Characterisation of the pressure term in the incompressible Navier-Stokes equations on the whole space

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Characterisation of the pressure term in the

incompressible Navier-Stokes equations on the whole space

Pierre Gilles Lemarié-Rieusset, Pedro Gabriel Fernández-Dalgo

To cite this version:

Pierre Gilles Lemarié-Rieusset, Pedro Gabriel Fernández-Dalgo. Characterisation of the pressure term

in the incompressible Navier-Stokes equations on the whole space. 2020. �hal-02456252�

Characterisation of the pressure term in the incompressible Navier–Stokes equations on the

whole space

Pedro Gabriel Fern´ andez-Dalgo ^∗† , Pierre Gilles Lemari´ e–Rieusset ^‡§

Abstract

We characterise the pressure term in the incompressible 2D and 3D Navier–Stokes equations for solutions defined on the whole space.

Keywords : Navier–Stokes equations, pressure term, Leray projection, extended Galilean invariance, suitable solutions

AMS classification : 35Q30, 76D05.

Introduction

In the context of the Cauchy initial value problem for Navier–Stokes equa- tions on R

(with d = 2 or d = 3)







∂

u = ∆u − (u · ∇)u − ∇p + ∇ · F

∇ · u = 0, u(0, .) = u

an important problem is to propose a formula for the gradient of the pressure, which is an auxiliary unknown (usually interpreted as a Lagrange multiplier for the constraint of incompressibility).

∗

LaMME, Univ Evry, CNRS, Universit´ e Paris-Saclay, 91025, Evry, France

†

e-mail : pedro.fernandez@univ-evry.fr

‡

LaMME, Univ Evry, CNRS, Universit´ e Paris-Saclay, 91025, Evry, France

§

e-mail : pierregilles.lemarierieusset@univ-evry.fr

As we shall not assume differentiability of u in our computations, it is better to write the equations as

(N S)







∂

u = ∆u − ∇ · (u ⊗ u) − ∇p + ∇ · F

∇ · u = 0, u(0, .) = u

Taking the Laplacian of equations (NS), since we have for a vector field w the identity

−∆w = ∇ ∧ (∇ ∧ w) − ∇(∇ · w) we get the equations

∂

∆u = ∆

u + ∇ ∧ (∇ ∧ (∇ · (u ⊗ u − F ))) and

0 = −∆∇p − ∇(∇ · (∇ · (u ⊗ u − F )) = −∆∇p − ∇( X

1≤i,j≤d

∂

∂

(u

u

− F

)).

Thus, the rotational-free unknown ∇p obeys a Poisson equation. If G

is the fundamental solution of the operator −∆ :

G

= 1 2π ln( 1

|x| ), G

= 1 4π|x|

(which satisifies −∆G

= δ), we formally have

∇p = G

∗ ∇( X

1≤i,j≤d

∂

∂

(u

u

− F

)) + H (1)

with ∆H = 0. In the litterature, one usually finds the assumption that ∇p vanishes at infinity and this is read as H = 0. Equivalently, this is read as

∂

u = G

∗ ∇ ∧ (∇ ∧ ∂

u);

the operator

P = G

∗ ∇ ∧ (∇ ∧ .)

is called the Leray projection operator and the decomposition (when justified ) w = P w + G

∗ ∇(∇ · w)

the Hodge decomposition of the vector field w.

Hence, an important issue when dealing with the Navier–Stokes equations is to study whether in formula (1) the first half of the right-hand term is well- defined, and if so which values the second half (the harmonic part H) may have.

In order to give some meaning to the formal convolution G

∗ ∇∂

∂

(u

u

) or to (∇∂

∂

G

) ∗ (u

u

), we should require u

to be locally L

L

(in order to define u

u

as a distribution) and to have small increase at infinity, since the distribution ∇∂

∂

G has small decay at infinity (it belongs to L

∩ L

far from the origin and is O(|x|

)). Thus, we will focus on solutions u that belong to L

((0, T ), L

( R

, w

dx)) where

w

(x) = (1 + |x|)

.

We shall recall various examples (from recent or older litterature) of solutions belonging to the space L

((0, T ), L

( R

, w

dx)) (with γ ∈ {d, d + 1}). As F

plays a role similar to u

u

, we shall assume that F ∈ L

((0, T ), L

( R

, w

dx)).

