Note on the Liouville type problem for the stationary Navier-Stokes equations in R3

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Note on the Liouville type problem for the stationary Navier-Stokes equations in R3 Dongho Chae a,b,∗ a Department of Mathematics, Chung-Ang University, Dongjak-gu Heukseok-ro 84, Seoul 06974, Republic of Korea b School of Mathematics, Korea Institute for Advanced Study, Dongdaemun-gu Hoegi-ro 85, Seoul 02455,

Republic of Korea Received 23 January 2019; revised 18 May 2019; accepted 19 August 2019

Abstract In this brief note we are concerned on the Liouville type problem for the stationary Navier-Stokes equations in R3 . We deduce an asymptotic formula for an integral involving the head pressure, Q = 12 |v|2 + p, and its derivative over domains enclosed by level surfaces of Q. This formula provides us with new sufficient condition for the triviality of solution to the Navier-Stokes equations. © 2019 Elsevier Inc. All rights reserved. MSC: 35Q30; 76D05; 76D03 Keywords: Steady Navier-Stokes equations; Head pressure; Liouville type theorem

1. Introduction In this paper we are concerned on the Liouville type problem for the n-dimensional steady Navier-Stokes equations in R3 .  v · ∇v = −∇p + v, (1.1) ∇ ·v=0 * Correspondence to: Department of Mathematics, Chung-Ang University, Dongjak-gu Heukseok-ro 84, Seoul 06974, Republic of Korea. E-mail address: [email protected].

https://doi.org/10.1016/j.jde.2019.08.027 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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equipped with the uniform decay condition at spatial infinity, v(x) → 0

as |x| → 0.

(1.2)

Here, v = (v1 (x), v2 (x), v3 (x)) is a vector field in R3 , and p = p(x) is a scalar field. Obviously (v, p) with v = 0 and p = constant is a trivial solution to (1.1)-(1.2). An important question is if there is other nontrivial solution. This uniqueness problem, or equivalently Liouville type problem is now hot issue in the community of mathematical fluid mechanics. In general we impose an extra condition, the finiteness of the Dirichlet integral,  |∇v|2 dx < +∞. (1.3) R3

The Liouville type problem for the solution of (1.1)-(1.2) together with (1.3) is posed explicitly in Galdi’s book [5, Remark X.9.4, pp. 729], where the triviality of solution is deduced under 9 stronger assumption v ∈ L 2 (R3 ). There are other numerous partial results (see e.g. [1–3,8–10,7, 4,6] and the references therein) proving the triviality of solution to (1.1)-(1.3) under various sufficient conditions. In the following theorem we derive an asymptotic blow-up rate for an integral of the head pressure over a region enclosed by level surfaces of the head pressure. The integral has the same scaling property as the Dirichlet integral. For k ∈ N let us define k−times

k−times

   expk (x) = exp(exp(· · · (exp x) · · · ),

   logk (x) = log(log(· · · (log x) · · · ),

and set expk (1) := ek , e0 (x) = log0 (x) := 1 for all x ∈ R. Theorem 1.1. Let (v, p) be a smooth solution of (1.1) satisfying (1.2)-(1.3), and Q = 12 |v|2 + p, ω = ∇ × v. We set infx∈R3 Q(x) = m. Then, for all k ∈ N ∪ {0} we have lim (logk+1 (1/λ))



−1

λ→0

{|Q|>λ}

|Q|

k

|∇Q|2

j =0 logj



ek |m| |Q|

dx =

|ω|2 dx

(1.4)

R3

Therefore, if there exists k ∈ N ∪ {0} such that  {|Q|>λ}

|Q|

k

|∇Q|2



j =0 logj

ek |m| |Q|



dx = o logk+1 (1/λ)

(1.5)

as λ → 0, then v = 0 on R3 . In order to appreciate the above theorem we consider the following simple case with k = 0. Corollary 1.1. Under the same assumptions as in Theorem 1.1 there holds lim (log(1/λ))−1

λ→0



{|Q|>λ}

|∇Q|2 dx = |Q|



R3

|ω|2 dx,

(1.6)

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and if  {|Q|>λ}

|∇Q|2 dx = o (log(1/λ)) |Q|

(1.7)

as λ → 0, then v = 0 on R3 .

√ 2 Remark 1.1. We note that R3 |∇ |Q||2 dx = 14 R3 |∇Q| |Q| dx has the same scaling as

2 3 |∇v| dx. As an immediate consequence of the above corollary we have the triviality, v = 0 if

R |∇Q|2 R3 |Q| dx < +∞. Indeed, if Q = constant, then (2.1) below implies ω = 0, and this together with ∇ · v = 0 leads to v = ∇h for a harmonic function h, which is zero from the condition (1.2) by the Liouville theorem for a harmonic function. It would be interesting to note the following inequality, which follows from the Sobolev embedding, ⎛ ⎜ ⎝



⎞1

⎛

6

⎟ ⎜ |Q| dx ⎠ ≤ C ⎝



3

R3

⎞1 

2

⎟ |∇ |Q|| dx ⎠ . 2

(1.8)

