THE GLOBAL SOLVABILITY OF THE HALL-MAGNETOHYDRODYNAMICS SYSTEM IN CRITICAL SOBOLEV SPACES

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THE GLOBAL SOLVABILITY OF THE

HALL-MAGNETOHYDRODYNAMICS SYSTEM IN CRITICAL SOBOLEV SPACES

Raphaël Danchin, Jin Tan

To cite this version:

Raphaël Danchin, Jin Tan. THE GLOBAL SOLVABILITY OF THE HALL-

MAGNETOHYDRODYNAMICS SYSTEM IN CRITICAL SOBOLEV SPACES. 2019. �hal-

02417517�

HALL-MAGNETOHYDRODYNAMICS SYSTEM IN CRITICAL SOBOLEV SPACES

RAPHA ¨ EL DANCHIN AND JIN TAN

Abstract. We are concerned with the 3D incompressible Hall-magnetohydro- dynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato’s theorem [10] for the Navier- Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita-Kato regularity class.

A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2

D flows for the Hall-MHD system, and prove the global existence of strong solutions, assuming only that the initial magnetic field is small.

1. Introduction

We are concerned with the following three dimensional incompressible resistive and viscous Hall-magnetohydrodynamics system:

∂ t u + u · ∇u + ∇P = (∇ × B) × B + µ∆u in R ⁺ × R ³ , (1.1)

div u = 0 in R ⁺ × R ³ , (1.2)

∂ t B − ∇ × (u × B) + ε∇ × ((∇ × B) × B) = ν∆B in R ⁺ × R ³ , (1.3)

(u, B)| t=0 = (u 0 , B 0 ) in R ³ , (1.4)

where the unknown functions u, B and P represent the velocity field, the magnetic field and the scalar pressure, respectively. The parameters µ and ν are the fluid viscosity and the magnetic resistivity, while the dimensionless positive number ε measures the magnitude of the Hall effect compared to the typical length scale of the fluid. For compatibility with (1.2), we assume that div u 0 = 0 and, for physical consistency, since a magnetic field is a curl, we suppose that div B 0 = 0, a property that is propagated by (1.3).

The above system is used to model the magnetic reconnection phenomenon, that cannot be explained by the classical MHD system where the Hall electric field E H := εJ × B (here the current J is defined by J := ∇ × B) is neglected. The study of the Hall-MHD system has been initiated by Lighthill in [16]. Owing to its importance in the theory of space plasma, like e.g. star formation, solar flares or geo-dynamo, it has received lots of attention from physicists (see [2, 11, 14, 19, 20]).

The Hall-MHD system has been considered in mathematics only rather recently.

In [1], Acheritogaray, Degond, Frouvelle and Liu formally derived the Hall-MHD system both from a two fluids system and from a kinetic model. Then, in [5], Chae,

2010 Mathematics Subject Classification. 35Q35; 76D03; 86A10.

Key words and phrases. Hall-magnetohydrodynamics; Well-posedness; Sobolev spaces.

1

Degond and Liu showed the global existence of Leray-Hopf weak solutions as well as the local existence of classical solutions pertaining to data with large Sobolev regularity. Weak solutions have been further investigated by Dumas and Sueur in [9] both for the Maxwell-Landau-Lifshitz system and for the Hall-MHD system. In [6], blow-up criteria and the global existence of smooth solutions emanating from small initial data have been obtained. In [21, 22], Weng studies the long-time behaviour and obtained optimal space-time decay rates of strong solutions. More recently, [4, 23, 24] established the well-posedness of strong solutions with improved regularity conditions for initial data in sobolev or Besov spaces.

Examples of smooth data with arbitrarily large L

norms giving rise to global unique solutions have been exhibited in [17]. Very recently, in [8], the authors of the present paper proved well-posedness in critical Besov spaces and pointed out that better results may be achieved if µ = ν.

In order to explain what we mean by critical regularity, let us first recall how it goes for the following incompressible Navier-Stokes equations:

(N S)



 

 

∂ t u + div(u ⊗ u) + ∇P = µ∆u in R + × R ³ ,

div u = 0 in R + × R ³ ,

u| t=0 = u 0 in R ³

where

div(v ⊗ w) ^j :=

3

X

k=1

∂ _k (v ^j w ^k ).

Clearly, (N S) is invariant for all positive λ by the rescaling

(u, P)(t, x) ; (λu, λ ² P)(λ ² t, λx) and u 0 (x) ; λu 0 (λx). (1.5) For that reason, global well-posedness results for (N S) in optimal spaces (in terms of regularity) may be achieved by means of contracting mapping arguments only if using norms for which u 0 and the solution (u, P ) have the above scaling invariance.

In contrast with the Navier-Stokes equations, the Hall-MHD system does not have a genuine scaling invariance (unless if the Hall-term is neglected). In a recent paper of ours [8], we observed that some scaling invariance does exist if one considers the current function J = ∇ × B to be an additional unknown. Then, the scaling invariance of (u, B, J ) is the same as the velocity in (1.5).

In [8], we also pointed out that the Hall-MHD system better behaves if µ = ν since, although being still quasi-linear, the Hall term disappears in the energy estimate involving the so-called velocity of electron v := u − εJ. Indeed, let us consider v as an additional unknown and look at the following extended formulation of the Hall-MHD system:



 

 

 



 

 

 



∂ t u − µ∆u = B · ∇B − u · ∇u − ∇π,

∂ _t B − µ∆B = ∇ × (v × B),

∂ t v − µ∆v = B · ∇B − u · ∇u − ε∇ × ((∇ × v) × B)

+ ∇ × (v × u) + 2ε∇ × (v · ∇B ) − ∇π, div u = div B = div v = 0.

(1.6)

That (redundant) system is still quasilinear but, owing to

(∇ × w | v) = (w | ∇ × v), (1.7)

where (· | ·) denotes the scalar product in L ² ( R ³ ), the most nonlinear term cancels out when performing an energy method, since

(∇ × ((∇ × v) × B) | v) = 0. (1.8)

Our primary goal is to provide an elementary proof of a Fujita-Kato type result for the Hall-MHD system in the spirit of the celebrated work [10] (see also [7]). In our context, this amounts to proving that System (1.1)–(1.3) supplemented with initial data (u 0 , B 0 ) such that (u 0 , B 0 , v 0 ) is small enough in the critical homogeneous Sobolev space ˙ H

( R ³ ) admits a unique global solution. In passing, we will obtain some informations on the long time behavior of the solutions, similar to those that are presented for the incompressible Navier-Stokes equations in e.g. [3, Chap. 5].

Our second purpose is to prove a weak-strong uniqueness result for the Leray- Hopf weak solutions of Hall-MHD, namely that all weak solutions coincide with the unique Fujita-Kato solution whenever the latter one exists. That result turns out to be less sensitive to the very structure of the system, and is valid for all values of µ, ν and ε.

Finally, as proposed by Chae and Lee in [6], we will consider the 2 ¹ ₂ D flows for the Hall-MHD system, that is, 3D flows depending only on two space variables.

