Weak solutions for Navier--Stokes equations with initial data in weighted $L^2$ spaces.

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Weak solutions for Navier–Stokes equations with initial data in weighted L

spaces.

Pedro Gabriel Fernández-Dalgo, Pierre Gilles Lemarié-Rieusset

To cite this version:

Pedro Gabriel Fernández-Dalgo, Pierre Gilles Lemarié-Rieusset. Weak solutions for Navier–Stokes equations with initial data in weightedL² spaces.. 2019. �hal-02164545�

Weak solutions for Navier–Stokes equations with initial data in weighted L ² spaces.

Pedro Gabriel Fern´ andez-Dalgo

, Pierre Gilles Lemari´ e–Rieusset

Abstract

We show the existence of global weak solutions of the 3D Navier- Stokes equations with initial velocity in the weighted spaces L²_wγ, where wγ(x) = (1 +|x|)^−γ and 0< γ≤2, using new energy controls.

As application we give a new proof of the existence of global weak discretely self-similar solutions of the 3D Navier–Stokes equations for discretely self-similar initial velocities which are locally square inte- grable.

Keywords : Navier–Stokes equations, weighted spaces, discretely self- similar solutions, energy controls

AMS classification : 35Q30, 76D05.

1 Introduction.

Infinite-energy weak Leray solutions to the Navier–Stokes equations were introduced by Lemari´e-Rieusset in 1999 [8] (they are presented more com- pletely in [9] and [10]). This has allowed to show the existence of local weak solutions for a uniformly locally square integrable initial data.

Other constructions of infinite-energy solutions for locally uniformly square integrable initial data were given in 2006 by Basson [1] and in 2007 by Kikuchi and Seregin [7]. These solutions allowed Jia and Sverak [6] to construct in 2014 the self-similar solutions for large (homogeneous of degree -1) smooth data. Their result has been extended in 2016 by Lemari´e-Rieusset [10] to

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

†e-mail : [email protected], [email protected]

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

§

solutions for rough locally square integrable data. We remark that an ho- mogeneous (of degree -1) and locally square integrable data is automatically uniformly locally L².

Recently, Bradshaw and Tsai [2] and Chae and Wolf [3] considered the case of solutions which are self-similar according to a discrete subgroup of dilations. Those solutions are related to an initial data which is self-similar only for a discrete group of dilations; in contrast to the case of self-similar solutions for all dilations, such an initial data, when locally L², is not nec- essarily uniformly locally L², therefore their results are no consequence of constructions described by Lemari´e-Rieusset in [10].

In this paper, we construct an alternative theory to obtain infinite-energy global weak solutions for large initial data, which include the discretely self- similar locally square integrable data. More specifically, we consider the weights

w_γ(x) = 1 (1 +|x|)^γ with 0< γ, and the spaces

L²_wγ =L²(w_γdx).

Our main theorem is the following one :

Theorem 1 Let 0 < γ ≤ 2. If u₀ is a divergence-free vector field such that u0 ∈ L²_wγ(R³) and if F is a tensor F(t, x) = (Fi,j(t, x))_1≤i,j≤3 such that F∈L²((0,+∞), L²_wγ), then the Navier–Stokes equations with initial valueu₀

(N S)







∂tu= ∆u−(u· ∇)u− ∇p+∇ ·F

∇ ·u= 0, u(0, .) = u₀ has a global weak solution u such that :

• for every 0< T < +∞, u belongs to L^∞((0, T), L²_wγ) and ∇u belongs to L²((0, T), L²_wγ)

• the pressure p is related to u and F through the Riesz transforms R_i =

∂i

√−∆ by the formula

p=

3

X

i=1 3

X

j=1

R_iR_j(u_iu_j −F_i,j) where, for every 0 < T < +∞, P3

i=1

P3

j=1R_iR_j(u_iu_j) belongs to

6/5 3 3

• the map t ∈ [0,+∞) 7→ u(t, .) is weakly continuous from [0,+∞) to L²_wγ, and is strongly continuous at t= 0 :

limt→0ku(t, .)−u₀k_L²_wγ = 0.

• the solution u is suitable : there exists a non-negative locally finite measure µ on (0,+∞)×R³ such that

∂_t(|u|²

2 ) = ∆(|u|²

2 )− |∇u|²− ∇ ·

(|u|²

2 +p)u

+u·(∇ ·F)−µ.