1 Main results.

First, we precise the meaning of ∇p in equations (NS) :

Lemma 1.1 Consider the dimension d ∈ {2, 3} and γ ≥ 0. Let 0 < T <

+∞. Let F be a tensor F (t, x) = (F

(t, x))

such that F belongs to L

((0, T ), L

( R

, w

dx)), and let u be a vector field u(t, x) = (u

(t, x))

such that u belongs to L

((0, T ), L

( R

, w

dx)) and ∇ · u = 0. Define the distribution S by

S = ∆u − ∇ · (u ⊗ u − F ) − ∂

u.

Then the following assertions are equivalent : (A) S is curl-free : ∇ ∧ S = 0.

(B) There exists a distribution p ∈ D

((0, T ) × R

) such that S = ∇p.

Theorem 1 Consider the dimension d ∈ {2, 3}. Let 0 < T < +∞. Let F be

a tensor F (t, x) = (F

(t, x))

such that F belongs to L

((0, T ), L

( R

, w

dx)).

Let u be a solution of the following problem







∂

u = ∆u − ∇ · (u ⊗ u) − S + ∇ · F

∇ · u = 0, ∇ ∧ S = 0,

(2)

such that : u belongs to L

((0, T ), L

( R

)), and S belongs to D

((0, T ) ×

R

).

Let us choose ϕ ∈ D( R

) such that ϕ(x) = 1 on a neighborhood of 0 and define

A

= (1 − ϕ)∂

∂

G

. Then, there exists g(t) ∈ L

((0, T )) such that

S = ∇p

+ ∂

g with

p

= X

i,j

(ϕ∂

∂

G

) ∗ (u

u

− F

)

+ X

i,j

Z

(A

(x − y) − A

(−y))(u

(t, y)u

(t, y) − F

(t, y)) dy.

Moreover,

• ∇p

does not depend on the choice of ϕ : if we change ϕ in ψ, then p

(t, x)−p

(t, x) = X

i,j

Z

(A

(−y)−A

(−y))(u

(t, y)u

(t, y)−F

(t, y)) dy.

• ∇p

is the unique solution of the Poisson problem

∆w = −∇(∇ · (∇ · (u ⊗ u − F )) with

τ→+∞

lim e

w = 0 in D

.

• if F belongs more precisely to L

((0, T ), L

d

( R

)) and u belongs to L

((0, T ), L

( R

)), then g = 0 and ∇p

= ∇p

where

p

= X

i,j

(ϕ∂

∂

G

) ∗ (u

u

− F

) + X

i,j

((1 − ϕ)∂

∂

G

) ∗ (u

u

− F

).

(p

does not actually depend on ϕ and could have been defined as p

= P

i,j

(∂

∂

G

) ∗ (u

u

− F

).)

When F = 0, the case g 6= 0 can easily be reduced to a change of referen-

tial, due to the extended Galilean invariance of the Navier–Stokes equations :

Theorem 2 Consider the dimension d ∈ {2, 3}. Let 0 < T < +∞. Let u be a solution of the following problem







∂

u = ∆u − ∇ · (u ⊗ u) − S

∇ · u = 0, ∇ ∧ S = 0, u(0, x) = u

(x)

(3)

such that : u belongs to L

((0, T ), L

( R

)), and S belongs to D

((0, T ) × R

).

Let us choose ϕ ∈ D( R

) such that ϕ(x) = 1 on a neighborhood of 0 and define

A

= (1 − ϕ)∂

∂

G

. We decompose S into

S = ∇p

+ ∂

g with

p

= X

i,j

(ϕ∂

∂

G

) ∗ (u

u

)

+ X

i,j

Z

(A

(x − y) − A

(−y))(u

(t, y)u

(t, y)) dy and

g(t) ∈ L

((0, T )).

Let us define

E(t) = Z

0

g(λ)dλ

and

w(t, x) = u(t, x − E(t)) + g(t).

