Global regularity of the axisymmetric Navier–Stokes model with logarithmically supercritical dissipation

Nonlinear Analysis 92 (2013) 24–29

Contents lists available at SciVerse ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Global regularity of the axisymmetric Navier–Stokes model with logarithmically supercritical dissipation✩ Jianli Liu a,∗ , Keyan Wang b a

Department of Mathematics, Shanghai University, Shanghai 200444, China

b

Department of Applied Mathematics, Shanghai Finance University, 201209, China

article

info

Article history: Received 8 April 2013 Accepted 28 June 2013 Communicated by Enzo Mitidieri MSC: 35B65 35Q30

abstract In this paper, we investigate the global regularity of a generalized axisymmetric model which was proposed in Hou and Lei (2009) [4] to study the role of convection in incompressible Navier–Stokes equations. By introducing a logarithmically supercritical dis7

sipation whose symbol satisfies m(ξ ) ≥ nondecreasing function g (s) satisfying

Keywords: Axisymmetric Navier–Stokes equations Global regularity Hyperdissipation

|ξ | 4

g (|ξ |) +∞ ds 1 sg 4 (s)



for all sufficiently large |ξ | and some

= +∞, we obtain the global regular-

ity of the generalized axisymmetric Navier–Stokes model. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction and main results Incompressible 3D Navier–Stokes equations are one of the most fundamental equations in fluid dynamics [1,2], which describe many interesting fluid phenomena. Precisely, it can be written in the following form: ut + (u · ∇)u + ∇ P − µ∆u = 0, ∇ · u = 0, u|t =0 = u0 (x), x = (x1 , x2 , z ).



(1.1)

In [3], Hou and Li investigated the stabilizing effect of convection. They showed that the convection term should play an essential role in canceling the destabilizing vortex term in an exact 1D model that can be used to construct a family of exact solutions of the 3D Euler or Navier–Stokes equations. The role of the convection for 3D Euler and Navier–Stokes equations have been investigated in [4], where the authors have constructed a 3D model for axisymmetric flows with swirl by neglecting the convection term from the reformulated axisymmetric 3D Navier–Stokes equations introduced in [3]. Using the

✩ This work was supported by NSFC (No. 11126058), Excellent Young Teachers Program of Shanghai and the First-class Discipline of Universities in Shanghai. ∗ Corresponding author. Tel.: +86 2166132182. E-mail addresses: [email protected] (J. Liu), [email protected] (K. Wang).

0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.06.018

J. Liu, K. Wang / Nonlinear Analysis 92 (2013) 24–29

25

cylindrical coordinates, we can get the following axisymmetric form of the Navier–Stokes equations

   ur uθ 1  θ θ r θ z θ  u − , ∂ u + u ∂ u + u ∂ u = µ ∆ −  t r z x   r2  r      1 (uθ )2 ur ω θ θ r θ z θ θ ∂t ω + u ∂r ω + u ∂z ω = µ ∆x − 2 ω + ∂z + ,  r r r      1   − ∆x − 2 ψ θ = ωθ .

(1.2)

r

In [3], the authors introduced the new variables

v=

uθ

,

w=

ωθ

,

φ=

r r and derived an equivalent system as

ψθ r

,

(1.3)

   3 r z 2 2   ∂t v + u ∂r v + u ∂z v = µ ∂r + r ∂r + ∂z v + 2∂z φv,      3 ∂t w + ur ∂r w + uz ∂z w = µ ∂r2 + ∂r + ∂z2 w + ∂z (v 2 ),  r      3  2 2  ∂ + ∂r + ∂ φ = w, z r

(1.4)

r

where 1

∂r (r 2 φ). r By dropping the convection term in the (1.4), the model follows as ur = −∂z (r φ),

uz =

(1.5)

   3 2 2   ∂t v = µ ∂r + r ∂r + ∂z v + 2∂z φv,      3 2 2 ∂t w = µ ∂r + ∂r + ∂z w + ∂z (v 2 ),  r      3  2 − ∂ + ∂r + ∂ 2 φ = w. z r

