Hajer Bahouri

THE LITTLEWOOD-PALEY THEORY : A COMMON THREAD OF MANY WORKS IN NONLINEAR ANALYSIS

In this article we present the Littlewood-Paley theory and illustrate the effectiveness of this microlocal analysis tool in the study of partial differential equations, in a context which is the least technical possible. As we shall see below, the Littlewood-Paley theory provides a robust approach not only to the separate study of the various regimes of solutions to nonlinear partial differential equations, but also to the fine study of functional inequalities, and to make them accurate.

The Littlewood-Paley theory : a tool that has become indispensable

The Littlewood-Paley theory is a localization procedure in the frequency space that, since about three decades ago, has established itself as a very powerful tool in harmonic analysis. The first goal of this text is to present it in a way as simple as possible 1 . Its basic idea is contained in two fundamental inequalities known as Bernstein's inequalities, that describe some properties of functions whose Fourier transform have compact support.

The first inequality says that, for a tempered distribution 2 in R d whose Fourier transform is supported in a annulus of size λ, to differentiate first and then take the L p norm is the same as to apply a homothety of ratio λ on the L p norm. In the L 2 setting this remarkable property is an easy consequence of the action of the Fourier transform on derivatives and of the Fourier-Plancherel formula. The proof in the case of general L p spaces uses Young's inequalities and the fact that the Fourier transform of a convolution is the product of the Fourier transforms.

In the other hand, the second inequality tells us that, for such a distribution, the change from the L p norm to the L q norm, with q ≥ p ≥ 1, costs λ

q , which must be understood as a Sobolev embedding. It is proved like the first inequality, using Young's inequalities and the relation between the Fourier transform and the convolution product.

Fourier Analysis is at the heart of the Littlewood-Paley theory, which has inspired a large number of my works. It was in conducting experiments on the propagation of heat that Joseph Fourier at the end of the 18th century opened the door to that theory, which was hugely expanded on the 20th century and intervenes in the majority of branches of Physics.

In this theory having the name of its creator, one performs the frequency analyis of a function f of L 1 (R d ) by the formula :

f (ξ) = R d e -ix•ξ f (x) dx .
1. For a more detailed presentation of this theory the reader can consult the monograph [?]. 2. A tempered distribution is an element of the topological dual of the Schwartz space S(R d ).

1 Under appropriate conditions, f the Fourier transform of f (also denoted Ff in the present text), allows the synthesis of f through the inversion formula :

f (x) = 1 (2π) d R d e ix•ξ f (ξ) dξ .
As a consequence, we obtain the Fourier-Plancherel identity

R d |f (x)| 2 dx = 1 (2π) d R d | f (ξ)| 2 dξ .
In fact, for all functions f of S(R d ), we have, due to Fubini's theorem,

R d f (x)f (x) dx = 1 (2π) d R d R d e ix•ξ f (ξ) dξ f (x) dx = 1 (2π) d R d f (ξ) R d e -ix•ξ f (x) dx dξ .
This representation created a true revolution in the way we think about functions. To give f is exactly equivalent to give f , and this duality between analysis in amplitude (in the physical space described by x) and analysis in frequency (in the frequency space described by ξ) is of extraordinary importance in Physics and in Mathematics.

A fundamental fact from the theory of distributions is that the Fourier transform can be extended to the space of tempered distributions S (R d ). The crucial point is the fact that F is a well-known isomorphism on the Schwartz space S(R d ) (the space of smooth functions that, together with all their derivatives, decrease faster than every polynomial) and its extension to S (R d ) is defined by duality [START_REF] Bahouri | Fourier analysis and applications to nonlinear partial differential equations[END_REF] .

Fourier transforms have a very large number of properties that we do not wish to list here. Let us just recall the two basic principles of this transforms that we cannot dissociate from the convolution product. The first principle of the Fourier transform is that regularity implies decreasing ; the second one is that decreasing leads to regularity. The usefulness of these properties, that play a crucial role in the study of Fourier transforms in S(R d ), will be clear very soon in what follows.

Fourier analysis allow us to explicitly solve linear equations with constant coefficients [START_REF] Bahouri | Refined Hardy inequalities[END_REF] . In particular, combining the Fourier transform with the convolution product we can explicitly determine the solutions of the Schrödinger equation, a fundamental equation in quantum mechanics :

(S) i ∂ t v + ∆v = 0 v |t=0 = v 0 ∈ S(R d ) .
In fact, taking the partial Fourier transform with respect to the variable x we obtain, for every

(t, ξ) in R × R d : i ∂ t v(t, ξ) -|ξ| 2 v(t, ξ) = 0 v(0, ξ) = v 0 (ξ) ,
and integrating we get v(t, ξ) = e -it|ξ| 2 v 0 (ξ).