In particular, we have the energy controls ku(t, .)k²_L2

wγ + 2 Z t

0

k∇u(s, .)k²_L2 wγds

≤ku₀k²_L2 wγ −

Z t 0

Z

∇|u|²· ∇w_γdx ds+ Z t

0

Z

(|u|²+ 2p)u· ∇(w_γ) dx ds

−2

3

X

i=1 3

X

j=1

Z t 0

Z

F_i,j(∂_iu_j)w_γ+F_i,ju_i∂_j(w_γ)dx ds

and ku(t, .)k²_L2

wγ ≤ ku₀k²_L2 wγ+C_γ

Z t 0

kF(s, .)k²_L2

wγds+C_γ Z t

0

ku(s, .)k²_L2

wγ+ku(s, .)k⁶_L2 wγ ds

A key tool for proving Theorem 1 and for applying it to the study of discretely self-similar solutions is given by the following a priori estimates for an advection-diffusion problem :

Theorem 2 Let 0 < γ ≤ 2. Let 0< T < +∞. Let u₀ be a divergence-free vector field such thatu₀ ∈L²_wγ(R³)andFbe a tensorF(t, x) = (F_i,j(t, x))_1≤i,j≤3 such that F ∈ L²((0, T), L²_wγ). Let b be a time-dependent divergence free vector-field (∇ ·b= 0) such that b∈L³((0, T), L³_w

3γ/2).

Let u be a solution of the following advection-diffusion problem

(AD)







∂_tu= ∆u−(b· ∇)u− ∇p+∇ ·F

∇ ·u= 0, u(0, .) = u₀ be such that :

• the pressure p is related to u, b and F through the Riesz transforms R_i = ^√∂_−∆ⁱ by the formula

p=

3

X

i=1 3

X

j=1

R_iR_j(b_iu_j −F_i,j)

whereP3 i=1

P3

j=1R_iR_j(b_iu_j)belongs toL³((0, T), L^6/5w6γ 5

)andP3 i=1

P3

j=1R_iR_jF_i,j belongs to L²((0, T), L²_wγ)

• the map t ∈ [0, T) 7→ u(t, .) is weakly continuous from [0, T) to L²_wγ, and is strongly continuous at t = 0 :

limt→0ku(t, .)−u₀k_L²_wγ = 0.

• there exists a non-negative locally finite measure µ on (0, T)×R³ such that

∂_t(|u|²

2 ) = ∆(|u|²

2 )−|∇u|²−∇·

|u|² 2 b

−∇·(pu)+u·(∇·F)−µ. (1) Then, we have the energy controls

ku(t, .)k²_L2 wγ + 2

Z t 0

k∇u(s, .)k²_L2 wγds

≤ku₀k²_L2 wγ −

Z t 0

Z

∇|u|²· ∇w_γdx ds+ Z t

0

Z

|u|²b· ∇(w_γ)dx ds + 2

Z t 0

Z

pu· ∇(wγ)dx ds−2

3

X

i=1 3

X

j=1

Z t 0

Z

Fi,j(∂iuj)wγ+Fi,jui∂j(wγ)dx ds and

ku(t, .)k²_L2 wγ +

Z t 0

k∇uk²_L2 wγ ds

≤ku₀k²_L2

wγ +C_γ Z t

0

kF(s, .)k²_L2

wγds+C_γ Z t

0

(1 +kb(s, .)k²_L3 w3γ/2

)ku(s, .)k²_L2 wγds

where C_γ depends only on γ (and not on T, and not on b, u, u₀ nor F).

In particular, we shall prove the following stability result :

Theorem 3 Let 0 < γ ≤ 2. Let 0 < T < +∞. Let u_0,n be divergence- free vector fields such that u_0,n ∈L²_wγ(R³) and Fn be tensors such that Fn ∈ L²((0, T), L²_wγ). Let b_n be time-dependent divergence free vector-fields such that b_n ∈L³((0, T), L³_w

3γ/2).

Let u_n be solutions of the following advection-diffusion problems

(AD_n)







∂_tu_n= ∆u_n−(b_n· ∇)u_n− ∇p_n+∇ ·Fⁿ

∇ ·u_n = 0, u_n(0, .) = u_0,n such that :

• u_n belongs to L^∞((0, T), L²_wγ) and ∇u_n belongs to L²((0, T), L²_wγ)

• the pressure p_n is related to u_n, b_n and Fn by the formula

p_n=

3

X

i=1 3

X

j=1

R_iR_j(b_n,iu_n,j −F_n,i,j)

• the map t ∈ [0, T) 7→ u_n(t, .) is weakly continuous from [0, T) to L²_wγ, and is strongly continuous at t = 0 :

limt→0ku_n(t, .)−u_0,nk_L²

wγ = 0.