R3

We see that v is trivial if the right hand side of (1.8) is finite. The left hand side of (1.8) is, however, easily shown to be finite from the hypothesis of Theorem  1.1. Indeed, using the well-known Calderon-Zygmund inequality for the pressure, p = − 3j,k=1 ∂j ∂k (vj vk ), we have ⎛ ⎜ ⎝



⎞1

⎛

6

⎟ ⎜ |Q|3 dx ⎠ ≤ C ⎝

R3



⎞1

⎛

6

⎟ ⎜ |v|6 dx ⎠ + C ⎝

R3

⎛ ⎜ ≤C⎝





⎞1

6

⎟ |p|3 dx ⎠

R3

⎞1

6

⎛

⎟ ⎜ |v|6 dx ⎠ ≤ C ⎝

R3



⎞1

2

⎟ |∇v|2 dx ⎠ < +∞.

(1.9)

R3

The following theorem refines the result in [1]. Theorem 1.2. Let v is a smooth solution to (NS) satisfying (1.2) and (1.3). We suppose at least one of the following belongs to L1 (R3 ), then v = 0. v · v,

|v|2 ,

Q,

v · ∇Q.

(1.10) 6

Remark 1.2. The sufficient condition for the Liouville theorem obtained in [1] is v ∈ L 5 (R3 ). In this case, since v ∈ L6 (R3 ) from the hypothesis (1.3) combined with the Sobolev inequality, we have  

1 

5 6 6 6 |v · v|dx ≤ |v|6 dx |v| 5 dx < +∞, R3

R3

R3

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and v · v ∈ L1 (R3 ). Therefore, the theorem of [1] follows from the above theorem. 2. Proof of the theorems Proof of Theorem 1.1. Given smooth solution (v, p) of (1.1) we set the head pressure Q = p + 12 |v|2 and the vorticity tensor ω = ∇ × v. Then, it is well-known that the following holds Q − v · ∇Q = |ω|2 .

(2.1)

Indeed, multiplying the first equation of (1.1) by v, we have |v|2 − |∇v|2 . 2

(2.2)

0 = p + Tr(∇v(∇v) ).

(2.3)

v · ∇Q =  Taking divergence of (1.1), we are led to

Adding (2.3) to (2.2), and observing |∇v|2 − Tr(∇v(∇v) ) = |ω|2 , we obtain (2.1). We also recall that it is known (see e.g. [2, Theorem X.5.1, p. 688]) that the condition (1.2) together with (1.3) implies that p(x) → p¯ for some constant p¯ as |x| → +∞. Therefore, re-defining Q − p¯ as new head pressure, we may assume Q(x) → 0

|x| → +∞.

as

(2.4)

In view of (2.4), applying the maximum principle to (2.1), we find Q ≤ 0 on R3 . Moreover, by the maximum principle again, either Q(x) = 0 for all x ∈ R3 , or Q(x) < 0 for all x ∈ R3 . Indeed, any point x0 ∈ R3 such that Q(x0 ) = 0 is a point of local maximum, which is not allowed unless Q ≡ 0 by the maximum principle. Let us assume Q(x) = 0 on R3 , then without the loss of generality we may assume Q(x) < 0 for all x ∈ R3 . Let us consider a continuous function f : R+ → R. Given λ ∈ (0, |m|), we multiply (2.1) by

|Q(x)| f (s)ds, and then integrating it over the set {x ∈ R3 | |Q(x)| > λ}, we have λ 

⎛ ⎜ 2 ⎝|ω|

{|Q|>λ}

|Q(x)| 

⎞

⎟ f (s)ds ⎠ dx =

⎛



⎜ ∇ · ⎝∇Q

{|Q|>λ}

λ

 |∇Q|2 f (|Q|)dx −

{|Q|>λ}

⎟ f (s)ds ⎠ dx

λ



+

⎞

|Q(x)| 

{|Q|>λ}

⎛

⎜ v · ∇Q ⎝

|Q(x)| 

⎞

⎟ f (s)ds ⎠ dx

λ

:= I1 + I2 + I3 . Let us define F (τ ) =

τ t 0

λ

(2.5) f (s)dtdτ . Using the divergence theorem, we compute

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I3 =

∇ · {vF (|Q|)} dx

v · ∇F (|Q|)dx =

{|Q|>λ}

{|Q|>λ}





v · νdS = F (λ)

= F (λ) {|Q|=λ}

∇ · vdx = 0,

(2.6)

{|Q|>λ}

where ν denotes the outward unit normal vector on the level surface. Applying the divergence theorem again, we deduce ⎛



⎟ f (s)ds ⎠ dσ

⎜ ⎝ν · ∇Q

I1 = {|Q|=λ}

λ

⎛



⎜ ⎝|∇Q|

=

⎞

|Q(x)| 

⎞

|Q(x)| 

{|Q|=λ}

⎟ f (s)ds ⎠ dσ = 0,

(2.7)