This issue is well known for the incompressible Navier-Stokes equations (see e.g.

the book by Bertozzi and Majda [18]). In our case, the corresponding system reads:

∂ t u + u e · ∇u e + ∇π e = B e · ∇B e + µ ∆u e in R ⁺ × R ² , (1.9)

div g u e = 0 in R + × R ² , (1.10)

∂ t B + u e · ∇B e + ε B e · ∇j e − εe j · ∇B e = ν ∆B e + B e · ∇u e in R ⁺ × R ² , (1.11)

(u, B)| t=0 = (u 0 , B 0 ) in R ² , (1.12)

where the unknowns u and B are functions from R + × R ² to R ³ , e u := (u ¹ , u ² ), B e := (B ¹ , B ² ), ∇ e := (∂ 1 , ∂ 2 ), div := g ∇·, e ∆ := e ∂ ₁ ² + ∂ ₂ ² and

j := ∇ × e B =





∂ 2 B ³

−∂ 1 B ³

∂ ₁ B ² − ∂ ₂ B ¹



 ·

A small modification of the proof of [5] allows to establish that for any initial data (u 0 , B 0 ) in L ² ( R ² ; R ³ ) with div g e u 0 = g div B e 0 = 0, there exists a global-in-time Leray-Hopf solution (u, B) of (1.9)–(1.12) that satisfies:

ku(t)k ² _L

(

) + kB(t)k ² _L

(

) + 2 w t

0 µk ∇uk e ² _L

(

) + ν k ∇Bk e ² _L

(

)

dτ

≤ ku 0 k ² _L

(

) + kB 0 k ² _L

(

) . (1.13) Whether that solution is unique and equality is true in (1.13) are open questions.

The difficulty here is that, unlike for the 2 ¹ ₂ D Navier-Stokes equations or for the 2 ¹ ₂ D MHD flows with no Hall term, the two-dimensional system satisfied by the first two components of the flow is coupled with the equation satisfied by the third component, through the term B e · ∇j e −e j · ∇B, e thus hindering any attempt to prove the global well-posedness for large data by means of classical arguments.

Our aim here is to take advantage of the special structure of the system so as

to get a global well-posedness statement for 2 ¹ ₂ D data such that only the initial

magnetic field is small. Since it has been pointed out in [6] that controlling just j

in the space L ² (0, T ; BMO( R ² )) prevents blow-up of a smooth solution at time T and because the space ˙ H ¹ ( R ² ) is continuously embedded in BMO( R ² ), it is natural to look for a control on j in the space L ² (0, T ; ˙ H ¹ ( R ² )) for all T > 0.

For 2 ¹ ₂ D flows, the critical Sobolev regularity corresponds to u 0 in L ² ( R ² ) and to B ₀ in H ¹ ( R ² ), and we shall see that, indeed, in the case of small data, one can establish rather easily a global well-posedness result at this level of regularity. For large data, owing to the coupling between the velocity and magnetic fields through the term ∇ × e (u × B), we do not know how to achieve global existence at this level of regularity. Then, our idea is to look at the equation satisfied by the vorticity ω := ∇ × e u, namely

∂ _t ω + ∇ × e (ω × u) = ∇ × e (j × B) + µ ∆w. e In the case µ = ν, the vector-field E := εω + B thus satisfies

∂ t E − E e · ∇u e + u e · ∇E e = µ ∆E. e (1.14) From that identity, obvious energy arguments and (1.13), it is easy to get a global control of E in the space L ² ( R + ; ˙ H ¹ ( R ² )), and, finally on j on L ² ( R + ; ˙ H ¹ ( R ² )) provided B 0 is small in H ¹ ( R ² ). We then get a global well-posedness statement, assuming only that the magnetic field is small.

We end this introductory part presenting a few notations. As usual, we denote by C harmless positive constants that may change from line to line, and A . B means A ≤ CB. For X a Banach space, p ∈ [1, ∞] and T > 0, the notation L ^p (0, T ; X) or L ^p _T (X ) designates the set of measurable functions f : [0, T ] → X with t 7→ kf (t)k X

in L ^p (0, T ), endowed with the norm k · k _L

T

(X) := kk · k X k _L

(0,T ) , and agree that C([0, T ]; X ) denotes the set of continuous functions from [0, T ] to X. Sometimes, we will use the notation L ^p (X) to designate the space L ^p ( R + ; X) and k · k L

(X) for the associated norm. We will keep the same notations for multi-component functions, namely for f : [0, T ] → X ^m with m ∈ N .

2. Main results

Our first result, that has to be compared with the Fujita-Kato theorem for the incompressible Navier-Stokes equations states that the Hall-MHD system is indeed globally well-posed if u ₀ , B ₀ and v ₀ are small in ˙ H

( R ³ ).

Theorem 2.1. Assume that µ = ν. Let (u ₀ , B ₀ ) ∈ H ˙

( R ³ ) with div u ₀ = div B ₀ = 0, and J 0 := ∇ × B 0 ∈ H ˙

( R ³ ). There exists a constant c > 0 such that, if

ku 0 k _˙

H

(

) + kB 0 k _˙

H

(

) + ku 0 − εJ 0 k _˙

H

(

) < cµ, (2.1) then there exists a unique global solution

(u, B) ∈ C _b ( R + ; ˙ H

( R ³ )) ∩ L ² ( R + ; ˙ H

( R ³ ))

to the Cauchy problem (1.1)-(1.4), such that J := ∇ × B ∈ L

( R + ; ˙ H

( R ³ )) ∩ L ² ( R + ; ˙ H

( R ³ )). Furthermore, for all t ≥ t ₀ ≥ 0, one has

k(u, B, u−εJ )(t)k ² _˙

H

+ µ 2

w t t

k(u, B, u−εJ )k ² _˙

H

dτ ≤ k(u, B, u−εJ )(t 0 )k ² _˙

H

. (2.2) In particular, the function

t 7→ ku(t)k ² _˙

H

+ kB(t)k ² _˙

H

+ ku(t) − εJ (t)k ² _˙

H

is nonincreasing.

If, in addition, u ₀ ∈ H ^s and B ₀ ∈ H ^s+1 for some s ≥ 0 with s 6= 1/2, then (u, B) ∈ C b ( R + ; H ^s × H ^s+1 ), ∇u ∈ L ² ( R + ; H ^s ) and ∇B ∈ L ² ( R + ; H ^s+1 ).

Remark 1. The first part of the above theorem has been proved in [8] by a different method that does not allow to get (2.2). A small variation on the proof yields local well-posedness if assuming only that ku 0 − εJ 0 k _˙

H

(

) is small. In contrast with the incompressible Navier-Stokes equations however, whether a local-in-time result may be proved without any smallness condition (or more regularity, see e.g. [8, Th.

2.2]) is an open question.

Corollary 2.2. If (u, B) denotes the solution given by Theorem 2.1 and if, in addition, (u 0 , B 0 ) is in L ² ( R ³ ), then (u, B) is continuous with values in L ² ( R ³ ), satisfies the following energy balance for all t ≥ 0 :

ku(t)k ² _L

+ kB(t)k ² _L

+ 2µ w t

0 k∇uk ² _L

+ k∇Bk ² _L

dτ = ku 0 k ² _L

+ kB 0 k ² _L

, (2.3) and we have

t→+∞ lim ku(t)k ² _˙

H

+ kB(t)k ² _˙

H

+ kJ (t)k ² _˙

H

= 0. (2.4)

The following corollary states that global solutions (even if large and with infinite energy) satisfying a suitable integrability property have to decay to 0 at infinity.