Then, w is a solution of the Navier–Stokes problem



 

 

 



 

 

 



∂

w = ∆w − ∇ · (w ⊗ w) − ∇q

∇ · w = 0, w(0, x) = u

(x) q

= X

i,j

(ϕ∂

∂

G

) ∗ (w

w

) + X

i,j

R (A

(x − y) − A

(−y))(w

(t, y)w

(t, y)) dy

(4)

2 Curl-free vector fields.

In this section we prove Lemma 1.1 with simple arguments : Proof. We take a partition of unity on (0, T )

X

j∈Z

ω

= 1

with ω

supported in (2

T, 2

T ) for j < 0, in (T /4, 3T /4) for j = 0 and in (T − 2

T, T − 2

T ) for j > 1. We define

V

= −ω

u + Z

0

ω

∆u − ω

∇ · (u ⊗ u − F ) + (∂

ω

)u ds.

Then V

is a sum of the form A + ∆B + ∇ · C + D with A, B, C and D in L

((0, T ), L

( R

, w

dx)); thus, by Fubini’s theorem, we may see it as a time-dependent tempered distribution. Moreover, ∂

V

= ω

S, V

is equal to 0 for t in a neighbourhood of 0, and ∇ ∧ V

= 0. Moreover, S = P

j∈Z

∂

V

. We choose Φ ∈ S( R

) such that the Fourier transform of Φ is compactly supported and is equal to 1 in the neighbourhood of 0. Then Φ ∗ V

is well-defined and ∇ ∧ (Φ ∗ V

) = 0. We define

X

= Φ ∗ V

and Y

= V

− X

. We have

Y

= ∇ 1

∆ ∇ · Y

and (due to Poincar´ e’s lemma) X

= ∇(

Z

x · X

(t, λx)dλ) We find S = ∇p with

p = ∂

X

j∈Z

( Z

0

x · X

(t, λx)dλ + 1

∆ ∇ · Y

).

3 The Poisson problem −∆U = ∂ _k ∂ _i ∂ _j h

We first consider a simple Poisson problem :

Proposition 3.1 Let h ∈ L

( R

, (1 + |x|)

dx) then

U = U

+ U

= (∂

(ϕ∂

∂

G

)) ∗ h + ∂

((1 − ϕ)∂

∂

G

) ∗ h.

is a distribution such that U

belongs to L

( R

, (1 + |x|)

dx) and U is a solution of the problem

− ∆U = ∂

∂

∂

h. (5) More precisely, U is the unique solution in S

such that lim

e

U = 0 in S

.

Proof. We may write ∂

G

as

∂

G

= − Z

0

∂

W

dt

where W

(x) is the heat kernel W

(x) = (4πt)

e

2

4t

, so that on R

\ {0}, we have

∂

G

= c

x

|x|

with c

= 1 2(4π)

Z

0

e

du u

The first part defining U, U

= (∂

(ϕ∂

∂

G

)) ∗ h, is well defined, since

∂

(ϕ∂

∂

G

) is a compactly supported distribution. To control U

, we write Z Z 1

(1 + |x|)

|∂

(1 − ϕ)∂

∂

G

(x − y)||h(y)|dydx

≤

Z Z 1 (1 + |x|)

C

(1 + |x − y|)

|h(y)|dydx

≤ C

Z 1

(1 + |y|)

|h(y)|dy since

Z 1 (1 + |x|)

1

(1 + |x − y|)

dx

≤ Z

|x|>^|y|₂

1 (1 + |x|)

1

(1 + |x − y|)

dx + Z

|x−y|>^|y|₂

1 (1 + |x|)

1

(1 + |x − y|)

dx

≤ 2

(1 + |y|)

Z 1

(1 + |x − y|)

dx + 2

(1 + |y|)

Z 1

(1 + |x|)

dx

≤ C 1

(1 + |y|)

.

Now that we know that U is well defined, we may compute −∆U . −∆U

is equal to

(−∆∂

(ϕ∂

∂

G

))∗h = ∂

(ϕ∂

∂

h)−∂

((∆ϕ)∂

∂

G

)∗h−2 X

1≤l≤d

∂

((∂

ϕ)∂

∂

∂

G

)∗h.

For computing −∆U

, we see that we can differentiate under the integration sign and find

−∆U

== ∂

((1−ϕ)∂

∂

h)+∂

((∆ϕ)∂

∂

G

)∗h+2 X

1≤l≤d

∂

((∂

ϕ)∂

∂

∂

G

)∗h.

Thus, U is a solution of the Poisson problem.