• there exists a non-negative locally finite measureµ_n on(0, T)×R³ such that

∂t(|u_n|²

2 ) = ∆(|u_n|²

2 )−|∇un|²−∇·

|u_n|² 2 bn

−∇·(pnun)+un·(∇·Fⁿ)−µn. Ifu_0,n is strongly convergent tou0,∞ inL²_wγ, if the sequenceFn is strongly convergent to F^∞ in L²((0, T), L²_wγ), and if the sequence b_n is bounded in L³((0, T), L³_w

3γ/2), then there exists p_∞, u_∞, b_∞ and an increasing sequence (n_k)k∈N with values in N such that

• u_nk converges *-weakly tou∞inL^∞((0, T), L²_wγ), ∇u_nk converges weakly to ∇u∞ in L²((0, T), L²_wγ)

• b_nk converges weakly to b∞ in L³((0, T), L³_w

3γ/2), p_nk converges weakly to p in L³((0, T), L^6/5 ) +L²((0, T), L² )

• u_nk converges strongly tou∞ in L²_loc([0, T)×R³): for every T₀ ∈(0, T) and every R >0, we have

k→+∞lim Z T0

0

Z

|y|<R

|u_nk(s, y)−u∞(s, y)|²ds dy = 0.

Moreover, u∞ is a solution of the advection-diffusion problem

(AD∞)







∂_tu∞= ∆u∞−(b∞· ∇)u∞− ∇p∞+∇ ·F^∞

∇ ·u∞ = 0, u∞(0, .) = u0,∞

and is such that :

• the map t ∈[0, T)7→u∞(t, .) is weakly continuous from [0, T) to L²_wγ, and is strongly continuous at t = 0 :

limt→0ku∞(t, .)−u0,∞k_L²

wγ = 0.

• there exists a non-negative locally finite measureµ∞ on(0, T)×R³ such that

∂_t(|u∞|²

2 ) = ∆(|u∞|²

2 )−|∇u_∞|²−∇·

|u∞|² 2 b_∞

−∇·(p_∞u_∞)+u_∞·(∇·F^∞)−µ_∞.

Notations.

All along the text, C_γ is a positive constant whose value may change from line to line but which depends only on γ.

2 The weights w

.

We consider the weights w_δ = _(1+|x|)¹ δ where 0 < δ and x ∈ R³. A very important feature of those weights is the control of their gradients :

|∇w_δ(x)|=δ w_δ(x)

1 +|x| (2)

Lemma 1 (Muckenhoupt weights) If 0< δ <3 and 1 < p <+∞, then w_δ belongs to the Muckenhoupt class A_p.

Proof : We recall that a weight w belongs to A_p(R³) for 1 < p < +∞ if and only if it satisfies the reverse H¨older inequality

sup

x∈R³,R>0

1

|B(x, R)|

Z

B(x,R)

w(y)dy ¹_p

1

|B(x, R)|

Z

B(x,R)

dy w(y)^p−11

!1−¹_p

<+∞. (3) For all 0 < R ≤ 1 the inequality |x−y|< R implies ¹₂(1 +|x|) ≤1 +|y| ≤ 2(1 +|x|), thus we can control the left side in (3) for wδ by 4^pδ.

For all R > 1 and |x| > 10R, we have that the inequality |x−y| < R implies ₁₀⁹ (1 +|x|) ≤ 1 +|y| ≤ ¹¹₁₀(1 +|x|), thus we can control the left side in (3) for w_δ by (¹¹₉ )^δp.

Finally, for R >1 and |x| ≤10R, we write 1

|B(x, R)|

Z

B(x,R)

w(y)dy ¹_p

1

|B(x, R)|

Z

B(0,R)

dy w(y)^p−11

!1−¹_p

≤

1

|B(0, R)|

Z

B(x,11R)

w(y)dy ¹_p

1

|B(0, R)|

Z

B(0,11R)

dy w(y)^p−11

!1−¹_p

= 1

R³ Z 11R

0

r² dr (1 +r)^δ

1 p

1 R³

Z 11R 0

r²(1 +r)^p−1δ dr ¹⁻

1 p

≤c_δ,p 1

R³ Z 11R

0

r²dr r^δ

¹_p 1 R³

Z 11R 0

r²dr ¹⁻¹_p

+ 1

R³ Z 11R

0

r^2+p−1δ dr

¹⁻¹_p!

=c_δ,p 11³ (3−δ)^1p





(11R)^−δp 3^1−p1

+ 1

(3 + _p−1^δ )^1−1p



.

The lemma is proved.