Corollary 2.3. Assume that (u ₀ , B ₀ ) ∈ H ˙

( R ³ ) with div u ₀ = div B ₀ = 0 and J ₀ ∈ H ˙

( R ³ ). Suppose in addition that the Hall-MHD system with µ = ν supplemented with initial data (u 0 , B 0 ) admits a global solution (u, B) such that

(u, B, ∇ × B) ∈ L ⁴ ( R + ; ˙ H ¹ ( R ³ )).

Then, (u, B) has the regularity properties of Theorem 2.1, and (2.4) is satisfied.

In particular, all the solutions constructed in Theorem 2.1 satisfy (2.4).

The next theorem states a weak-strong uniqueness property of the solution. It is valid for all positive coefficients µ, ν and ε.

Theorem 2.4. Consider initial data (u 0 , B 0 ) in L ² ( R ³ ) with div u 0 = div B 0 = 0.

Let (u, B) be any Leray-Hopf solution of the Hall-MHD system associated with ini- tial data (u 0 , B 0 ). Assume in addition that u and J := ∇×B are in L ⁴ (0, T ; ˙ H ¹ ( R ³ )) for some time T > 0. Then, all Leray-Hopf solutions associated with (u 0 , B 0 ) coin- cide with (u, B) on the time interval [0, T ].

Our last result states the existence of global strong solutions for the 2 ¹ ₂ D Hall- MHD system. Two cases are considered : either the data are small and have just critical regularity, or the velocity field is more regular and only the magnetic field has to be small accordingly.

Theorem 2.5. Assume that µ = ν. Let (u 0 , B 0 ) be divergence free vector-fields with u 0 in L ² ( R ² ) and B 0 in H ¹ ( R ² ). There exists a constant c > 0 such that, if

ku 0 k L

(

) + kB 0 k L

(

) + ku 0 − ε ∇ × e B ₀ k L

(

) < cµ, (2.5) then there exists a unique global solution (u, B) to the Cauchy problem (1.9)-(1.12), with (u, B) ∈ C b ( R + ; L ² ( R ² )) ∩ L ² ( R + ; ˙ H ¹ ( R ² )) and

∇B ∈ L

( R + ; L ² ( R ² )) with ∇ ² B ∈ L ² ( R + ; L ² ( R ² )).

If both u 0 and B 0 are in H ¹ ( R ² ), then there exists a constant C 0 depending only on the L ² norm of u 0 , ∇u 0 , and on µ, ε such that if

kB 0 k _H

(

) ≤ C 0 , (2.6) then there exists a unique global solution (u, B) to (1.9)-(1.12), with

(u, B) ∈ C b ( R + ; H ¹ ( R ² )) and (∇u, ∇B) ∈ L ² ( R + ; H ¹ ( R ² )).

Moreover, in the two cases, (1.13) becomes an equality.

The next section is devoted to the proof of Theorem 2.1 and of its two corollaries.

In section 4, we establish the weak-strong uniqueness result. Section 5 is dedicated to the proof of Theorem 2.5. A few definitions and technical results are recalled in Appendix.

3. Small data global existence in critical Sobolev spaces Throughout this section, we shall assume for simplicity that µ = ν = ε = 1 (the general case µ = ν > 0 and ε > 0 may be deduced after suitable rescaling). We shall use repeatedly the fact that, as B is divergence free, one has the following equivalence of norms for any s ∈ R :

k∇Bk H ˙

∼ kJk H ˙

, (3.1) and also that we have B = curl

(u − v), where the −1-th order homogeneous Fourier multiplier curl

is defined on the Fourier side by

F(curl

J )(ξ) := iξ × J b

|ξ| ² · (3.2)

Proving the existence part of Theorem 2.1 is based on the following result : Proposition 3.1. Let (u, B) be a smooth solution of the 3D Hall-MHD system with ε = µ = ν = 1, on the time interval [0, T ]. Let v := u − ∇ × B. There exists a universal constant C such that on [0, T ], we have

d dt (kuk ²

H ˙

+ kBk ²

H ˙

+ kvk ²

H ˙

) + (kuk ²

H ˙

+ kB k ²

H ˙

+ kvk ²

H ˙

)

≤ C q kuk ²

H ˙

+ kBk ²

H ˙

+ kvk ²

H ˙

(kuk ² _˙

H

+ kBk ² _˙

H

+ kvk ² _˙

H

). (3.3) Proof. Applying operator Λ

to both sides of System (1.6) and taking the L ² scalar product with Λ

u, Λ

B and Λ

v, respectively, we get

1 2

d dt kuk ² _˙

H

+ kuk ² _˙

H

= (Λ

(u ⊗ u) | ∇Λ

u) − (Λ

(B ⊗ B) | ∇Λ

u) = A 1 + A 2 , 1

2 d dt kBk ² _˙

H

+ kB k ² _˙

H

= (Λ

(v × B) | ∇ × Λ

B) = A 3 , 1

2 d dt kvk ² _˙

H

+ kvk ² _˙

H

= A ₄ + A ₅ + · · · + A ₈ , where

A 4 := (Λ

(u ⊗ u) | ∇Λ

v), A ₅ := −(Λ

(B ⊗ B) | ∇Λ

v),

A ₆ := −(Λ

((∇ × v) × B) − (Λ

∇ × v) × B | Λ

∇ × v),

A 7 := (Λ

(v × u) | ∇ × Λ

v), A 8 := 2(Λ

(v · ∇B) | ∇ × Λ

v).

By Lemma A.3 and Sobolev embedding (A.2), we get

|A 1 | ≤ CkΛ

uk _L

kuk _L

k∇Λ

uk _L

≤ Ckuk ² _{H ˙}

kuk _˙

H

,

|A 6 | ≤ C(k∇Bk L

kΛ

vk L

+ kΛ

B k L

k∇ × vk L

)kvk _˙

H

≤ C k∇Bk H ˙

kvk H ˙

+ k∇Bk _˙

H

kvk _˙

H

kvk _˙

H

,

|A 8 | ≤ C(kΛ

vk L

k∇Bk L

+ kvk L

k∇Λ

B k L

)kvk _˙

H

≤ Ck∇B k H ˙

kvk H ˙

kvk _˙

H

.

Terms A 2 , A 3 , A 4 , A 5 and A 7 may be bounded similarly as A 1 :

|A ₂ | ≤ CkBk ² _{H ˙}

kuk _˙

H

,

|A ₃ | ≤ Ckvk H ˙

kBk H ˙

kB k _˙

H

,

|A 4 | ≤ Ckuk ² _{H ˙}

kvk _˙

H

,

|A ₅ | ≤ CkBk ² _{H ˙}

kvk _˙

H

,

|A 7 | ≤ Ckvk H ˙

kuk H ˙

kvk _˙

H

. Hence, using repeatedly the fact that

kzk H ˙

≤ q kzk _˙

H

kzk _˙

H

and Young inequality and, sometimes, (3.1), it is easy to deduce (3.3) from the

above inequalities.

Now, combining Proposition 3.1 with Lemma A.4 (take α = 1, W ≡ 0) implies that there exists a constant c > 0 such that if

ku 0 k _˙

H

+ kB 0 k _˙

H

+ kv 0 k _˙

H

< c, (3.4) then we have for all time t ≥ 0,

k(u, B, v)(t)k ²

H ˙

+ 1 2

w t

0 k(u, B, v)k ²

H ˙

dτ ≤ k(u, B, v)(0)k ²

H ˙

. (3.5) That inequality obviously implies that Condition (3.4) is satisfied for all time t ₀ . Hence, repeating the argument, we get (2.2) for all t ≥ t ₀ ≥ 0.