Computing e

U , we find that

e

U = (e

∂

∂

∂

G

) ∗ h and thus

|e

U (x)| ≤ C

Z 1

( √

τ + |x − y|)

|h(y)| dy.

By the dominated convergence theorem, we get that lim

e

U = 0 in L

( R

, (1 + |x|)

dx). If V is another solution of the same Poisson prob- lem with V ∈ S

and lim

e

V = 0 in S

, then ∆(U − V ) = 0 and U − V ∈ S

, so that U − V is a polynomial; with the assumption that lim

e

(U − V ) = 0, we find that this polynomial is equal to 0.

If we have better integrability on h, then of course we have better inte- grability of U

. For instance, we have :

Proposition 3.2 Let h ∈ L

( R

, (1 + |x|)

dx)) then U

= ∂

((1 − ϕ)∂

∂

G

) ∗ h.

belongs to L

( R

, (1 + |x|)

)).

Proof.

We write Z Z

1

(1 + |x|)

|∂

((1 − ϕ)∂

∂

G

(x − y))||h(y)| dy dx

≤ C Z Z

1 (1 + |x|)

1

(1 + |x − y|)

|h(y)| dy dx.

For |y| < 1, we have Z 1

(1 + |x|)

1

(1 + |x − y|)

dx ≤

Z 1

(1 + |x|)

dx ≤ C and for |y| > 1, as the real number R

|x|<¹₂ 1

|x|^d−1 1

|x−_|y|^y |²

dx is finite and does not depend on y, we can write

Z 1 (1 + |x|)

1

(1 + |x − y|)

dx

≤ Z

|x|>^|y|₂

1 (1 + |x|)

1

(1 + |x − y|)

dx + Z

|x|<^|y|₂

1 (1 + |x|)

1

(1 + |x − y|)

dx

≤ 2

(1 + |y|)

Z 1

(1 + |x − y|)

dx + 2

(1 + |y|)

Z

|x|<^|y|₂

1

|x|

1

|x − y|

dx

≤ C 1

(1 + |y|)

+ C 1 (1 + |y|)

1

|y|

Z

|x|<¹₂

1

|x|

1

|x −

|

dx

≤ C

1 (1 + |y|)

.

This concludes the proof.

4 The Poisson problem −∆V = ∂ _i ∂ _j h

Proposition 4.1 Let h ∈ L

((1 + |x|)

dx) and A

= (1 − ϕ)∂

∂

G

then V = V

+ V

= (ϕ∂

∂

G

) ∗ h +

Z

(A

(x − y) − A

(−y))h(y)dy is a distribution such that V

belongs to L

((1 + |x|)

), for γ > d + 1, and V is a solution of the problem

− ∆V = ∂

∂

h. (6)

Proof. We know that V

is well defined since ϕ∂

∂

G

is a supported compactly distribution, and we will verify that V

is well defined.

We have

Z Z

1

(1 + |x|)

|A

(x − y) − A

(−y)|dx

≤ CkA

k

Z 1

(1 + |x|)

dx

For |y| > 1, we have by the mean value inequality Z

|x|<^|y|

2

1

(1 + |x|)

|A

(x − y) − A

(−y)|dx ≤ C 1

|y|

Z

|x|<^y

2

|x|

(1 + |x|)

dx and we can control the other part as follows

Z

|x|>^|y|₂

1

(1 + |x|)

|A

(−y)|dx ≤ C 1

|y|

Z

|x|>^y₂

1

(|x|)

≤ C 1

|y|

and for ε > 0 such that γ − ε ≥ d + 1, we have

Z

|x|>^|y|₂

1

(1 + |x|)

|A

(x − y)|dx ≤ C Z

|x|>^|y|₂

1

|x|

1

(1 + |x − y|)

dx

≤ C Z

|x|>^|y|₂

1

|x|

1

|x − y|

dx

≤ C 1

|y|

Z

|x|>¹

2

1

|x|

1

|x −

|

dx

≤ C

1

|y|

.

5 Proof of Theorems 1 and 2.

We may now prove Theorem 1 : Proof. Taking the divergence of

∂

u = ∆u − ∇ · (u ⊗ u) − S + ∇ · F , we obtain

− X

i,j

∂

∂

(u

u

) + X

i,j

∂

∂

F

− ∇ · S = 0 and

−∆S = ∇( X

i,j

∂

∂

(u

u

− F

)).

