Diego Chamorro
email: diego.chamorro@univ-evry.fr

David Llerena
email: david.llerena@univ-evry.fr

A crypto-regularity result for the micropolar fluids equations

Keywords: Micropolar Equation, Morrey spaces, Local Regularity. MSC2020: 35B65, 35Q35, 76D03

In the analysis of PDEs, regularity of often measured in terms of Sobolev, Hölder, Besov or Lipschitz spaces, etc. However, sometimes a gain of regularity can also be expressed just in terms of Lebesgue spaces, by passing from a singular setting to a less singular one. In this article we will obtain a gain of integrability for weak solutions of the micropolar fluid equations using as general framework Morrey spaces, which is a very useful language to study regularity in PDEs. An interesting point is that the two variables of the micropolar fluid equations can be studied separately.

Introduction

In this article we study some integrability results for weak solutions of the perturbed 3D micropolar fluid equations given by the following system:

           ∂ t u = ∆ u -( u • ∇) u -∇p + 1 2 ∇ ∧ ω + f + ( a • ∇) u + ( u • ∇) a, div( u) = div( a) = 0, ∂ t ω = ∆ ω + ∇div( ω) -ω -( u • ∇) ω + 1 2 ∇ ∧ u, div( f ) = 0,
u(0, x) = u 0 (x), div( u 0 ) = 0, ω(0, x) = ω 0 (x), x ∈ R 3 .

(1.1)

In the previous equations u is the fluid velocity field, ω is the field of microrotation representing the angular velocity of the rotation of the fluid particles, p is the scalar pressure, the quantity f represents some given divergence free external force and u 0 and ω 0 are the initial data. The perturbation a ∈ L 6 t L 6

x which appears in the first equation above in the term ( a • ∇) u + ( u • ∇) a is a given divergence free vector field and the presence of this particular type of perturbation is inspired by some applications (see [START_REF] Barker | Quantitative regularity for the Navier-Stokes equations via spatial concentration[END_REF] and [START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF]Section 12.6]) and it is linked to a previous work [START_REF] Chamorro | Interior epsilon-regularity theory for the solutions of the magnetomicropolar equations with a perturbation term[END_REF]. Of course, if ω = a = 0 this system reduces to the classical Navier-Stokes equations.

Micropolar equations were first introduced in 1966 by Eringen [START_REF] Eringen | Theory of micropolar fluids[END_REF] and they have been recently studied by many authors, see e.g. [START_REF] Chamorro | Interior epsilon-regularity theory for the solutions of the magnetomicropolar equations with a perturbation term[END_REF], [START_REF] Galdi | A note on the existence and uniqueness of solutions of the micropolar fluid equations[END_REF], [START_REF] Gu | Global regularity of 3D magneto-micropolar fluid equations[END_REF], [START_REF] Lorenz | Regularity criteria for weak solutions of the Magneto-micropolar equations[END_REF] and the references there in. One very interesting feature of the micropolar fluid system is that the variable ω is not a divergence free vector field and this fact makes its study quite different from other systems of PDEs based on the Navier-Stokes equations (such as the Magneto-Hydrodyamic equations for example).

The main result of this article (see Theorem 1.1 below) states that, if we have some local control of the variable u in terms of parabolic Morrey spaces (see formula (1.2 below for a definition of these spaces), then we can deduce a local control for the variables u and ω in terms of Lebesgue spaces. Let us remark that Morrey spaces are generalization of Lebesgue spaces and they can contain quite singular objects (see the books [START_REF] Adams | Morrey spaces in harmonic analysis[END_REF][START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF][START_REF] Triebel | Local Function Spaces, Heat and Navier-Stokes Equations[END_REF] for a detailed treatment of Morrey spaces) thus, this gain of integrability can be seen as a gain of crypto-regularity as we pass from a singular framework to a less singular framework (and this motivates the title of the article). To be more precise, recall that in the setting of the 3D Navier-Stokes equations it is known that if we have some suitable local integrability information then, following the work of Serrin [START_REF] Serrin | On the interior regularity of weak solutions of the Navier-Stokes equations[END_REF] in the case 2 p + 3 q < 1 with q > 3 and Struwe [START_REF] Struwe | On partial regularity results for the Navier-Stokes equations[END_REF] and Takahashi [START_REF] Takahashi | On interior regularity criteria for weak solutions of the Navier-Stokes equations[END_REF] in the case 2 p + 3 q = 1 with q > 3, it is possible to deduce a (usual) gain of regularity in terms of Sobolev spaces. These results were generalized by O'Leary [START_REF] O'leary | Conditions for the local boundedness of solutions of the Navier-Stokes system in three dimensions[END_REF] to the framework of Morrey spaces and it is shown there that a local gain of integrability implies, by the Serrin local regularity theory (see [START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF]Chapter 13]), a gain of regularity. Note that, to the best of our knowledge, in the setting of the perturbed equation (1.1) all the different results that constitute the Serrin theory are not available, thus we will follow here the general spirit of this theory, but with some modifications. Indeed, due to the structure of the system (1.1) we will perform our study in two separated steps: first we will study the variable u in order to obtain a gain of integrability on u and only then we will use this result to study the variable ω and doing so, we will obtain a gain of integrability for the two variables, note that at each step we will also need to deal with the perturbation term and the external forces. Finally, we will show how to obtain a classical Hölder regularity result for the couple ( u, ω). However, we will not look for a Serrin type regularity theory and instead we will use the more sophisticated theory of Caffarelli-Kohn-Nirenberg which was displayed in our previous article [START_REF] Chamorro | Interior epsilon-regularity theory for the solutions of the magnetomicropolar equations with a perturbation term[END_REF]: doing so we will mix these two theories in order to obtain a different and new regularity framework for the system (1.1).

Let us recall now the definition of parabolic Morrey spaces. For 1 < p ≤ q < +∞, parabolic Morrey spaces M p,q t,x are defined as the set of measurable functions ϕ : R × R 3 -→ R 3 that belong to the space (L p t,x ) loc such that ϕ M p,q t,x < +∞ where ϕ M p,q t,x = sup

x 0 ∈R 3 ,t 0 ∈R,r>0 1 r 5(1-p q ) |t-t 0 |<r 2 B(x 0 ,r) | ϕ(t, x)| p dxdt 1 p . (1.2)
Note that we have M p,p t,x = L p t,x . See [START_REF] Adams | Morrey spaces in harmonic analysis[END_REF] for more details on these spaces or [START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF] for a general theory concerning the Morrey spaces and Hölder continuity applied to the analysis of PDEs from fluid mechanics. See also Section 2 below for a list of useful facts concerning these spaces.

As we are interested in the study of local properties of a weak solution ( u, p, ω) to the micropolar fluid equations (1.1), we will fix once and for all an open parabolic ball Q ⊂]0, +∞[×R 3 of the form

Q =]a, b[×B(x 0 , r), with 0 < a < b < +∞, x 0 ∈ R 3 and 0 < r < +∞, (1.3)
note in particular that (0, 0) / ∈ Q. We are now ready to state our main result in which we obtain a gain of integrability for u and for ω.

Theorem 1.1. Let ( u, p, ω) be a weak solution over Q of the perturbed micropolar fluid equations (1.1) where Q is the parabolic ball given in (1.3)

. Assume that u, ω ∈ L ∞ t L 2 x ∩ L 2 t Ḣ1 x (Q), p ∈ D t,x (Q), f ∈ L 2 t Ḣ1
x (Q) and a ∈ L 6 t L 6

x (Q). If moreover we have the following local hypothesis

1 Q u ∈ M p 0 ,q 0 t,x (R × R 3 ) with 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6, (1.4) then 1) for a parabolic ball Q 1 =]a 1 , b 1 [×B(x 0 , r 1 )
, with a < a 1 < b 1 < b and 0 < r 1 < r (we thus obtain the inclusion Q 1 ⊂ Q), we have that

1 Q 1 u ∈ L q 0 t,x (R × R 3 ), 5 < q 0 ≤ 6. (1.5)
2) for a parabolic ball

Q 2 =]a 2 , b 2 [×B(x 0 , r 2 ), with a 1 < a 2 < b 2 < b 1 and 0 < r 2 < r 1 (note the inclusion of parabolic balls Q 2 ⊂ Q 1 )
, we have that

1 Q 2 ω ∈ L q 0 t,x (R × R 3 ), (1.6)
for some 5 < q 0 ≤ 6.

Some remarks are in order here. Note first that we only impose the Morrey-type condition (1.4) to the variable u and we do not assume any further assumption over ω. Indeed, with the sole information

ω ∈ L ∞ t L 2 x ∩ L 2 t Ḣ1
x (Q) we can perform our computations for the variable u and we can obtain for this variable the local gain of integrability given in (1.5). Now, once we have this information over u at hand, we can focus on the variable ω and we can prove (1.6) and this two-step procedure shows that, when studying the gain of integrability problem, the velocity field u "dominates" the angular velocity ω.

Remark now that the pressure p is a very general object as we only ask p ∈ D , this is a general feature of the Serrin local regularity theory and the usual trick to get rid of the pressure consists in applying the Curl operator to the first equation in (1.1). However, if we want to mix our results with other regularity theories (such as the Caffarelli-Kohn-Nirenberg criterion), then some information over the pressure will be needed, see Corollary 1.1 below. To end the remarks, it is worth noting here that the upper bound for the parameter q 0 in (1.4), i.e. the range 5 < q 0 ≤ 6, is mainly technical and it is related to the information available on the perturbation term a ∈ L 6 t L 6

x . Let us mention that this control over a is very useful to obtain existence of Leray-type solution to the problem (1.1) and we do not claim any optimality over these parameters.

Let us now explain how the previous crypto-regularity result (Theorem 1.1) can help to deduce a real gain of regularity in terms of usual spaces. The following result heavily relies in our previous work done in [START_REF] Chamorro | Interior epsilon-regularity theory for the solutions of the magnetomicropolar equations with a perturbation term[END_REF].

Corollary 1.1. Let ( u, p, ω) be a weak solution over Q of the perturbed micropolar fluid equations (1.1) where Q is the parabolic ball given in (1.3)

. Assume that u, ω ∈ L ∞ t L 2 x ∩ L 2 t Ḣ1 x (Q), p ∈ D t,x (Q), f ∈ L 2 t Ḣ1
x (Q) and a ∈ L 6 t L 6 x (Q). Assume moreover the following points:

1) 1 Q u ∈ M p 0 ,q 0 t,x (R × R 3 ) with 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6,
2) The pressure p belongs to the space L

3 2 t,x (Q) ∩ L 5 2 t L 1 x (Q),
3) The external force satisfies

1 Q f ∈ M 10 7 ,τ t,x (R × R 3 ) for some τ > 5 2-α with 0 < α 1,
4) The weak solution ( u, p, ω) is suitable in the sense of the Caffarelli-Kohn-Nirenberg partial regularity theory (see the Definition 1.1 of [START_REF] Chamorro | Interior epsilon-regularity theory for the solutions of the magnetomicropolar equations with a perturbation term[END_REF], see also the Section 13.8 of the book [START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF]).

5) There exists a positive constant * which depends only on τ such that, if for some (t 0 , x 0 ) ∈ Q, we have

lim sup r→0 1 r ]t 0 -r 2 ,t 0 +r 2 [×B(x 0 ,r) | ∇ ⊗ u| 2 + | ∇ ⊗ ω| 2 dxds < * ,
then ( u, ω) is Hölder regular (in the time and space variables) of exponent α in a neighborhood of (t 0 , x 0 ) for some small α in the interval 0 < α 1.

Note in particular that, to obtain a Hölder regularity result for the variables u and ω, we only need to impose an integrability hypothesis on u.

Proof of the Corollary 1.1. Remark that the first hypothesis here corresponds to the condition (1.4) above and thus, applying Theorem 1.1 we obtain the conclusion (1.6), i.e. 1 Q 2 ω ∈ L q 0 t,x (R × R 3 ) for some 5 < q 0 ≤ 6. Now, if we fix 0 < α 1 small enough in order to verify the condition 5 q 0 < 1 -2α, we actually satisfy a local condition of integrability for ω needed in order to apply the Caffarelli-Kohn-Nirenberg regularity theory, see Remark 4.2 of [START_REF] Chamorro | Interior epsilon-regularity theory for the solutions of the magnetomicropolar equations with a perturbation term[END_REF]. Thus, following this theory displayed in the article [START_REF] Chamorro | Interior epsilon-regularity theory for the solutions of the magnetomicropolar equations with a perturbation term[END_REF] we obtain the wished Hölder regularity result for the variables u and ω.

The plan of the article is the following: in Section 2 we recall some useful facts related to the Morrey spaces. In Section 3 we study the gain of integrability for the variable u while in Section 4 we obtain the gain of integrability for the variable ω.