In order to prove rigorously the existence part of Theorem 2.1, one can resort to the following classical procedure:

(1) smooth out the initial data and get a sequence (u ⁿ , B ⁿ ) _n∈

of smooth solutions to Hall-MHD system on the maximal time interval [0, T ⁿ );

(2) apply (3.5) to (u ⁿ , B ⁿ ) _n∈N and prove that T ⁿ = ∞ and that the sequence (u ⁿ , B ⁿ , v ⁿ ) _n∈

with v ⁿ := u ⁿ − ∇ × B ⁿ is bounded in L

( R + ; ˙ H

) ∩ L ² ( R + ; ˙ H

);

(3) use compactness to prove that (u ⁿ , B ⁿ ) _n∈N converges, up to extraction, to a solution of Hall-MHD system supplemented with initial data (u 0 , B 0 );

(4) prove stability estimates in L ² to get the uniqueness of the solution.

To proceed, let us smooth out the initial data as follows:

u ⁿ ₀ := F

(1

c u 0 ) and B ₀ ⁿ := F

(1

B c 0 ),

where C n stands for the annulus with small radius n

and large radius n. Clearly, u ⁿ ₀ and B ₀ ⁿ belong to all Sobolev spaces, and

k(u ⁿ ₀ , B ₀ ⁿ , v ₀ ⁿ )k _˙

H

≤ k(u 0 , B ₀ , v ₀ )k _˙

H

· (3.6)

The classical well-posedness theory in Sobolev spaces (see e.g. [5]) ensures that the Hall-MHD system with data (u ⁿ ₀ , B ₀ ⁿ ) has a unique maximal solution (u ⁿ , B ⁿ ) on [0, T ⁿ ) for some T ⁿ > 0, belonging to C([0, T ]; H ^m ) ∩ L ² (0, T ; H ^m+1 ) for all T < T ⁿ . Since the solution is smooth, we have according to (3.5) and (3.6),

k(u ⁿ , B ⁿ , v ⁿ )k ²

L

( ˙ H

) + 1

2 k(u ⁿ , B ⁿ , v ⁿ )k ²

L

( ˙ H

) ≤ k(u 0 , B 0 , v 0 )k ² _˙

H

· Now, using (3.1) and the embedding ˙ H

( R ³ ) , → BM O( R ³ ), we get

w T

0 k(u ⁿ , ∇B ⁿ )k ² _{BM O} dt . k(u ⁿ , v ⁿ )k ²

L

( ˙ H

) .

Hence, the continuation criterion of [6] guarantees that T ⁿ = +∞. This means that the solution is global and that, furthermore,

k(u ⁿ , B ⁿ , v ⁿ )k ²

L

( ˙ H

) + 1

2 k(u ⁿ , B ⁿ , v ⁿ )k ²

L

( ˙ H

) ≤ k(u 0 , B 0 , v 0 )k ² _˙

H

· (3.7) We claim that, up to extraction, the sequence (u ⁿ , B ⁿ ) _n∈

converges in D

( R + × R ³ ) to a solution (u, B) of (1.1)–(1.3) supplemented with data (u ₀ , B ₀ ). The definition of (u ⁿ ₀ , B ⁿ ₀ ) and the fact that (u 0 , B 0 , v 0 ) belongs to ˙ H

already entails that

(u ⁿ ₀ , B ₀ ⁿ , v ⁿ ₀ ) → (u 0 , B 0 , v 0 ) in H ˙

·

Proving the convergence of (u ⁿ , B ⁿ , v ⁿ ) _n∈

can be achieved from compactness arguments, after exhibiting bounds in suitable spaces for (∂ t u ⁿ , ∂ t B ⁿ , ∂ t v ⁿ ) _n∈

. Then, combining with compact embedding will enable us to apply Ascoli’s theorem and to get the existence of a limit (u, B, v) for a subsequence. Furthermore, the uniform bound (3.7) will provide us with additional regularity and convergence properties so that we will be able to pass to the limit in the Hall-MHD system.

To proceed, let us introduce (u L , B L , v L ) := e ^t∆ (u 0 , B 0 , v 0 ), (u ⁿ _L , B ⁿ _L , v _L ⁿ ) :=

F

1

( u b L , B b L , b v L )

and ( e u ⁿ , B e ⁿ , e v ⁿ ) := (u ⁿ − u ⁿ _L , B ⁿ − B ⁿ _L , v ⁿ − v ⁿ _L ).

It is clear that (u ⁿ _L , B _L ⁿ , v _L ⁿ ) tends to (u L , B L , v L ) in C( R + ; ˙ H

) ∩ L ² ( R + ; ˙ H

), which implies that v L = u L − ∇ × B L , since v ⁿ _L = u ⁿ _L − ∇ × B _L ⁿ for all n ∈ N .

Proving the convergence of ( u e ⁿ , B e ⁿ , e v ⁿ ) _n∈N relies on the following lemma:

Lemma 3.2. Sequence ( u e ⁿ , B e ⁿ , e v ⁿ ) _n∈

is bounded in C _loc

( R ⁺ ; ˙ H

).

Proof. Observe that ( u e ⁿ , B e ⁿ , e v ⁿ )(0) = (0, 0, 0) and that



 

 

 

 

 

 

∂ _t u e ⁿ = ∆ e u ⁿ + P div (B ⁿ ⊗ B ⁿ ) − div (u ⁿ ⊗ u ⁿ ) ,

∂ t B e ⁿ = ∆ B e ⁿ + ∇ × (v ⁿ × B ⁿ ),

∂ _t e v ⁿ = ∆ e v ⁿ + P

div (B ⁿ ⊗ B ⁿ ) − div (u ⁿ ⊗ u ⁿ )

− ∇ × ((∇ × v ⁿ ) × B ⁿ ) + ∇ × (v ⁿ × u ⁿ ) + 2∇ × (v ⁿ · ∇B ⁿ ).

(3.8)

Using the uniform bound (3.7), the product laws:

ka bk _L

. kak _˙

H

kbk H ˙

and k(∇ × a) × bk _L

. kak H ˙

kbk L

,

and Gagliardo-Nirenberg inequality (A.4) with s = 1 and s

= 2, we discover that the right-hand side of (3.8) is uniformly bounded in L _loc

( R + ; ˙ H

). Then applying H¨ older inequality completes the proof of the lemma.

One can now turn to the proof of the existence of a solution. Let (φ _j ) _j∈

be a sequence of C _c

( R ³ ) cut-off functions supported in the ball B (0, j + 1) of R ³ and equal to 1 in a neighborhood of B(0, j).

Lemma 3.2 ensures that ( u e ⁿ , B e ⁿ , e v ⁿ ) _n∈N is uniformly equicontinuous in the space C([0, T ]; ˙ H

) for all T > 0, and (3.7) tells us that it is bounded in L

( R + ; ˙ H

).

