 , ψ → t + ∇ ϕ = μ ψ → ( 1 + ∣ x ∣ ) λ .

Therefore, (43) becomes

(44) d d t ∫ R 3 e − t G 1 ( ρ − 1 ) d x + ∫ R 3 e − t G 2 → ⋅ ( ρ u ) d x = ∫ R 3 e − t ∇ G 2 → ⋅ ( ρ u ⊗ u ) d x + ∫ R 3 e − t ( ∇ ⋅ G 2 → ) ( ρ − 1 ) 2 2 d x .

Before going on, we first claim that there exists a positive constant R 1 such that for r ≥ R 1

(45) ∇ G 2 → ( u ⊗ u ) ≥ 1 2 ∇ 2 F ( u ⊗ u ) ≥ 1 2 ∂ r r F ( x ⋅ u ) 2 r 2 , ∇ ⋅ G 2 → ≥ 1 2 F .

According to (16), we have

(46) ∇ G 2 → u i u j = ∂ i ∂ j F u i u j + ∂ i g j F u i u j + g i ∂ j F u i u j .

Also, it is easy to obtain from (30) that

(47) ∂ i g j F = h ″ ( r ) x i x j r 2 + h ′ ( r ) δ i j r − x i x j r 3 F 1 + Q + h ′ ( r ) ∂ r F x i x j r 2 1 + Q + h 1 + Q − 1 F ″ ( r ) x i x j r 2 + F ′ ( r ) δ i j r − x i x j r 3 + 1 − h 1 + Q ( ∂ r F ) 2 F x i x j r 2 + λ Q ( 1 + Q ) 2 r ( 1 + r ) ( h ′ ( r ) F + h ∂ F ) x i x j r 2 ≜ I 1 + I 2 + I 3 + I 4 + I 5 , g i ∂ j F = h ′ ( r ) F ′ ( r ) 1 + Q x i x j r 2 + h 1 + Q − 1 ( F ′ ( r ) ) 2 F x i x j r 2 ≜ I 6 + I 7 .

Noting the fact

lim r → ∞ h ( r ) = 1 , lim r → ∞ h ′ ( r ) = lim r → ∞ h ″ ( r ) = lim r → ∞ Q ( r ) = 0 , F ′ ( r ) ≲ F , F ( r ) ∼ F ″ ( r ) for r ≥ 0 , F ( r ) ≤ 2 F ′ ( r ) for r ≥ 2 ,

there exists a positive constant R 1 ( ≥ 2 ) such that for r ≥ R 1

(48) I 1 ⋅ u i u j , I 3 ⋅ u i u j ≤ 1 14 ∇ 2 F ( u ⊗ u )

and

(49) ( I 2 , I 4 , I 5 , I 6 , I 7 ) ⋅ u i u j ≤ 1 14 ∂ r r F ( x ⋅ u ) 2 r 2 ,

then claim (45)1 follows. For claim (45)2, it follows from the fact

(50) ∇ ⋅ G 2 → = F ( r ) + F ′ ( r ) g i ( r ) x i r + F g i ′ ( r ) x i r

and

lim r → ∞ g i ( r ) = lim r → ∞ g i ′ ( r ) = 0 , 0 ≤ F ′ ( r ) ≲ F ( r ) .

Conversely, we may conclude from (46) and (50) that for ∣ x ∣ ≤ R 1

(51) ∇ G 2 → ≤ C ∇ 2 F , ∇ ⋅ G 2 → ≤ C F ,

where C denotes some positive constant and the first inequality means that the difference between the matrices is semi-positive.

