A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces

Journal Pre-proof A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces

Oscar Jarrín

PII:

S0022-247X(20)30033-0

DOI:

https://doi.org/10.1016/j.jmaa.2020.123871

Reference:

YJMAA 123871

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

4 October 2019

Please cite this article as: O. Jarrín, A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces, J. Math. Anal. Appl. (2020), 123871, doi: https://doi.org/10.1016/j.jmaa.2020.123871.

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A remark on the Liouville problem for stationary Navier-Stokes equations in Lorentz and Morrey spaces OSCAR JARR´IN∗1 1

Direcci´ on de investigaci´ on y desarrollo (DIDE). Universidad T´ecnica de Ambato, campus Huachi, Avenida de los Chasquis y rio Payamino, 180207, Ambato, Ecuador.

January 13, 2020

Abstract The Liouville problem for the stationary Navier-Stokes equations on the whole space is a challenging open problem who has know several recent contributions. We prove here some Liouville type theorems for these equations provided the velocity field belongs to some Lorentz spaces and then in the more general setting of Morrey spaces. Our theorems correspond to a improvement of some recent results on this problem and contain some well-known results as a particular case. Keywords: Navier–Stokes equations; stationary system; Liouville theorem; Lorentz spaces; Morrey spaces

1

Introduction

In this article we review some recent results on the Liouville problem for the stationary and incompressible Navier-Stokes equations in the whole space R3 :  + (U  · ∇)  U  + ∇P  = 0, − ΔU

 ) = 0, div(U

(1)

 : R3 −→ R3 is the velocity and P : R3 −→ R is the pressure. Recall that a weak solution of where U  , P ) ∈ L2 (R3 ) × D (R3 ). Moreover, since the pressure P is always related to these equations is a couple (U loc    .  · ∇)  U   by the identity P = 1 div (U then we can concentrate our study in the variable U the velocity U −Δ The classical Liouville problem for the stationary Navier-Stokes equations states that the unique solution of equations (1) which verifies   ⊗U  (x)|2 dx < +∞, |∇ R3

and  (x)| −→ 0, |U

as

|x| −→ +∞,

 = 0, see the book [8], the PhD thesis [10] and the articles [3, 4, 5, 16, 17] for more is the trivial solution U references. Even though an answer to this question is not yet available, great efforts have been invested to understand this open problem. More precisely, the  main idea is2 to give some a priori conditions on the  ⊗U  (x)| dx < +∞, and with this information at  which allow us to prove that 3 |∇ decaying of solution U R ∗

[email protected]

1

 , we look for the identity U  = 0. hand, and sometimes with supplementary hypothesis on the solution U In this setting, one of the first results is due to G. Galdi, see Theorem X.9.5 (page 729 ) of the book [8],   ∈ L 92 (R3 ) then we have 3 |∇  ⊗U  |2 dx ≤ cU  3 9 , and moreover, it is proven where it is proven that if U R L2 the following local estimate for all R > 0 and c > 0 a constant independent of R:   ⊗U  |2 dx ≤ cU  3 9 |∇ , L 2 (C(R/2,R))

BR/2

where BR = {x ∈ R3 : |x| < R} and C(R/2, R) = {x ∈ R3 :  ∈ L 92 (R3 ). provided that U

R 2

 =0 ≤ |x| ≤ R}, which yields the identity U 9

Galdi’s result was thereafter extend to the Lorentz space L 2 ,∞ (R3 ) by H. Kozono et. al. in [13], but in  this more general space some supplementary hypothesis were needed to obtain U 1.2  = 0. Indeed,2 in Theorem 9 ,∞ 3  U  (x)| dx ≤ cU  3 9  of the article [13] it is proven that if U ∈ L 2 (R ) then we have the estimate R3 |∇⊗ ,∞ L2

 = 0 is then obtained under the hypothesis and the desired identity U  3   ⊗U  (x)|2 dx, U  9 ,∞ ≤ δ |∇ L2

(2)

R3

 = 0 we may with δ > 0 small enough. Although this supplementary hypothesis allow us to prove that U observe that it is a quite strong hypothesis and one of the aims of the article [18] by G. Seregin & W.   9 ,∞ . For this purpose, in Theorem 1.1 of the Wang is to relax the restriction imposed on the quantity U L2  is a smooth solution of equations (1) and if for a parameter article [18] the following result is proven: if U 3 < r < +∞ we have 2 3  Lr,∞ (C(R/2,R)) < +∞, (3) M (r) = sup R 3 − r U 

R>1

 ⊗U  (x)|2 dx ≤ cM 3 (r), and moreover, if for δ > 0 small enough we have the then we get the estimate R3 |∇ supplementary a priori control   ⊗U  (x)|2 dx, |∇ M 3 (r) ≤ δ R3

 = 0. Remark that for the value r = 9 the condition M 3 (9/2) ≤ δ then we get U 2 regarded as a relaxation of the condition (2) given in [13].