Lemma 2 If 0 < δ < 3 and 1 < p < +∞, then the Riesz transforms R_i and the Hardy–Littlewood maximal function operator are bounded on L^p_w

δ = L^p(w_δ(x)dx) :

kRjfk_L^p_wδ ≤Cp,δkfk_L^p_wδ and kMfk_L^p_wδ ≤Cp,δkfk_L^p_wδ.

Proof : The boundedness of the Riesz transforms or of the Hardy–Littlewwod maximal function onL^p(w_γdx) are basic properties of the Muckenhoupt class

A_p [5].

We will use strategically the next corollary, which is specially useful to obtain

Corollary 1 (Non-increasing kernels) Let θ ∈L¹(R³) be a non-negative radial function which is radially non-increasing. Then, if 0 < δ < 3 and 1< p <+∞, we have, for f ∈L^p_w

δ, the inequality kθ∗fk_L^p

wδ ≤C_p,δkfk_L^p

wδkθk₁.

Proof : We have the well-known inequality for radial non-increasing kernels [4]

|θ∗f(x)| ≤ kθk₁M_f(x)

so that we may conclude with Lemma 2.

We illustrate the utility of Lemma 2 with the following corollaries:

Corollary 2 Let 0 < γ < ⁵₂ and 0< T < +∞. Let F be a tensor F(t, x) = (F_i,j(t, x))_1≤i,j≤3 such that F ∈ L²((0, T), L²_w

γ). Let b be a time-dependent divergence free vector-field (∇ ·b= 0) such that b∈L³((0, T), L³_w

3γ/2).

Let u be a solution of the following advection-diffusion problem







∂_tu= ∆u−(b· ∇)u− ∇q+∇ ·F

∇ ·u= 0,

(4) be such that : ubelongs toL^∞((0, T), L²_w

γ)and∇ubelongs toL²((0, T), L²_w

γ), and the pressure q belongs to D⁰((0, T)×R³).

Then, the gradient of the pressure ∇q is necessarily related to u, b and F through the Riesz transforms R_i = ^√∂_−∆ⁱ by the formula

∇q=∇

3

X

i=1 3

X

j=1

RiRj(biuj −Fi,j)

!

andP3 i=1

P3

j=1R_iR_j(b_iu_j)belongs toL³((0, T), L^6/5w6γ 5

)andP3 i=1

P3

j=1R_iR_jF_i,j belongs to L²((0, T), L²_wγ).

Proof : We define p=

3

X

i=1 3

X

j=1

R_iR_j(b_iu_j −F_i,j)

! .

As 0< γ < ⁵₂ we can use Lemma 2 and (2) to obtain P3 i=1

P3

j=1R_iR_j(b_iu_j)

6/5 3 3

Taking the divergence in (4), we obtain ∆(q −p) = 0. We take a test function α ∈ D(R) such that α(t) = 0 for all |t| ≥ ε, and a test function β ∈ D(R³); then the distribution∇q∗(α⊗β) is well defined on (ε, T−ε)×R³.

We fix t∈(ε, T −ε) and define

A_α,β,t = (∇q∗(α⊗β)− ∇p∗(α⊗β))(t, .).

We have

A_α,β,t=(u∗(−∂_tα⊗β+α⊗∆β) + (−u⊗b+F)·(α⊗ ∇β))(t, .)

−(p∗(α⊗ ∇β))(t, .). (5)

Convolution with a function in D(R³) is a bounded operator on L²_wγ and on L^6/5w_6γ/5 (as, for ϕ∈ D(R³) we have |f∗ϕ| ≤C_ϕM_f). Thus, we may conclude from (5) that A_α,β,t ∈ L²_w

γ +L^6/5w_6γ/5. If max{γ,^γ+2₂ } < δ < 5/2 , we have A_α,β,t ∈L^6/5w6δ/5.

In particular,A_α,β,t is a tempered distribution. As we have

∆A_α,β,t= (α⊗β)∗(∆(q−p))(t, .) = 0,

we find that Aα,β,t is a polynomial. We remark that for all 1< r <+∞ and 0 < δ < 3, L^r_w

δ does not contain non-trivial polynomials. Thus, A_α,β,t = 0.

We then use an approximation of identity ¹4α(^t)β(^x) and conclude that

∇(q−p) = 0.

Actually, we can answer a question posed by Bradshaw and Tsai in [2]

about the nature of the pressure for self-similar solutions of the Navier–Stokes equations. In effect, we have the next corollary:

Corollary 3 Let 1 < γ < ⁵₂ and 0< T < +∞. Let F be a tensor F(t, x) = (F_i,j(t, x))_1≤i,j≤3 such that F∈L²((0, T), L²_wγ).