3. For a complete presentation of the theory of distributions we can, for instance, see the fundamental references [?, ?].

4. Linear equations with variable coefficients and nonlinear equations require different methods.

Combining the inverse Fourier transform together with the properties of the Fourier transform and the convolution product, we deduce that the solution of (S) for t = 0 can be written as

v(t, •) = e i |x| 2 4t (4πit) d 2 v 0 •
By Young's inequality, it follows the fundamental dispersion property

v(t, •) L ∞ (R d ) ≤ 1 |4πt| d 2 v 0 L 1 (R d ) •
This technique of explicit representation of solutions can be adapted to all linear evolution equation with constant coefficients. However, it is not always straightforward to deduce the dispersion effects. In fact, for example, to establish dispersive estimations for the wave equation in R d requires more elaborate techniques involving oscillating integrals, which necessitate an hypothesis of spectral localization in a annulus of Cauchy data.

The analysis of dispersion, a central problem in linear wave mechanics, provides a framework of formidable effectiveness for solving and analyzing nonlinear dispersive partial differential equations. It is thanks to the remarkable work of Robert Strichartz [?], in the late 1970s, that we have been able to transcribe dispersion phenomena, which correspond to a pointwise inequality, into robust inequalities. The idea of these estimates, known as Strichartz estimates, is to pass from a pointwise in time decay estimate to a spatial integrability gain after an appropriate time average. These Strichartz estimates, which have known a big boom these last few years, go along with the Littlewood-Paley theory : they can be expressed equally in Lebesgue spaces and in Besov spaces which we will define next.

The Littlewood-Paley theory was introduced by John Edensor Littlewood and Raymond Paley [?, ?] in the 1930s for the harmonic analysis of L p spaces, but its systematic use in the analysis of partial differential equations is more recent. In fact, the main breakthrough of this theory was made after the seminal paper [?] by Jean-Michel Bony, in 1981, on the paradifferential calculus that connects nonlinear functions and the Littlewood-Paley decomposition.

The main idea of this theory consists in sampling the frequencies by means of a decomposition of the frequency space in annulus of size 2 j , thus allowing the decomposition of a function into a sum of a countable number of regular functions whose Fourier transform is supported in an annulus of size 2 j :

(1) f = j∈Z ∆j f ,
where the homogeneous dyadic blocks of f , ∆j f, are defined by the filtering of f at frequencies of order 2 j . Observe that this so called homogeneous Littlewood-Paley decomposition is valid modulo polynomials P . In fact, since the Fourier transform of every polynomial is supported at the origin, the identity (??) cannot be applied to polynomials. This restriction on the lower frequencies is overcome in the case of the inhomogeneous Littlewood-Paley decomposition :

(2)

f = j≥-1 ∆ j f ,
where ∆ j f := ∆j f for j varying in N and ∆ -1 f is an operator filtering the lower frequencies, that is : it only preserves the frequencies in a ball centered at the origin.

The Littlewood-Paley decompositions (??) and (??) introduced above are obtained by a decomposition in the space of frequencies arising from dyadic partitions of unity. More precisely, if we are given a radial function χ belonging to D(B(0, 4/3)), identically equal to 1 in B(0, 3/4), we have the following identities

χ + j≥0 ϕ(2 -j •) = 1 in R d , and j∈Z ϕ(2 -j •) = 1 in R d \ {0} ,
where ϕ is the function defined by ϕ(ξ) = χ(ξ/2) -χ(ξ).

With this normalization ϕ is a radial function of D(C) where C is the annulus centered at the origin with inner radius 3/4 and outer radius 8/3 and we define the homogeneous dyadic blocks ∆j by 5

∆j f := ϕ(2 -j D)f := F -1 (ϕ(2 -j •)Ff ) = 2 jd h(2 j •) f with h = F -1 ϕ
and the inhomogeneous dyadic blocks ∆ j by ∆ j f :

= ∆j f = 2 jd h(2 j •) f if j ≥ 0 and ∆ -1 f := χ(D)f := F -1 (χFf ) = h f , where h = F -1 χ .
In a similar way, we also introduce the low-frequency cut-off operators

Ṡj f := k≤j-1 ∆k f := F -1 (χ(2 -j •)Ff ) = 2 jd h(2 j •) f for j ∈ Z and S j f := k≤j-1 ∆ k f = 2 jd h(2 j •) f for j ∈ N .
It is worth noticing that the dyadic blocks that are frequency cut-off operators are convolution operators. This property, which is a trivial consequence of the fact that the Fourier transform changes the convolution product to the pointwise product of functions, plays a central role in the techniques arising from Littlewood-Paley theory. In particular, all these operators act in the spaces L p in a uniform way with respect to p and j.