Using the fact that the application z 7→ φ j z is compact from ˙ H

into ˙ H

, combin- ing Ascoli’s theorem and Cantor’s diagonal process enables to conclude that there exists some triplet ( u, e B, e e v) such that for all j ∈ N ,

(φ j u e ⁿ , φ j B e ⁿ , φ j e v ⁿ ) → (φ j e u, φ j B, φ e j e v) in C( R + ; ˙ H

). (3.9) This obviously entails that ( u e ⁿ , B e ⁿ , e v ⁿ ) tends to ( e u, B, e e v) in D

( R + × R ³ ), which is enough to pass to the limit in all the linear terms of (1.6) and to ensure that e v = e u − ∇ × B, e and thus v = u − ∇ × B.

From the estimates (3.7), interpolation and classical functional analysis argu- ments, we gather that ( e u, B, e e v) belongs to L

(0, T ; ˙ H

) ∩ L ² (0, T ; ˙ H

) and to C

([0, T ]; ˙ H

) for all T > 0, and better properties of convergence like, for in- stance,

φ j ( u e ⁿ , B e ⁿ , e v ⁿ ) → φ j ( e u, B, e v) e in L ² _loc ( R + ; ˙ H ¹ ) for all j ∈ N . (3.10) As an example, let us explain how to pass to the limit in the term ∇×((∇×v ⁿ )×B ⁿ ).

Let θ ∈ C _c

( R + × R ³ ; R ³ ) and j ∈ N be such that Supp θ ⊂ [0, j] × B(0, j). We write

h∇ × ((∇ × v ⁿ ) × B ⁿ ), θi − h∇ × ((∇ × v) × B), θi

= h(∇ × v ⁿ ) × φ _j (B ⁿ − B), ∇ × θi + h(∇ × φ _j (v ⁿ − v)) × B, ∇ × θi.

Now, we have for all T > 0,

k(∇ × v ⁿ ) × φ j (B ⁿ − B)k L

(0,T ;L

) . k∇ × v ⁿ k

L

(0,T; ˙ H

) kφ j (B ⁿ − B)k _L

(0,T; ˙ H

) , k(∇ × φ j (v ⁿ − v)) × Bk

L

(0,T ;L

) . k(∇ × φ j (v ⁿ − v))k _L

(0,T×

) kBk _L

(0,T;L

) . Thanks to (3.7) and to (3.10), we see that the right-hand sides above converge to 0. Hence

∇ × ((∇ × v ⁿ ) × B ⁿ ) → h∇ × ((∇ × v) × B) in D

( R + × R ³ ).

Arguing similarly to pass to the limit in the other nonlinear terms, one may conclude

that (u, B, v) satisfies the extended formulation (1.6). Besides, as we know that

v = u − ∇ × B, the couple (u, B) satisfies the Hall-MHD system for some suitable

pressure function P.

To prove that (u, B) is continuous in ˙ H

, it suffices to notice that the properties of regularity of the solution ensure that u and B satisfy a heat equation with initial data in ˙ H

and right-hand side in L ² ( R + ; ˙ H

) (we do not know how to prove the time continuity with values in ˙ H

for ∇B or, equivalently, v, though).

Let us next prove the uniqueness part of the theorem. Let (u 1 , B 1 ) and (u 2 , B 2 ) be two solutions of the Hall-MHD system on [0, T ] × R ³ , supplemented with the same initial data (u 0 , B 0 ) and such that, denoting v i = u i − ∇ × B i for i = 1, 2,

(u i , B i , v i ) ∈ L

([0, T ]; ˙ H

) and (∇u i , ∇B i , ∇v i ) ∈ L ² (0, T ; ˙ H

).

In order to prove the result, we shall estimate the difference (δu, δB, δv) := (u 1 − u 2 , B 1 − B 2 , v 1 − v 2 ) in the space C([0, T ]; L ² ) ∩ L ² (0, T ; ˙ H ¹ ). In order to justify that, indeed, (δu, δB, δv) belongs to that space, one can observe that



 

 

∂ _t δu − ∆δu := R ₁ ,

∂ t δB − ∆δB := R 2 ,

∂ _t δv − ∆δv := R ₁ + R ₃ + R ₄ + R ₅ ,

(3.11)

where

R 1 := P (B 1 · ∇δB + δB · ∇B 2 − u ¹ · ∇δu − δu · ∇u 2 ), R ₂ := ∇ × (v ₁ × δB + δv × B ₂ ),

R 3 := −∇ × ((∇ × v 1 ) × δB + (∇ × δv) × B 2 ), R 4 := ∇ × (v 1 × δu + δv × u 2 ),

R ₅ := 2∇ × (v ₁ · ∇δB + δv · ∇B 2 ).

Since (δu, δB, δv)| t=0 =0, in order to achieve our goal, it suffices to prove that R ₁ to R 5 belong to the space L ² (0, T ; ˙ H

). Now, since (δu, δB, δv) ∈ L

(0, T ; ˙ H

) ∩ L ² (0, T ; ˙ H

), we have, by interpolation and H¨ older inequality that (δu, δB, δv) ∈ L ⁴ (0, T ; ˙ H ¹ ). Hence, using repeatedly the fact that the numerical product of func- tions may be continuously extended to ˙ H ¹ × H ˙ ^1/2 → L ² , one can write that

kR 1 k _L

(0,T; ˙ H

) . kB 1 ⊗ δBk _L

(0,T;L

) + kB 2 ⊗ δBk _L

(0,T;L

)

+ ku 1 ⊗ δuk L

(0,T;L

) + ku 2 ⊗ δuk L

(0,T;L

)

. T

k(u 1 , u ₂ , B ₁ , B ₂ )k

L

(0,T; ˙ H

) k(δu, δB)k _L

(0,T; ˙ H

) , kR 2 k _L

(0,T; ˙ H

) . kv ¹ × δBk L

(0,T;L

) + kδv × B 2 k L

(0,T ;L

)

. T

k(v 1 , B 2 )k

L

(0,T; ˙ H

) k(δv, δB)k _L

(0,T; ˙ H

) ,

kR 3 k _L

(0,T; ˙ H

) . k(∇ × v 1 ) × δBk _L

(0,T;L

) + k(∇ × δv) × B 2 k _L

(0,T;L

)

. kv 1 k

L

(0,T; ˙ H

) kδBk _L

(0,T; ˙ H

) + kδvk

L

(0,T; ˙ H

) kB 2 k _L

(0,T; ˙ H

)

. k(v 1 , δv)k

L

(0,T ; ˙ H

) k(B 2 , δB)k _L

(0,T; ˙ H

) ,

Note that our assumptions ensure that B i and ∇B i are in L

(0, T ; ˙ H

) and thus

we do have, by interpolation inequality (A.1), B _i in L

(0, T ; ˙ H ¹ ) for i = 1, 2. Terms

R ₄ and R ₅ may be treated similarly.