We write h

= u

u

− F

, and A

= (1 − ϕ)∂

∂

G

. By Proposition 4.1, we can define

p

= X

i,j

(ϕ∂

∂

G

) ∗ h

+ X

i,j

Z

(A

(x − y) − A

(−y))h

(y)dy

and

U = U

+ U

= ∇ X

i,j

(ϕ∂

∂

G

) ∗ h

+ ∇ X

i,j

((1 − ϕ)∂

∂

G

) ∗ h

= ∇p

. Let ˜ U = S − U. First, we remark that ∆U = ∆S so that ∆ ˜ U = 0, hence ˜ U is harmonic in the space variable.

On the other hand, for a test function α ∈ D( R ) such that α(t) = 0 for all |t| ≥ ε, and a test function β ∈ D( R

), and for t ∈ (ε, T − ε), we have U ˜ (t) ∗

(α ⊗ β) =(u ∗ (−∂

α ⊗ β + α ⊗ ∆β) + (−u ⊗ u + F ) · ∗(α ⊗ ∇β))(t, ·)

− X

i,j

((h

) ∗ (∇(ϕ∂

∂

G

) ∗ (α ⊗ β)))(t, ·) − (U

∗ (α ⊗ β))(t, ·).

By Proposition 3.1, we conclude that ˜ U ∗ (α ⊗ β)(t, .) belongs to the space L

( R

, (1 + |x|

). Thus, it is a tempered distribution; as it is harmonic, it must be polynomial. The integrability in L

( R

, (1 + |x|

) implies that this polynomial is constant.

If F belongs more precisely to L

((0, T ), L

( R

)) and u belongs to L

((0, T ), L

( R

)), we find that this polynomial belongs to L

( R

, w

dx), hence is equal to 0.

Then, using the identity approximation Φ

=

α(

)β(

) and letting go to 0, we obtain a similar result for ˜ U . Thus S = ∇p

+ f(t), with f(t) = 0 if F belongs to L

((0, T ), L

( R

)) and u belongs to L

((0, T ), L

( R

)).

As f does not depend on x, we may take a function β ∈ D( R

) with R β dx = 1 and write f = f ∗

β; we find that

f (t) = ∂

(u

∗ β − u ∗ β + Z

0

u ∗ ∆β − (u ⊗ u − F ) · ∗∇β − p

∗ ∇β ds) = ∂

g.

As ∂

∂

g = ∂

f = 0 and ∂

g(0, .) = 0, we find that g depends only on t;

moreover, the formula giving g proves that g ∈ L

((0, T )).

The proof of Theorem 2 is classical and the result is known as the ex- tended Galilean invariance of the Navier–Stokes equations :

Proof. Let us suppose that

∂

u = ∆u − (u · ∇)u − ∇p

− d dt g(t), with g ∈ L

((0, T )). We define

E(t) = Z

0

g(λ)dλ and w = u(t, x − E(t)) + g(t).

We have

∂

w =∂

u(t, x − E(t)) − g(t) · ∇u(t, x − E(t)) + d dt g(t)

=∆u(t, x − E(t)) − [(u · ∇)u](t, x − E(t)) − ∇p

(t, x − E(t)) − d dt g(t)

− g(t) · ∇u(t, x − E(t)) + d dt g(t)

=∆w − (w · ∇)w − ∇p

(t, x − E(t)).

If we define q

(t, x) = p

(t, x − E(t)), we find that we have q

= X

i,j

(ϕ∂

∂

G

)∗(w

w

)+ X

i,j

Z

(A

(x−y)−A

(−y))(w

(t, y)w

(t, y)) dy.

The theorem is proved.

6 Applications

A consequence of Proposition 3.1 is that we may define the Leray projection operator on the divergence of tensors that belong to L

((0, T ), L

( R

, w

dx)) : Definition 6.1 Let H ∈ L

((0, T ), L

( R

, w

dx)) and w = ∇ · H . The Leray projection P (w) of w on solenoidal vector fields is defined by

P w = w − ∇p

where ∇p

is the unique solution of

−∆∇p = ∇(∇ · w) such that

τ→+∞

lim e

∇p = 0.