(1.6)

r

This model shares many properties of the 3D incompressible Euler and Navier–Stokes equations (more Refs. [4,5]), including an energy identity for its classical solutions, a non-blowup criterion of Beale–Kato–Majda type [6] as well as a nonblowup criterion of Prodi–Serrin type [7,8]. In [9], the authors further established a new partial regularity result for the model which is an analogue of the Caffarelli–Kohn–Nirenberg theory [10] for the full Navier–Stokes equations. The main difference between the above 3D model and the reformulated 3D Euler and Navier–Stokes equations is that the convection term is neglected. As has been pointed out in [11], it is much more convenient to post this 3D model in the five-dimensional setting to perform the well-posedness and related analysis. In the following we will work this problem in five dimensional space. We propose the generalized Hou–Lei model which has slightly supercritical dissipation as following:

 ∂ v = −D2 v + 2∂z φv,   t ∂t w = −D2 w + ∂z (v 2 ),  −△φ = w, t = 0 : v = v0 (x), w = w0 (x),

(1.7)

φ = φ0 (x),

for v : R+ × R5 → R5 , φ : R+ × R5 → R5 , x = (x1 , x2 , x3 , x4 , z ), µ = 1 and ∆ = 5

4

+

i =1

∂x2i + ∂z2 . D is a Fourier multiplier

whose symbol m : R → R is non-negative. We will obtain the following global regularity result of (1.7) with large initial data: Theorem 1.1. Suppose the initial data satisfy

v0 ∈ H 4 (R5 ),

φ0 ∈ H 5 (R5 ),

−△φ0 = w0

(1.8)

and m obeys the lower bound 7

m(ξ ) ≥

|ξ | 4 g (|ξ |)

for all sufficiently large ξ and some nondecreasing function g : R+ → R+ , such that global classical solution (u, w, φ).

(1.9)

 +∞ 1

ds sg 4 (s)

= +∞. Then system (1.7) has a

26

J. Liu, K. Wang / Nonlinear Analysis 92 (2013) 24–29

As we known, for n ≥ 3, the global regularity of the Navier–Stokes system is of course a difficult unsolved problem. This supercriticality can be avoided by the strengthening the dissipative symbol m(ξ ), for instance setting m(ξ ) = |ξ |α for 2 some α > 1. This hyperdissipation variant of the Navier–Stokes equation becomes subcritical for α > n+ and critical for 4

α=

n +2 . It is known that global regularity can be recovered in these cases, see [12]. For 1 4

≤α<

n +2 , only partial regularity 4

results are known; see [10] for the α = 1 case and [12] for the α > 1 case. Recently, Tao obtained the global regularity of the generalized Navier–Stokes system under the same logarithmically supercritical hyperdissipation in [13]. For the inviscid Hou–Lei model, Tao and Wu obtained the global regularity to include dissipation given by a general fractional Laplacian in [5]. In [14], the authors proved that the 3D inviscid Hou–Lei model can develop a finite time singularity starting from smooth initial data on a rectangular domain. A global well-posedness result was also proved for a class of smooth initial data with some smallness condition. In [11], they give that the 3D inviscid model with an appropriate Neumann–Robin boundary condition will develop a finite time singularity starting from smooth initial data in an axisymmetric domain and has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. In particular, the results apply for the logarithmically supercritical dissipation m(ξ ) =

7 |ξ | 4 1 log 4 (e+|ξ |2 )

. Analogous ‘‘barely

supercritical’’ global regularity results were established for nonlinear wave equation [15–17], MHD system [18] and related topics [19,20]. 2. The proof of Theorem 1.1 In this section, we will give the proof of Theorem 1.1 using the standard energy method arguments [21]. Firstly, we rewrite system (1.7) as follows:

 ∂t v = −D2 v + 2(∂z φ)v, (2.1) ∂ (△φ) = −D2 (△φ) − ∂z (v 2 ),  t t = 0 : v = v0 (x), φ = φ0 (x).  Basic energy estimate. Denote E (t ) = R5 |v|2 + 2|∇φ|2 dx. For any fixed T , we multiply v to the first equation of system

(2.1) and integrate it in R5 to obtain



d dt

|v|2 dx = −2



R5

R5

|Dv|2 dx + 4

 R5

|v|2 ∂z φ dx.