Useful results related to parabolic Morrey spaces

In this section we state without proof some results on parabolic Morrey spaces and we refer to the books [START_REF] Adams | Morrey spaces in harmonic analysis[END_REF][START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF] for a proof of these facts and for a more detailed study of these functional spaces. We start recalling the parabolic framework given by the homogeneous space (R × R 3 , d, µ) where d is the parabolic quasi-distance given by

d (t, x), (s, y) = |t -s| 1 2 + |x -y|, (2.1)
and where µ is the usual Lebesgue measure dµ = dtdx. Remark that the homogeneous dimension is N = 5. See [START_REF] Folland | Hardy spaces on homogeneous groups[END_REF] for more details concerning the general theory of homogeneous spaces.

Lemma 2.1 (Hölder inequalities).

1) If f , g : R × R 3 -→ R 3 are two functions such that f ∈ M p,q t,x (R × R 3 ) and g ∈ L ∞ t,x (R × R 3 ), then for all 1 ≤ p ≤ q < +∞ we have f • g M p,q t,x ≤ C f M p,q t,x g L ∞ t,x . 2) If f , g : R × R 3 -→ R 3 are two functions that belong to the space M p,q t,x (R × R 3 ) then we have the inequality f • g

M p 2 , q 2 t,x ≤ C f M p,q
t,x g M p,q t,x . 3) More generally, let 1 ≤ p 0 ≤ q 0 < +∞, 1 ≤ p 1 ≤ q 1 < +∞ and 1 ≤ p 2 ≤ q 2 < +∞.

If 1 p 1 + 1 p 2 ≤ 1 p 0 and 1 q 1 + 1 q 2 = 1 q 0 , then for two functions f , g : R × R 3 -→ R 3 such that f ∈ M p 1 ,q 1 t,x (R × R 3 ) and g ∈ M p 2 ,q 2 t,x (R × R 3 ), we have the following Hölder inequality in Morrey spaces

f • g M p 0 ,q 0 t,x ≤ f M p 1 ,q 1 t,x g M p 2 ,q 2 t,x . Lemma 2.2 (Localization). Let Q be a bounded set of R × R 3 of the form given in (1.
3). If we have 1 ≤ p 0 ≤ p 1 and 1 ≤ p 0 ≤ q 0 ≤ q 1 < +∞ and if the function f : R × R 3 -→ R 3 belongs to the space M p 1 ,q 1 t,x (R × R 3 ) then we have the following localization property

1 Q f M p 0 ,q 0 t,x ≤ C 1 Q f M p 1 ,q 1 t,x ≤ C f M p 1 ,q 1 t,x
. Now, for 0 < a < 5 we can define the (parabolic) Riesz potential I a of a locally integrable function f : R × R 3 -→ R 3 by

I a ( f )(t, x) = R R 3 1 (|t -s| 1 2 + |x -y|) 5-a f (s, y)dyds. (2.2)
Then, we have the following property Lemma 2.3 (Adams-Hedberg inequality). If 0 < a < 5 q , 1 < p ≤ q < +∞ and f ∈ M p,q t,x (R × R 3 ), then for ν = 1 -aq 5 we have the following boundedness property in Morrey spaces:

I a ( f ) M p ν , q ν t,x ≤ f M p,q t,x .
We finish this section with two simple corollaries of the previous lemma obtained in [START_REF] Chamorro | On the local regularity theory for the Magnetohydrodynamic equations[END_REF] and [START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF] which will be helpful in our computations Corollary 2.1 (Estimates for I 1 ). Let Q be a bounded set of R × R 3 of the form given in (1.3). If we have 2 < p ≤ q, 5 < q and if the function f : R × R 3 -→ R 3 belongs to the space M p 2 , q 2 t,x (R × R 3 ) then we have 1)

1 Q I 1 ( f ) ∈ M p ν , q ν t,x (R × R 3 ), with ν = 1 -q-5 5q (remark that 0 < ν < 1). 2) 1 Q I 1 ( f ) ∈ M σ,q t,x (R × R 3 )
, where σ = min{ p ν , q} with the same ν as before.

Corollary 2.2 (Estimates for I 2 ). Let Q be a bounded set of R × R 3 of the form given in (1.3). If we have 2 < p ≤ q, 5 < q and if the function

f : R × R 3 -→ R 3 belongs to the space M p 2 , q 2 t,x (R × R 3 ) then we have 1 Q I 2 (1 Q f ) ∈ M σ,q t,x (R × R 3 ),
where σ = min{ p ν , q} with ν = 1 -q-5 5q .

3 Gain of integrability for the variable u

Let Q 0 , Q 1 be two parabolic balls of the form (1.3) such that we have the inclusions

Q 1 ⊂ Q 0 ⊂ Q.
We will start showing a small gain of integrability stated in terms of Morrey spaces. Indeed, by the hypothesis (1.4) we have 1 Q u ∈ M p 0 ,q 0 t,x (R × R 3 ) with 2 < p 0 ≤ q 0 and 5 < q 0 ≤ 6 and thus we can define a technical parameter 0 < ν 0 < 1 such that

ν 0 = 1 - q 0 -5 5q 0 , (3.1)
and we will prove that we have

1 Q 1 u ∈ M σ 0 ,q 0 t,x (R × R 3 ), (3.2)
where

σ 0 = min{ p 0 ν 0 , q 0 }. (3.3)
Note that since σ 0 > p 0 we will obtain a (small) gain of integrability with respect of the first index of Morrey spaces and we will see later on how to iterate this process in order to deduce the first point of Theorem 1.1.

We introduce now two test functions ψ, φ : R × R 3 -→ R that belong to the space C ∞ 0 (R × R 3 ) and such that

ψ ≡ 1 on Q 0 and supp(ψ) ⊂ Q, φ ≡ 1 on Q 1 and supp(φ) ⊂ Q 0 ⊂ Q. (3.4)
It is clear that we have φ(0, •) = ψ(0, •) = 0 since the sets Q 0 , Q 1 are subsets of the parabolic ball Q given in (1.3) and (0, 0) / ∈ Q, note moreover that we have the identity ψφ = φ in the whole space R × R 3 . With the help of these technical functions we can consider the function

U = φ u, (3.5)
hence, verifying the condition (3.2) amounts to prove that U ∈ M σ 0 ,q 0 t,x (R×R 3 ) as we are only interested in the behavior of u inside the parabolic ball Q 1 . Now, using the localizing properties of the functions φ and ψ, we have in S the identity

U = ψ 1 ∆ ∆(φ u) , (3.6)
which can be rewritten as

U = ψ 1 ∆ φ∆ u -(∆φ) u + 2 3 i=1 ∂ i (∂ i φ) u = ψ 1 ∆ (φ∆ u) -ψ 1 ∆ (∆φ) u + 2 3 i=1 ψ 1 ∆ ∂ i (∂ i φ) u := U 1 -U 2 + U 3 . (3.7)
Then, we shall prove that each term U 1 , U 2 and U 3 of the expression above belong to the Morrey space M σ 0 ,q 0 t,x and we will deduce from this that we have (3.2). We start with the terms U 2 and U 3 which are easier to study than U 1 : Proposition 3.1. Under the hypotheses of Theorem 1.1, we have

U 2 , U 3 ∈ M σ 0 ,q 0 t,x (R × R 3 ), (3.8)
where σ 0 = min{ p 0 ν 0 , q 0 } and ν 0 = 1 -q 0 -5 5q 0 with 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6.

Proof. We claim first that

U 2 , U 3 ∈ L ∞ ([0, +∞[, L 6 (R 3 )). Indeed, since u ∈ L ∞ t L 2 x ∩ L 2 t Ḣ1 x (Q), it is easy to see that (∆φ) u ∈ L ∞ ([0, +∞[, L 6 5 (R 3
)) as by the Hölder inequalities we can write

(∆φ) u L ∞ t L 6 5 x ≤ ∆φ L ∞ t L 3 x 1 supp(φ) u L ∞ t L 2 x ≤ C 1 Q u L ∞ t L 2 x < +∞. Now, the embedding L 6 5 (R 3 ) ⊂ Ḣ-1 (R 3 ) implies that (∆φ) u ∈ L ∞ ([0, +∞[, Ḣ-1 (R 3 )). (3.9)
Moreover, as for any 1

≤ i ≤ 3 we have (∂ i φ) u ∈ L ∞ ([0, +∞[, L 2 (R 3
)), we obtain

3 i=1 ∂ i ((∂ i φ) u) ∈ L ∞ ([0, +∞[, Ḣ-1 (R 3 )). (3.10)
Thus, with the informations (3.9) and (3.10) above we have

U 2 = ψ 1 ∆ ((∆φ) u) ∈ L ∞ ([0, +∞[, Ḣ1 (R 3
)) and

U 3 = 2 3 i=1 ψ 1 ∆ ∂ i ((∂ i φ) u) ∈ L ∞ ([0, +∞[, Ḣ1 (R 3 )),
and using the classical Sobolev embedding Ḣ1 (R 3 ) ⊂ L 6 (R 3 ) we have

U 2 , U 3 ∈ L ∞ ([0, +∞[, Ḣ1 (R 3 )) ⊂ L ∞ ([0, +∞[, L 6 (R 3 )).
We remark now that due to the properties of the auxiliar function

ψ we have supp( U 2 ), supp( U 3 ) ⊂ Q and since Q is a bounded (parabolic) ball, we obtain that U 2 , U 3 ∈ L 6 t,x (R × R 3 ) = M 6,6 t,x (R × R 3
). Finally due to the fact that q 0 ≤ 6, we can conclude (3.8) by using the first inequality of Lemma 2.2 as we have for i = 2, 3:

U i M σ 0 ,q 0 t,x = 1 Q U i M σ 0 ,q 0 t,x ≤ 1 Q U i M 6,6 t,x ≤ U i L 6 t L 6 x < +∞.
The proof of Proposition 3.1 is finished.

Remark 3.1. Note that for the terms U 2 and U 3 we easily obtain the best control stated the expression (1.5) of the Theorem 1.1 as we have

U 2 , U 3 ∈ L 6 t,x (R × R 3 ).
The most technical part of the article deals with the remaining term U 1 and for this we have the following proposition Proposition 3.2. Under the hypotheses of Theorem 1.1, we have

U 1 ∈ M σ 0 ,q 0 t,x (R × R 3 ), (3.11)
where

U 1 = 1 ∆ (φ∆ u) is given in (3.7
) and where σ 0 = min{ p 0 ν 0 , q 0 } and ν 0 = 1 -q 0 -5 5q 0 is a small parameter with 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6.

Proof. In order to proof this proposition we need to study the evolution of the variable U 1 which is intimately associated to the evolution of the variable u. We note that in the first equation of (1.1), which is satisfied by u, the pressure p is assumed to be a general object (a distribution) and to get rid of this variable p in a local framework it is usual to apply the curl operator " ∇∧". This fact motivates the use of the following expression for U 1

U 1 = ψ 1 ∆ (φ∆ u) = -ψ 1 ∆ φ( ∇ ∧ [ψ ∇ ∧ u]) , (3.12)
where we used the property div( u) = 0 and the properties of the auxiliary functions φ and ψ defined in (3.4). See Lemma 4.1 in the Appendix below for a proof of this identity.