Estimating (δu, δB, δv) in the space L

(0, T ; L ² ) ∩ L ² (0, T ; ˙ H ¹ ) follows from a standard energy method applied on (3.11), H¨ older’s inequality and Sobolev embed- ding. More precisely, we have

1 2

d

dt kδuk ² _L

+kδuk ² _{H ˙}

. (kB 1 ⊗ δBk _L

+kB 2 ⊗ δBk _L

+ ku 2 ⊗ δuk _L

)k∇δuk _L

. (kB 1 k H ˙

kδBk _˙

H

+kB 2 k H ˙

kδBk _˙

H

+ku 2 k H ˙

kδuk _˙

H

)kδuk H ˙

, 1

2 d

dt kδBk ² _L

+kδBk ² _{H ˙}

. (kv 1 × δBk _L

+ kδv × B 2 k _L

)k∇δBk _L

. (kv ₁ k H ˙

kδBk _˙

H

+ kδvk _˙

H

kB ₂ k H ˙

)kδBk H ˙

and, using (3.1), 1

2 d

dt kδvk ² _L

+ kδvk ² _{H ˙}

. (kB 1 ⊗ δBk _L

+ kB 2 ⊗ δBk _L

+ ku 1 ⊗ δuk _L

+ ku 2 ⊗ δuk _L

)k∇δvk _L

+ k(∇ × v 1 ) × δBk _L

k∇ × δvk _L

+ (kv 1 × δuk L

+ kδv × u ₂ k L

)k∇ × δvk L

+ (kv 1 · ∇δBk _L

+ kδv · ∇B 2 k _L

)k∇ × δvk _L

.

kB 1 k H ˙

kδBk _˙

H

+ kB 2 k H ˙

kδBk _˙

H

+ ku 1 k H ˙

kδuk _˙

H

+ ku 2 k H ˙

kδuk _˙

H

+ k∇×v 1 k _˙

H

(kδuk _L

+kδvk _L

)+kv 1 k H ˙

kδuk _˙

H

+kδvk H ˙

ku 2 k _˙

H

+ kv 1 k H ˙

(kδuk _˙

H

+ kδvk _˙

H

) + kδvk _˙

H

k∇B 2 k H ˙

kδvk H ˙

. At this stage, interpolation and Young’s inequality imply that

kB 1 k H ˙

kδBk _˙

H

kδuk H ˙

. kB 1 k H ˙

kδBk _L

kδBk _H

_˙

kδuk H ˙

≤ 1

10 k(δB, δu)k ² _{H ˙}

+ CkδBk ² _L

kB 1 k ⁴ _{H ˙}

,

and similar inequalities for all the terms of the right-hand sides of the above in- equalities, except for the one with k∇ × v ₁ k _˙

H

that we bound as follows:

k∇ × v 1 k _˙

H

k(δu, δv)k _L

kδvk H ˙

≤ 1

10 kδvk ² _{H ˙}

+ Ckv 1 k ² _˙

H

k(δu, δv)k ² _L

. In the end, we get for all t ∈ (0, T ),

1 2

d

dt k(δu, δB, δv)k ² _L

+ k(δu, δB, δv)k ² _{H ˙}

. V (t)k(δu, δB, δv)k ² _L

with

V (t) := k(u 1 , u 2 , B 1 , B 2 , v 1 , v 2 )(t)k ⁴ _{H ˙}

+ kv 1 (t)k ² _˙

H

.

Since our assumptions ensure that V is integrable on [0, T ] and (δu, δB, δv)(0) = 0, applying Gronwall’s inequality yields

(δu, δB, δv) ≡ 0 in L

(0, T ; L ² ( R ³ )).

Let us finally explain the propagation of higher Sobolev regularity if the initial

data (u 0 , B 0 ) are, additionally, in H ^s × H ^s+1 for some s ≥ 0. Our aim is to prove

that the solution (u, B) constructed above is in C b ( R + ; H ^s × H ^s+1 ), and such that

(∇u, ∇B) ∈ L ² ( R + ; H ^s × H ^s+1 ).

For the time being, let us assume that (u, B) is smooth and explain how to perform estimates in Sobolev spaces. First, we multiply (1.1) and (1.3) by u and B, respectively, integrate and add up the resulting equations. Using the fact that

(∇ × (J × B) | B) = (J × B | J) = 0, one gets the following energy balance:

1 2

d

dt (kuk ² _L

+ kBk ² _L

) + k∇uk ² _L

+ k∇Bk ² _L

= 0. (3.12) Let Λ ^s denote the fractional derivative operator defined in the Appendix. Since

kuk H

+ kBk _H

≈ k(u, B)k L

+ k(Λ ^s u, Λ ^s B, Λ ^s v)k L

,

in order to prove the desired Sobolev estimates, it suffices to get a suitable control on kΛ ^s uk L

and on kΛ ^s+1 Bk L

. To this end, apply Λ ^s to (1.1), then take the L ² scalar product with Λ ^s u. We get:

1 2

d

dt kΛ ^s uk ² _L

+ kΛ ^s ∇uk ² _L

= (Λ ^s (B ⊗B) | Λ ^s ∇u)− (Λ ^s (u⊗u) | Λ ^s ∇u) =: E 1 +E 2 . In order to control kΛ ^s+1 Bk _L

, one has to use the cancellation property (1.8).

Then, applying Λ ^s to the second and third equation of (1.6) and taking the L ² scalar product with Λ ^s B, Λ ^s v, respectively, yields:

1 2

d

dt kΛ ^s Bk ² _L

+ kΛ ^s ∇Bk ² _L

= (Λ ^s (v × B) | Λ ^s ∇ × B) =: E 3 , 1

2 d

dt kΛ ^s vk ² _L

+ kΛ ^s ∇vk ² _L

= (Λ ^s (B ⊗ B) | Λ ^s ∇v) − (Λ ^s (u ⊗ u) | Λ ^s ∇v)

−(Λ ^s ((∇ × v) × B) − (Λ ^s ∇ × v) × B | Λ ^s ∇ × v) + (Λ ^s (v × u) | Λ ^s ∇ × v) +2(Λ ^s (v · ∇B) | Λ ^s ∇ × v) =: E ₄ + E ₅ + · · · + E ₈ . Sobolev embedding, Young’s inequality and Lemma A.3, imply that

|E 1 | . kΛ ^s Bk L

kBk L

kΛ ^s ∇uk L

. kB k _˙

H

(kΛ ^s ∇Bk ² _L

+ kΛ ^s ∇uk ² _L

),

|E 2 | . kuk _˙

H

kΛ ^s ∇uk ² _L

,

|E 3 | . (kΛ ^s vk _L

kBk _L

+ kΛ ^s Bk _L

kvk _L

)kΛ ^s ∇ × Bk _L

. k(B, v)k _˙

H

(kΛ ^s ∇Bk ² _L

+ kΛ ^s ∇vk ² _L

),

|E 4 | . kB k _˙

H

(kΛ ^s ∇Bk ² _L

+ kΛ ^s ∇vk ² _L

),

|E 5 | . kuk _˙

H

(kΛ ^s ∇uk ² _L

+ kΛ ^s ∇vk ² _L

),

|E 6 | . (k∇Bk L

kΛ ^s−1 ∇ × vk L

+ kΛ ^s Bk L

k∇ × vk L

)kΛ ^s ∇vk L

≤ (Ck∇B k _˙

H

+ ¹ ₂ )kΛ ^s ∇vk ² _L

+ CkΛ ^s ∇Bk ² _L

kvk ² _˙

H

,

|E 7 | . (kΛ ^s vk _L

kuk _L

+ kΛ ^s uk _L

kvk _L

)kΛ ^s ∇vk _L

. k(u, v)k _˙

H

(kΛ ^s ∇uk ² _L

+ kΛ ^s ∇vk ² _L

),

|E 8 | . (kΛ ^s vk L

k∇Bk L

+ kΛ ^s ∇Bk L

kvk L

)kΛ ^s ∇vk L

. k(∇B, v)k _˙

H

(kΛ ^s ∇Bk ² _{H ˙}

+ kΛ ^s ∇vk ² _L

).