Hence, we have

(52) ∫ R 3 e − t ∇ G 2 → ⋅ ( ρ u ⊗ u ) d x = ∫ ∣ x ∣ ≥ R 1 e − t ∇ G 2 → ⋅ ( ρ u ⊗ u ) d x + ∫ ∣ x ∣ ≤ R 1 e − t ∇ G 2 → ⋅ ( ρ u ⊗ u ) d x ≥ 1 2 ∫ ∣ x ∣ ≥ R 1 e − t ρ ∂ r r F ( x ⋅ u ) 2 r 2 d x − C e − t ∫ ∣ x ∣ ≤ R 1 ∇ 2 F ⋅ ( ρ u ⊗ u ) d x = 1 2 ∫ R 3 e − t ρ ∂ r r F ( x ⋅ u ) 2 r 2 d x − 1 2 ∫ ∣ x ∣ ≤ R 1 e − t ρ ∂ r r F ( x ⋅ u ) 2 r 2 d x − C e − t ∫ ∣ x ∣ ≤ R 1 ∇ 2 F ⋅ ( ρ u ⊗ u ) d x ≥ 1 2 ∫ R 3 e − t ρ ∂ r r F ( x ⋅ u ) 2 r 2 d x − C 3 e − t ∫ ∣ x ∣ ≤ R 1 ρ u 2 d x ≥ 1 2 ∫ R 3 e − t ρ ∂ r r F ( x ⋅ u ) 2 r 2 d x − C 3 e − t ∫ R 3 ρ u 2 d x .

In the same way, we can obtain

(53) ∫ R 3 e − t ( ∇ ⋅ G 2 → ) ( ρ − 1 ) 2 2 d x = 1 2 ∫ ∣ x ∣ ≥ R 1 e − t ( ∇ ⋅ G 2 → ) ( ρ − 1 ) 2 d x + ∫ ∣ x ∣ ≤ R 1 e − t ( ∇ ⋅ G 2 → ) ( ρ − 1 ) 2 2 d x ≥ 1 4 ∫ ∣ x ∣ ≥ R 1 e − t F ( ρ − 1 ) 2 d x − C e − t ∫ ∣ x ∣ ≤ R 1 F ( ρ − 1 ) 2 2 d x = 1 4 ∫ R 3 e − t F ( ρ − 1 ) 2 d x − 1 2 + C e − t ∫ ∣ x ∣ ≤ R 1 F ( ρ − 1 ) 2 2 d x ≥ 1 4 ∫ R 3 e − t F ( ρ − 1 ) 2 d x − C 4 e − t ∫ R 3 ( ρ − 1 ) 2 2 d x .

Substituting the above two inequalities into (44) and combining the energy inequality (6) in Lemma 2.2, we obtain

(54) d d t ∫ R 3 e − t G 1 ( ρ − 1 ) d x + ∫ R 3 e − t G 2 → ⋅ ( ρ u ) d x ≥ 1 2 ∫ R 3 e − t ρ ∂ r r F ( x ⋅ u ) 2 r 2 d x + 1 4 ∫ R 3 e − t F ( ρ − 1 ) 2 d x − C 5 e − t ∫ R 3 ρ u 2 + ( ρ − 1 ) 2 2 d x ≥ 1 2 ∫ R 3 e − t ρ ∂ r r F ( x ⋅ u ) 2 r 2 d x + 1 4 ∫ R 3 e − t F ( ρ − 1 ) 2 d x − C 5 e − t ε 2 .

Set Z ( t ) = e − t ∫ R 3 G 1 ( ρ − 1 ) d x + ∫ R 3 G 2 → ⋅ ( ρ u ) d x . Since h ( x ) ∈ C 2 ( R 3 ) and lim ∣ x ∣ → ∞ h ( x ) = 1 , then ∣ G 1 ( x ) ∣ = ∣ h ( x ) F ( x ) ∣ ≲ F ( x ) . Recalling the support property of the solutions in Proposition 2.1, we have

(55) Z 2 ( t ) ≤ 2 e − 2 t ∫ R 3 G 1 ( ρ − 1 ) d x 2 + 2 e − 2 t ∫ R 3 G 2 → ⋅ ( ρ u ) d x 2 ≲ e − 2 t ∫ R 3 F ( ρ − 1 ) 2 d x ∫ ∣ x ∣ ≤ t + 1 F d x + e − 2 t ∫ R 3 G 2 → ⋅ ( ρ u ) d x 2 ≲ e − 2 t ∫ R 3 F ( ρ − 1 ) 2 d x ∫ 0 t + 1 e r ( 1 + r ) d r + e − 2 t ∫ R 3 G 2 → ⋅ ( ρ u ) d x 2 ≲ e − t ( 1 + t ) ∫ R 3 F ( ρ − 1 ) 2 d x + e − 2 t ∫ R 3 G 2 → ⋅ ( ρ u ) d x 2 .