For what follows, it is also important to underline that the properties of the supports of the functions ϕ and χ give rise to quasi-orthogonal relations for the Littlewood-Paley decomposition, namely ∆j ∆k = 0 and

∆ j ∆ k = 0 if |j -k| > 1 ,
which easily implies that

(3) ∀ξ ∈ R d , 1 2 ≤ χ 2 (ξ) + j≥0 ϕ 2 (2 -j ξ) ≤ 1, and (
4) ∀ξ ∈ R d \ {0} , 1 2 ≤ j∈Z ϕ 2 (2 -j ξ) ≤ 1.
Littlewood-Paley analysis allows the exact characterization of the regularity of a function f in terms of the decay properties of its dyadic blocks with respect the summation 5. By F -1 we denote the inverse Fourier transform in R d and F(ϕ(2 -j D)f )(ξ) = ϕ(2 -j ξ) f (ξ) which shows that F( ∆jf ) is supported in the annulus 2 j C. index j. We thus recover, in a more precise way, the idea already present in Fourier analysis that space regularity is translated into frequency decay.

In particular, using the Fourier-Plancherel formula and the quasi-orthogonality properties (??)-(??), it is easy to observe that we can characterize a function f as an element of L 2 (R d ) in terms of the sequence ( ∆j f L 2 (R d ) ) j∈Z in 2 (Z), and the same for its inhomogeneous dyadic blocks. More precisely, thanks to an elementary Hilbertian analysis lemma, we can show the existence of a constant C such that we have

C -1 j∈Z ∆j f 2 L 2 (R d ) ≤ f 2 L 2 (R d ) ≤ C j∈Z ∆j f 2 L 2 (R d ) ,
and

C -1 j≥-1 ∆ j f 2 L 2 (R d ) ≤ f 2 L 2 (R d ) ≤ C j≥-1 ∆ j f 2 L 2 (R d ) .
Similarly, several classic norms can be written in terms of the Littlewood-Paley decomposition. This is, for example, the case of the Sobolev and Hölder norms. In particular the fact that some function belongs to some Sobolev (resp. Hölder) space is related with properties of decay with respect to j of the L 2 (resp. L ∞ ) norm of ∆j u or ∆ j u according to whether they are homogeneous or nonhomogeneous spaces.

Let us recall that the nonhomogeneous Sobolev spaces H s (R d ) that naturally show up in a large number of Mathematical Physics problems are, in the case when s = m ∈ N, the subspaces of functions f of L 2 (R d ) for which all derivatives (in the sense of distributions) of order smaller than or equal to m belong to L 2 (R d ). It is then clear, given the quasi-orthogonality of the Littlewood-Paley decomposition and the action of the Fourier transform on the derivatives, that the fact that a function is in

H m (R d ) is characterized as follows : f H m (R d ) ∼ (2 jm ∆ j f L 2 (R d ) ) 2 (j≥-1) .
A similar equivalence holds in the case of homogeneous Sobolev spaces Ḣm (R d ) which are more appropriate to study scale invariant problems such as the incomprehensible Navier-Stokes system 6 and several variants of this system in meteorology and oceanography, or nonlinear wave equations that we have studied in [?, ?, ?], and many other equations such as those dealt with in [?, ?].

In general, to say that a function f belongs to H s (R d ) means, roughly speaking, that f has s derivatives (fractional derivatives if s is noninteger) in L 2 (R d ), and, as before, we can prove that there exists a constant C such that

C -1 j≥-1 2 2js ∆ j f 2 L 2 (R d ) ≤ f 2 H s (R d ) ≤ C j≥-1 2 2js ∆ j f 2 L 2 (R d ) .
This heuristic idea can also be applied to the homogeneous Sobolev norms, giving rise to the following correspondence in the setting of the Littlewood-Paley theory :