 R3

 ⊗U  (x)|2 dx can be |∇

The first purpose of this article is to review these results on the Liouville problem for stationary NavierStokes equations in the setting of Lorentz spaces. More precisely, we will prove that if we consider a slight smaller space than L9/2,∞ (R3 ): the space L9/2,q (R3 ) with 9/2 ≤ q < +∞, then the information  ∈ L9/2,q (R3 ) is sufficient to derive the identity U  = 0 and we do not need any additional control on the U  quantity U L9/2,q contrary to the result given in [13]. Moreover, we will see that the space L9/2,q (R3 ) seems to be a critical space to obtain the uniqueness of trivial solution in the sense that if we have the information  ∈ Lr,q (R3 ) for the values 9 < r ≤ q < +∞ then a faster decay condition on the solution U  is required to U 2  obtain U = 0.   ⊗U  |2 dx and this approach allows us Our methods are based on a local estimate on the quantity B |∇ R/2 to consider more general spaces than the Lorentz spaces. Thus, the second purpose of this article is to study  = 0 in a the framework of the Morrey spaces M˙ p,r (R3 ) with 3 ≤ p < r < +∞, generalizing the identity U in this way some recent results. This article is organized as follows: in Section 2 we state all the results obtained. In Section 3 we prove a local estimate on the quantity above from which we will able to study the Liouville problem in the setting 2

of Lorentz space and this will be done in Section 4. Finally, in Section 5 we extend our study to the setting of Morrey spaces.

2

Statement of the results

Recall first the definition of Lorentz spaces. Let f : R3 −→ R be a measurable function, the distribution function df (α) is defined as   df (α) = dx {x ∈ R3 : |f (x)| > α} , where dx denotes de Lebesgue measure. By definition, for 1 ≤ r < +∞ and 1 ≤ q ≤ +∞ the Lorentz space Lr,q (R3 ) is the space of measurable functions f : R3 −→ R such that f Lr,q < +∞, where f Lr,q =

⎧  ⎪ ⎪ 1/q ⎪ ⎨ r

q 1/r α df (α)

+∞  0

  ⎪ ⎪ ⎪ ⎩ sup α d1/r f (α) ,

dα α

1/q , if q < +∞, if q = +∞.

α>0

This space is a homogeneous space of degree − 3r and we have the continuous embedding Lr (R3 ) = Lr,r (R3 ) ⊂ Lr,q (R3 ) for r < q ≤ +∞. In the framework of Lorentz spaces our first result is stated as follows:  ∈ L2 (R3 ) be a weak solution of the stationary Navier-Stokes equations (1). Theorem 1 Let U loc  = 0.  ∈ L9/2,q (R3 ), with 9/2 ≤ q < +∞, then we have U 1) If U  ∈ Lr,q , with 9/2 < r ≤ q < +∞, and moreover, if 2) If U 9

 Lr,q (C(R/4,2R)) < +∞, sup R2− r U

(4)

R>1

 = 0. then we have U Several remarks follow from this result. First, as mentioned in the introduction, the result given in point 1) is of particular interest since this result can be regarded as a improvement of the results given in [13] and [18]. Moreover, due to the embedding L9/2 (R3 ) ⊂ L9/2,q (R3 ), Galdi’s result [8] follows from this theorem.  ∈ Lr,q (R3 ) seems to Now, in point 2) we may observe that for the values 92 < r < +∞ the information U  = 0 and then it is necessary a faster decay of the solution which is given in be not enough to prove that U expression (4). In this expression we may observe that as long as the parameter r is larger than the critical value 92 the solution must have a faster decaying at infinity. As pointed out in the introduction, we also generalize our results to the framework of Morrey spaces and we start by recalling their definition. For 1 < p < r < +∞ the homogeneous Morrey space M˙ p,r (R3 ) is the set of functions f ∈ Lploc (R3 ) such that f M˙ p,r =

sup

R>0, x0 ∈R3

R

3 − p3 r

1



p

B(x0 ,R)

|f (x)| dx

p

< +∞,

(5)

where B(x0 , R) denotes the ball centered at x0 and with radio R. This is a homogeneous space of degree − 3r and moreover we have the following chain of continuous embeddings Lr (R3 ) ⊂ Lr,q (R3 ) ⊂ Lr,∞ (R3 ) ⊂ M˙ p,r (R3 ). In the framework of Morrey spaces our second result is the following: 3

 ∈ L2 (R3 ) be a weak solution of the stationary Navier-Stokes equations (1). If U  ∈ Theorem 2 Let U loc 9 p,r 3 ˙  M (R ) with 3 ≤ p < r < 2 , then U = 0. Observe that this result contains as particular case the uniqueness of the trivial solution of equations (1) in the setting of Lebesgue spaces Lr (R3 ) and Lorentz spaces Lr,∞ (R3 ) with the values 3 < r < 92 , and this fact extend to a more general framework some recent results obtained in the article [7]. Now, It is natural to ask what happens for the values 92 ≤ r < +∞. For those values of parameter r, following some ideas of the articles [13] and [18] exposed in the introduction, in our third result we prove    (x)|2 dx by means of the quantity U   ˙ p,r (where 3 ≤ p < r and some estimates of the quantity R3 |∇ ⊗ U M 9 p,r  ˙  ∈ H˙ 1 (R3 ). ≤ r < +∞), and thus, the information U ∈ M allow us to derive the fact that U 2  ∈ L2 (R3 ) be a weak solution of the stationary Navier-Stokes equations (1). Suppose Theorem 3 Let U loc p,r 3  ˙ that U ∈ M (R ) with 3 ≤ p < r and 92 ≤ r < +∞.  9  ⊗U  (x)|2 dx ≤ cU  3 9 . |∇ 1) For the limit value r = 2 we have ˙ p, M

R3

2) For the values

9 2

2

< r < +∞, if moreover ⎛ N (r) = sup R R>1

2− 9r

⎝R

3 − p3 r

1 ⎞

 C(R/2,R)