It is worth noting here that in the expression (3.12) it will be enough to start our study with the quantity

U := ∇ ∧ [ψ ∇ ∧ u], (3.13)
indeed, if we obtain some integrability information over U, then it would be easy to deduce the wished Morrey information for U 1 as we have the formula

U 1 = ψ 1 ∆ φ U . (3.14)
Having this remark in mind, we will deduce the equation satisfied by U given in the expression (3.13). Thus, applying the operator " ∇∧" to the first equation of (1.1) and since we have the identity ∇ ∧ ∇p ≡ 0 we obtain

∂ t ( ∇ ∧ u) = ∆( ∇ ∧ u) -∇ ∧ ( u • ∇) u + 1 2 ∇ ∧ ( ∇ ∧ ω) + ∇ ∧ f + ∇ ∧ ( a • ∇) u + ∇ ∧ ( u • ∇) a.
We introduce now the localizing function ψ and from the previous equation we have

∂ t [ψ ∇ ∧ u] = (∂ t ψ) ∇ ∧ u + ψ ∂ t ( ∇ ∧ u) = (∂ t ψ) ∇ ∧ u + ψ ∆( ∇ ∧ u) -ψ ∇ ∧ ( u • ∇) u + ψ 2 ∇ ∧ ( ∇ ∧ ω) + ψ ∇ ∧ f +ψ ∇ ∧ ( a • ∇) u + ψ ∇ ∧ ( u • ∇) a .
Noting that we have the identity

ψ∆( ∇ ∧ u) = ∆(ψ ∇ ∧ u) + (∆ψ) ∇ ∧ u -2 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u , (3.15)
we can rewrite the equation above as

∂ t [ψ ∇ ∧ u] = ∆(ψ ∇ ∧ u) + (∂ t ψ + ∆ψ)( ∇ ∧ u) -2 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u -ψ ∇ ∧ ( u • ∇) u + ψ 2 ∇ ∧ ( ∇ ∧ ω) + ψ ∇ ∧ f + ψ ∇ ∧ ( a • ∇) u + ψ ∇ ∧ ( u • ∇) a,
and applying again the operator " ∇∧" we deduce

∂ t ( ∇ ∧ [ψ ∇ ∧ u]) = ∆( ∇ ∧ [ψ ∇ ∧ u]) + ∇ ∧ (∂ t ψ + ∆ψ)( ∇ ∧ u) -2 ∇ ∧   3 j=1 ∂ j (∂ j ψ) ∇ ∧ u   -∇ ∧ ψ ∇ ∧ ( u • ∇) u + ∇ ∧ ψ 2 ∇ ∧ ( ∇ ∧ ω) + ψ ∇ ∧ f + ∇ ∧ ψ ∇ ∧ ( a • ∇) u + ψ ∇ ∧ ( u • ∇) a ,
and since by (3.13) we have

U = ∇ ∧ [ψ ∇ ∧ u]: ∂ t U = ∆ U + ∇ ∧ (∂ t ψ + ∆ψ)( ∇ ∧ u) -2 ∇ ∧   3 j=1 ∂ j (∂ j ψ) ∇ ∧ u   -∇ ∧ ψ ∇ ∧ ( u • ∇) u + ∇ ∧ ψ 2 ∇ ∧ ( ∇ ∧ ω) + ψ ∇ ∧ f + ∇ ∧ ψ ∇ ∧ ( a • ∇) u + ψ ∇ ∧ ( u • ∇) a ,
and this equation can be rewritten as

∂ t U = ∆ U + ∇ ∧ R, (3.16)
where the vector R is given by the terms

R = (∂ t ψ + ∆ψ)( ∇ ∧ u) -2 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u + ψ 2 ∇ ∧ ( ∇ ∧ ω) + ψ ∇ ∧ f -ψ ∇ ∧ ( u • ∇) u + ψ ∇ ∧ ( a • ∇) u + ψ ∇ ∧ ( u • ∇) a.
For our purposes we need to give another expression of the three last terms above and we will use the following generic formula which is valid for b, c two divergence free vector fields:

ψ ∇ ∧ ( b • ∇) c = ∇ ∧ 3 j=1 ∂ j (ψb j c) -∇ ∧ 3 j=1 (∂ j ψ)b j c - 3 j=1 ∂ j ( ∇ψ ∧ (b j c)) + 3 j=1 (∂ j ∇ψ) ∧ (b j c),
see the Lemma 4.2 in the Appendix below for a proof of this vectorial identity. Thus, the vector R can be written in the following manner:

R = (∂ t ψ + ∆ψ)( ∇ ∧ u) -2 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u + ψ 2 ∇ ∧ ( ∇ ∧ ω) + ψ ∇ ∧ f -∇ ∧ 3 j=1 ∂ j (ψu j u) + ∇ ∧ 3 j=1 (∂ j ψ)u j u + 3 j=1 ∂ j ( ∇ψ ∧ (u j u)) - 3 j=1 (∂ j ∇ψ) ∧ (u j u) (3.17) + ∇ ∧ 3 j=1 ∂ j (ψa j u) -∇ ∧ 3 j=1 (∂ j ψ)a j u - 3 j=1 ∂ j ( ∇ψ ∧ (a j u)) + 3 j=1 (∂ j ∇ψ) ∧ (a j u) + ∇ ∧ 3 j=1 ∂ j (ψu j a) -∇ ∧ 3 j=1 (∂ j ψ)u j a - 3 j=1 ∂ j ( ∇ψ ∧ (u j a)) + 3 j=1 (∂ j ∇ψ) ∧ (u j a).
Note at this point that, by the properties of the localizing function ψ defined in (3.4), we have

U(0, x) = ∇ ∧ [ψ(0, x) ∇ ∧ u(0, x)] = 0,
and thus with the equation (3.16) we can consider the following problem

   ∂ t U = ∆ U + ∇ ∧ R, U(0, x) = 0,
now, by the Duhamel formula we can write

U(t, x) = t 0 e (t-s)∆ ( ∇ ∧ R)(s, x)ds = ∇ ∧ t 0 e (t-s)∆ R(s, x)ds. (3.18)
With the help of this integral expression above for U and using the formula (3.14) we will deduce the wished Morrey information. Indeed we have

U 1 = ψ 1 ∆ φ U = ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ R(s, x)ds ,
and using the identity (3.17) for R we have the following lengthy expression:

U 1 = ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ (∂ t ψ + ∆ψ)( ∇ ∧ u)ds (1) -2 ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u ds   (2) + ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ψ 2 ∇ ∧ ( ∇ ∧ ω)ds (3)
+ ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ψ ∇ ∧ f ds (4) (3.19) -ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ 3 j=1 ∂ j (ψu j u)ds   (5) + ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ 3 j=1 (∂ j ψ)u j uds   (6) + ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 ∂ j ( ∇ψ ∧ (u j u))ds   (7) -ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 (∂ j ∇ψ) ∧ (u j u)ds   (8) + ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ 3 j=1 ∂ j (ψa j u)ds   (9)
-

ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ 3 j=1 (∂ j ψ)a j uds   (10)
-

ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 ∂ j ( ∇ψ ∧ (a j u))ds   (11)
+ ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 (∂ j ∇ψ) ∧ (a j u)ds   (12)
+ ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ 3 j=1 ∂ j (ψu j a)ds   (13) -ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ 3 j=1 (∂ j ψ)u j ads   (14) -ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 ∂ j ( ∇ψ ∧ (u j a))ds   (15) + ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 (∂ j ∇ψ) ∧ (u j a)ds   . ( 16
)
With this expression for U 1 , we only need to prove that each one of the previous terms belong to the Morrey space M σ 0 ,q 0 t,x (R × R 3 ) and in our study we will gather the terms that have similar structure.

• We remark first that since u, f ∈ L 2 t Ḣ1 x (Q) then we have ∇ ∧ u, ∇ ∧ f ∈ L 2 t L 2
x (Q) and thus the terms (1) and (4) in equation (3.19) share the same structure and will be treated with the help of the following lemma: Lemma 3.1. Consider the parabolic ball Q given in (1.3) and let A be a function in L 2 t L 2 x (Q). Assume that ψ, φ are the localizing function given in (3.4) and consider Φ a smooth function supported over the set Q. Then for 1 < p ≤ q ≤ 6 we have

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ Φ A(s, •)ds ∈ M p,q t,x (R × R 3 ).
Proof. For the sake of simplicity let us denote A = t 0 e (t-s)∆ Φ Ads. Since we have 1 < p ≤ q ≤ 6 and since the function ψ is supported in the parabolic ball Q, by the localization property given in Lemma 2.2 and by the space identification M p,p t,x = L p t,x , we can write

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ Φ A(s, •)ds M p,q t,x = ψ 1 ∆ φ ∇ ∧ A M p,q t,x ≤ C ψ 1 ∆ φ ∇ ∧ A M 6,6 t,x ≤ C ψ 1 ∆ φ ∇ ∧ A L 6 t L 6 x (3.20) ≤ C ψ L ∞ t,x 1 ∆ φ ∇ ∧ A L 6 t,x
. Now, with the embedding Ḣ1 (R 3 ) ⊂ L 6 (R 3 ) and the properties of the negative powers of the Laplacian we obtain

≤ C 1 ∆ φ ∇ ∧ A L 6 t Ḣ1 x ≤ C φ ∇ ∧ A L 6 t Ḣ-1 x ≤ C φ ∇ ∧ A L 6 t L 6 5 x
, where we used the dual embedding L 6 5 (R 3 ) ⊂ Ḣ-1 (R 3 ) in the last estimate above. Using the fact that the function φ is supported in the (fixed and bounded) parabolic ball Q we obtain by the Hölder inequality in the space variable the estimate:

φ ∇ ∧ A L 6 t L 6 5 x ≤ C φ ∇ ∧ A L ∞ t L 6 5 x ≤ C φ L ∞ t L 3 x ∇ ∧ A L ∞ t L 2 x .
By the definition of A we only have to study the quantity

∇ ∧ A L ∞ t L 2 x = sup t>0 ∇ ∧ t 0 e (t-s)∆ Φ A(s, •))ds L 2 ≤ C sup t>0 t 0 e (t-s)∆ Φ A(s, •))ds Ḣ1 ≤ C Φ A L 2 t L 2 x ,
where in the last line we used the properties of the heat kernel (see Lemma 7.2 of [START_REF] Lemarié-Rieusset | The Navier-Stokes problem in the 21st century[END_REF]). To finish, it is enough to recall that the function Φ is supported in the parabolic ball Q and that we have

Φ A L 2 t L 2 x ≤ C A L 2 t L 2
x (Q) < +∞. We have thus proved that the quantity

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ Φ A(s,
•)ds belongs to the space M p,q t,x (R × R 3 ) and the proof of Lemma 3.1 is finished.

With this result at hand, to study the term (1) in (3.19) we only need to apply this lemma with 1 < σ 0 ≤ q 0 ≤ 6, Φ = (∂ t + ∆)ψ and A = ∇ ∧ u and for the term (4) we use Φ = ψ and A = ∇ ∧ f . We have proven that (1) + (4) ∈ M σ 0 ,q 0 t,x (R × R 3 ).

• The term (2) in (3.19) will be studied with the following result Lemma 3.2. Under the general hypotheses of Theorem 1.1 we have for 1 < p ≤ q ≤ 6 we have

ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u ds   ∈ M p,q t,x (R × R 3 ). ( 3
ψ 1 ∆   φ ∇ ∧ t 0 e (t-s)∆ 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u ds   M σ 0 ,q 0 t,x = ψ 1 ∆ (φ ∇ ∧ B) M σ 0 ,q 0 t,x ≤ C ψ 1 ∆ (φ ∇ ∧ B) M 6,6 t,x = C ψ 1 ∆ (φ ∇ ∧ B) L 6 t L 6 x .
Let us define now ∇ ∧ B = ∆ B, where B = -

3 j=1 t 0 e (t-s)∆ 1 ∆ ∇ ∧ ∂ j (∂ j ψ) ∇ ∧ u (s, •) ds and we use the identity φ(∆ B) = ∆(φ B) + (∆φ) B -2 3 i=1 ∂ i ((∂ i φ) B) to obtain ψ 1 ∆ φ ∇ ∧ B = ψφ B + ψ 1 ∆ (∆φ) B -2 3 j=1 ψ ∂ j ∆ (∂ j φ) B . (3.22)
We will now prove that each term of (3.22) belongs to the space L 6 t L 6

x . Indeed, for the first term above, by using the support properties of the functions ψ and φ given in (3.4) and by the Sobolev embedding Ḣ1 (R 3 ) ⊂ L 6 (R 3 ) and a standard heat kernel estimate, we get

ψφ B L 6 t L 6 x ≤ C B L ∞ t L 6 x ≤ C B L ∞ t Ḣ1 x ≤ C 3 j=1 t 0 e (t-s)∆ 1 ∆ ∇ ∧ ∂ j (∂ j ψ) ∇ ∧ u (s, •) ds L ∞ t Ḣ1 x ≤ C 3 j=1 1 ∆ ∇ ∧ ∂ j (∂ j ψ) ∇ ∧ u L 2 t L 2 x ≤ C 3 j=1 (∂ j ψ) ∇ ∧ u L 2 t L 2 x ≤ C u L 2 t Ḣ1 x (Q) < +∞. (3.23)
For the second term on the right-hand side of (3.22), we use the embedding Ḣ1 (R 3 ) ⊂ L 6 (R 3 ) and the embedding L 6 5 (R 3 ) ⊂ Ḣ-1 (R 3 ), as well as the support properties of the function φ to get

ψ 1 ∆ (∆φ) B L 6 t L 6 x ≤ C 1 ∆ (∆φ) B L 6 t Ḣ1 x ≤ C (∆φ)1 Q B L 6 t Ḣ-1 x ≤ C (∆φ)1 Q B L 6 t L 6 5 x
. Now, by the Hölder inequality we have (using again the support properties of the function φ):

(∆φ)1 Q B L 6 t L 6 5 x ≤ C (∆φ) L ∞ t L 3 x 1 Q B L 6 t L 2 x ≤ C B L ∞ t L 6 x ,
and from the estimates displayed in (3.23) we obtain

ψ 1 ∆ (∆φ) B L 6 t L 6 x ≤ C u L 2 t Ḣ1
x (Q) < +∞. For the last term of (3.22), by the Sobolev embedding, the Hölder inequality and the inequalities (3.23) we can write

3 i=j ψ ∂ j ∆ (∂ j φ) B L 6 t L 6 x ≤ C 3 j=1 ∂ j ∆ (∂ j φ) B L 6 t Ḣ1 x ≤ C 3 j=1 (∂ j φ) 1 Q B L 6 t L 2 x ≤ C 3 j=1 ∂ j φ L ∞ t L 3 x 1 Q B L 6 t L 6 x ≤ C B L ∞ t L 6 x ≤ C u L 2 t Ḣ1 x (Q) < +∞.
Thus gathering all the L 6 t L 6

x estimates for (3.22), we finally obtain ψ 1 ∆ (φ( ∇∧ B)) ∈ M σ 0 ,q 0 t,x (R×R 3 ) which is the desired result.