Let u be a solution of the following problem







∂_tu= ∆u−(u· ∇)u− ∇p+∇ ·F

∇ ·u= 0,

be such that : ubelongs toL^∞([0,+∞), L²)_locand∇ubelongs toL²([0,+∞), L²)_loc, and the pressure q is in D⁰((0, T)×R³).

We suppose that there exists λ >1 such that λ²F(λ²t, λx) = F(t, x) and λu(λ²t, λx) = u(t, x). Then, the gradient of the pressure ∇q is necessarily related to u and F through the Riesz transforms R_i = ^√∂_−∆ⁱ by the formula

∇q =∇

3

X

i=1 3

X

j=1

R_iR_j(u_iu_j−F_i,j)

!

andP3 i=1

P3

j=1R_iR_j(u_iu_j)belongs toL⁴((0, T), L^6/5w6γ 5

)andP3 i=1

P3

j=1R_iR_jF_i,j belongs to L²((0, T), L²_w

γ).

Proof : We shall use Corollary 2, and thus we need to show thatubelongs to L^∞((0, T), L²_wγ ∩L³((0, T), L³_3γ/2)) and ∇u belongs to L²((0, T), L²_wγ). In fact,

kuk_L^∞_((0,T),L²

wγ)≤ sup

0≤t≤T

Z

|x|<1

|u(t, x)|²dx+c sup

0≤t≤T

X

k∈N

Z

λ^k−1<|x|<λ^k

|u(t, x)|² λ^γk dx

and sup

0≤t≤T

X

k≥1

Z

λ^k−1<|x|<λ^k

|u(t, x)|²

λ^γk dx≤ sup

0≤t≤T

X

k∈N

λ^(1−γ)k Z

λ⁻¹<|x|<1

|u( t

λ^2k, x)|²dx

≤c sup

0≤t≤T

Z

λ⁻¹<|x|<1

|u(t, x)|²dx <+∞.

For ∇u, we compute for k∈N, Z T

0

Z

λ^k−1<|x|<λ^k

|∇u(t, x)|²dt dx=λ^k Z ^T

λ2k

0

Z

1 λ<|x|<1

|∇u(t, x)|²dx dt.

We may conclude that∇ubelongs toL²((0, T), L²_wγ), since forγ >1 we have P

k∈Nλ^(1−γ)k<+∞.

Now, we use the Sobolev embeddings described in next Lemma (Lemma 3) to get that u belongs toL²((0, T), L⁶_w3γ), and thus (by interpolation with L^∞((0, T), L²_wγ)) to L⁴((0, T), L³_w

3γ/2).

In particular, P3 i=1

P3

j=1R_iR_j(u_iu_j) belongs to L⁴((0, T), L^6/5w6γ 5

), since we have

k(u⊗u)w_γk_L6/5 ≤ k√

w_γuk_L²k√

w_γuk_L³ ≤ k√

w_γuk_L³²2k√

w_γuk_L¹²6.

Lemma 3 (Sobolev embeddings) Let δ > 0. If f ∈ L²_wδ and ∇f ∈ L²_wδ then f ∈L⁶_w

3δ and

kfk_L⁶_w

3δ ≤C_δ(kfk_L²

wδ +k∇fk_L²

wδ).

Proof : Since both f and w_δ/2 are locally in H¹, we write

∂_i(f w_δ/2) = w_δ/2∂_if +f ∂_i(w_δ/2) = w_δ/2∂_if −δ 2

x_i

|x|w_δ/2f

and thus

kw_δ/2fk²₂+k∇(w_δ/2f)k²₂ ≤(1 + δ²

2)kw_δ/2fk²₂+ 2kw_δ/2∇fk²₂. Thus, w_δ/2f belongs to L⁶ (sinceH¹ ⊂L⁶), or equivalently f ∈L⁶_w

3δ.

3 A priori estimates for the advection-diffusion problem.

3.1 Proof of Theorem 2.

Let 0 < t₀ < t₁ < T. We take a functionα ∈ C^∞(R) which is non-decreasing, with α(t) equal to 0 for t < 1/2 and equal to 1 for t > 1. For 0 < η <

min(^t₂⁰, T −t₁), we define

α_η,t0,t1(t) = α(t−t₀

η )−α(t−t₁ η ).