 (x)|p dx |U

p

⎠ < +∞,

(6)

 then we have

R3

 ⊗U  (x)|2 dx ≤ cU  2˙ p,r N (r). |∇ M

Comparing this result with the results obtained in [13] and [18] (in the setting of Lorentz spaces) we may observe that point 1) below generalizes to Morrey spaces of the result given in [13], whereas if we compare the expression (3) with the expression (6) below then we may see that point 2) is in a certain sense a generalization to Morrey spaces of the result given in [18].  = 0 in the framework of this result, and to the best of Now, in order to obtain the desired identity U  . Following always our knowledge, it is still necessary to make supplementary hypothesis on the solution U   p, 9 and N (r) by the ideas of [13] and [18] we could suppose an additional control on the quantities U ˙ 2 M   ⊗U  (x)|2 dx, however we will use here a different approach. means of 3 |∇ R

 = 0.  ∈ B˙ −1 (R3 ) then we have U Corollary 1 Within the framework of Theorem 3. If U ∞,∞ −1 (R3 ), which is characterized as the set of distributions f ∈ S  (R3 ) such Recall that the Besov space B˙ ∞,∞ 1

= sup t 2 ht ∗ f L∞ < +∞ and where ht denotes the heat kernel, plays a very important role that f B˙ ∞,∞ −1 t>0

in the analysis on the Navier-Stokes equations (stationary and non stationary) since this is the largest space which is invariant under scaling properties of these equations (see the article [1] and the books [14] and [15] −1 (R3 ) which is  = 0, we have supposed U  ∈ B˙ ∞,∞ for more references). Thus, in order obtain the identity U  less restrictive compared to those made in [13] and [18]. a condition on U

3

A local estimate

 ∈ L2 (R3 ) will be a weak solution of the stationary Navier-Stokes equations (1). Our results From now on U loc deeply relies on the following technical estimate (also known as a Caccioppoli type inequality): 4

p

 verifies U  ∈ Lp (R3 ) and ∇  ⊗U  ∈ L 2 (R3 ) with 3 ≤ p < +∞, then for Proposition 3.1 If the solution U loc loc all R > 1 we have ⎛

 BR/2

 ⊗U  |2 dx ≤ c ⎝ |∇

2



p 2

C(R/2,R)

 ⊗U  | dx |∇

2 ⎞



p

p 2

+ C(R/2,R)

×R

2− p9

 ⊗U  | dx |U

p

⎠ (7)

1

 C(R/2,R)

 |p dx |U

p

.

 R as follows: for a fixed R > 1, we define first Proof. We start by introducing the test functions ϕR and W ∞ 3 the function ϕR ∈ C0 (R ) by 0 ≤ ϕR ≤ 1 such that for |x| ≤ R2 we have ϕR (x) = 1, for |x| ≥ R we have ϕR (x) = 0, and  R L∞ ≤ c . ∇ϕ (8) R  R as the solution of the problem Next we define the function W  R ) = ∇ϕ  R·U , div(W

over BR ,

and

 R = 0 over ∂BR ∪ ∂B R , W 2

(9)

 R is assured by Lemma III.3.1 (page 162) where ∂BR = {x ∈ R3 : |x| = R}. Existence of such function W  R ∈ W 1,q (BR ) with supp (W  R) ⊂ of the book [8] and where it is proven that for 1 < q < +∞ we have W  R is extended by zero outside the set C(R/2, R)) and C(R/2, R) (the function W  ⊗W  R Lq (C(R/2,R)) ≤ c∇ϕ  R·U  Lq (C(R/2,R)) . ∇

(10)

 R above, we consider now the function ϕR U  −W  R and we Once we have defined the functions ϕR and W write       + (U  · ∇)  U  + ∇P   −W  R dx = 0. −ΔU · ϕR U (11) BR

 ∈ Lp (R3 ) (with 3 ≤ p < +∞) then U  ∈ L3 (R3 ) and by Theorem X.1.1 (page 658) of Remark that as U loc loc ∞ 3 ∞ 3  ∈ C (R ) and P ∈ C (R ), thus every term in the last identity above is well defined. the book [8] we have U In identity (11), we start by studying the third term in the left-hand side and integrating by parts we obtain           R·U  + ϕR div(U  ) − div(W  R ) dx, ∇P · ϕR U − WR dx = − P ∇ϕ BR

BR

 R is a solution of problem (9) and since div(U  ) = 0 we can write but since W  BR

   · ϕR U  −W  R dx = 0, ∇P

and thus identity (11) can be written as  BR

      −ΔU · ϕR U − WR dx +

 BR

    −W  R dx = 0.  · ∇)  U  · ϕR U (U

(12)

In this identity we study now the first term in the left-hand side and always by integration by parts we 5

have  BR

=

3        −ΔU · ϕR U − WR dx =

i,j=1 BR

3   i,j=1 BR

=

3  

i,j=1 BR

(∂j Ui )(∂j ϕR )Ui dx +

3  

i,j=1 BR

(∂j Ui )∂j (ϕR Ui − (WR )i )dx

ϕR (∂j Ui )2 dx −

 (∂j Ui )(∂j ϕR )Ui dx +

BR

 ⊗U  |2 dx − ϕR | ∇

3   i,j=1 BR

3  

i,j=1 BR

(∂j Ui )∂j (WR )i dx

(∂j Ui )∂j (WR )i dx.