We have proven, for 1 < σ 0 ≤ q 0 ≤ 6, that (2) ∈ M σ 0 ,q 0 t,x (R × R 3 ).

• The term (3) in (3.19) can be easily studied with the previous results. Indeed, we use the vectorial

formula ψ ∇ ∧ ( ∇ ∧ ω) = ∇ ∧ (ψ ∇ ∧ ω) -( ∇ψ) ∧ ( ∇ ∧ ω) to obtain ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ψ ∇ ∧ ( ∇ ∧ ω)ds = ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (ψ ∇ ∧ ω)ds -ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ( ∇ψ) ∧ ( ∇ ∧ ω)ds . (3.24)
Recall that by hypothesis we have ∇ ∧ ω ∈ L 2 t L 2 x (Q), thus the first term in the right-hand side above can be treated using the Lemma 3.2 since this expression shares the same structure as the formula (3.21) above. The second term of the previous identity can be studied with the help of Lemma 3.1. We finally obtain, for 1 < σ 0 ≤ q 0 ≤ 6, that

(3) ∈ M σ 0 ,q 0 t,x (R × R 3 ).
It remains to study the terms ( 5)-( 16) of (3.19). Recall that we have 1 Q u ∈ M p 0 ,q 0 t,x (R × R 3 ) with 2 < p 0 ≤ q 0 and 5 < q 0 ≤ 6 but we also have 1 Q a ∈ M p 0 ,q 0 t,x (R × R 3 ) since by the Lemma 2.2 we have

1 Q a M p 0 ,q 0 t,x ≤ C 1 Q a M 6,6 t,x = C 1 Q a L 6 t L 6
x < +∞. Thus, since we have the same information over the variables u and a we will perform a similar treatment of all the terms (5)-( 16) following their inner structure.

• For the terms (5), ( 9) and ( 13) of (3.19) we will use the following general statement. Lemma 3.3. Consider the auxiliar functions ψ and φ defined in (3.4). If A, B : R × R 3 -→ R 3 are two vector fields such that 1 Q A ∈ M p 0 ,q 0 t,x (R × R 3 ) and 1 Q B ∈ M p 0 ,q 0 t,x (R × R 3 ) with indexes 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6, then for 1 ≤ j ≤ 3 we have

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ ∂ j (ψA j B)ds ∈ M σ 0 ,q 0 t,x (R × R 3 ),
with σ 0 = min{ p 0 ν 0 , q 0 } where ν 0 is given in (3.1).

Proof. We consider the function C = t 0 e (t-s)∆ ∇ ∧ ∂ j (ψA i B)ds and we define C by the formula

C = t 0 e (t-s)∆ 1 ∆ ∇ ∧ ∇ ∧ ∂ j (ψA j B)ds
Now, as ∇ ∧ C = ∆ C and by the classical identity φ∆

C = ∆(φ C) + (∆φ) C -2 3 j=1 ∂ j ((∂ j φ) C), we can write ψ 1 ∆ φ ∇ ∧ C M σ 0 ,q 0 t,x ≤ ψφ C M σ 0 ,q 0 t,x + ψ 1 ∆ ((∆φ) C) M σ 0 ,q 0 t,x + 2 3 j=1 ψ ∂ j ∆ ((∂ j φ) C) M σ 0 ,q 0 t,x . (3.25)
For the first term above, using the properties of the heat kernel and the definition of the parabolic Riesz potentials (2.2) we write

|ψφ C(t, x)| = ψφ t 0 e (t-s)∆ 1 ∆ ∇ ∧ ∇ ∧ ∂ j (ψA j B)ds ≤ |(ψφ)(t, x)| t 0 (∂ j e (t-s)∆ ) 1 ∆ ∇ ∧ ∇ ∧ (ψA j B)ds ≤ C|(ψφ)(t, x)| R R 3 1 (|t -s| 1 2 + |x -y|) 4 1 ∆ ∇ ∧ ∇ ∧ (ψA j B) (s, y)dyds ≤ C|(ψφ)(t, x)|I 1 1 ∆ ∇ ∧ ∇ ∧ (ψA j B) (t, x),
and using the support properties of the functions ψ and φ we deduce

ψφ C M σ 0 ,q 0 t,x ≤ C 1 Q I 1 1 ∆ ∇ ∧ ∇ ∧ (ψA j B) M σ 0 ,q 0 t,x . (3.26)
Now, we apply the second point of the Corollary 2.1 with σ 0 = min{ p 0 ν 0 , q 0 } where ν 0 is given in (3.1) and where 5 < q 0 ≤ 6 and we obtain

1 Q I 1 1 ∆ ∇ ∧ ∇ ∧ (ψA j B) M σ 0 ,q 0 t,x ≤ C 1 ∆ ∇ ∧ ∇ ∧ (ψA j B) M p 0 2 , q 0 2 t,x ≤ C ψA j B M p 0 2 , q 0 2 t,x
, where in the last estimate above we used the boundedness of the Riesz transforms in Morrey spaces. Using the Hölder inequalities we thus obtain (by the support properties of the function ψ):

ψφ

C M σ 0 ,q 0 t,x ≤ C 1 Q A M p 0 ,q 0 t,x 1 Q B M p 0 ,q 0 t,x < +∞. (3.27)
For the second and the third term of (3.25) we will proceed as follows. First recall that the operators 1 ∆ and

∂ j
∆ are given by convolution with the kernels 1 |x-y| and

x j -y j |x-y| 3 . Thus if we define the operators T 1 ( C) and T 2,j ( C) for 1 ≤ j ≤ 3 by

T 1 ( C)(t, x) = ψ 1 ∆ ((∆φ) C)(t, x) = ψ(t, x) R 3 1 |x -y| ∆φ(t, y) C(t, y)dy (3.28) T 2,j ( C)(t, x) = ψ ∂ j ∆ ((∂ j φ) C)(t, x) = ψ(t, x) R 3
x j -y j |x -y| 3 ∂ j φ(t, y) C(t, y)dy.

(3.29)

we remark that the kernels associated to the operators T 1 and T 2,j are bounded in L 1 (R 3 ) due to the support properties of the functions ψ and φ. Indeed, for T 1 , we have

ψ(t, x) R 3 1 |x -y| ∆φ(t, y)dy = ψ(t, x) Q 1 |x -y| ∆φ(t, y)dy ≤ C,
for almost all x ∈ R 3 and ∆φ(t, y)

R 3 ψ(t, x) 1 |x -y| dx = ∆φ(t, y) Q ψ(t, x) 1 |x -y| dx ≤ C,
for almost all y ∈ R 3 . Thus, by the Schur test, we obtain that T 1 L 1 →L 1 ≤ C. Moreover, by the same ideas above we have T 2,j L 1 →L 1 ≤ C for 1 ≤ j ≤ 3. Now, since the norm of M σ 0 ,q 0 t,x is translation invariant, we deduce (taking into account the property supp(φ) ⊂ Q):

T 1 ( C) M σ 0 ,q 0 t,x = T 1 (1 Q C) M σ 0 ,q 0 t,x ≤ C 1 Q C M σ 0 ,q 0 t,x T 2,j ( C) M σ 0 ,q 0 t,x = T 2,j (1 Q C) M σ 0 ,q 0 t,x ≤ C 1 Q C M σ 0 ,q 0 t,x .
(3.30) Thus, coming back to the last terms of (3.25) we can write

ψ 1 ∆ ((∆φ) C) M σ 0 ,q 0 t,x + 2 3 j=1 ψ ∂ j ∆ ((∂ j φ) C) M σ 0 ,q 0 t,x ≤ C 1 Q C M σ 0 ,q 0 t,x
.

We only need to apply the computations performed in (3.26)-(3.27) to obtain

ψ 1 ∆ ((∆φ) C) M σ 0 ,q 0 t,x + 2 3 j=1 ψ ∂ j ∆ ((∂ j φ) C) M σ 0 ,q 0 t,x ≤ C 1 Q A M p 0 ,q 0 t,x 1 Q B M p 0 ,q 0 t,x
< +∞, which ends the proof of Lemma 3.3.

It is straightforward to apply this results to the terms ( 5), ( 9) and ( 13) of (3.19) as we have by the hypothesis (1.4) the information 1 Q u ∈ M p 0 ,q 0 t,x (R × R 3 ) with 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6 and we have

1 Q a ∈ L 6 t L 6 x (R × R 3
) from which we easily deduce that 1 Q a ∈ M p 0 ,q 0 t,x (R × R 3 ) with the same indexes as above.

We have proven that ( 5) + ( 9) + ( 13) ∈ M σ 0 ,q 0 t,x (R × R 3 ). • The terms ( 6), ( 10) and ( 14) of (3.19) we will be treated with the help of the next lemma: Lemma 3.4. Consider the functions ψ and φ given in (3.4)

. If A, B : R × R 3 -→ R 3 are two vector fields such that 1 Q A ∈ M p 0 ,q 0 t,x (R × R 3 ) and 1 Q B ∈ M p 0 ,q 0 t,x (R × R 3
) with indexes 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6, then for 1 ≤ j ≤ 3 we have

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (∂ j ψ)A j Bds ∈ M σ 0 ,q 0 t,x (R × R 3 ),
with σ 0 = min{ p 0 ν 0 , q 0 } where ν 0 is given in (3.1).

Proof. Let us write D j (t, x) = t 0 e (t-s)∆ ∇ ∧ (∂ j ψ)A j Bds, using the properties of the heat kernel, we thus have

| D j (t, x)| ≤ C R R 3 1 (|t -s| 1 2 + |x -y|) 4 |(∂ j ψ)A j B(s, y)|dyds ≤ CI 1 (|(∂ j ψ)A j B|)(t, x), (3.31)
where I 1 is the (parabolic) Riesz potential defined in (2.2). We observe now that we have

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (∂ j ψ)A j Bds M σ 0 ,q 0 t,x = ψ 1 ∆ φ ∇ ∧ D M σ 0 ,q 0 t,x ≤ ψ 1 ∆ ∇ ∧ φ D M σ 0 ,q 0 t,x + ψ 1 ∆ ∇φ ∧ D M σ 0 ,q 0 t,x
.

At this point we remark that the operator "ψ 1 ∆ ∇ ∧ (•)" in the first term above is of the same structure of the operator defined in (3.29) while the operator "ψ 1 ∆ ∇φ ∧ • " given in the second term is of the same nature than the operator given in (3.28), thus by the same arguments that leaded us to (3.30) we obtain (taking into account the support properties of the localizing function φ):

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (∂ j ψ)A j Bds M σ 0 ,q 0 t,x ≤ C 1 Q D M σ 0 ,q 0 t,x
. Now, with the help of the pointwise estimate (3.31) and by the support properties of the function (∂ j ψ) we have

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (∂ j ψ)A j Bds M σ 0 ,q 0 t,x ≤ C 1 Q I 1 (|1 Q A j B|) M σ 0 ,q 0 t,x
.

We apply now the second point of Corollary 2.1 with σ 0 = min{ p 0 ν 0 , q 0 } and ν 0 = 1 -q 0 -5 5q 0 to obtain

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (∂ j ψ)A j Bds M σ 0 ,q 0 t,x ≤ C 1 Q A j B M p 0 2 , q 0 2 t,x ≤ C 1 Q A M p 0 ,q 0 t,x 1 Q B M p 0 ,q 0 t,x < +∞,
where in the last line above we used the Hölder inequality in Morrey spaces. The proof of Lemma 3.4 is thus finished.

Since by Lemma 2.2 we have

1 Q a M p 0 ,q 0 t,x ≤ C 1 Q a L 6 t L 6
x < +∞ and since we have by the hypothesis (1.4) that 1 Q u M p 0 ,q 0 t,x < +∞, we can easily apply the previous lemma to the cases when A = u, a and B = u, a and we have proven that ( 6) + ( 10) + ( 14) ∈ M σ 0 ,q 0 t,x (R × R 3 ).