We take as well a non-negative function φ ∈ D(R³) which is equal to 1 for

|x| ≤ 1 and to 0 for |x| ≥ 2. For R > 0, we define φ_R(x) = φ(_R^x). Finally, we define, for > 0, w_γ, = ¹

(1+√

²+|x|²)^δ. We have α_η,t0,t1(t)φ_R(x)w_γ,(x) ∈ D((0, T)×R³) and α_η,t0,t1(t)φ_R(x)w_γ,(x)≥0. Thus, using the local energy

balance (1) and the fact that µ≥0, we find

−

Z Z |u|²

2 ∂_tα_η,t0,t1φ_Rw_γ,dx ds

≤ −

3

X

i=1

Z Z

∂_iu·uα_η,t0,t1(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds

− Z Z

|∇u|² α_η,t0,t1φ_Rw_γ,dx ds

+

3

X

i=1

Z Z |u|²

2 biαη,t0,t1(wγ,∂iφR+φR∂iwγ,)dx ds +

3

X

i=1

Z Z

α_η,t0,t1pu_i(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds

−

3

X

i=1 3

X

j=1

Z Z

F_i,ju_jα_η,t0,t1(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds

−

3

X

i=1 3

X

j=1

Z Z

F_i,j∂_iu_j α_η,t0,t1φ_Rw_γ,dx ds.

We remark that, independently from R >1 and >0, we have (for 0< γ ≤ 2)

|w_γ,∂_iφ_R|+|φ_R∂_iw_γ,| ≤C_γ w_γ(x)

1 +|x| ≤C_γw_3γ/2(x).

Moreover, we know thatubelongs toL^∞((0, T), L²_w

γ)∩L²((0, T), L⁶_w

3γ) hence to L⁴((0, T), L³_w

3γ/2). Since T <+∞, we have as well u ∈L³((0, T), L³_w

3γ/2).

(This is the same type of integrability as required for b). Moreover, we have pu_i ∈L¹_w

3γ/2 sincew_γp∈L²((0, T), L^6/5+L²) andw_γ/2u ∈L²((0, T), L²∩L⁶).

All those remarks will allow us to use dominated convergence.

We first letη go to 0. We find that

−lim

η→0

Z Z |u|²

2 ∂_tα_η,t0,t1φ_Rw_γ,dx ds

≤ −

3

X

i=1

Z t1

t0

Z

∂_iu·u(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds

− Z t1

t0

Z

|∇u|² φ_Rw_γ,dx ds

+

3

X

i=1

Z t1

t0

Z |u|²

2 b_i(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds +

3

X

i=1

Z t1

t0

Z

pui(wγ,∂iφR+φR∂iwγ,)dx ds

−

3

X

i=1 3

X

j=1

Z t1

t0

Z

F_i,ju_j(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds

−

3

X

i=1 3

X

j=1

Z t1

t0

Z

F_i,j∂_iu_j φ_Rw_γ,dx ds.

Let us define

A_R,(t) = Z

|u(t, x)|²φ_R(x)w_γ,(x)dx.

As we have

−

Z Z |u|²

2 ∂_tα_η,t0,t1φ_Rw_γ,dx ds =−1 2

Z

∂_tα_η,t0,t1A_R,(s)ds

we find that, when t₀ and t₁ are Lebesgue points of the measurable function A_R,

limη→0−

Z Z |u|²

2 ∂_tα_η,t0,t1φ_Rw_γ,dx ds = 1

2(A_R,(t₁)−A_R,(t₀)).

Then, by continuity, we can let t₀ go to 0 and thus replace t₀ by 0 in the inequality. Moreover, if we lett₁go tot, then by weak continuity, we find that A_R,(t) ≤ lim_t1→tA_R,(t₁), so that we may as well replace t₁ by t ∈ (0, T).

Thus we find that for every t∈(0, T), we have

Z |u(t, x)|²

2 φ_Rw_γ,dx

≤

Z |u₀(x)|²

2 φ_Rw_γ,dx

−

3

X

i=1

Z t 0

Z

∂iu·u(wγ,∂iφR+φR∂iwγ,)dx ds

− Z t

0

Z

|∇u|² φ_Rw_γ,dx ds

+

3

X

i=1

Z t 0

Z |u|²

2 b_i(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds +

3

X

i=1

Z t 0

Z

pui(wγ,∂iφR+φR∂iwγ,)dx ds

−

3

X

i=1 3

X

j=1

Z t 0

Z

F_i,ju_j(w_γ,∂_iφ_R+φ_R∂_iw_γ,)dx ds

−

3

X

i=1 3

X

j=1

Z t 0

Z

F_i,j∂_iu_j φ_Rw_γ,dx ds.