C -1 j∈Z 2 2js ∆j f 2 L 2 (R d ) ≤ f 2 Ḣs (R d ) ≤ C j∈Z 2 2js ∆j f 2 L 2 (R d ) .
In examining these inequalities we observe that three parameters play a role : the regularity parameter s, the exponent of the Lebesgue norm used to measure the dyadic 6. Recall that for the incomprehensible Navier-Stokes system the question of eventual creation of singularities after a finite time is one of the Millenium problems proposed by the Clay Mathematics Institute.

blocks ∆j f or ∆ j f , and the type of sum preformed, either over Z or for j ≥ -1. This observation allows, more generally, to efficiently characterize the norms of homogeneous or nonhomogeneous Besov spaces, respectively Ḃs p,r (R d ) and B s p,r (R d ). The norms of these spaces, which can be defined in terms of finite differences or using the heat kernel (as we can see, for example, in [?, ?]) can be expressed in terms of Littlewood-Paley decompositions 7 :

f B s p,r (R d ) ∼ j≥-1 2 rjs ∆ j f r L p (R d ) 1 r , and
f Ḃs p,r (R d ) ∼ j∈Z 2 rjs ∆j f r L p (R d ) 1 r .
Even if scale invariant, the homogeneous Sobolev spaces (and more generally the homogeneous Besov spaces) have to be manipulated with care since, as was mentioned above, the homogeneous Littlewood-Paley decomposition (??) is only defined modulo polynomials of arbitrary degree. There is no consensus about the definition of these spaces. In certain references, such as [?], they are defined modulo polynomials of arbitrary degree. In others, such as [?], they are define subject to a condition on the low frequencies. This condition requires limiting oneself to tempered distributions f satisfying (in the sense of distributions

) Ṡj f L ∞ (R d ) j→-∞ -→ 0 .
The dyadic decompositions provide not only the possibility of characterizing a function as an element of almost all the classical spaces (Hölder, Sobolev, Besov, Lebesgue, Triebel-Lizorkin) by conditions concerning only its dyadic blocks, but they also allow to define a plethora of functional spaces.

Littlewood-Paley decompositions and more simply the decomposition of functions into low and high frequency components are techniques that have proved their usefulness in the study of functional inequalities and in the analysis of nonlinear partial differential equations.

Sobolev embeddings are among the most celebrated of all functional inequalities. They provide key tools for the study of linear and nonlinear partial differential equations, in the elliptic, parabolic, or hyperbolic framework. Sobolev inequalities express a strong integrability or regularity property for a function f in terms of integrability properties of some derivatives of f . Among those inequalities, we can mention the Sobolev inequalities in Lebesgue spaces :

(5) Ḣs (R d ) → L p (R d ) , with 0 ≤ s < d/2 and p = 2d/(d -2s).
Let us observe that the value p = 2d/(d-2s) can be easily deduced using an homogeneity argument. In fact, if for every function v defined in R d and all λ > 0 we define a function v λ by v λ (x) = v(λx), it is easy to verify that

v λ L p (R d ) = λ -d p and v λ Ḣs (R d ) = λ s-d 2 v λ Ḣs (R d ) .
7. Observe that the Besov spaces are independent of the dyadic blocks ∆j and ∆j.

Since both quantities • L p (R d ) and • Ḣs (R d ) have the same homogeneity degree when the Lebesgue index p = 2d/(d -2s) (which means that they behave in the same way under a change of the unit of length), it is thus natural to compare them and we can assume in what follows that f Ḣs (R d ) = 1.

We know that for all real number p ≥ 1 and all measurable function f , we have, due to Fubini's theorem,

f p L p (R d ) = p ∞ 0 λ p-1 µ |f | > λ dλ .
To establish the Sobolev embedding (??), we decompose f into low and high frequency components in the following way :

f = f ,A + f h,A with f ,A = F -1 (1 B(0,A) f ) .
Since the support of the Fourier transform of f ,A is a compact set, the function f ,A is bounded and, more precisely, by using the inversion formula and the Cauchy-Schwarz inequality, we have

f ,A L ∞ (R d ) ≤ (2π) -d f ,A L 1 (R d ) ≤ (2π) -d R d |ξ| s |ξ| -s | f ,A (ξ)| dξ ≤ C s A d 2 -s f Ḣs (R d ) .
Now, the triangle inequality implies, for all A > 0,

|f | > λ ⊂ |f ,A | > λ/2 ∪ |f h,A | > λ/2 •

Consequently, by choosing

A = A λ def = λ 4C s p d ,
we deduce that

f p L p (R d ) ≤ p ∞ 0 λ p-1 µ |f h,A λ | > λ/2 dλ .
Since, by the Bienaymé-Tchebychev inequality,