We are in a position to deal with the last term of (55). Direct computation implies

(56) ∫ R 3 G 2 → ⋅ ( ρ u ) d x = ∫ R 3 ( ρ u ) ⋅ ∂ r F x r + F ∣ h ′ ( r ) ∣ + h ∂ r F 1 + Q − ∂ r F x r d x ≲ ∫ R 3 ρ ( u ⋅ x ) 2 r 2 e r 1 + r d x 1 2 ∫ ∣ x ∣ ≤ t + 1 ρ e r 1 + r d x 1 2 ≲ ∫ R 3 ρ ( u ⋅ x ) 2 r 2 ∂ r r F d x 1 2 ∫ ∣ x ∣ ≤ t + 1 e r 1 + r d x + ∫ ∣ x ∣ ≤ t + 1 ( ρ − 1 ) e r 1 + r d x 1 2 ≲ ∫ R 3 ρ ( u ⋅ x ) 2 r 2 ∂ r r F d x 1 2 e t ( 1 + t ) + ∫ ∣ x ∣ ≤ t + 1 ( ρ − 1 ) e r 1 + r d x 1 2 ,

where we used (35), (36) in Lemma 2.6, the asymptotic behavior of h and h ′ ( r ) and the support property in Proposition 2.1. For the last term in (56), by Hölder inequality, we obtain

(57) ∫ ∣ x ∣ ≤ t + 1 ( ρ − 1 ) e r 1 + r d x ≲ ∫ R 3 ( ρ − 1 ) 2 d x 1 2 ∫ ∣ x ∣ ≤ t + 1 e 2 r ( 1 + r ) 2 d x 1 2 ≲ e t ε ,

here ε comes from the energy inequality (6) in Lemma 2.2. We then plug (56), (57) into (55), and combine (54) to obtain

(58) Z 2 ( t ) ≲ e − t ( 1 + t ) ∫ R 3 F ( ρ − 1 ) 2 d x + e − 2 t ∫ R 3 ρ ( u ⋅ x ) 2 r 2 ∂ r r F d x ( e t ( 1 + t ) + e t ε ) ≲ e − t ( 1 + t ) ∫ R 3 F ( ρ − 1 ) 2 d x + ∫ R 3 ρ ( u ⋅ x ) 2 r 2 ∂ r r F d x ≲ e − t ( 1 + t ) e t d Z ( t ) d t + C 6 ε 2 ,

which means

(59) d Z ( t ) d t ≥ C 7 Z 2 ( t ) 1 + t − C 8 e − t ε 2 .

Integrating (59) over [ 0 , t ] yields

(60) Z ( t ) ≥ ∫ 0 t C 7 Z 2 ( τ ) 1 + τ d τ + Z ( 0 ) − C 8 ε 2 ≥ 1 2 C 0 ε − C 8 ε 2 ≥ C 9 ε

for small enough ε , and C 0 is defined in Theorem 1.3. Hence, there exists t 0 > 0 independent of ε such that

(61) 1 2 C 7 Z 2 ( t ) 1 + t ≥ 1 2 C 7 C 9 2 ε 2 1 + t ≥ C 8 e − t ε 2 for t ≥ t 0 .

Let Z ˜ ( t ) satisfy the following differential system:

(62) d Z ˜ ( t ) d t = 1 2 C 7 Z ˜ 2 ( t ) 1 + t , t ≥ t 0 , Z ˜ ( t 0 ) = C 9 ε ,

then Z ˜ ( t ) will blow up in a finite time and the lifespan estimate satisfies

(63) T ( ε ) ≤ exp ( C ε − 1 ) .

However, Z ( t ) satisfies

(64) d Z ( t ) d t ≥ 1 2 C 7 Z 2 ( t ) 1 + t , t ≥ t 0 , Z ( t 0 ) ≥ C 9 ε ,

which means that Z ( t ) ≥ Z ˜ ( t ) for t ≥ t 0 , and hence, the lifespan for Z ( t ) also satisfies (63), we complete the proof of Theorem 1.3.