(6)

Thus, letting R go to +∞ and then go to 0, we find by dominated convergence that, for every t ∈(0, T), we have

ku(t, .)k²_L2 wγ + 2

Z t 0

k∇u(s, .)k²_L2 wγds

≤ku₀k²_L2 wγ −

Z t 0

Z

∇|u|²· ∇w_γdx ds+ Z t

0

Z

(|u|²b+ 2pu)· ∇(w_γ)dx ds

−2

3

X

i=1 3

X

j=1

Z t 0

Z

F_i,j(∂_iu_j)w_γ+F_i,ju_i∂_j(w_γ)dx ds.

Now we write

Z t 0

Z

∇|u|²· ∇w_γds ds

≤2γ Z t

0

Z

|u||∇u|w_γdx ds

≤1 4

Z t 0

k∇uk²_L2

wγ ds+ 4γ² Z t

0

kuk²_L2 wγ ds.

Writing

p₁ =

3

X

i=1 3

X

j=1

R_iR_j(b_iu_j) and p₂ =−

3

X

i=1 3

X

j=1

R_iR_j(F_i,j) and using the fact that w_6γ/5 ∈ A_6/5 and wγ∈ A2, we get

Z t 0

Z

(|u|²b+ 2p₁u)· ∇(w_γ)dx ds

≤γ Z t

0

Z

(|u|²|b|+ 2|p₁| |u|)w^3/2_γ dx ds

≤γ Z t

0

kw_γ^1/2uk₆(kw_γ|b||u|k_6/5+kw_γp₁k_6/5)ds

≤C_γ Z t

0

kw_γ^1/2uk₆kw_γ|b||u|k_6/5ds

≤C_γ Z t

0

kw_γ^1/2uk₆kw_γ^1/2bk₃kw_γ^1/2uk₂ds

≤C_γ⁰ Z t

0

(k∇uk_L²_wγ +kuk_L²_wγ) kbk_L³_w

3γ/2kuk_L²_wγds

≤ 1 4

Z t 0

k∇uk²_L2

wγds+C_γ⁰⁰ Z t

0

kuk²_L2

wγ(kbk_L³_w

3γ/2 +kbk²_L3 w3γ/2

)ds and

Z t 0

Z

2p2u· ∇(wγ)dx ds

≤2γ Z t

0

Z

|p2| |u|wγdx ds

≤γ Z t

0

kuk²_L2

wγ +kp2k²_L2 wγ ds

≤C_γ Z t

0

kuk²_L2

wγ +kFk²_L2 wγds.

Finally, we have

2

3

X

i=1 3

X

j=1

Z t 0

Z

F_i,j(∂_iu_j)w_γ+F_i,ju_i∂_j(w_γ)dx ds

≤2 Z t

0

Z

|F|(|∇u|+γ|u|)w_γdx ds

≤ 1 4

Z t 0

k∇uk²_L2

wγds+C_γ Z t

0

kuk²_L2

wγ +kFk²_L2 wγ ds.

We have obtained ku(t, .)k²_L2

wγ + Z t

0

k∇uk²_L2 wγ ds

≤ku₀k²_L2 +C_γ Z t

kF(s, .)k²_L2 ds+C_γ Z t

(1 +kb(s, .)k²_L3 )ku(s, .)k²_L2 ds

and Theorem 2 is proven.

3.2 Passive transportation.

From inequality (7), we have the following direct consequence : Corollary 4 Under the assumptions of Theorem 2, we have

sup

0<t<T

kuk_L²_wγ ≤(ku₀k_L²_wγ +C_γkFk_L²_((0,T),L²_wγ)) e

Cγ(T+T^1/3kbk²

L3((0,T),L3 w3γ/2))

and

k∇uk_L²_((0,T),L²

wγ) ≤(ku₀k_L²_wγ +C_γkFk_L²_((0,T),L²_wγ)) e

Cγ(T+T^1/3kbk²

L3((0,T),L3 w3γ/2))

where the constant C_γ depends only on γ.

Another direct consequence is the following uniqueness result for the advection- diffusion problem with a (locally in time), bounded b :

Corollary 5 . Let 0< γ ≤2. Let0< T <+∞. Letu0 be a divergence-free vector field such thatu₀ ∈L²_wγ(R³)andFbe a tensorF(t, x) = (F_i,j(t, x))_1≤i,j≤3 such that F ∈ L²((0, T), L²_wγ). Let b be a time-dependent divergence free vector-field (∇ ·b = 0) such that b ∈ L³((0, T), L³_w

3γ/2). Assume moreover that b belongs to L²_tL^∞_x (K) for every compact subset K of (0, T)×R³.