Since k(u, B, v)k _˙

H

is small, putting the above estimates and (3.12) together, and using (3.1), one gets after time integration that

ku(t)k ² _{H ˙}

+ kB(t)k ² _{H ˙}

+ kv(t)k ² _{H ˙}

+ w t

0 (k∇uk ² _{H ˙}

+ k∇Bk ² _{H ˙}

+ k∇vk ² _{H ˙}

) dτ

≤ ku 0 k ² _{H ˙}

+ kB 0 k ² _{H ˙}

+ kv 0 k ² _{H ˙}

+ C w t

0 (kuk ² _{H ˙}

+ kvk ² _{H ˙}

)kvk ² _˙

H

dτ.

By Gronwall inequality, we then get for all t ≥ 0, k(u, B, v)(t)k ² _{H ˙}

+ w t

0 k(∇u, ∇B, ∇v)k ² _{H ˙}

dτ

≤ k(u ₀ , B ₀ , v ₀ )k H ˙

exp

C w t 0 kvk ²

H ˙

dτ

· Putting together with (3.12) and using that r t

0 kv(τ)k ²

H ˙

dτ is bounded thanks to the first part of the theorem, we get a global-in-time control of the Sobolev norms.

Of course, to make the proof rigorous, one has to smooth out the data. For that, one can proceed exactly as in [8]. This completes the proof of Theorem 2.1.

Proof of Corollary 2.2. As the solution (u, B) belongs to C b ( R + ; ˙ H

( R ³ )) ∩ L ² ( R + ; ˙ H

( R ³ ))

and J ∈ C _b ( R + ; ˙ H

( R ³ )) ∩ L ² ( R + ; ˙ H

( R ³ )), the interpolation inequality between Sobolev norms implies that (u, B, J) belongs to the space L ⁸ _loc ( R + ; ˙ H

( R ³ )), which, in view of Sobolev embedding, is a subspace of L ⁴ _loc ( R + ; L ⁴ ( R ³ )). Now, the right- hand sides of the first two equations of (1.6) belongs to L ² _loc ( R + ; ˙ H

( R ³ )), which ensures time continuity.

In order to prove that the energy balance is fulfilled, one can use the same approximation scheme as in the proof of existence (the energy balance is clearly satisfied by (u n , B n )) then observe that (u n , B n ) _n∈

is actually a Cauchy sequence in L

( R ⁺ ; L ² ( R ³ ))∩L ² ( R ⁺ ; ˙ H ¹ ( R ³ )), as may be checked by arguing as in the proof of uniqueness.

Let us next prove that (u, B, v) goes to 0 in ˙ H

( R ³ ) when t → +∞. Inequality (2.2) and interpolation guarantee that B ∈ L ⁴ ( R + ; ˙ H ¹ ( R ³ )). Hence one can find some t 0 ≥ 0 so that v(t 0 ) ∈ L ² ( R ³ ). Then, performing an energy argument on the equation satisfied by v, we get for all t ≥ t 0 ,

kv(t)k ² _L

+ w t

t

k∇vk ² _L

dτ ≤ kv(t 0 )k ² _L

+ w t t

kB ⊗ B − u ⊗ uk _L

+ kv × uk _L

+ kv · ∇Bk _L

² dτ.

Using repeatedly the product law ˙ H ¹ ( R ³ ) × H ˙

( R ³ ) → L ² ( R ³ ), the equivalence (3.1) and adding up to (2.3) yields

k(u, B, v)(t)k ² _L

+ w t

t

k(u, B, v)k ² _{H ˙}

≤ k(u, B, v)(t 0 )k ² _L

+C w t

t

k(u, B, v)k ² _{H ˙}

k(u, B, v)k ² _˙

H

dτ.

Since k(u, B, v)(t)k _˙

H

is small for all t ≥ 0, the last term may be absorbed

by the left-hand side, and one can conclude (by interpolation) that (u, B, v) ∈

L ⁴ (t ₀ , +∞; ˙ H

( R ³ )). Therefore, for all σ > 0 one may find some t ₁ ≥ t ₀ so

that k(u, B, v)(t 1 )k _˙

H

≤ σ. Combining with (2.2) allows to conclude the proof

of (2.4).

Proof of Corollary 2.3. We shall argue as in [3] and [13], splitting the data into a small part in ˙ H

( R ³ ) and a (possibly) large part in H

( R ³ ). More precisely, we set

v ₀ = u ₀ − ∇ × B ₀ , u ₀ = u _{0,`</sub> + u <sub>0,h</sub> , B <sub>0</sub> = B <sub>0,`} + B _0,h , v ₀ = v _0,` + v _0,h with

u _0,` := F

(1

c u ₀ ), B _0,` := F

(1

B c ₀ ) and v _0,` := F

(1

v b ₀ ).

Fix some η ∈ (0, c) (with c being the constant of (2.1)) and choose ρ such that k(u _{`,0</sub> , B <sub>`,0} , v _`,0 )k _˙

H

< η 2 ·

By Theorem 2.1, we know that there exists a unique global solution (u ` , B ` ) to the Hall-MHD system supplemented with data (u `,0 , B `,0 ), that satisfies

k(u ` , B ` , v ` )k ²

L

( ˙ H

) + 1

2 k(u ` , B ` , v ` )k ²

L

( ˙ H

) < η

2 with v ` := u ` −∇×B ` · (3.13) Let (u h , B h , v h ) := (u − u ` , B − B ` , v − v ` ). We have (u h , B h , v h ) ∈ C( R + ; ˙ H

) ∩ L ⁴ ( R + ; ˙ H ¹ ) since that result holds for both (u, B, v) and (u ` , B ` , v ` ) (use interpo- lation). Furthermore, (u 0,h , B h,0 , J h,0 ) is in L ² ( R ³ ) owing to the high-frequency cut-off and we have



 

 

 

 

 

 

∂ t u h − ∆u h := R e 1 ,

∂ _t B _h − ∆B _h := R e ₂ ,

∂ t v h − ∆v h := R e 1 + R e 3 + R e 4 + R e 5 , (u _h , B _h , v _h )| _t=0 =(u _0,h , B _h,0 , J _h,0 ),

(3.14)

where

R e 1 := P (B · ∇B h + B h · ∇B ` − u · ∇u h − u h · ∇u ` ), R e 2 := ∇ × (v × B h + v h × B ` ),

R e ₃ := −∇ × ((∇ × v) × B _h + (∇ × v _h ) × B <sub>`</sub> ), R e 4 := ∇ × (v × u h + v h × u ` ),

R e 5 := 2∇ × (v · ∇B h + v h · ∇B ` ).

Let us bound the terms R e 1 , R e 2 , R e 4 and R e 5 as in the proof of the uniqueness part of Theorem 2.1, and estimate R e 3 as follows:

k R e 3 k _L

T

( ˙ H

) ≤ k(∇ × v) × B h k _L

T

(L

) + k(∇ × v h ) × B ` k _L

(L

)

≤ kvk _L

T

( ˙ H

) kB h k _L

T

(L

) + kv h k _L

T

( ˙ H

) kB ` k _L

(L

) .