A special form of the Navier–Stokes equations is then given by (M N S) ∂

u = ∆u − P ∇ · (u ⊗ u − F ), u(0, .) = u

. This leads to the integro-differential equation

u = e

u

− Z

0

e

P ∇ · (u ⊗ u − F ) ds.

The kernel of the convolution operator e

P ∇· is called the Oseen kernel;

its study is the core of the method of mild solutions of Kato and Fujita [12].

Thus, we will call equations (MNS) a mild formulation of the Navier–Stokes equations.

The mild formulation together with the local Leray energy inequality has been as well a key tool for extending Leray’s theory of weak solutions in L

to the setting of weak solutions with infinite energy. We may propose a general definition of suitable Leray-type weak solutions :

Definition 6.2 (Suitable Leray-type solution)

Let F ∈ L

((0, T ), L

( R

,

)) and u

∈ L

( R

,

) with ∇ · u

= 0. We consider the Navier–Stokes problem on (0, T ) × R

:

∂

u =∆u − P (u ⊗ u − F ),

∇ · u = 0, u(0, .) = u

.

A suitable Leray-type solution u of the Navier–Stokes equations is a vector field u defined on (0, T ) × R

such that :

• u is locally L

H

on (0, T ) × R

• sup

R |u(t, x)|

dx < +∞

• RR

(0,T)×R^d

|∇ ⊗ u(t, x)|

dx dt < +∞

• the application t ∈ [0, T ) 7→ R

u(t, x) · w(x) dx is continuous for every smooth compactly supported vector field w

• for every compact subset K of R

, lim

R

K

|u(t, x) − u

(x)|

dx = 0.

• defining p

as (the) solution of −∆p

= P

i,j

∂

∂

(u

u

− F

) given by Proposition 4.1, u is suitable in the sense of Caffarelli, Kohn and Nirenberg : there exists a non-negative locally bounded Borel measure µ on (0, T ) × R

such that

∂

( |u|

2 ) = ∆( |u|

2 ) − |∇ ⊗ u|

− ∇ · (( |u|

2 + p

)u) + u · (∇ · F ) − µ Remarks :

a) With those hypotheses, p

belongs locally to L

and u belongs locally to L

so that the distribution (

+ p

)u is well-defined.

b) Suitability is a local assumption. It has been introduced by Caffarelli,

Kohn and Nirenberg in 1982 [6] to get estimates on partial regularity for

weak Leray solutions. If we consider a solution of the Navier–Stokes equa- tions on a small domain with no specifications on the behaviour of u at the boundary, the estimates on the pressure (and the Leray projection opera- tor) are no longer available. However, Wolf described in 2017 [21] a local decomposition of the pressure into a term similar to the Leray projection of

∇ · (u ⊗ u) and a harmonic term; he could generalize the notion of suitability to this new description of the pressure. On the equivalence of various notions of suitability, see the paper by Chamorro, Lemari´ e-Rieusset and Mayoufi [8].

c) The relationship between the system (NS) and its mild formulation (MNS) described in Theorem 1 has been described by Furioli, Lemari´ e–Rieusset and Terraneo in 2000 [13, 16] in the context of uniformly locally square integrable solutions. See the paper by Dubois [11], as well.

We list here a few examples to be found in the litterature :

1. Solutions in L

: in 1934, Leray [18] studied the Navier–Stokes problem (NS) with an initial data u

∈ L

and a forcing tensor F ∈ L

L

. He then obtained a solution u ∈ L

L

∩ L

H ˙

. Remark that this solution is automatically a solution of the mild formulation of the Navier–Stokes equations (MNS). Leray’s construction by mollification provides suit- able solutions.

2. Solutions in L

: in 1999, Lemari´ e-Rieusset [15, 16] studied the Navier–Stokes problem (MNS) with an initial data u

∈ L

(and, later in [17], a forcing tensor F ∈ (L

L

)

). He obtained (local in time) existence of a suitable solution u on a small strip (0, T

) × R

such that

sup

x0∈R^d

sup

0<t<T0

Z

B(x0,1)

|u(t, x)|

dx < +∞

and

sup

x0∈R^d

Z

0

Z

B(x0,1)

|∇ ⊗ u(t, x)|

dx < +∞.

Remark that we have u ∈ L

((0, T

), L

( R

,

dx)) but u does not belong to L

((0, T

), L

( R

,

dx)); thus, in this setting, prob- lems (NS) and (MNS) are not equivalent.