Let(u₁, p₁)and(u₂, p₂)be two solutions of the following advection-diffusion problem

(AD)







∂_tu= ∆u−(b· ∇)u− ∇p+∇ ·F

∇ ·u= 0, u(0, .) = u₀ be such that, for k = 1 and k = 2, :

• u_k belongs to L^∞((0, T), L²_wγ) and ∇u_k belongs to L²((0, T), L²_wγ)

• the pressure p_k is related to u_k, b and F through the Riesz transforms R_i = ^√∂_−∆ⁱ by the formula

p_k =

3

X

i=1 3

X

j=1

R_iR_j(b_iu_k,j−F_i,j)

• the map t ∈ [0, T) 7→ u_k(t, .) is weakly continuous from [0, T) to L²_wγ, and is strongly continuous at t = 0 :

Then u₁ =u₂.

Proof : Letv=u1−u2 and q=p1−p2. Then we have







∂_tv= ∆v−(b· ∇)v− ∇q

∇ ·v= 0, v(0, .) = 0

Moreover on every compact subset K of (0, T)×R³,b⊗v is in L²_tL²_x, while it belongs globally to L³_tL^6/5w_6γ/5. Writing, forϕ, ψ ∈ D((0, T)×R³) such that ψ = 1 on the neigborhood of the support of ϕ,

ϕq =q₁+q₂ =ϕ

3

X

i=1 3

X

j=1

R_iR_j(ψb_iv_j) +ϕ

3

X

i=1 3

X

j=1

R_iR_j((1−ψ)b_iv_j) we find that kq₁k_L²_L² ≤C_ϕ,ψkψb⊗vk_L²_L² and

kq₂k_L³_L^∞ ≤C_ϕ,ψkb⊗vk_L3L^6/5w6γ/5

with

Cϕ,ψ ≤Ckϕk∞k1−ψk∞ sup

x∈Suppϕ

Z

y∈Supp (1−ψ)

(1 +|y|)^γ

|x−y|³

6!1/6

<+∞.

Thus, we may take the scalar product of ∂_tv with v and find that

∂_t(|v|²

2 ) = ∆(|v|²

2 )− |∇v|²− ∇ · |v|²

2 b

− ∇ ·(qv).

Thus we are under the assumptions of Theorem 2 and we may use Corollary

4 to find that v= 0.

3.3 Active transportation.

We begin with the following lemma :

Lemma 4 Let α be a non-negative bounded measurable function on [0, T) such that, for two constants A, B ≥0, we have

α(t)≤A+B Z t

0

α(s) +α(s)³ds.

If T₀ > 0 and T₁ = min(T, T₀, ¹ ), we have, for every t ∈ [0, T₁],

Proof : We write α≤1 +α³. We define Φ(t) = A+BT0+B

Z t 0

α³ds and Ψ(t) =A+BT0+B Z t

0

Φ³(s)ds.

We have, for t∈[0, T₁], α≤Φ≤Ψ. Since Ψ is C¹, we may write Ψ⁰(t) =BΦ(t)³ ≤BΨ(t)³

and thus

1

Ψ(0)² − 1

Ψ(t)² ≤2Bt.

We thus find

Ψ(t)² ≤ Ψ(0)²

1−2BΨ(0)²t ≤2Ψ(0)².

The lemma is proven.

Corollary 6 Assume thatu0, u, p, Fandb satisfy assumptions of Theorem 2, Assume moreover that b is controlled by u : for every t ∈(0, T),

kb(t, .)k_L³_w

3γ/2 ≤C₀ku(t, .)k_L³_w

3γ/2.

Then there exists a constant Cγ ≥1 such that if T0 < T is such that C_γ(1 +C₀⁴)

1 +C₀⁴+ku₀k²_L2 wγ +

Z T0

0

kFk²_L2 wγ ds

²

T₀ ≤1 then

sup

0≤t≤T₀

k u(t, .)k²_L2

wγ ≤C_γ(1 +C₀⁴+ku₀k²_L2 wγ +

Z T0

0

kFk²_L2 wγ ds)

and

Z T0

0

k∇uk²_L2

wγds ≤C_γ(1 +C₀⁴+ku₀k²_L2 wγ +

Z T0

0

kFk²_L2 wγ ds).

Proof : We start from inequality (7) : ku(t, .)k²_L2

wγ + Z t

0

k∇uk²_L2 wγ ds

≤ku₀k²_L2

wγ +C_γ Z t

0

kF(s, .)k²_L2

wγds+C_γ Z t

0

(1 +kb(s, .)k²_L3 w3γ/2

)ku(s, .)k²_L2 wγds