(2.2)

Multiplying φ to the second equation of system (2.1) and integrating by parts, we have



d dt

|∇φ|2 dx = −2 R5



|D∇φ|2 dx + 2



R5

R5

∂z (|v|2 )φ dx.

(2.3)

Then



d

2

2



|v| + 2|∇φ| dx = −2

dt

R5

where a(t ) =

R5

. |Dv|2 + 2|D∇φ|2 dx = −2a(t ),

|Dv|2 + 2|D∇φ|2 dx. Integrating (2.4) in time, we obtain the energy dissipation bound  T  a(t )dt ≤ |v0 |2 + 2|∇φ0 |2 dx ≤ C , 

(2.4)

R5

(2.5)

R5

0

where C denotes some positive constant, possibly depending on E (0), T and g. Higher energy estimate. Now we consider the higher energy E4 ( t ) =

4   j=0

R5

|∇ j v|2 + 2|∇ j+1 φ|2 dx.

(2.6)

Differentiating (2.6) and integrating by parts, we obtain dE4 (t ) dt

=

 4  2

R5

j =0

=−

2

R5

 4  4

j =0

 4  4

R5

j =0

 4  j =0

−

(∇ j v)(∇ j vt )dx +

R5

|∇ j Dv|2 dx + 4

4   j =0

|∇ j+1 Dφ|2 dx +

R5

(∇ j+1 φ)(∇ j+1 φt )dx

∇ j v∇ j ((∂z φ)v)dx

 4  4

j =0

R5

∇ j+1 φ∇ j+1 △−1 (∂z (v 2 ))dx

(2.7)

J. Liu, K. Wang / Nonlinear Analysis 92 (2013) 24–29

=−

 4  2

R5

j =0

+4

4  

|∇ j Dv|2 + 2|∇ j+1 Dφ|2 dx

∇

j+1

R5

j =0

27

φ∇ ((∂z φ)v)dx + 8 j

4  

∇ j+1 φ∇ j+1 △−1 (v∂z v)dx.

R5

j=0

(2.8)

We apply the Leibniz rule to the second term of (2.8) and the contribution of integration by parts to get 4 



j=0 0≤j1 ,j2 ≤j,j1 +j2 =j

 R5

O(∇ j v∇ j1 ∂z φ∇ j2 v)dx

(2.9)

where O(∇ j v∇ j1 ∂z φ∇ j2 v) denotes some constant coefficient trilinear combination of the components of ∇ j v, ∇ j1 ∂z φ, ∇ j2 v . We can integrate by parts using D and D−1 and using the Cauchy–Schwartz inequality to bound

 R5

O(∇ j v∇ j1 ∂z φ∇ j2 v)dx ≤ ∥(1 + D)∇ j v∥L2 (R5 ) ∥(1 + D)−1 O(∇ j1 ∂z φ∇ j2 v)∥L2 (R5 ) C

≤ ε∥(1 + D)∇ j v∥2L2 +

ε

∥(1 + D)−1 (∇ j1 ∂z φ∇ j2 v)∥2L2

(2.10)

for any positive ε . Using the triangle inequality and the fact that D commutes with ∇ j , we can get

∥(1 + D)∇ j v∥2L2 (R5 ) ≤ C (∥∇ j Dv∥2L2 (R5 ) + E4 (v)). In the following we introduce a parameter N to be determined later, and divide (1 + D)−1 = (1 + D)−1 P≤N + (1 + D)−1 P>N , where P≤N and P>N are the Fourier projections to the regions {ξ : |ξ | ≤ N } and {ξ : |ξ | > N }. We first deal with the low-frequency contribution. Applying Sobolev embedding, the Hölder inequality and the Gagliardo–Nirenberg inequality, for 1 ≤ j1 , j2 , we can get 7