With this identity at hand, we get back to equation (12) and we can write 3   i,j=1 BR



 (∂j Ui )(∂j ϕR )Ui dx +

BR



+ BR

 ⊗U  |2 dx − ϕR | ∇

    −W  R dx = 0,  · ∇)  U  · ϕR U (U

3   i,j=1 BR

(∂j Ui )∂j (WR )i dx

hence we have  BR

 ⊗U  |2 dx = − ϕR | ∇

3   i,j=1 BR



(∂j Ui )(∂j ϕR )Ui dx +



3   i,j=1 BR

(∂j Ui )∂j (WR )i dx

    · ∇)  U  · ϕR U  −W  R dx. (U

+ BR

But recall the fact that test function ϕR verifies ϕR (x) = 1 if |x| < 

R 2,

and then we have

 BR/2

 ⊗U  |2 dx ≤ |∇

BR

 ⊗U  |2 dx. ϕR | ∇

Thus by this inequality and the identity above we can write the following estimate:  BR/2

 ⊗U  |2 dx ≤ − |∇

3   i,j=1 BR



(∂j Ui )(∂j ϕR )Ui dx +



+ BR

3   i,j=1 BR

(∂j Ui )∂j (WR )i dx

    −W  R dx  · ∇)  U  · ϕR U (U

= I1 + I 2 + I 3 .

(13)

We study now these three terms above. In term I1 remark that we have the function ∂i ϕR , but since the  R ) ⊂ C(R/2, R), test function ϕR verifies ϕR (x) if |x| < R2 and ϕR (x) = 0 if |x| > R then we have supp (∇ϕ and thus we can write 3   I1 = − ∂j Ui (∂j ϕR )Ui dx. i,j=1 C(R/2,R)

6

Then, applying the Hold¨er inequalities with the relation 1 = 3 

I1 ≤



|∂j Ui | dx



≤ c  ≤ c

we write 1

C(R/2,R)

 ⊗U  | dx |∇

 R L∞ ∇ϕ 2

p

p 2

q

|(∂j ϕR )Ui | dx 1



p

p 2

C(R/2,R)

1 q

q

2 C(R/2,R)

+

 2  p

p 2

C(R/2,R)

i,j=1

2 p

1 R 

 ⊗U  | dx |∇

q

q

C(R/2,R)

 | dx |U

1

 C(R/2,R)

q

 |q dx |U

,



(14)



(a)

where the last estimate is due to (8). We need to study now the term (a). Remark the fact that as 3 ≤ p < +∞ and by the relation 1 = p2 + 1q then we have q ≤ 3 ≤ p, and thus we can write (a) ≤

R

3( 1q − p1 )

1



R

p

p

C(R/2,R)

 | dx |U

R

≤

1



3((1− p2 )− p1 )

R

p

p

C(R/2,R)

 | dx |U

≤R

2− p9

1



p

p

C(R/2,R)

 | dx |U

. (15)

With this estimate at hand we write  I1 ≤ c

2

p

p 2

C(R/2,R)

 ⊗U  | dx |∇

R

2− p9



1 C(R/2,R)

 |p dx |U

p

(16)

 R ) ⊂ C(R/2, R), hence we get In order to study the term I2 in (13), recall that the have supp (W   supp (∇ ⊗ WR ) ⊂ C(R/2, R) and then we can write I2 =

3   i,j=1 BR

(∂j Ui )∂j (WR )i dx =

3   i,j=1 C(R/2,R)

(∂j Ui )∂j (WR )i dx.

Now, we apply the H¨ older inequalities always with the relation 1 =  I2 ≤ c

p 2

C(R/2,R)

 2 

+

1 q

and we write 1

p

 ⊗U  | dx |∇

2 p

C(R/2,R)

 ⊗W  R |q dx |∇

q

,

where it remains to study the second term in the right-hand. For this, applying first the estimate (10), then applying the estimate (8) and finally by estimate (15) we can write 

1 q

C(R/2,R)



q

 ⊗W  R | dx |∇

≤ c

C(R/2,R)

≤ cR and thus we have

 I2 ≤ c

q

2− p9

 R·U  | dx |∇ϕ

 R L∞ ≤ c∇ϕ

C(R/2,R)

 |p dx |U

p

 ⊗U  | p2 dx |∇

R 7

2− p9

1 q

C(R/2,R)

1



2 C(R/2,R)



1 q

q

 | dx |U

p

, 1

 C(R/2,R)

 |p dx |U

p

.

(17)

 ) = 0 then in this term we write (∇  ·U  )U  = Finally we study the term I3 in (13). As we have div(U   div(U ⊗ U ) and we obtain 3   I3 = ∂j (Ui Uj )(ϕR Ui − (WR )i )dx, i,j=1 BR

integrating by parts we write I3 = − = −

3   i,j=1 BR 3  

i,j=1 BR

(Ui Uj )((∂j ϕR )Ui + ϕR (∂j Ui ) − ∂j (WR )i )dx (Ui Uj )((∂j ϕR )Ui dx −

3   i,j=1 BR

3  

(Ui Uj )ϕR (∂j Ui )dx +

i,j=1 BR

(Ui Uj )∂j (WR )i )dx

= I3,a + I3,b + I3,c ,

(18)

 R ) ⊂ C(R/2, R) then where we will study these three terms separately. In term I3,a , as we have supp (∇ϕ we write 3   (Ui Uj )((∂j ϕR )Ui dx, I3,a = − i,j=1 C(R/2,R)

then, applying first the H¨older inequalities (with the same relation 1 = estimate (14) and then estimate (15) we have  I3,a ≤ c  ≤ c

 ⊗U  | dx |U p

 ⊗U  | dx |U

1

R

q

q

C(R/2,R)

2 p 2

C(R/2,R)

+ 1q ) and thereafter, applying first

 2  p

p 2

C(R/2,R)

2 p

2− p9

 R·U  | dx |∇ϕ 1



C(R/2,R)

p

 |p dx |U

.