4 Remark for γ > 2

If γ > 2 , the last nonlinear term in (44) will change to

(65) ∫ R 3 e − t ( ∇ ⋅ G 2 → ) N ( ρ ) γ d x

with

(66) N ( ρ ) ≜ ρ γ − 1 − γ ( ρ − 1 ) ∼ ∣ ρ − 1 ∣ γ if  ρ → + ∞ ∣ ρ − 1 ∣ 2 if  ρ → 1 C if  ρ → 0 ≳ ∣ ρ − 1 ∣ 2 ,

and then the computation in (53) will be replaced by

(67) ∫ R 3 e − t ( ∇ ⋅ G 2 → ) N ( ρ ) γ d x = 1 γ ∫ ∣ x ∣ ≥ R 1 e − t ( ∇ ⋅ G 2 → ) N ( ρ ) d x + 1 γ ∫ ∣ x ∣ ≤ R 1 e − t ( ∇ ⋅ G 2 → ) N ( ρ ) d x ≥ 1 2 γ ∫ ∣ x ∣ ≥ R 1 e − t F N ( ρ ) d x − C e − t ∫ ∣ x ∣ ≤ R 1 F N ( ρ ) d x ≳ ∫ R 3 e − t F ( ρ − 1 ) 2 d x − e − t ∫ R 3 N ( ρ ) d x ≳ ∫ R 3 e − t F ( ρ − 1 ) 2 d x − e − t ε 2 ,

where the energy equality (14) is used. With (67) in hand, the left steps of the proof for γ > 2 are exactly the same as that for γ = 2 .

Acknowledgement

The authors would like to express their sincere thanks to Prof. Chengbo Wang, Prof. Naqing Xie, and Prof. Yi Zhou for the helpful discussion. The authors also would like to express their sincere gratitude to the reviewers for the helpful comments and suggestions.

  1. Funding information: J. B. Geng and N.-A Lai were partially supported by the NSFC (12271487), K. Hu and N.-A. Lai were partially supported by the NSFC (12171097), and M. W. Yuen was supported by the Dean’s Research Fund (IRS-4/7th round), the Education University of Hong Kong.

  2. Author contributions: 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. All authors contributed equally to this article.

  3. Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.

  4. Ethical approval: Compliance with ethical standards.

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Received: 2023-08-02
Revised: 2024-04-07
Accepted: 2024-09-17
Published Online: 2024-10-17

© 2024 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/anona-2024-0043
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compressible Euler equations; damping; blow-up; lifespan; test function method
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  43. Boussinesq's equation for water waves: Asymptotics in Sector I
  44. Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
  45. Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
  46. Normalized solutions for the double-phase problem with nonlocal reaction
  47. Normalized solutions for Sobolev critical fractional Schrödinger equation
  48. Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
  49. Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
  50. Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
  51. Quasiconvex bulk and surface energies: C1,α regularity
  52. Time decay estimates of solutions to a two-phase flow model in the whole space
  53. Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
  54. The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
  55. A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
  56. Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
  57. Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
  58. The second Yamabe invariant with boundary
  59. Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
  60. Normalized solutions for the Choquard equations with critical nonlinearities
  61. Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
  62. Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
  63. Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
  64. Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
  65. Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
  66. Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
  67. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
  68. On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
  69. Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
  70. Uniqueness and nondegeneracy of ground states for − Δ u + u = ( I α ⋆ u 2 ) u in R 3 when α is close to 2
  71. Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
  72. Weighted Hardy-Adams inequality on unit ball of any even dimension
  73. Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
  74. Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
  75. Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
  76. The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
  77. Nonlocal heat equations with generalized fractional Laplacian
  78. Choquard equations with recurrent potentials
  79. Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
  80. The Riemann problem for two-layer shallow water equations with bottom topography
  81. Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
  82. On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
  83. Energy-variational solutions for viscoelastic fluid models
  84. Stability on 3D Boussinesq system with mixed partial dissipation
  85. Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
  86. Erratum
  87. Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
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Heruntergeladen am 7.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/anona-2024-0043/html
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