• For the terms ( 7), ( 11) and ( 15) of (3.19) we use the following generic result.

Lemma 3.5. If A, B : R × R 3 -→ R 3 are two vector fields such that 1 Q A ∈ M p 0 ,q 0 t,x (R × R 3 ) and 1 Q B ∈ M p 0 ,q 0 t,x (R × R 3 ) with 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6, then for 1 ≤ j ≤ 3 we have

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ ∂ j ( ∇ψ ∧ (A i B))ds ∈ M σ 0 ,q 0 t,x (R × R 3 ),
with σ 0 = min{ p 0 ν 0 , q 0 } where ν 0 is given in (3.1).

Proof. Consider the vector field E = t 0 e (t-s)∆ ∂ j ( ∇ψ ∧ (A i B))ds. By the decay properties of the heat kernel and by the support properties of the test function ψ, we have

E(t, x) ≤ t 0 R 3 ∂ j e (t-s)∆ [ ∇ψ ∧ (A i B)(s, y)] dyds ≤ C R R 3 1 (|t -s| 1 2 + |x -y|) 4 ∇ψ ∧ (A i B)(s, y) dy ds.
Using the definition of the Riesz potential given in (2.2) and by the properties of the test function ψ defined in (3.4), we obtain the point-wise estimate:

| E(t, x)| ≤ CI 1 (1 Q | ∇ψ ∧ (A i B)(t, x)|) ≤ CI 1 (1 Q | A(t, x) ⊗ B(t, x)|). (3.32)
Recalling the identity φ ∇ ∧ E = ∇ ∧ φ E -∇φ ∧ E, we have

ψ 1 ∆ φ ∇ ∧ E M σ 0 ,q 0 t,x ≤ ψ 1 ∆ ∇ ∧ φ E M σ 0 ,q 0 t,x + ψ 1 ∆ ∇φ ∧ E M σ 0 ,q 0 t,x
, but since the operator 1 ∆ is given by convolution, we can proceed in the same fashion as in (3.29) for the first term above and as in (3.28) for the second term above and then, with the help of the estimate (3.30), we can deduce the inequalities

ψ 1 ∆ ∇ ∧ φ E M σ 0 ,q 0 t,x ≤ C 1 Q 1 E M σ 0 ,q 0 t,x and
ψ 1 ∆ ∇φ ∧ E M σ 0 ,q 0 t,x ≤ C 1 Q 1 E M σ 0 ,q 0 t,x
. It follows that

ψ 1 ∆ φ ∇ ∧ E M σ 0 ,q 0 t,x ≤ C 1 Q 1 E M σ 0 ,q 0 t,x ≤ C 1 Q 1 I 1 (1 Q | A ⊗ B|) M σ 0 ,q 0 t,x
, where in the last control we used the estimate (3.32) above. Thus, using the second point of the Corollary 2.1 we have

ψ 1 ∆ φ ∇ ∧ E M σ 0 ,q 0 t,x ≤ 1 Q 1 I 1 (1 Q | A ⊗ B|) M σ 0 ,q 0 t,x (3.33)
≤ C 1 Q | A ⊗ B| M p 0 2 , q 0 2 t,x ≤ C 1 Q A M p 0 ,q 0 t,x 1 Q B M p 0 ,q 0 t,x < +∞
where we applied the Hölder inequality in the framework of Morrey spaces and with this estimate the proof of Lemma 3.5 is finished.

Again, as we have

1 Q a M p 0 ,q 0 t,x ≤ C 1 Q a L 6 t L 6 x < +∞ and 1 Q u M p 0 ,q 0 t,x
< +∞, we can easily apply the previous lema to the cases when A = u, a and B = u, a and we have proven that ( 7) + ( 11) + ( 15) ∈ M σ 0 ,q 0 t,x (R × R 3 ).

• For the terms (8), ( 12) and ( 16) of (3.19) we will use the following result:

Lemma 3.6. If A, B : R × R 3 -→ R 3 are two vector fields such that 1 Q A ∈ M p 0 ,q 0 t,x (R × R 3 ) and 1 Q B ∈ M p 0 ,q 0 t,x (R × R 3 ) with 2 < p 0 ≤ q 0 , 5 < q 0 ≤ 6, then for 1 ≤ j ≤ 3 we have

ψ 1 ∆ φ ∇ ∧ t 0 e (t-s)∆ (∂ j ∇ψ) ∧ A j Bds ∈ M σ 0 ,q 0 t,x (R × R 3 ),
with σ 0 = min{ p 0 ν 0 , q 0 } where ν 0 is given in (3.1).

Proof. As before and for the sake of simplicity we define F = t 0 e (t-s)∆ ( ∇∂ j ψ) ∧ A j Bds. Using the decay properties of the heat equation and the definition of the parabolic Riesz potential I 2 given in (2.2) we obtain the estimate

| F(t, x)| ≤ CI 2 (1 Q |( ∇∂ j ψ) ∧ A j B(t, x)|) ≤ CI 2 (1 Q (| A(t, x) ⊗ B(t, x)|)). (3.34)
Thus, with the same arguments displayed in the previous lemma we can write

ψ 1 ∆ φ ∇ ∧ F M σ 0 ,q 0 t,x ≤ ψ 1 ∆ ∇ ∧ φ F M σ 0 ,q 0 t,x + ψ 1 ∆ ∇φ ∧ F M σ 0 ,q 0 t,x ≤ C 1 Q 1 F M σ 0 ,q 0 t,x
, and it follows that

ψ 1 ∆ φ ∇ ∧ F M σ 0 ,q 0 t,x ≤ C 1 Q 1 I 2 (1 Q | A ⊗ B|) M σ 0 ,q 0 t,x . (3.35)
At this point we invoque the Corollary 2.2 to estimate the last term above and we can write

ψ 1 ∆ φ ∇ ∧ F M σ 0 ,q 0 t,x ≤ C 1 Q | A ⊗ B| M p 0 2 , q 0 2 t,x ≤ C 1 Q A M p 0 ,q 0 t,x 1 Q B M p 0 ,q 0 t,x
< +∞, and this finishes the proof of Lemma 3.6.

As before, we can easily apply this lema to the cases when A = u, a and B = u, a and we have proven that (8) + ( 12) + (16) ∈ M σ 0 ,q 0 t,x (R × R 3 ).

End of the proof of Proposition 3.2. With all the previous lemmas, we have proven that all the terms of (3.19) belong to the Morrey space M σ 0 ,q 0 t,x (R × R 3 ) and so does U 1 : we have proven (3.11) and this ends the proof of Proposition 3.2. Proposition 3.3. Under the hypotheses of Theorem 1.1, and recalling that U is given in the formula (3.7), we have that U ∈ L q 0 t,x (R × R 3 ), with 5 < q 0 ≤ 6.

Proof. Recall that following (3.7) we have U = U 1 -U 2 + U 3 . Note also that by Remark 3.1 the terms U 2 and U 3 belong to the space L 6 t,x (R × R 3 ) and by the local properties of these vector fields we deduce that U 2 , U 3 ∈ L q 0 t,x (R × R 3 ) for 5 < q 0 ≤ 6. It only remains to study the term U 1 . For this, once we obtained that U 1 ∈ M σ 0 ,q 0 t,x (R × R 3 ) with σ 0 = min{ p 0 ν 0 , q 0 }, then we can iterate the ideas of the Proposition 3.2 to obtain that

U 1 ∈ M σ 0 ,q 0 t,x (R × R 3 ) with σ 0 = min{ σ 0 ν 0 , q 0 } = min{ p 0 ν 2 0 , q 0 }. Since lim n→+∞ p 0 ν n 0 = +∞, we easily obtain that U 1 ∈ M q 0 ,q 0 t,x (R × R 3 ) = L q 0 t,x (R × R 3 ) from which we deduce that U ∈ L q 0 t,x (R × R 3 ).
To end this section, it is enough to recall that by the formula (3.5) we have the identity U = φ u and we have proven that φ u ∈ L q 0 t,x (R × R 3 ) with 5 < q 0 ≤ 6 and this conclusion corresponds with the point (1.5) of Theorem 1.1: we have thus obtained the wished gain of integrability for the variable u.

Gain of integrability for the variable ω

We have proven in the previous section that, under the hypotheses of Theorem 1.1, we have the information

1 Q 1 u ∈ L q 0 t,x (R × R 3
) for some 5 < q 0 ≤ 6. We will now use this information in order to study the integrability properties of the second equation of (1.1), i.e.:

∂ t ω = ∆ ω + ∇div( ω) -ω -( u • ∇) ω + 1 2 ∇ ∧ u. Note that, from the generic information ω ∈ L ∞ t L 2 x ∩L 2 t Ḣ1
x (Q), by an interpolation argument we easily obtain that

1 Q ω ∈ L 10 3 t,x (R × R 3 ) = M 10 3 , 10 3 t,x (R × R 3 ). ( 4.1)
Thus, with these (local) informations on u and ω we will first prove that a (local) gain in integrability for ω is possible. For this, we introduce two auxiliar smooth functions ϕ, : R × R 3 -→ R such that for two parabolic balls Q a , Q 2 of the general form (1.3) that satisfy

Q 2 ⊂ Q a ⊂ Q 1 , we have ϕ ≡ 1 on Q a , and supp(ϕ) ⊂ Q 1 (4.2) ≡ 1 on Q 2 , and supp( ) ⊂ Q a . (4.3)
We recall that by construction we have ϕ(0, •) = (0, •) = 0 and we have the identity ϕ = in the whole space. In the following proposition we will show how to obtain a first gain of integrability on ω:

Proposition 4.1. Under the hypotheses of Theorem 1.1 and considering the local framework stated above in (4.2)-( 4.3) we have

1 Q 2 ω ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 . (4.4)
Proof. Due to the properties of the localizing function given in (4.3), if we define W = ω and if we prove that W ∈ M p,q t,x (R × R 3 ) then we will easily deduce (4.4) and we will focus our study in the function W . Thus, just as in the expressions (3.6)-(3.7) above, by the properties of the functions ϕ and given in (4.2) and ( 4.3) we have the identity

W = ϕ 1 ∆ ∆( ω) = ϕ 1 ∆ ( ∆ ω) -ϕ 1 ∆ ((∆ ) ω) + 2 3 i=1 ϕ 1 ∆ ∂ i (∂ i ) ω = W 1 -W 2 + W 3 . Recall that we have ω ∈ L ∞ t L 2 x ∩ L 2 t Ḣ1
x (Q) and following the same arguments given in the proof of the Lemma 3.8 we obtain that W 2 , W 3 ∈ L 6 t,x (R × R 3 ) = M 6,6 t,x (R × R 3 ) which is the strongest expected result (see Remark 3.1 above): we thus only need to treat the term W 1 . For this, using the Lemma 4.1, we can rewrite the vector field W 1 in the following manner:

W 1 = ϕ 1 ∆ ∆( ω) = -ϕ 1 ∆ ( ∇ ∧ [ϕ ∇ ∧ ω]) + ϕ 1 ∆ ( ∇div( ω)) = -ϕ 1 ∆ ( ∇ ∧ [ϕ ∇ ∧ ω]) W 1,a + ϕ 1 ∆ ∇( div( ω)) W 1,b -ϕ 1 ∆ ( ∇ )div( ω) W 1,c , (4.5)
where in the last line we used the vectorial identity ( ∇div( ω)) = ∇( div( ω)) -( ∇ )div( ω).