(19)

In order to estimate the term I3,b we write I3,b = −

3   i,j=1 BR

(Ui Uj )ϕR (∂j Ui )dx = −

3   i,j=1

3  1  Uj ϕR ((∂j Ui )Ui )dx = − Uj ϕR ∂j (Ui2 )dx, 2 BR BR i,j=1

 ) = 0 and since the function ∇ϕ  R is then, by integration by parts, and moreover, using the fact that div(U localized at the set C(R/2, R), then we get: −

3  3  3  1  1  1  Uj ϕR ∂j (Ui2 )dx = ∂j (Uj ϕR )Ui2 dx = (∂j Uj )ϕR Ui2 dx 2 2 2 BR BR BR i,j=1

+

=

1 2

1 2

3 

i,j=1 BR

3 

i,j=1

 Uj (∂j ϕR )Ui2 dx =

1 2



i,j=1 C(R/2,R)

i,j=1



Uj (∂j ϕR )Ui2 dx =

BR

1 2

 )ϕR |U  |2 dx + div(U 

3 

i,j=1 C(R/2,R)

1 2

3 



i,j=1 C(R/2,R)

Uj (∂j ϕR )Ui2 dx

Ui2 (∂j ϕR )Uj dx.

With this identity at hand and following the same estimates done for the term I3,a in (19) we have  I3,b ≤ c

2 C(R/2,R)

 ⊗U  | p2 dx |U 8

p

R

2− p9

1

 C(R/2,R)

 |p dx |U

p

.

(20)

Now, in order to study term I3,c remark that using the inequality (10) and following always the estimates done for term I3,a (see (19)) we have  I3,c ≤ c

2

p

p 2

C(R/2,R)

 ⊗U  | dx |U

R

2− p9

1

 C(R/2,R)

p

 |p dx |U

.

(21)

.

(22)

With estimates (19), (20) and (21) we get back to identity (18) hence we have  I3 ≤ c

2 C(R/2,R)

p

 ⊗U  | p2 dx |U

R

1



2− p9

C(R/2,R)

p

 |p dx |U

Finally, once we dispose of estimates (16), (17) and (22), applying these estimates in each term in the right-hand side of (13) we obtain the desired estimate (7). 

4

The Lorentz spaces: proof of Theorem 1

 ∈ L2 (R3 ) of equations (1) verifies U  ∈ Lr,q (R3 ) with 9 ≤ r ≤ q < +∞. The Suppose that the solution U loc 2  verifies the hypothesis of Proposition 3.1, and for this recall the following first think to do is to prove that U estimate: for 1 < p < r ≤ q < +∞ and for R > 1 we have   |p dx ≤ c R3(1− pr ) U  p r,∞ ≤ c R3(1− pr ) U  p r,q , |U (23) L L BR

 ∈ see Proposition 1.1.10, page 22 of the book [6] for a proof of this fact. From this estimate we have U p  ⊗U  ∈ L 2 (R3 ) for 3 ≤ p < +∞. Indeed, since U  verifies the Lploc (R3 ) and then it remains to prove that ∇ loc  ) = 0 then this solution can be written as follows equations (1) and since div(U 3      1    · ∇)  U   = − 1 P (U ) , = P ∂j (Uj U U − Δ Δ j=1

where P is the Leray’s projector. Then, for i = 1, 2, 3 we have  =− ∂i U

3 3      1   ) = ) , P Ri Rj (Uj U P ∂i ∂j (Uj U Δ j=1

(24)

j=1

i where recall that Ri = √∂−Δ denotes the i-th Riesz transform. Thus, by continuity of the operator P(Ri Rj ) r,q 3 on Lorentz spaces L (R ) for the values 1 < r ≤ q (see the article [2]) and applying the H¨older inequalities we obtain the following estimate:

 ⊗U  ∇

r q L2,2

≤c

3 

 )) P(Ri Rj (Uj U

r q

L2,2

i,j=1

 ⊗U  ≤ cU

r q

L2,2

 2Lr,q . ≤ cU

With this estimate at hand we can use now the last estimate in (23) (with 1 < 

p

BR

 ⊗ U | 2 dx ≤ c R |∇ 9

p/2

3(1− r/2 )

p 2

<

r 2

< +∞) to write

p

 ⊗U  2 r q , ∇ , L2

2

(25)

p

 ⊗U  ∈ L 2 (R3 ). hence we obtain ∇ loc  verifies (7) and by this estimate we can write for all R > 1 Thus, by Proposition 3.1 the solution U ⎛  2  2 ⎞ 1    p p p 9 p p 2− 2 p  ⊗U  | dx ≤ ⎝  ⊗U  | 2 dx  ⊗U  | 2 dx ⎠ R p  | dx |∇ |∇ + |U |U BR

C(R/2,R)

2

⎛

1 ≤ c⎝ R3

C(R/2,R)

2



p

p 2

C(R/2,R)



 ⊗U  | dx |∇

+

1 R3



p

p 2

C(R/2,R)

C(R/2,R)