λ

where we used the fact ν = ∇Q/|∇Q|. Therefore we have an identity ⎛



⎜ 2 ⎝|ω|

⎞

|Q(x)| 



⎟ f (s)ds ⎠ dx =

{|Q|>λ}

|∇Q|2 f (|Q|)dx,

(2.8)

{|Q|>λ}

λ

for any continuous function f on [0, |m|]. Choosing 1

f (s) =  k s j =0 logj ek s|m| in (2.8), we find  |Q| 

{|Q|>λ}

=

|∇Q|2



dx ek |m| log j j =0 |Q|   ek |m| ek |m| 2 |ω| log(k+1) − log(k+1) dx λ |Q|

k

{|Q|>λ}

Dividing the both sides of (2.10) by log(k+1) λ1 , and passing λ → 0, we have   1 −1 lim log(k+1) λ→0 λ



{|Q|>λ}

{|Q|>λ}

|∇Q|2

j =0 logj



 =

|Q|

k

|ω| dx − lim 2

λ→0 {|Q|>λ}

|ω|

2



ek |m| |Q|

dx

ek |m| |Q| log(k+1) λ1

log(k+1)

dx

(2.9)

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 =

|ω|2 dx,

(2.10)

R3

where we used the fact  lim

λ→0 {|Q|>λ}

|ω|2

ek |m| |Q| log(k+1) λ1

log(k+1)

 dx ≤ lim

λ→0 R3

|ω|2

ek |m| |Q| log(k+1) λ1

log(k+1)

dx = 0,

which follows from the hypothesis ω ∈ L2 (R3 ) and the dominated convergence theorem. 2 Proof of Theorem 1.2. We first observe  Q − 

|v|2 − v · v = |∇v|2 ∈ L1 (R3 ), 2

3  |v|2 ∂j vk ∂k vj ∈ L1 (R3 ), = p = − 2 j,k=1

and Q − v · ∇Q = |ω|2 ∈ L1 (R3 ). Therefore, we see that if one of the four belongs to L1 (R3 ), then all of functions in (1.10) belongs to L1 (R3 ). For r ∈ (0, ∞) let ηr = η(|x|) ∈ Cc∞ (B(2r)) be a standard cut-off function with 0 ≤ η ≤ 1 on R3 , η = 1 on B(r) and |∇η| ≤ cr −1 in B(2r) \ B(r). Then, since Q ∈ L1 (R3 ), we find that          Qdx  =    R3

                lim  Qηr dx  = lim  Qηr dx  r→+∞   r→+∞     R3 R3

  ≤ c lim  r→+∞

1

 |Q| dx 3

3

→0

(2.11)

B(2r)\B(r)

as r → +∞, since Q L3 ≤ c v 2L6 ≤ c ∇v 2L2 < +∞. From the fact Q < 0 on R3 , we find that {x ∈ R3 | Q(x) < λ}  R3 as λ  0. Hence by the dominated convergence theorem and (2.6) with F (s) = s, we obtain 

 v · ∇Qdx = lim

R3

λ0 {Q<λ}

Summarizing (2.11) and (2.12). We have shown

v · ∇Qdx = 0.

(2.12)

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 Qdx −

R3

7

v · ∇Qdx = 0.

R3

Therefore, Q − v · ∇Q = |ω|2 implies R3 |ω|2 dx = 0, and ω = 0 on R3 . Combining this with ∇ · v = 0, we find that v = ∇h for a harmonic function h on R3 . By the Liouville theorem of for a harmonic function the condition (1.2) implies v = 0. 2 Acknowledgments The author was partially supported by NRF grants 2016R1A2B3011647. The author would like to thank to the anonymous referee for helpful suggestion. References [1] D. Chae, Liouville-type theorem for the forced Euler equations and the Navier-Stokes equations, Commun. Math. Phys. 326 (2014) 37–48. [2] D. Chae, T. Yoneda, On the Liouville theorem for the stationary Navier-Stokes equations in a critical space, J. Math. Anal. Appl. 405 (2) (2013) 706–710. [3] D. Chae, J. Wolf, On Liouville type theorems for the steady Navier-Stokes equations in R3 , J. Differ. Equ. 261 (2016) 5541–5560. [4] D. Chamorro, O. Jarrin, P-G. Lemarié-Rieusset, Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces, arXiv preprint, arXiv:1806.03003. [5] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, second edition, Springer Monographs in Mathematics, Springer, New York, 2011, xiv+1018 pp. [6] D. Gilbarg, H.F. Weinberger, Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 5 (1978) 381–404. [7] H. Kozono, Y. Terasawa, Y. Wakasugi, A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions, J. Funct. Anal. 272 (2017) 804–818. [8] G. Seregin, Liouville type theorem for stationary Navier-Stokes equations, Nonlinearity 29 (2016) 2191–2195. [9] G. Seregin, Remarks on Liouville type theorems for steady-state Navier-Stokes equations, Algebra Anal. 30 (2) (2018) 238–248. [10] G. Seregin, W. Wang, Sufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations, arXiv preprint, arXiv:1805.02227.