∥(1 + D)−1 P≤N (∇ j1 ∂z φ∇ j2 v)∥L2 (R5 ) ≤ Cg (N )∥⟨∇⟩− 4 (∇ j1 ∂z φ∇ j2 v)∥L2 (R5 ) ≤ Cg (N )|∥∇ j1 ∂z φ∥∇ j2 v| ∥ ≤ Cg (N )(∥∇∂z φ ∥ ≤ Cg (N )(∥∇∂z φ∥

20

L 7 (R5 )

20

L 17 (R5 )

∥∇ j−1 v ∥L2 (R5 ) +∥∇v ∥

20

L 7 (R5 )

∥∇ j−1 ∂z φ ∥L2 (R5 ) )

1

20 L 7 (R5 )

+ ∥∇v∥

20 L 7 (R5 )

)Ek2 .

(2.11)

From Sobolev embedding and Plancherel, we have

∥∇∂z φ∥

20

≤ ∥∇∇φ∥

L 7

20

L 7 (R5 )

≤ ∥∇ P≤N ∇φ∥

20

L 7 (R5 )

+ ∥∇ P>N ∇φ∥ 1

7

≤ ∥⟨∇⟩ 4 P≤N ∇φ∥L2 (R5 ) +

20

L 7 (R5 )

1 2

E4

N

7

≤ ∥(1 + D)−1 ⟨∇⟩ 4 (1 + D)P≤N ∇φ∥L2 (R5 ) + ≤ Cg (N )∥(1 + D)∇φ∥L2 + 1 2

≤ Cg (N )(1 + a(t )) + ∥∇ u∥

20

L 7

≤ ∥∇ P≤N v∥

20

L 7

1 N

+ ∥∇ P>N v∥

7

≤ C ∥⟨∇⟩ 4 P≤N v∥L2 + 1

1 N

≤ Cg (N )(1 + a(t )) 2 +

1 N

1 2

1

1

E42

N

E4

1 2

E4 .

(2.12)

20

L 7

∥v∥L2 1 N

1

E42 .

(2.13)

Otherwise, if j1 = 0 or j2 = 0, we have 7

∥(1 + D)−1 P≤N (∇ j1 ∂z φ∇ j2 v)∥L2 (R5 ) ≤ Cg (N )∥⟨∇⟩− 4 (∇ j1 ∂z φ∇ j2 v)∥L2 (R5 ) ≤ Cg (N )(|∥∂z φ∥∇ j v| ∥ ≤ Cg (N )(∥∂z φ∥ ≤ Cg (N )(∥∂z φ∥

20

L 7 (R5 )

20

L 17 (R5 )

+|∥∇ j ∂z φ∥v|∥

∥∇ j v∥L2 (R5 ) + ∥∇v∥

20

L 17 (R5 )

20

L 7 (R5 )

)

∥∇ j ∂z φ∥L2 (R5 ) )

1

20 L 7 (R5 )

+ ∥v∥

20 L 7 (R5 )

)E42 .

(2.14)

28

J. Liu, K. Wang / Nonlinear Analysis 92 (2013) 24–29

From Sobolev embedding and Plancherel, we have

∥∂z φ∥

20

≤ ∥∇φ∥

L 7

20

L 7 (R5 )

≤ ∥P≤N ∇φ∥

+ ∥P>N ∇φ∥

20

L 7 (R5 )

1

3

≤ ∥⟨∇⟩ 4 P≤N ∇φ∥L2 (R5 ) +

20

L 7 (R5 )

1 2

E4

N

3

≤ ∥(1 + D)−1 ⟨∇⟩ 4 (1 + D)P≤N ∇φ∥L2 (R5 ) + ≤ Cg (N )∥(1 + D)∇φ∥L2 + 1

1 2

≤ Cg (N )(1 + a(t )) + ∥ u∥

20

L 7

≤ ∥P≤N v∥

20

L 7

+ ∥P>N v∥

3 4

≤ C ∥⟨∇⟩ P≤N v∥L2 +

N

1 N

1 2

1 N

1

E42

E4

1 2

(2.15)

E4

20

L 7

1

∥v∥L2

N

1

1

≤ Cg (N )(1 + a(t )) 2 +

1

E42 .