2 ⎞

 ⊗U  | dx |U



⎠ R2

1 R3

1

 C(R/2,R)

 |p dx |U

p

,

(26)

hence we have ⎛

 BR



 ⊗U  |2 dx ≤ c ⎝R |∇

2

6 r



1 R3

2

 C(R/2,R)



p

p 2

 ⊗U  | dx |∇

+R

6 r

 (a)

1 R3

⎛ 9

2 ⎞



 3

×R2− r ⎝R r 

p

p 2

C(R/2,R)

 ⊗U  | dx |U

 1 ⎞



1 R3

⎠

C(R/2,R)

 |p dx |U

p



(27)

⎠, 

(b)

where we will estimate the terms (a) and (b). For this we introduce the cut-off function θR ∈ C0∞ (R3 )  R L∞ ≤ c ; and we consider the localized such that θR = 1 on C(R/2, R), supp (θR ) ⊂ C(R/4, 2R) and ∇θ R  and θR (∇  ⊗U  ). functions θR U Now, as we have θR = 1 on the set C(R/2, R) then for the first term in (a) we can write     ⊗U  | p2 dx =  ⊗U  )| p2 dx ≤  ⊗U  )| p2 dx, |∇ |θR (∇ |θR (∇ C(R/2,R)

C(R/2,R)

B2R

 ⊗U  ) (and with 1 < p < and applying the estimate (25) with the function θR (∇ 2  p p/2  ⊗U  )| p2 dx ≤ c R3(1− r/2 ) θR (∇  ⊗U  ) 2 r q , |θR (∇ , L2

B2R

r 2

< +∞) we have

2

hence the first term in expression (a) is estimated as  R

6 r

1 R3

2

 C(R/2,R)

 ⊗U  | p2 dx |∇

p

 ⊗U  ) ≤ cθR (∇

The second term in (a) treated in a similar way: first we write     ) ⊗ (θR U  )| p2 dx ≤  ⊗U  | p2 dx = |U |(θR U C(R/2,R)

r q

L2,2

C(R/2,R)

.

p

B2R

 ) ⊗ (θR U  )| 2 dx, |(θR U

 ) ⊗ (θR U  ) (always with 1 < p < r < +∞) and by the then we apply estimate (25) with the function (θR U 2 2 H¨ older inequalities we have  p p/2 p/2  ) ⊗ (θR U  )| p2 dx ≤ cR3(1− r/2 ) (θR U  ) ⊗ (θR U  ) 2 r q ≤ cR3(1− r/2 ) θR U  p r,q , |(θR U L , L2

B2R

10

2

hence we can write

 R

6 r

1 R3

2

 C(R/2,R)

p

 ⊗U  | p2 dx |U

 2Lr,q . ≤ cθR U

With these inequalities, the term (a) above is estimated as follows:   2Lr,q . + θR U

  ⊗U  ) (a) ≤ c θR (∇

r q L2,2

(28)

We study now the term (b). Following similar estimates done for term (a): applying always estimate (23) and as supp (θR ) ⊂ C(R/4, 2R), we can write  C(R/2,R)

 |p dx ≤ c |U

 B2R

hence we obtain

 p r,q  |p dx ≤ cR3(1− pr ) θR U  p r,q ≤ cR3(1− pr ) U |θR U L L (C(R/4,R)) ,

 (b) ≤ R

3 r

1 R3

1



p

p

C(R/2,R)

 | dx |U

 Lr,q (C(R/4,R)) . ≤ cU

(29)

Once we dispose of estimates (28) and (29) we get back to (27) and we write  BB

  ⊗U  (x)|2 dx ≤ c θR (∇  ⊗U  ) |∇

r q L2,2

 2Lr,q + θR U



9  Lr,q (C(R/4,R)) , R2− r U

(30)

2

and at this point we will consider two cases for the value of the parameters r and q: 1) For r =

9 2

and 9/2 ≤ q < +∞. By (30) we can write  BB

  ⊗U  (x)|2 dx ≤ c θR (∇  ⊗U  ) |∇

9 q L4,2

   2 9/2,q U + θR U L

9

L 2 ,q

.

2

Now, recall that we have supp (θR ) ⊂ C(R/4, 2R) and then we obtain

lim

R−→+∞

 = 0 a.e. in R3 and θR U

since we have q < +∞ we can apply the dominated convergence theorem in Lorentz spaces (see Theorem   9/2,q = 0.  ⊗U  ) 9/4,q/2 = 0 and lim θR U 1.2.8, page 74 of the book [6]) to obtain lim θR (∇ L L R−→+∞

Thus, taking the limit

lim

R−→+∞

R−→+∞

 ⊗U  2 2 = 0. Moreover, by the in the estimate above we obtain ∇ L

 L6 ≤ c∇  ⊗U  L2 , hence we get the desired identity U  = 0. Hardy-Littlewood-Sobolev we also have U 2) For

9 2

< r ≤ q < +∞. In this case by estimate (30) we have  BB

  ⊗U  |2 dx ≤ c θR (∇  ⊗U  ) |∇

r q L2,2

 2Lr,q + θR U

2



9  Lr,qs (C(R/4,2R)) , sup R2− r U

R>1

where by formula (4) we know that the last term in the right-hand side is bounded. Thus, always by  ⊗U  ) r , q = 0 and lim θR U  Lr,q = 0 and taking the limit lim in the fact that lim θR (∇ 2 2 R−→+∞