Now, we will prove that each term W 1,a , W 1,b and W 1,c belong to the Morrey space M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

In order to study the term W 1,a of (4.5) we introduce the variable

W a = ϕ ∇ ∧ ω. (4.6)
Note that by the localizing properties of the function ϕ we have W a (0, •) = 0 and we want now to study the dynamic of this variable W a . Thus, following the second equation of (1.1) and applying the curl operator (since we have the vectorial identity ∇ ∧ ∇div( ω) = 0), we obtain:

∂ t ( ∇ ∧ ω) = ∆( ∇ ∧ ω) -∇ ∧ ω -∇ ∧ ( u • ∇) ω + 1 2 ∇ ∧ ( ∇ ∧ u).
Now, introducing the localizing function ϕ in the equation above we have

∂ t [ϕ ∇ ∧ ω] = ∂ t ϕ( ∇ ∧ ω) + ϕ ∆( ∇ ∧ ω) -∇ ∧ ω -∇ ∧ ( u • ∇) ω + 1 2 ∇ ∧ ( ∇ ∧ u) ,
and using the identity (3.15) for the term ϕ∆( ∇ ∧ ω) we write

∂ t [ϕ ∇ ∧ ω] = ∂ t ϕ( ∇ ∧ ω) + ∆[ϕ ∇ ∧ ω] + (∆ϕ)( ∇ ∧ ω) -2 3 j=1 ∂ j (∂ j ϕ) ∇ ∧ ω -ϕ ∇ ∧ ω -ϕ ∇ ∧ ( u • ∇) ω + 1 2 ϕ ∇ ∧ ( ∇ ∧ u).
which, by the definition of the variable W a given in (4.6) can be rewritten as

∂ t W a = ∆ W a + (∂ t + ∆ϕ)( ∇ ∧ ω) -2 3 j=1 ∂ j (∂ j ϕ) ∇ ∧ ω -ϕ ∇ ∧ ω -ϕ ∇ ∧ ( u • ∇) ω + 1 2 ϕ ∇ ∧ ( ∇ ∧ u).
We need to modify the term ϕ ∇ ∧ ( u • ∇) ω and for this we write (using the fact that div( u) = 0):

ϕ ∇ ∧ ( u • ∇) ω = ∇ ∧ 3 j=1 ∂ j (ϕu j ω) -∇ ∧ 3 j=1 (∂ j ϕ)u j ω - 3 j=1 ∂ j ( ∇ϕ ∧ u j ω) + 3 j=1 ( ∇∂ j ϕ) ∧ u j ω,
thus, with this formula at hand we can write

∂ t W a = ∆ W a + (∂ t + ∆ϕ)( ∇ ∧ ω) -2 3 j=1 ∂ j (∂ j ϕ) ∇ ∧ ω -ϕ ∇ ∧ ω -∇ ∧ 3 j=1 ∂ j (ϕu j ω) + ∇ ∧ 3 j=1 (∂ j ϕ)u j ω + 3 j=1 ∂ j ( ∇ϕ ∧ u j ω) - 3 j=1 ( ∇∂ j ϕ) ∧ u j ω + 1 2 ϕ ∇ ∧ ( ∇ ∧ u),
and since we have by construction that W a (0, •) = 0, by the Duhamel formula we can write

W a = t 0 e (t-s)∆ (∂ t + ∆ϕ)( ∇ ∧ ω) -2 3 j=1 ∂ j (∂ j ϕ) ∇ ∧ ω -ϕ ∇ ∧ ω -∇ ∧ 3 j=1 ∂ j (ϕu j ω) + ∇ ∧ 3 j=1 (∂ j ϕ)u j ω + 3 j=1 ∂ j ( ∇ϕ ∧ u j ω) + 3 j=1 ( ∇∂ j ϕ) ∧ u j ω + 1 2 ϕ ∇ ∧ ( ∇ ∧ u) ds,
and thus, using the definition of the variable W 1,a given in (4.5) we finally obtain the representation formula

W 1,a = ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ (∂ t + ∆ϕ)( ∇ ∧ ω)ds (1)
-2

3 j=1 ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ∂ j (∂ j ϕ) ∇ ∧ ω ds (2) -ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ϕ ∇ ∧ ωds (3)
-

3 j=1 ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ ∂ j (ϕu j ω)ds (4)
(4.7)

+ 3 j=1 ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (∂ j ϕ)u j ωds (5)
+ 3 j=1 ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ∂ j ( ∇ϕ ∧ u j ω)ds (6)
-

3 j=1 ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ( ∇∂ j ϕ) ∧ u j ωds (7) + 1 2 ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ϕ ∇ ∧ ( ∇ ∧ u)ds (8)
and thus we need to prove that all the previous terms belong to the Morrey space M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 and we will study them in the following points:

• The terms (1) and ( 3) of (4.7) can be treated in the same manner since we have ∇∧ ω ∈ L 2 t L 2 x (Q). Indeed, by applying the Lemma 3.1 with p = q = 6 we easily obtain that the terms (1) and (3) belong to the space L 6 t L 6

x (R × R 3 ) (see the estimate (3.20) above) and due to the localizing properties of the function ϕ given in (4.2) we easily deduce (by Lemma 2.2) that (1) + (3) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

• For the term (2) of (4.7), since ∇ ∧ ω ∈ L 2 t L 2

x (Q) we can apply Lemma 3.2 and we easily deduce that (2) belongs to the space L 6 t L 6

x (R × R 3 ), from which we obtain

(2) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

• The term (4) of (4.7) will be treated as follows. Following the same ideas given in the Lemma 3.3 (see in particular the estimates (3.26)-(3.30)), by the support properties of the functions ϕ and we can write

ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ ∂ j (ϕu j ω)ds M p,q t,x ≤ C 1 Q 2 I 1 1 ∆ ∇ ∧ ∇ ∧ (ϕ u ⊗ ω) M p,q t,x . (4.8)
Since 10 3 < p ≤ q ≤ 15 4 , if we consider 2 < q 1 ≤ 15 7 then we have 1 < 5 q 1 and if we define ν 1 = 1 -q 1 5 , we have 10 3 < q 1 ν 1 ≤ 15 4 . Thus, by properties given in Lemma 2.2 we can write

1 Q 2 I 1 1 ∆ ∇ ∧ ∇ ∧ (ϕ u ω) M p,q t,x ≤ 1 Q 2 I 1 1 ∆ ∇ ∧ ∇ ∧ (ϕ u ⊗ ω) M q 1 ν 1 , q 1 ν 1 t,x
, and using the Lemma 2.3 we deduce the estimate

1 Q 2 I 1 1 ∆ ∇ ∧ ∇ ∧ (ϕ u ⊗ ω) M q 1 ν 1 , q 1 ν 1 t,x ≤ 1 ∆ ∇ ∧ ∇ ∧ (ϕ u ⊗ ω) M q 1 ,q 1 t,x ≤ ϕ u ⊗ ω L q 1 t,x
, where we used the boundedness of the Riesz transforms in Morrey spaces and the identification between Morrey and Lebesgue spaces. Thus, by the usual Hölder inequalities with 1 q 1 = 1 q 0 + 3 10 (note that the condition 2 < q 1 ≤ 15 7 stated above is related to the previous identity and the condition 5 < q 0 ≤ 6) we obtain (since

Q 1 ⊂ Q) 1 Q 1 u ⊗ ω L q 1 t,x ≤ 1 Q 1 u L q 0 t,x 1 Q ω L 10 3 t,x < +∞, (4.9)
which is a bounded quantity since we have already proven

1 Q 1 u L q 0 t,x
< +∞ and we have by interpolation the information 1 Q ω L 10 3 t,x < +∞. Thus, we obtain that (4) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

• For the term (5) of (4.7), by the same arguments displayed in the Lemma 3.5 (see the inequality (3.33) above) we obtain the estimate

ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ∇ ∧ (∂ j ϕ)u j ωds M p,q t,x ≤ C 1 Q 2 I 1 (1 Q 1 | u ⊗ ω|) M p,q t,x , (4.10)
following the ideas of the previous point (see formula (4.9)) we can write

1 Q 2 I 1 (1 Q 1 | u ⊗ ω|) M p,q t,x ≤ I 1 (1 Q 1 | u ⊗ ω|) M q 1 ν 1 , q 1 ν 1 t,x ≤ 1 Q 1 u L q 0 t,x 1 Q ω L 10 3 t,x
< +∞, and we have proven (5) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

• The term (6) of (4.7) shares the same structure of the term (5) and thus, by the same arguments we obtain (6) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 . • For the term (7) of (4.7), following the same ideas displayed in the proof of the Lemma 3.6 (see in particular the estimates (3.34)-(3.35)) we can write

ϕ 1 ∆ ∇ ∧ t 0 e (t-s)∆ ( ∇∂ j ϕ) ∧ u j ωds M p,q t,x ≤ C 1 Q 2 I 2 (|1 Q 1 u ⊗ ω|) M p,q t,x . (4.11)
Again, since 10 3 < p ≤ q ≤ 15 4 and 2 < q 1 ≤ 15 7 , we have 2 < 5 q 1 and if we define ν 1 = 1 -2q 1 5 then we have 10 < q 1 ν 1 ≤ 15 (note that with these values we obtain directly the wished indexes of integrability). Thus by the Lemma 2.2 and by Lemma 2.3 we have

C 1 Q 2 I 2 (|1 Q 1 u⊗ ω|) M p,q t,x ≤ C I 2 (|1 Q 1 u⊗ ω|) M q 1 ν 1 , q 1 ν 1 t,x ≤ C 1 Q 1 u⊗ ω M q 1 ,q 1 t,x = C 1 Q 1 u⊗ ω L q 1 t,x
, and once we have this estimate, recalling the inequalities (4.9) we can easily conclude that (7) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

• The term ( 8) of (4.7) can be treat just as in the formula (3.24) above and since

∇ ∧ u ∈ L 2 t L 2 x (Q) and we obtain (8) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .
With the estimates of the terms (1)-( 9) of (4.7) we have proven that the term W 1,a given in (4.5) belongs to the space M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

We treat now the term W 1,b defined in (4.5) and for this we define the variable

W b = div( ω),
which satisfies the following system (derived from the second equation of (1.1)):

∂ t W b = 2∆W b + (∂ t + 2∆ -)div( ω) -4 3 j=1 ∂ j ((∂ j )div( ω)) -div(( u • ∇) ω),
where we used the identity (3.15) for the term ∆div( ω) and the fact that div( u) = 0. Noting that, by the support properties of the function , we have W b (0, •) = 0, we can write by the Duhamel formula:

W b = t 0 e 2(t-s)∆ (∂ t + 2∆ -)div( ω) -4 3
-ϕ 1 ∆ ∇ t 0 e 2(t-s)∆ div(( u • ∇) ω) ds

, (4.13) and we will prove that each term above belongs to the Morrey space M p,q t,x (R × R 3 ) with indexes 10 3 < p ≤ q ≤ 15 4 . • For the first term of (4.13) we denote by φ = (∂ t + 2∆ -). Since we have p ≤ q ≤ 15 4 < 6, by Lemma 2.2 we can write

ψ 1 ∆ ∇ t 0
e 2(t-s)∆ φ)div( ω)ds M p,q t,x ≤ ψ 1 ∆ ∇ t 0 e 2(t-s)∆ φdiv( ω)ds

L 6 t,x ≤ ψ L 6 t L ∞ x 1 ∆ ∇ t 0
e 2(t-s)∆ φdiv( ω)ds

L ∞ t L 6 x ,
and by the Sobolev embedding Ḣ1 (R 3 ) ⊂ L 6 (R 3 ) we obtain

ψ 1 ∆ ∇ t 0
e 2(t-s)∆ φ)div( ω)ds

M p,q t,x ≤ C 1 ∆ ∇ t 0
e 2(t-s)∆ φdiv( ω)ds

L ∞ t Ḣ1 x ≤ C
t 0 e 2(t-s)∆ φdiv( ω)ds

L ∞ t L 2 x .
Then, using the properties of the heat kernel, the dual embedding Ḣ-1 (R 3 ) ⊂ L 6 5 (R 3 ), the Hölder inequality and the properties of the function φ, we have t 0 e 2(t-s)∆ φdiv( ω)ds

L ∞ t L 2 x ≤ C φdiv( ω) L 2 t Ḣ-1 x ≤ C φdiv( ω) L 2 t L 6 5 x ≤ C φ L ∞ t L 3 x 1 Q div( ω) L 2 t L 2
x < +∞, from which we deduce that the quantity (1) of (4.13) belongs to the wished space M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 . • For the second term of (4.13), by the same arguments as above we obtain ϕ 1 ∆ ∇ t 0 e 2(t-s)∆ ∂ j ((∂ j )div( ω)) ds M p,q t,x ≤ C t 0 e 2(t-s)∆ ∂ j ((∂ j )div( ω)) ds

L ∞ t L 2 x ≤ C
t 0 e 2(t-s)∆ (∂ j )div( ω)ds

L ∞ t Ḣ1 x ≤ (∂ j )div( ω) L 2 t L 2 x ≤ C 1 Q div( ω) L 2 t L 2
x < +∞, and thus (2) ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 . • For the last term of (4.13), we need to study the quantity

ϕ 1 ∆ ∇ t 0 e 2(t-s)∆ div(( u • ∇) ω) ds M p,q t,x , (4.14)
but we remark that since we have div( u) = 0 we can write div(( u • ∇) ω) = div(div( ω ⊗ u)) and then we should deal with terms of the form ∂ i (∂ j (u k w )) for 1 ≤ i, j, k, ≤ 3, which can we rewritten in the following manner , which is of the same shape of the left-hand side of the formula (4.8) and thus, by the same arguments displayed in (4.8)-(4.9) we easily obtain that this quantity is bounded in the wished Morrey space M p,q t,x (R × R 3 ). Since the second and the third term of (4.15) are of the same structure, we only study one of them and we write ϕ 1 ∆ ∇ t 0 e 2(t-s)∆ ∂ i ((∂ j )u k w )ds M p,q t,x

∂ i (∂ j (u k w )) = ∂ i (∂ j ( u k w )) -∂ i ((∂ j )u k w ) -∂ j ((∂ i )u k w ) -(∂ i ∂ j )u k w .
.