(2.16)

N Then, we get the low-frequency contribution

  1 1 1 1 ∥(1 + D)−1 P≤N (∇ j1 ∂z φ∇ j2 v)∥L2 (R5 ) ≤ Cg (N )E42 g (N )(1 + a(t )) 2 + E42 . N

Next we turn to the high-frequency contribution of the last term in (2.10). From Plancherel, the Hölder inequality, we have 7

∥(1 + D)−1 P>N (∇ j1 ∂z φ∇ j2 v)∥L2 (R5 ) ≤ Cg (N )N − 4 |∥∇ j1 ∂z φ ∥ ∇ j2 v|∥L2 (R5 ) 1

7

≤ Cg (N )N − 4 (∥∇φ∥L∞ + ∥v∥L∞ )E42 7

≤ Cg (N )N − 4 E4 .

(2.17)

Then, the second term of (2.8) can be controlled by



Cg 2 (N )E4 g 2 (N )(1 + a(t )) +



1 N

E . 2 4

Lastly, we will estimate last term in (2.8) and use a similar method to obtain

 R5

∇ j+1 φ∇ j+1 △−1 (v∂z v)dx =

 R5

∇ j ∂z φ∇ j+2 △−1 (v 2 )dx

≤ ∥(1 + D)∇ j ∂z φ∥L2 (R5 ) ∥(1 + D)−1 ∇ j+2 △−1 (v 2 )∥L2 (R5 ) C

≤ ε∥(1 + D)∇ j ∂z φ∥2L2 (R5 ) +

ε

∥(1 + D)−1 ∇ j (v 2 )∥2L2 (R5 )

≤ ε(∥∇ j+1 Dφ∥2L2 (R5 ) + E4 (v)) +

C

ε

∥(1 + D)−1 ∇ j (v 2 )∥2L2 (R5 ) .

(2.18)

Now we will estimate the term ∥(1 + D)−1 ∇ j (u2 )∥L2 (R5 ) . It can be rewritten as

∥(1 + D)−1 P≤N ∇ j (v∂z v)∥L2 + ∥(1 + D)−1 P>N ∇ j (v∂z v)∥L2 .

(2.19)

Then, the first part of (2.19) can be estimated as

∥(1 + D)−1 P≤N ∇ j (v∂z v)∥L2 ≤ Cg (N )

7



∥⟨∇⟩− 4 (∇ j1 v∇ j2 ∂z v)∥L2

0≤j1 ,j2 ≤j,j1 +j2 =j

≤ Cg (N )



|∥∇ j1 v∥(∇ j2 ∂z v)| ∥

≤ Cg (N )∥∇ j v∥L2 ∥∇v∥ ≤ Cg (N )∥∇v∥

20

L 17

0≤j1 ,j2 ≤4,j1 +j2 =j 20

L 7

1

20

L 7

E42 .

(2.20)

Noting (2.16), we can get the low-frequency contribution of (2.19) as

  1 1 1 1 ≤ Cg (N ) g (N )a(t ) 2 + E42 E42 . N

(2.21)

J. Liu, K. Wang / Nonlinear Analysis 92 (2013) 24–29

29

Next, we turn to the high-frequency contribution of (2.19) to get 7

1

∥(1 + D)−1 P>N (∇ j1 v∇ j2 ∂z v)∥L2 ≤ Cg (N )N − 4 ∥∇v∥L∞ E42 7

≤ Cg (N )N − 4 E4 .