µ |f h,A λ | > λ/2 ≤ 4 f h,A λ 2 L 2 (R d ) λ 2 ,
we obtain

f p L p (R d ) ≤ 4p ∞ 0 λ p-3 f h,A λ 2 L 2 (R d ) dλ .
Finally, by the Fourier-Plancherel identity,

f h,A λ 2 L 2 (R d ) = (2π) -d (|ξ|≥A λ ) | f (ξ)| 2 dξ ,
which implies, due to Fubini's theorem, that for all p > 2

f p L p (R d ) ≤ 4p (2π) -d R d 4Cs|ξ| d p 0 λ p-3 dλ | f (ξ)| 2 dξ ≤ C p R d |ξ| d(p-2) p | f (ξ)| 2 dξ ,
where

C p = (2π) -d 4p p-2 (4C s ) p-2 . Since s = d 1 2 - 1 p
, this concludes the proof of the Sobolev embedding.

The proof presented above is borrowed from [?]. We have other previous proofs of this estimate, namely one based on the Hardy-Littlewood-Sobolev inequality, which is for instance presented in [?]. We should note that the arguments of the above proof have inspired a number of other works, among them we can point out the paper [?] where the authors considered Sobolev embeddings in the Lorentz spaces L p,q . Recall that the Lorentz spaces [START_REF] Bahouri | Lack of compactness in the 2D critical Sobolev embedding[END_REF] were introduced in the 1950s by Lorentz so that L p,∞ are the weak spaces introduced by Marcinkiewicz in the 1930s, and L p,p are the usual Lebesgue spaces L p . This technique of decomposition into low and high frequencies was also relevant for the study of nonlinear partial differential equations, namely to establish that some Cauchy problems are globally well posed. Among these works we can refer to the article of Fujita-Kato [?] on the Navier-Stokes equations. In this type of approach, the idea is to decompose the Cauchy data (assumed here, for simplicity, in some Sobolev space Ḣs ) into low and high frequencies in such a way that the high frequency part has rather small norm in Ḣs . If we have a global existence theorem for small initial data, then this high frequency part will give rise to a global solution to the problem, whereas the low frequency part (that will be regular) will satisfy a modified equation, and all we need to do is to prove that we can solve this perturbed equation.

The Sobolev embedding (??) is invariant by translation and scaling, but it is not invariant by oscillations, that is, by multiplication by oscillating functions, namely by those of the type u (x) = e i (x|ω) ϕ(x), where ω is a unit vector of R d , and ϕ is a function in S(R d ). Revisiting the proof of the Sobolev embedding presented above we can establish the following inequality due to Gérard-Meyer-Oru [?] :

(6) u L p (R d ) ≤ C (p -2) 1 p u 1-2 p Ḃs-d 2 ∞,∞ (R d ) u 2 p Ḣs (R d ) •
This Sobolev inequality is sharp, as the oscillatory example u (x) = e i (x|ω) ϕ(x) shows. Many other examples show the optimality of the estimate (??), in particular a fractal example constructed in [?], supported in a Cantor type set, and the example of the chirp signal :

f (x) = x -α sin 1 x , α > 0 , investigated in [?].
The refined estimate (??) is one of the key arguments in [?] where Patrick Gérard gave a characterization of the defect of compactness of the critical Sobolev embedding (??) by means of profile decompositions [START_REF] Bahouri | Lack of compactness in the 2D critical Sobolev embedding, the general case[END_REF] . We recall that the study of the defect of compactness of Sobolev embeddings of functional spaces, which goes back to the seminal works of Pierre-Louis Lions [?, ?], provides a useful tool in the study of geometric problems and the understanding of the behavior of solutions to nonlinear partial differential equations.

Nonlinear analysis progressed substantially in the last decades due to profile decomposition techniques. This type of decomposition has been generalized, by different approaches, to other functional settings. In particular, we can point out the recent works [?, ?] about the description of the defect of compactness of the critical Sobolev embedding of H 1 (R 2 ) in L(R 2 ), where L(R 2 ), the so called Orlicz space 10 , is the space of measurable functions u : R 2 → C for which there exists a real number λ > 0 such that

R 2 e |u(x)| 2 λ 2 -1 dx < ∞ ,
as well as its generalization to higher dimensions in [?]. This Sobolev embedding, which is based on the Trudinger-Moser inequalities, deal with the limiting case of the Sobolev embedding (??) and intervenes in numerous geometrical and physical problems, namely in the propagation of laser beams in different media. The study of this embedding is done in [?] by Fourier analysis arguments that highlight the fact that the elements responsible for the lack of compactness are, in this case and in contradistinction to the case of the Sobolev embedding (??), spread over the frequencies.