This term can be treated just as the left-hand side of (4.10) and we obtain that this quantity is bounded. Finally, for the last term of (4.15) we write

ϕ 1 ∆ ∇ t 0 e 2(t-s)∆ (∂ i ∂ j )u k w ds M p,q t,x
, and following the computations made in (4.11) we also obtain that it is bounded in the corresponding Morrey space. We have studied all the terms of (4.15) and then we deduce that the quantity (4.14) belongs to the Morrey space M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 .

We need now to study the last term of (4.5), which is given by W 1,c = ϕ 1 ∆ ( ∇ )div( ω) , and we study the quantity for all 1 ≤ j, k, ≤ 3. For the first term above, using the support properties of the function ϕ, the fact that p ≤ q < 6 and the Sobolev embedding Ḣ1 (R 3 ) ⊂ L 6 (R 3 ), we write

ϕ 1 ∆ (∂ k (φ j w )) M p,q t,x ≤ C ϕ 1 ∆ (∂ k (φ j w )) L 6 t L 6 x ≤ C 1 ∆ (∂ k (φ j w )) L ∞ t Ḣ1 x ≤ C φ j w L ∞ t L 2
x < +∞, since we have that

1 Q ω L ∞ t L 2
x < +∞. For the second term of (4.16) we have

ϕ 1 ∆ ((∂ k φ j )w ) M p,q t,x ≤ C ϕ 1 ∆ ((∂ k φ j )w ) L 6 t L 6 x ≤ C 1 ∆ ((∂ k φ j )w ) L ∞ t Ḣ1 x ≤ C (∂ k φ j )w L ∞ t Ḣ-1 x ,
and by the embedding L 6 5 (R 3 ) ⊂ Ḣ-1 (R 3 ), we can write

ϕ 1 ∆ ((∂ k φ j )w ) M p,q t,x ≤ C (∂ k φ j )w L ∞ t L 6 5 x ≤ ∂ k φ i L ∞ t L 3 x 1 Q w L ∞ t L 2
x < +∞.

We have proven that all the terms of (4.16) belong to the Morrey space M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 and so does the quantity W 1,c .

  .21) Proof. We define B = t 0 e (t-s)∆ 3 j=1 ∂ j (∂ j ψ) ∇ ∧ u ds and thus by the same previous ideas we can write

j=1∂

  j ((∂ j )div( ω)) -div(( u • ∇) ω) ds, t-s)∆ (∂ t + 2∆ -)div( ω) ds

  t-s)∆ ∂ j ((∂ j )div( ω)) ds

(4. 15 ) 1 ∆ ∇ t 0 e 2

 15102 Thus, to study the quantity (4.14) we first consider the expressionϕ (t-s)∆ ∂ i (∂ j ( u k w ))ds M p,q t,x

.

  Let us introduce the function φ = ∇ , and by the identityφ j ∂ k (w ) = ∂ k (φ j w ) -(∂ k φ j )w , it is enough to treat the quantities

We have proven so far that all the term in the expression (4.5) belong to the Morrey space M p,q t,x (R × R 3 ) and this ends the proof of the Proposition 4.1

We now return to the proof of the second point of Theorem 1.1: we have proven so far the local information 1 Q 2 ω ∈ M p,q t,x (R × R 3 ) with 10 3 < p ≤ q ≤ 15 4 and in particular, due to the Lemma 2.2 and the identification of Morrey and Lebesgue spaces, we have

which is a gain of information with respect to (4.1). It is then enough to replace this information in the previous arguments (reducing if necessary the support of the auxiliary functions) to obtain a higher integrability control, and by suitable iterations we finally obtain that

), for some 5 < q 0 ≤ 6, which corresponds with the wished estimate (1.6). The proof of Theorem 1.1 is now complete.

Appendix

Lemma 4.1. If φ, ψ are the test functions given in (3.4), and u is a regular enough vector field, we have

Proof. We have

Moreover, by the support properties of φ and ψ we have φ ∇ψ ≡ 0 and ψφ = φ on the support of φ.

Then, the second term in the identity above disappear when we multiply by φ and we obtain

and from this identity it is easy to recover the wished result.

Lemma 4.2. Assume that b and c are two vector fields such that div( b) = div( c) = 0 and assume that ψ is the smooth function given in (3.4). Then we have the following vectorial identity

Proof. We start with the formula

Using now the fact that div( b) = div( c) = 0 in the second and the third term of the right-hand side above we obtain


Raphaël Baena

Lucas Drumetz

Vincent Gripon

A Local Mixup to Prevent Manifold Intrusion

Keywords: Mixup, Regularization, Manifold Intrusion

Deployed in the context of supervised learning, Mixup is a data-dependent regularization technique that consists in linearly interpolating input samples and associated outputs. It has been shown to improve accuracy when used to train on standard machine learning datasets. However, authors have pointed out that Mixup can produce out-of-distribution virtual samples and even contradictions in the augmented training set, potentially resulting in adversarial effects. In this paper, we introduce Local Mixup in which distant input samples are weighted down when computing the loss. In constrained settings we demonstrate that Local Mixup can create a trade-off between bias and variance, with the extreme cases reducing to vanilla training and classical Mixup. Using standardized computer vision benchmarks, we also show that Local Mixup can improve accuracy.

I. INTRODUCTION

Deep Learning has become the golden standard for many tasks in the fields of machine learning and signal processing. Using a large number of tunable parameters, Deep Neural Networks (DNNs) are able to identify subtle dependencies in large training datasets to be later leveraged to perform accurate predictions on previously unseen data. Without constraints or enough samples, many models can fit the training data (high variance) and it is difficult to find the ones that would generalize correctly (low bias).

Regularization techniques have been deployed with the aim of improving generalization [START_REF] Goodfellow | Deep learning[END_REF]. A popular data-dependent regularization technique consists of artificially increasing the size of the training set, which is referred to as data augmentation [START_REF] Simard | Transformation invariance in pattern recognition -tangent distance and tangent propagation[END_REF], [START_REF] Guo | Mixup as locally linear out-of-manifold regularization[END_REF]. In the field of computer vision, for example, it is very common to generate new samples using basic classinvariant transformations [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], [START_REF] He | Deep residual learning for image recognition[END_REF].

In [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF], the authors introduce Mixup, a data augmentation technique in which artificial training samples (x, ỹ), called virtual samples, are generated through linear interpolations between two training samples (x i , y i ) and (x j , y j ). The associated output is computed as the corresponding linear interpolation on the respective outputs. Mixup improves generalization error of state-of-the-art models on ImageNet, CIFAR, speech, and tabular datasets [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF]. This method is also used in the context of few shot learning [START_REF] Mangla | Charting the right manifold: Manifold mixup for few-shot learning[END_REF], [START_REF] Dhillon | A baseline for few-shot image classification[END_REF].

By using linear interpolation, virtual samples can in some cases contradict each other, or even generate out-ofdistribution inputs. This phenomenon has been described in [START_REF] Guo | Mixup as locally linear out-of-manifold regularization[END_REF] where the authors use the term manifold intrusion. As such, it is not clear if Mixup is always desirable. More generally, the question arises of whether Mixup could be amended to reduce the risk of generating spurious interpolations.

In this paper we introduce Local Mixup, where virtual samples are weighted in the training loss. The weight of each possible virtual sample depends on the distance between the endpoints of the corresponding segment (x i , x j ). In particular, this method can be implemented to forbid interpolations between samples that are too distant from each other in the input domain, reducing the risk of generating spurious virtual samples, as we shall discuss later in this paper.

Here are our main contributions: [START_REF] Bishop | Pattern recognition[END_REF]. We also consider that both input and output data lie in metric spaces (X , d X ) and (Y, d Y ). Typically, X and Y are assumed to be Euclidean spaces with the usual metrics. We denote by f : X → Y the parametric model to be trained and by F the hypothesis set, i.e. the set containing all candidate parametrizations of the model f ∈ F.

To train our model, we use an error function L that measures the discrepancy between the model outputs and expected ones. Training the model amounts to minimizing the training loss while generalization may be quantitavely evaluated by the test loss, resulting in the following simplified equation:

L vanilla = (x,y)∈D L(f (x), y).

Data augmentation and mixup:

To improve generalization one can use regularization techniques [START_REF] Goodfellow | Deep learning[END_REF]. Among them, data augmentation is a form of data-dependent regularization [START_REF] Guo | Mixup as locally linear out-of-manifold regularization[END_REF]. Recently, the use of data-dependent methods relying on some sort of mixing has emerged [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF], [START_REF] Verma | Manifold mixup: Better representations by interpolating hidden states[END_REF]- [START_REF] Rame | Mixmo: Mixing multiple inputs for multiple outputs via deep subnetworks[END_REF]. The pioneering mixing method is Mixup [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF], whose mixed samples (x, ỹ) are generated by linear interpolations between pairs of samples, Fig. 1: Illustration of the proposed Local Mixup method. On the left, only vanilla samples are used, without data augmentation. Ground truth is depicted in filled regions. On the middle we depict Local Mixup where we only interpolate samples which are close enough, leading to no contradiction with ground thuth. On the right we depict Mixup in which we interpolate all samples, leading to contradictory virtual samples.

i.e. xi,j,λ = λx i + (1 -λ)x j and ỹi,j,λ = λy i + (1 -λ)y j for some training samples (x i , y i ) and (x j , y j ) and some λ ∈ [0, 1] . The Mixup training criterion is defined as:

Definition II.1 (Mixup Criterion). Let λ ∼ Beta[α, β],
i, j discrete variables uniformly drawn with repetitions in {0, . . . , n -1}. f * minimizes the Mixup criterion if:

f * = argmin f ∈F 1 n 2 E λ        D 2 train L (ỹ i,j,λ , f (x i,j,λ )) Lmixup        .
As defined above Mixup encourages linear behavior [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF]. The positive effect of this linear behavior questioned several authors who aimed at explaining theoretically and empirically Mixup. In [START_REF] Carratino | On mixup regularization[END_REF] they shows that Mixup can be interpreted as the combination of a data transformation and a data perturbation; In [START_REF] Gyawali | Enhancing mixup-based semisupervised learning with explicit lipschitz regularization[END_REF] they highlight that Mixup impacts the Lipschitz constant L of the gradient of the network.

In [START_REF] Guo | Mixup as locally linear out-of-manifold regularization[END_REF] they described and introduced the term manifold intrusion. This phenomenon is depicted in Figure 1 on the right, where we see that virtual samples created through Mixup between distant red samples lie outside the manifold domain for the red class. The method of [START_REF] Guo | Mixup as locally linear out-of-manifold regularization[END_REF] called Adamixup uses an additional neural network to remove such interpolations.

III. MIXUP IN DIMENSION 1

The detailed proofs of each theoretical result can be found in a longer version of this paper [START_REF] Baena | Preventing manifold intrusion with locality: Local mixup[END_REF]. Let us consider the simple case where our model f is defined on R. Without loss of generality, let us consider that the training set D train = {x i , y i } is ordered by increasing input, i.e, x i ≤ x i+1 .

For a given x, Mixup's loss implies that the output f * (x) of the model is determined by the set E(x) of all convex combinations that can be obtain x from two training inputs x i and x j :

E(x) = {i, j, λ i,j |x = λ ij x i + (1 -λ ij ) x j }. It is clear that for any x ∈ [x 0 , x n-1 ], E(x)
is non empty and finite. In practice, the distribution of λ can be uniform [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF], [START_REF] Verma | Manifold mixup: Better representations by interpolating hidden states[END_REF] λ ∼ Beta(α = 1, β = 1) = U(0, 1). In this case, we show the output f * (x) of x ∈ [x 0 , x n ] is the barycenter of the target values corresponding to the points of E(x).

Lemma III.1. ∀x ∈ [x 0 , x n-1 ], f * (x) = 1 card(E(x)) (i,j,λi,j )∈E(x) λ i,j y i + (1 -λ i,j )y j . (1)
A consequence of this lemma is the following theorem:

Theorem III.2. The function f * that minimizes the loss on the training set is piecewise linear on [x 0 , x n-1 ], linear on each segment [x i , x i+1 ] and defined by Equation (1).

In practice inferring a function f * that minimizes that the Mixup Criterion is usually not desired in machine learning, and one looks for f with a sufficiently small loss to have a regularizing effect. Indeed f * is not likely to generalize well. Still, we note that it tends to an average of convex combinations and thus leads to a model with a low variance.

IV. LOCAL MIXUP

A. Locality graphs

Consider a (training) dataset D made of pairs (x, y). We propose to build a graph from D as follows. We define G D = ⟨V, W⟩ where V = {x | ∃y, (x, y) ∈ D}. The symmetric real matrix W is based on D, where D is the pairwise distance matrix D[i, j] = d X (x i , x j ).