L

R−→+∞

 = 0. Theorem 1 is proven. this estimate we obtain the identity U

11

R−→+∞



5

The Morrey spaces

 ∈ L2 (R3 ) of equations (1) verifies U  ∈ M˙ p,r (R3 ) with 3 ≤ p < r < +∞. Suppose now the solution U loc  satisfies the hypothesis of Proposition 3.1: Before to prove our results we need to verify that the solution U p p  ∈ L (R3 ) and ∇  ⊗U  ∈ L 2 (R3 ) with 3 ≤ p < +∞, but, as U  ∈ M˙ p,r (R3 ) then we have U  ∈ Lp (R3 ) U loc loc loc p 3 2   (see Definition (5) of Morrey spaces) so it remains to verify that ∇ ⊗ U ∈ Lloc (R ) and for this we will prove  ⊗U  ∈ M˙ p2 , r2 (R3 ). Indeed, by identity (24), the continuity of the operator P(Ri Rj ) on the Morrey that ∇ spaces M˙ r,p (R3 ) with the values 1 < p < r < +∞ (see Lemma 4.2 of the article [11]) and applying the H¨ older inequalities we can write  ⊗U  ∇

˙ M

p r 2,2

≤c

3 

 )) P(Ri Rj (Uj U

i,j=1

p r

˙ 2,2 M

 ⊗U  ≤ cU

p r

˙ 2,2 M

 2˙ p,r . ≤ cU M

(31)

p

 ∈ Lp (R3 ) and ∇  ⊗U  ∈ L 2 (R3 ), by Proposition 3.1 we dispose of the Once we have the information U loc loc inequality (7) and with this estimate at hand we will consider the following cases of the values of parameters p and r.

5.1

Proof of Theorem 2

We consider here the values 3 ≤ p < r < 92 . By estimate (7) and following the same computations done in estimate (26) we can write ⎛ 2  2 ⎞    p p p p 1 1  ⊗U  |2 dx ≤ c ⎝  ⊗U  | 2 dx  ⊗U  | 2 dx ⎠ |∇ | ∇ + | U R3 C(R/2,R) R3 C(R/2,R) BR 2    (a)

 × R2 

1 R3

1

 C(R/2,R)

 |p dx |U



p

, 

(b)

 ∈ M˙ p,r (R3 ) then by (31) we have where will estimate the terms (a) and (b). In term (a) remark that as U p r , 3  ⊗U  ∈ M˙ 2 2 (R ) and moreover by the H¨  ⊗U  ∈ M˙ p2 , r2 (R3 ). Thus, by definition ∇ older inequalities we have U of Morrey spaces (see (5)) we can write    ⊗U   p , r + U  ⊗U   p , r R− 6r . (a) ≤ c ∇ ˙ 2 2 ˙ 2 2 M

M

 ∈ M˙ p,r (R3 ) for term (b) we write moreover, always by the fact that U     ˙ p,r R− 3r ≤ U   ˙ p,r R2− r3 , (b) ≤ R2 U M M and with this estimates on terms (a) and (b) we obtain    ⊗U  |2 dx ≤ c ∇  ⊗U   p , r + U  ⊗U  |∇ ˙ 2 2

˙ M

M

BR

 p r 2,2

9

  ˙ p,r R2− r . U M

2

But recall that we have 3 < r <

9 2

hence we get −1 < 2 −

9 r

< 0 and then, taking the limit

 ⊗U  2 2 = 0 hence we obtain the identity U  = 0. Theorem 2 is now proven. have ∇ L

12

lim

R−→+∞

we 

5.2

Proof of Theorem 3

 ∈ M˙ p, 92 . Following the same computations 1) For the values 3 ≤ p < 92 and r = 92 . In this case we have U done in estimate (27) and moreover, always by definition of the Morrey spaces given in (5) we get the uniform bound ⎛ 2 2 ⎞      p p 4 1 1  ⊗U  |2 dx ≤ c ⎝R 43  ⊗U  | p2 dx  ⊗U  | p2 dx ⎠ 3 |∇ | ∇ + R | U R3 C(R/2,R) R3 C(R/2,R) BR 2 ⎛  1 ⎞  p 2 1  |p dx ⎠ × ⎝R 3 | U R3 C(R/2,R)    ⊗U   p , 9 + U  ⊗U   p , 9 U  3 9 ,   p, 9 ≤ cU ≤ c ∇ ˙ 2 4 ˙ 2 4 ˙ 2 ˙ p, M

M

M

M

 and taking the limit

2) For the values 3 ≤ p ≤  BR

lim

R−→+∞ 9 2

and

9 2

⎛



 ⊗U  |2 dx ≤ c ⎝R |∇

2

we obtain

R3

2

 ⊗U  |2 dx ≤ cU  3 9 . |∇ ˙ p, M

2

< r < +∞. Always by estimate (27) for all R > 1 we write

6 r



2



1 R3

C(R/2,R)



p

p 2

 ⊗U  | dx |∇

+R

6 r

 (a)

×R

1 R3

⎛

2− 9r

⎝R





 3 r

2 ⎞ p  ⊗U  | dx ⎠ |U p 2

C(R/2,R)

1 R3

 C(R/2,R)

 1 ⎞ p  |p dx ⎠, |U





(b)