(2.22)

Then, we can control the second term of (2.18) by



Cg (N )E4 g (N )a(t ) + 2

2

1 N2



E4 .

Choosing a small ε and setting N := 1 + Ek , we can conclude that



∂t E4 ≤ Cg (N )E4 g (N )a(t ) + 2

2

1 N

≤ Cg (1 + E4 )4 E4 (1 + a(t )).

 E4 (2.23)

Noting Ek (0) ≤ C and using the standard ODE comparison argument, we can get Ek (t ) ≤ C (T ) for all 0 ≤ t ≤ T . Then, we obtain the proof of Theorem 1.1. Acknowledgments This work was done when Jianli Liu was visiting the Department of Mathematics of Penn State University during 2012. He would like to thank Prof. Qiang Du, Prof. Chun Liu and the institute for their invitation and warmly hospitality. This work also owes much to Prof. Zhen Lei for his guidance and encouragement. References [1] O.A. Ladyenskaja, Mathematical Questions of the Dynamics of a Viscous Incompressible Fluid, second revised and supplemented ed., Nauka, Moscow, 1970. [2] P.L. Lions, Mathematical Topics in Fluid Mechanics: Compressible Models, in: Oxford Lecture Series in Mathematics and Applications, vol. 2 (10), Oxford Univ. Press, 1998. [3] T.Y. Hou, C.M. Li, Dynamic stability of the 3D axisymmtric Navier–Stokes equations with swirl, Comm. Pure Appl. Math. 61 (2008) 661–697. [4] T.Y. Hou, Z. Lei, On the stabilizing effect of convection for three dimensional incompressible flows, Comm. Pure Appl. Math. 62 (4) (2009) 501–564. [5] L. Tao, J. Wu, A study on the global regularity for a model of the 3D axisymmetric Navier–Stokes equations, Nonlinear Anal. 75 (2012) 3092–3098. [6] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1) (1984) 61–66. [7] G. Prodi, Un teorema di unicità per le equazioni di Navier–Stokes, Ann. Mat. Pura Appl. 48 (1959) 173–182. [8] J. Serrin, The Initial Value Problem for the Navier–Stokes Equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, 1963, pp. 69–98. [9] T.Y. Hou, Z. Lei, On partial regularity of a 3D model of Navier–Stokes equations, Commun. Math. Phys. 287 (2) (2009) 589–612. [10] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions to the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (1982) 771–831. [11] T.Y. Hou, Z. Lei, S. Wang, C. Zou, On finite time singularity and global regularity of an axisymmetric model for the 3D Euler equations, http://arxiv.org/ abs/1203.2980v1. [12] N. Katz, N. Pavlovic, A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyperdissipation, Geom. Funct. Anal. 12 (2002) 355–379. [13] T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation, Anal. PDE 2 (2009) 361–367. [14] T.Y. Hou, Z. SHi, S. Wang, On singularity formation of a 3D model for incompressible Navier–Stokes equations, http://arxiv.org/abs/0912.1316. [15] T. Roy, Global existence of smooth solutions of a 3D log–log energy-supercritical wave equation, Anal. PDE 2 (3) (2009) 261–280. [16] T. Roy, One Remark on Barely H sp supercritical wave equations, arXiv:0906.0044. [17] T. Tao, Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data, J. Hyperbolic Differ. Equ. 4 (2007) 259–266. [18] J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech. 13 (2011) 295–305. [19] D. Chae, J. Wu, Logarithmically regularized inviscid models in borderline Sobolev spaces, J. Math. Phys. 53 (2012) 115601. [20] D. Chae, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math. 226 (2011) 1803–1822. [21] T. Kato, Abstract Differential Equations and Nonlinear Mixed Problems, in: Lezioni Fermiane, [Fermi Lectures] Scuola Normale Superiore, Pisa, Accademia Nazionale dei Lincei, Rome, 1985.