Various reformulations of local Leray solutions in L

have been pro-

vided, such as Kikuchi and Seregin in 2007 [14] or Bradshaw and Tsai

in 2019 [4]. The formulas proposed for the pressure, however, are ac-

tually equivalent, as they all imply that u is solution to the (MNS)

problem.

In the case of dimension d = 2, Basson [1] proved in 2006 that the solution u is indeed global (i.e. T

= T ) and that, moreover, the solution is unique.

3. Solutions in a weighted Lebesgue space : in 2019, Fern´ andez-Dalgo and Lemari´ e–Rieusset [9] considerered data u

∈ L

( R

, w

dx) and F ∈ L

((0, +∞), L

( R

, w

dx)) with 0 < γ ≤ 2. They proved (global in time) existence of a suitable solution u such that, for all T

< +∞,

sup

0<t<T0

Z

|u(t, x)|

w

(x) dx < +∞

and

Z

0

Z

|∇ ⊗ u(t, x)|

w

(x), dx < +∞.

[Of course, for such solutions, (NS) and (MNS) are equivalent.] They showed that, for

< γ ≤ 2, this frame of work is well adapted to the study of discretely self-similar solutions with locally L

initial value, providing a new proof of the results of Chae and Wolf in 2018 [7] and of Bradshaw and Tsai in 2019 [3].

4. Homogeneous Statistical Solutions : in 1977, Vishik and Fursikov [19]

considered the (MNS) problem with a random initial value u

(ω).

The statistics of the initial distributions were supposed to be invari- ant though translation of the arguments of u

: for every Borel subset B of L

( R

) and every x

∈ R

,

P r(u

(· − x

) ∈ A) = P r(u

∈ A).

Another assumption was that u

has a bounded mean energy density : e

= E

R

|x|≤1

|u

|

dx R

|x|≤1

dx

!

< +∞.

Then

P r(u

∈ L

and u 6= 0) = 0 while, for any > 0,

P r(

Z

|u

|

1

(1 + |x|)

dx < +∞) = 1.

In [20], they constructed a solution u(t, x, ω) that solved the Navier–

Stokes equation for almost every initial value u

(ω), and the solution

belonged almost surely to L

L

(

dx) with ∇⊗u ∈ L

L

(

dx).

In 2006, Basson [2] gave a precise description of the pressure in those equations (which is equivalent to our description through the Leray projection operator) and proved the suitability of the solutions.

7 The space B _γ ² .

Instead of dealing with weighted Lebesgue spaces, one may deal with a kind of local Morrey space, the space B

.

Definition 7.1 For γ ≥ 0, define w

(x) =

and L

= L

( R

, w

(x) dx).

For 1 ≤ p < +∞, we denote B

the Banach space of all functions u ∈ L

such that :

kuk

= sup

R≥1

( 1 R

Z

B(0,R)

|u|

dx)

< +∞.

Similarly, B

L

(0, T ) is the Banach space of all functions u ⊂ (L

L

)

such that

kuk

= sup 1

R

Z

0

Z

|u|

dx ds.

Lemma 7.1 Let γ ≥ 0 and γ < δ < +∞, we have the continuous embedding L

, → B

, → B

, → L

δ

, where B

⊂ B

is the subspace of all functions u ∈ B

such that lim

R^γ

R

B(0,R)

|u(x)|

dx = 0.

Proof. Let u ∈ L

. We verify easily that kuk

≤ 2

kuk

and we see that

1 R

Z

|x|≤R

|u|

dx = Z

|x|≤R

|u|

(1 + |x|)

(1 + |x|)

R

dx

converges to zero when R → +∞ by dominated convergence, so L

, → B

. To demonstrate the other part, we estimate

Z |u|

(1 + |x|)

dx = Z

|x|≤1

|u|

(1 + |x|)

dx + X

n∈N

Z

2ⁿ⁻¹≤|x|≤2ⁿ

|u|

(1 + |x|)

dx

≤ Z

|x|≤1

|u|

dx + X

n∈N

1 (1 + 2

)

Z

2ⁿ⁻¹≤|x|≤2ⁿ

|u|

dx

≤ Z

|x|≤1

|u|

dx + c X

n∈N

1 2

Z

2ⁿ⁻¹≤|x|≤2ⁿ

|u|

dx

≤ (1 + c X

n∈N

1

2

) sup

R≥1

1 R

Z

|x|≤R

|u|

dx,

thus, B

⊂ L

. Remark : Similarly, for all δ > γ , B

L

(0, T ) ⊂ L

((0, T ), L

δ

).