 ∈ M˙ p,r (R3 ) then the term (a) is uniformly bounded as follows: where, as we have U  ⊗U  (a) ≤ c(∇

p r

˙ 2,2 M

 ⊗U  + U

˙ 2,2 ) M p r

 2˙ p,r , ≤ cU M

moreover, the term (b) is uniformly bounded as ⎛ (b) ≤ R

2− 9r

⎝R

 3 r

1 R3

1 ⎞

 C(R/2,R)

 |p dx |U

p

⎠ ≤ N (r),

where the quantity N (r) < +∞ is defined in formula (6).  With these estimates we can write BR

 obtain

5.3



 ⊗U  |2 dx ≤ cU  2˙ p,r N (r), and taking the limit |∇ M

lim

R−→+∞

we

2

R3

 ⊗U  |2 dx ≤ cU  2˙ p,r N (r). Theorem 3 is proven. |∇ M



Proof of Corollary 1

 ⊗U  |2 dx < +∞ we get U  ∈ H˙ 1 (R3 ), and with the information U  ∈ B˙ −1 (R3 ) we can apAs R3 |∇ ∞,∞ ply the improved Sobolev inequalities (see the article [9] for a proof of these inequalities) and we write 13

1

1

 L4 ≤ cU   2 U   2 −1 . Once we dispose of the information U  ∈ L4 (R3 ) we can derive now the identity U H˙ 1 B˙ ∞,∞

 = 0 as follows: multiplying equation (1) by U  and integrating on the whole space R3 we have U     ·U  dx, )·U  dx =  · ∇)  U )·U  dx + ∇P (−ΔU ((U R3

R3

R3

 ∈ H˙ 1 ∩ L4 (R3 ) each term in this identity is well-defined. Indeed, for the term in where due to the fact U  ∈ H˙ 1 (R3 ) then we have −ΔU  ∈ H˙ −1 (R3 ). Then, for the first term in the left-hand side remark that as U  ) = 0 we write (U  · ∇)  U  = div(U  ⊗U  ) where, as U  ∈ L4 (R3 ) by the H¨older the right-hand side, as div(U 2 3 −1 3     ˙ inequalities we have U ⊗ U ∈ L (R ) and then div(U ⊗ U ) ∈ H (R ). Finally, in order to study the second 1  ⊗U  )) hence we get P ∈ L2 (R3 ) div(div(U term in the right-hand side we write the pressure P as P = −Δ  ⊗U  ∈ L2 (R3 )) and then ∇P  ∈ H˙ −1 (R3 ). (since we have U   )·U  dx = 3 |∇  ⊗U  |2 dx, Now, integrating by parts each term in the identity above we have that R3 (−ΔU    R  · ∇)  U  )·U  dx = 0 and 3 ∇P  ⊗U  |2 dx = 0  ·U  dx = 0. With this identities we get 3 |∇ and moreover R3 ((U R R  = 0. and thus we have U 

References [1] J. Bourgain & N. Pavlovi´c. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. Journal of Functional Analysis: 255, 2233–2247 (2008). [2] C. Bjorland, L. Brandolese, D. Iftimie & M.E. Schonbek. Lp solutions of the stady-state Navier-Stokes equations with rough external forces. Comm. Part. Diff. Equ., 36: 216–246 (2011). [3] D. Chae & T. Yoneda. On the Liouville theorem for the stationary Navier-Stokes equations in a critical space. J. Math. Anal. Appl. 405: 706-710 (2013). [4] D. Chae & J. Wolf. On Liouville type theorems for the steady Navier-Stokes equations in R3 . arXiv:1604.07643 (2016). [5] D. Chae & S. Weng. Liouville type theorems for the steady axially symmetric Navier-Stokes and magneto-hydrodynamic equations. Discrete And Continuous Dynamical Systems, Volume 36, Number 10: 5267-5285 (2016). [6] D. Chamorro. Espacios de Lebesgue y de Lorentz. Vol. 3. hal-01801025v1 (2018). [7] D. Chamorro, O. Jarr´ın & P.G. Lemari´e-Rieusset. Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces. arXiv:1806.03003 (2018). [8] 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). [9] P. G´erard, Y. Meyer & F. Oru. Improved Sobolev inequalities. Seminary of partial differential equations. Vol. 1-8 (1996-1997). [10] O. Jarr´ın. Descriptions d´eterministes de la turbulence dans les ´equations de Navier-Stokes. Ph.D. thesis at Paris-Saclay University, Paris-France (2018). [11] T. Kato. Strong Solutions of the Navier-Stokes Equation in Morrey Spaces. Bol. Soc. Bras. Mat., Vol. 22, No. 2: 127-155 (1992). [12] G. Koch, N. Nadirashvili, G. Seregin & V. Sverak. Liouville theorems for the Navier-Stokes equations and applications. Acta Mathematica, 203: 83- 105 (2009). [13] H. Kozono, Y. Terasawab & Y. Wakasugib. A remark on Liouville-type theorems for the stationary Navier-Stokes equations in three space dimensions. Journal of Functional Analysis, 272: 804-818 (2017). 14

[14] P.G. Lemari´e-Rieusset. The Navier-Stokes Problem in the 21st Century, Chapman & Hall/CRC, 2016. [15] P.G. Lemari´e-Rieusset. Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC, (2002). [16] G. Seregin. Liouville Type Theorem for Stationary Navier-Stokes Equations. Nonlinearity, 29 : 21912195 (2015). [17] G. Seregin. A Liouville type theorem for steady-state Navier-Stokes equations. arXiv :1611.01563 (2016). [18] G. Seregin & W. Wang. Sufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations. arXiv:1805.02227 (2018).

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