Note that our assumptions ensure that B _` and B _h are in L ⁴ (0, T ; ˙ H ¹ ∩ H ˙ ² ), thus in L ⁴ (0, T ; L

) owing to the Gagliardo-Nirenberg inequality (A.4) with s = 1 and s

= 1. Then one can conclude by a straightforward energy argument that

k(u h , B h , v h )(t)k ² _L

+ 2 w t

0 k(u h , B h , v h )k ² _{H ˙}

dτ ≤ k(u h , B h , v h )(0)k ² _L

+C w t

0 V e k(u h , B h , v h )k ² _L

dτ with V e (t) := k(u, u ` , B, B ` , v, v ` )(t)k ⁴ _{H ˙}

and Gronwall’s lemma thus implies that k(u h , B h , v h )(t)k ² _L

+ w t

0 k(u h , B h , v h )(τ)k ² _{H ˙}

dτ

≤ k(u _h,0 , B _h,0 , v _h,0 )k ² _L

exp C w t

0 V e (τ ) dτ

· Since V e is globally integrable on R + thanks to our assumptions and (3.13), we see by interpolation that (u _h , B _h , v _h ) is in L ⁴ ( R + ; ˙ H

). This in particular im- plies that there exists some t ₀ ≥ 0 such that k(u _h , B _h , v _h )(t ₀ )k _˙

H

< η/2. Hence k(u, B, v)(t 0 )k _˙

H

< η and Theorem 2.1 thus ensures that k(u, B, v)(t)k _˙

H

< η for all t ≥ t 0 . This completes the proof of the corollary.

4. Weak-strong uniqueness

This section is devoted to the proof of Theorem 2.4. Let us underline that, in contrast with the other parts of the paper, the proof works for any positive coefficients µ, ν and ε. Furthermore, it could be adapted to the 2 ¹ ₂ D flows of the next section. For expository purpose however, we focus on the 3D case.

Throughout, we shall repeatedly use the following result.

Lemma 4.1. Let a, b, c ∈ L

(0, T ; L ² ( R ³ )) ∩ L ² (0, T ; ˙ H ¹ ( R ³ )) be three divergence free vector fields in R ³ . The following inequalities hold:

• If, in addition, b belongs to L ⁴ (0, T ; ˙ H ¹ ( R ³ )), then

w T

0 (a · ∇b | c) dτ

. kak _L

T

(L

) kak

L

( ˙ H

) kbk _L

T

( ˙ H

) kck _L

T

( ˙ H

) . (4.1)

• If, in addition, c belongs to L ⁴ (0, T ; ˙ H ¹ ( R ³ )), then

w T

0 (a · ∇b | c) dτ

. kak _L

T

(L

) kak

L

( ˙ H

) kbk _L

T

( ˙ H

) kck _L

T

( ˙ H

) . (4.2)

• If, in addition, ∇ × c belongs to L ⁴ (0, T ; ˙ H ¹ ( R ³ )), then

w T

0 ∇× ((∇× a)×b) | c dτ

. kak _L

T

( ˙ H

) kbk _L

T

(L

) kbk

L

( ˙ H

) k∇×ck _L

T

( ˙ H

) . (4.3) Proof. To prove the first inequality, we use the fact that a · ∇b = div (b ⊗ a) and the duality inequality between ˙ H ¹ and ˙ H

so as to write

w T

0 (a · ∇b | c) dτ ≤ w T

0 kb ⊗ ak _L

kck H ˙

dτ.

Hence, thanks to H¨ older and Gagliardo-Nirenberg inequalities, and Sobolev embed- ding (A.2),

w T

0 (a · ∇b | c) dτ ≤ w t

0 kak _L

kbk _L

k∇ck _L

dτ . w T

0 kak _˙

H

kbk H ˙

kck H ˙

dτ . kak

L

( ˙ H

) kbk _L

T

( ˙ H

) kck _L

( ˙ H

) .

Using the interpolation inequality (A.1) yields (4.1).

Proving (4.2) is similar. To get the last inequality, we take advantage of (1.7), of H¨ older and Gagliardo-Nirenberg inequalities, and Sobolev embedding:

w T

0 ∇ × ((∇ × a) × b) | c dτ

=

w T

0 (∇ × a) × b | ∇ × c dτ

≤ w T

0 k∇ × ak L

kbk L

k∇ × ck L

dτ . w T

0 kak H ˙

kbk _˙

H

k∇ × ck H ˙

dτ . kak _L

T

( ˙ H

) kbk

L

( ˙ H

) k∇ × ck _L

( ˙ H

) . Using the interpolation inequality (A.1) completes the proof.

One can now start the proof of Theorem 2.4. Let us recall our situation: we are given two Leray-Hopf solutions (u, B) and (¯ u, B) in ¯ L

_T (L ² )∩L ² _T ( ˙ H ¹ ) corresponding to the same initial data (u 0 , B 0 ) ∈ L ² with div u 0 = div B 0 = 0, and assume in addition that u and J := ∇ × B (and thus also B) are in L ⁴ _T ( ˙ H ¹ ). We want to prove that the two solutions coincide on [0, T ], that is to say

(δu, δB) ≡ (0, 0) with (δu, δB) := (u − u, B ¯ − B ¯ ). (4.4) By definition of what a Leray-Hopf solution is, both (u, B) and (¯ u, B) satisfy the ¯ energy inequality (1.13) on [0, T ], which implies that for all t ≥ T,

k(δu, δB)(t)k ² _L

+ 2µ w t

0 k∇δuk ² _L

dτ + 2ν w t

0 kδBk ² _L

dτ

≤ 2k(u 0 , B 0 )k ² _L

− 2(u(t) | u(t)) ¯ − 2(B(t) | B(t)) ¯

− 4µ w t

0 (∇u | ∇¯ u) dτ − 4ν w t

0 (∇B | ∇ B) ¯ dτ. (4.5) Then, the key to proving (4.4) is the following lemma, which is an adaptation to our setting of a similar result for the Navier-Stokes equations in [12].

Lemma 4.2. Under the assumptions of Theorem 2.4, we have for all time t ≤ T, (u(t) | ¯ u(t)) + (B(t) | B(t)) + 2µ ¯ w t

0 (∇u | ∇¯ u) dτ + 2ν w t

0 (∇B | ∇ B) ¯ dτ

= ku 0 k ² _L

+ kB 0 k ² _L

+ w t 0

(δu · ∇B | δB) + (δu · ∇u | δu) − (δB · ∇B | δu)

−(δB · ∇u | δB) − ε(∇ × ((∇ × δB) × δB) | B) dτ.

Proof. The result is obvious if (u, B) and (¯ u, B) are smooth and decay at infinity. In ¯ our setting where the solutions are rough, it requires some justification. Therefore, we consider two sequences (u _n , B _n ) _n∈

and (¯ u _n , B ¯ _n ) _n∈

of smooth and divergence free vector fields, such that

n→∞ lim (u n , B n , ∇ × B n ) = (u, B, ∇ × B) in L ⁴ _T ( ˙ H ¹ ) (4.6) and lim

n→∞ (u _n , B _n , u ¯ _n , B ¯ _n ) = (u, B, u, ¯ B) ¯ in L ² _T ( ˙ H ¹ ) ∩ L

_T (L ² ). (4.7) Since our assumptions on (u, B) also ensure that (∂ _t u, ∂ _t B) is in L ² _T ( ˙ H

), one can require in addition that

n→∞ lim (∂ t u n , ∂ t B n ) = (∂ t u, ∂ t B) in L ² _T ( ˙ H

). (4.8)