In this work, we consider various ways to obtain W, but the rationale is always the same: to obtain a similarity matrix where large weights correspond to closest pairs of samples. Namely, we consider K-nearest neighbors graphs, where we set to 1 weights of target vertices corresponding to the K closest samples for a given source vertex and 0 otherwise; thresholded graphs where W[i, j] = ϕ(D[i, j]) and ϕ(d) = 1 d≤ε ; smooth decreasing exponential graphs where W[i, j] = exp(-αD[i, j]). The loss is then weighted using W:

L local mixup = D 2 train W[i, j]L (ỹ i,j,λ , f (x i,j,λ )) . (2)
For computational cost considerations, we compute a graph for each batch (random subset) of samples during stochastic gradient descent. As such, the weights associating two samples can vary depending on the chosen graph and random batch.

In the extreme case where some weights are 0, the corresponding virtual samples are discarded during gradient descent, resulting in only considering local interpolations of samples, hence the name Local Mixup.

B. Low dimension

In this section, we are interested in proving that Local Mixup allows to tune a trade-off between bias and variance on trained models. For this purpose, we simplify the problem to dimension 1 and only consider K-nearest neighbor graphs.

In this case, note that varying K can create a range of settings where K = 0 boils down to vanilla training and K ≥ n where n is the number of training samples boils down to classical Mixup where all combinations are allowed.

1) Local Mixup and the bias/variance trade-off: Let us first recall the definitions of the bias and variance in the context of a machine learning problem.

Definition IV.1 (Bias and Variance). Let us consider a training set D train and a function f from X to Y. We define Bias and Variance as follow:

• Bias: Bias(f ) 2 = E train [(f (x) -y) 2 ]. • Variance: V ar(f ) = E train [(f -E train [f ]) 2 ].
We consider two settings. In the first one, the input domain Z/nZ is periodic and thus the number of samples is finite. In the second one, the input domain Z is infinite and outputs are independent and identically distributed (i.i.d).

Periodic setting

Let us consider that the training set D train is made of pairs (x, y), where {x | ∃y, (x, y) ∈ D train } = Z/nZ. We also consider d

X (x, x ′ ) = |x -x ′ | ∈ {0, • • • , n -1}.
In this case, we can write explicit formulations of f * K , the function that minimizes the Local Mixup criterion for Knearest neighbors graphs. Following similar arguments to those used to obtain Equation (1): for a given x i we know that the optimal value for f * K (x i ) would be an average of the the ỹ that correspond to the possible interpolations. we obtain:

∀x i ∈ Z/nZ, f * K (x i ) = 1 K(K + 3)/2 (2Ky i + S K (x i )), (3)
where S K (x i ) is defined recursively as follows:

S K+1 = 0 if K = 0 S K (x i ) + A K+1 (x i ) ∀K ≥ 1 . (4)
and:

A K (x i ) = 1 K K-1 k=1 (K -k) • y i-k + k • y i+K-k .
On Figure 2 we depicted for a given x i the different interpolations and ỹ that contribute to f K (x i ). In blue the interpolation between x i and its direct neighbors, in red the interpolation between points other than x i that happen to intersect x i . As we increase K, the influence of S K (red points) increases. We obtain the following theorem:

x i-2 x i-1 x i x i+1 x i+2 x i-2 x i-1 x i x i+1 x i+2
Fig. 2: We depict here the terms of f * K (x i ) given by Eq (3) for different values of K. In blue the interpolations corresponding to 2Ky i and in red the terms of the sum S K . On the right, K = 2 and on the left K = 3.

Theorem IV.1 (Convergence of f * K in the periodic setting). As K grows, it holds that:

∀x i ∈ Z/nZ, f * K (x i ) → E Dtrain [y],
(5)

Bias 2 (f * K ) → E train [(y i -E train [y]) 2 ], (6)
V ar(

f * K ) = E train [(f * K (x i ) -E train [f * K (x i ]) 2 ] → 0, (7)
V ar(f * K ) is eventually nonincreasing. This theorem states two main results: 1) in the case of Mixup the function that minimizes the loss f * has zero variance and converges to E train Theorem IV.2. For a signal with i.i.d outputs, the variance is eventually bounded by:

4 2 σ 2 K 2 ≤ V ar(f K (x i )) ≤ 8σ 2 K . (8)
2) Invariance of linear models: Interestingly, we can show that both Mixup and Local Mixup lead to the same optimal linear models, as stated in the following theorem:

Theorem IV.3.

C. High Dimension and Lipschitz constraint

The proofs given in low dimension have some limitations. Basically, the averaging effect happens since any point x within the interval [x 1 , x n ] can be written as at least one convex combination of pairs from the training set. Contradictions may occur as illustrated above when several combinations corresponds to x. In higher dimension such explicit contradictions are not necessarily expected. Still, the more segments we allow to interpolate, the larger becomes the virtual training set, eventually leading to potentially smaller margins between classes and thus to harder variance/trade-off compromises. We will illustrate this point in the experiments. V. EXPERIMENTS Our method depends on an hyperparameter α or ϵ. The value of these parameters has been chosen for each experiments according to the ablations studies that we provide in [START_REF] Baena | Preventing manifold intrusion with locality: Local mixup[END_REF].

A. Low dimension

As stated in the introduction and [START_REF] Guo | Mixup as locally linear out-of-manifold regularization[END_REF], Mixup leads to interpolations that may be misleading for the model. To illustrate this effect, we consider a 2d toy dataset of two coiling spirals where such interpolations occur frequently. The two coiling spirals is a binary classification dataset: each spiral corresponds to a different class. We expect to retrieve better performance for Local Mixup compared to Mixup: local interpolations are likely to stay in the same spiral and therefore avoid manifold intrusion. For this experiment we use a thresholded graph with parameter ε.

We use a fully connected neural network made of two hidden layers with 100 neurons and ReLU function as non linear function. For small values of ε many weights of the graph are zero and thus the corresponding interpolations are disregarded into the loss. This means that for a given batch only a small proportion of samples are actually considered. To avoid side effects, we vary the batch size so that in average the same number of samples are used to update the parameters. In Figure 3, we depict the evolution of the average error rate as a function of the parameter ε. One can note the significant benefit of Mixup and Local Mixup over Vanilla. As expected, Local Mixup presents a minimum error rate which is significantly smaller than Mixup's error rate. We can note that the minimum is reached with a value of ε smaller than the first quantile. This means that for this dataset Mixup interpolations given above this threshold are either useless or misleading for the network's training.

B. Lipschitz lower bound

To illustrate the impact of ε on the optimal Lipschitz constant, we use the dataset CIFAR-10 [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF]. In Figure 4, we depict the evolution of Q(D) a lower bound of the Lipschitz constant that we obtain in [START_REF] Baena | Preventing manifold intrusion with locality: Local mixup[END_REF]. We observe that ε can be used to smoothly tune the lower bound Q(D). In practice, a lower Q(D) is preferable, but this only accounts for the optimal Lipschitz constant. Larger values of ε lead to larger training sets and thus potentially better generalization.

C. High dimension: Experiments on classification dataset

The previous toy dataset is particularly suitable to generate contradictory virtual samples. We delve into more complex and real world datasets in the following subsection to illustrate the negative impact of some interpolations.

We tested our proposition on different classification datasets and architectures and report the results in table 1. We consider the datasets CIFAR-10, CIFAR-100 [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] Fashion-MNIST [START_REF] Xiao | Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms[END_REF] and SVHN [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF]. Fashion-MNIST is composed of clothes images of size 28x28 pixels (grayscale) . There are 60,000 images in the training set corresponding to 10 classes. SVHN is a real-world image dataset made of small cropped digits of size 32x32 pixels and 3 colors. There are 73257 digits in the training set corresponding to 10 classes. For these tests, we use a smooth decreasing exponential graph tuned by α.

We observed that Local Mixup showed a smaller error rate on these datasets. For Fashion MNIST, we note that Mixup impacts negatively the error rate, suggesting that on this dataset Mixup creates spurious interpolations as discussed in [START_REF] Guo | Mixup as locally linear out-of-manifold regularization[END_REF], which might be caused by a low intrinsic dimensionality.

For these experiments, we also tried to use a K-nearest neighbor graph or a thresholded graph but without being able to achieve smaller error rates compared to Mixup or even Vanilla. This may indicate that some segments generated by Mixup are important to act as a regularizer during training even if some of them may generate manifold intrusions. By tuning α, we weight the importance of this regularization.

D. Comparaison with Adamixup

We compare our proposed approach with Adamixup, another method presented as capable of preventing manifold intrusion. We use the GitHub Repository of the author and we changed only the Mixup part to implement our method. On CIFAR-10 we used 1400 epochs as well for Local Mixup and Adamixup since the authors used this number for Adamixup. We observe a slight advantage for Adamixup on MNIST: 0.49% ± 0.03 for Adamixup, 0.54%±0.02 for Local Mixup (error rates averaged over 10 runs). Still, on CIFAR-10 our method outperforms Adamixup: 4.11% ± 0.12 for Adamixup and 3.89% ± 0.12 for Local Mixup (averaged over 5 runs).

Note that Local Mixup does not completely discard interpolations but weighs them, contrary to Adamixup that prevents the interpolations which are considered as causing manifold intrusions. Thus, this may indicates that even interpolated samples causing manifold intrusion could be beneficial as long as their are not predominant in the loss.

We also note that our method seems to converge faster on MNIST. One benefit of our proposed approach is the simplicity and the small number of parameters (CIFAR-10): 836522 for Local Mixup and 11171146 for Adamixup.

VI. CONCLUSION

In this paper, we introduced a method called Local Mixup, in which pairs of samples are interpolated and weighted in the loss depending on the distance between them in the input domain. This method comes with a hyper-parameter that allows to provide a continuous range of solutions between Vanilla and classical Mixup. Using a simple framework, we showed that Local Mixup can control the bias/variance tradeoff of trained models. In more general settings, we showed that Local Mixup can tune a lower bound on the Lipschitz constant of the trained model. We used real world datasets to prove the ability of Local Mixup to achieve better generalization, as measured using the test error rate, than Vanilla and classical Mixup.

Overall, our methodology introduces a simple way to incorporate locality notions into Mixup. We believe that such a notion of locality is beneficial and could be leveraged to a greater level in future work, or could be incorporated to the various Mixup extensions that have been proposed in the community. In future work, we would like to investigate further the choice of the graph, the choice of the hyperparameter that comes with it, and trainable versions of Local Mixup. We would like to rely on quantitative information given on the topology such as the histogram of the distance or persistence diagrams [START_REF] Wasserman | Topological data analysis[END_REF] to tune these hyper-parameters. There are also many possibilities to improve over using the Euclidean metric, in particular the pullback metrics given by the Euclidean distance between the samples once in the feature space corresponding to the penultimate layer. Extending the theoretical results to more general contexts would definitely allow to gain intuition on the effect of locality on Mixup.

  [y]. 2). Eventually the variance of the function that minimizes the Local Mixup criterion is decreasing, showing that the proposed Local Mixup can indeed tune the trade-off between the bias and variance. i.i.d random output setting Let us now consider that the training set is made of inputs {x | ∃y, (x, y) ∈ D train } = Z and y i are i.i.d. according to a random variable R of variance σ 2 .

  For a linear model: f (x) = ax + b, a, b ∈ R, the function f * that minimizes the loss of Mixup and Local Mixup is the same.

Fig. 3 :

 3 Fig.3: Error rate (averaged over 1000 runs) as a function of ε for the two coiling spirals dataset. 1000 samples per class generated with a Gaussian noise of standard deviation σ = 1.5. Extremes correspond to Vanilla (ε = 0) and Mixup (ε > 4).

Fig. 4 :

 4 Fig. 4: Evolution of Q(D) on the dataset CIFAR10. ε = 0 corresponds to Vanilla. ε = 50 corresponds to classical Mixup.

  In supervised Machine Learning, a training dataset D train made of pairs of inputs and corresponding outputs is used to learn the model's parameters, and a test one D test is used to evaluate the performance of the model on previously unseen inputs

• We introduce Local Mixup, a mixup method depending on a single parameter whose extremes correspond to classical Mixup and Vanilla. • In dimension one, we prove that Local Mixup allows to select a bias/variance trade-off. • In higher dimensions, we show that Local Mixup can help achieve more accurate models than classical Mixup using standard vision datasets. • Our work contributes more broadly to better understanding the impact of Mixup during training. II. RELATED WORK Introducing notations:

TABLE I :

 I Error rates (%) on CIFAR-10, Fashion-MNIST and SVHN. Values are averaged on 100 runs for CIFAR-10 and 10 runs for Fashion-MNIST and SVHN. Mean errors with their confidence intervals are given.