Proposition 7.1 The space B

can be obtained by interpolation, B

= [L

, L

δ

]

δ,∞

for all 0 < γ < δ < ∞, and the norms

k · k

and k · k

are equivalent.

Proof. Let f ∈ B

. For A < 1, we write f

= 0 and f

= f , then we have f = f

+ f

and kf

k

≤ CA

kf k

.

For A > 1, we let R = A

> 1. We write f

= f 1

and f

= f 1

, then

kf

k

≤ Ckf k

R

= CA

kfk

and

kf

k

= X

n∈N

Z

2ⁿ⁻¹R≤|x|≤2ⁿR

|u|

(1 + |x|)

dx

≤ CR

X

n∈N

1

2

kf k

= CA

kf k

Thus, B

, → [L

, L

δ

]

δ,∞

. Let f ∈ [L

, L

δ

]

δ,∞

, then there exist c > 0 such that for all A > 0, there exist f

∈ L

and f

∈ L

so that f = f

+ f

,

kf

k

≤ cA

and kf

k

wδ

≤ cA

.

For j ∈ N we take A = 2

, then 1

2

Z

|x|<2^j

|f |

dx

≤ C 1

2

Z

|x|<2^j

|f

|

dx + 1 2

Z

|x|<2^j

|f

|

dx

≤ C 1

2

kf

k

+ C 2

Z

|x|≤1

|f

|

(1 + |x|)

dx + C

j

X

k=1

2

2

Z

2^k−1<|x|<2^k

|f

|

(1 + |x|)

dx

!

≤ C 1

2

kf

k

+ C

2

kf

k

≤ C

which implies sup

R

|x|<2^j

|f |

dx < +∞, so sup

R

|x|<R

|f |

dx <

+∞.

Thus, we can see that the local Morrey spaces B

are very close to the weighted Lebesgue spaces L

. Indeed, the methods and results of Fern´ andez-Dalgo and Lemari´ e–Rieusset [9] can be easily extended to the setting of local Morrey spaces in dimension d = 2 or d = 3 : considering data u

∈ B

( R

) and F ∈ (B

L

)(0, T )( R

) with 0 < γ ≤ 2, one gets (local in time) existence of a suitable solution u for the (MNS) system on a small strip (0, T

) × R

such that u ∈ L

((0, T

), B

) and ∇ ⊗ u ∈ (B

L

)(0, T

).

The case of γ = 2 deserves some comments. In the case d = 3, the results is slightly more general than the results in [9], as the class B

is larger than the space L

. Equations in B

have been very recently discussed by Bradshaw, Kukavica and Tsai [5]. The case d = 2 is more intricate. Indeed, while the Leray projection operator is bounded on B

( R

) (by interpolation with L

and L

δ

with 2 < δ < 3, the Riesz transforms being bounded on L

δ

by the theory of Muckenhoupt weights), this is no longer the fact on B

( R

).

Thus, one must be careful in the handling of the pressure. This has been done by Basson in his Ph. D. thesis in 2006 [1].

Local Morrey spaces B

occur naturally in the setting of homogeneous statistical solutions. By using an ergodicity argument, Dostoglou [10] proved in 2001 that, under the assumptions of Vishik and Fursikov [19], we have

P r(u

(., ω) ∈ B

( R

)) = 1.

Thus, the solutions of Vishik and Fursikov live in a smaller space than L

d+

.

References

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[2] A. Basson, Homogeneous Statistical Solutions and Local Energy In- equality for 3D Navier–Stokes Equations, Commun. Math. Phys. 266 (2006), 17–35.

[3] Z. Bradshaw and T.P. Tsai, Discretely self-similar solutions to the Navier-Stokes equations with data in L

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[5] Z. Bradshaw, I. Kukavica and T.P. Tsai, Existence of global weak solutions to the Navier-Stokes equations in weighted spaces, arXiv:1910.06929v1

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´

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