It is also noteworthy that an approach started by Stéphane Jaffard in [?] has allowed the extension of Patrick Gérard's result in [?] to the setting of the Triebel-Lizorkin spaces and has inspired the abstract analysis in [?]. This approach was based on the theory of wavelets, which, for its part, was inspired by the Littlewood-Paley, and will be discuss later.

As was referred to above, the second Bernstein inequality must be understood as a Sobolev embedding. In fact, it is easy to deduce from this second inequality that for all real numbers s, and for all 1

≤ p 1 ≤ p 2 ≤ ∞ and 1 ≤ r 1 ≤ r 2 ≤ ∞, we have (7) Ḃs p 1 ,r 1 (R d ) → Ḃs-d( 1 p 1 -1 p 2 ) p 2 ,r 2 (R d ) ,
and analogously for the nonhomogeneous case.

Observe that these Sobolev embeddings are strict, as is shown, in the particular case of the Sobolev embedding Ḣs (R d ) → Ḃs 2,∞ (R d ), by the following example based on the idea of lacunar series. Given a function χ of S(R d ) whose Fourier transform is supported in a small ball centered at 0 with radius 0 , and given a vector ω ∈ R d with Euclidean norm 3/2, we consider the sequence of functions (f n ) n∈N defined by

f n (x) = √ n j≥n 2 -js 1 j + 1 e i2 j (x|ω) χ(x).
It is easy to observe that

   ∆j f n = 0 if j ≤ n -1 and ( ∆j f n )(x) = √ n 2 -js j + 1 e i2 j (x|ω) χ(x) if j ≥ n .
By an elementary computation, we conclude that

f n 2 Ḣs (R d ) ∼ n j≥n 1 (j + 1) 2 ∼ 1 and f n Ḃs 2,∞ (R d ) 1 √ n ,
which clearly shows the strict inclusion of Ḣs (R d ) into Ḃs 2,∞ (R d ). The techniques arising from the Littlewood-Paley theory allow also the analysis of the product of two tempered distributions (if it exists) by means of J.-M. Bony's paradifferential calculus. It does so in the following way : given two tempered distributions u and v, we write

u = p ∆ p u and v = q ∆ q v .
Formally, if the product exists it is written as

uv = p,q ∆ p u ∆ q v .
The idea consists in decomposing the product uv into three parts : a first one with terms where the frequencies of u are large compared with those of v, a second one with terms where the frequencies of v are large compared with those of u, and a third one for which the frequencies of u and v have comparable sizes. This leads to the following definition, first introduced by Jean-Michel Bony in [?] : we write

uv = T u v + T v u + R(u, v) with T v def = p≤q-2 ∆ p u∆ q v = q S q-1 u∆ q v and R(u, v) def = |q-p|≤1 ∆ q u∆ p v .
This so called Jean-Michel Bony's decomposition is fundamental in the study of product laws as well as in the study of nonlinear partial differential equations. Clearly, it admits an homogeneous version. Let us recall that the bilinear operator T u v is called the paraproduct of v by u whereas the symmetric bilinear operator R(u, v) is called the remainder.

From the detailed study of the way the paraproduct and the remainder act on Sobolev, Hölder, and, more generally, Besov spaces, one can identify some principles :

-For two compactly supported distributions, the paraproduct is always defined, and the regularity of T u v is determined, mainly, by the regularity of v. -In the other hand, the remainder is not always defined, but when it is the regularities of u and v add up to determine its regularity.

Jean-Michel Bony's paradifferential calculus has proven to be very effective in the study of evolution equations, which describe the behaviour of a physical phenomenon dependent of time. This method's relevance will be illustrated by presenting a method of microlocal decomposition we have introduced in [?, ?], in collaboration with Jean-Yves Chemin (see also [?, ?]), for the study of quasilinear wave equations of the type

(E) ∂ 2 t u -∆u -∂(G(u)∂u) = Q(∇u, ∇u) (u, ∂ t u) |t=0 = (u 0 , u 1 ) with ∂(G∂u) = 1≤j,k≤d ∂ j (G j,k ∂ k u),
where Q is a quadratic form on R 1+d ; and G is a C ∞ function vanishing on 0, which, together with all its derivatives is bounded from R into the space of symmetric matrices on R d , and takes its values in a compact set K such that Id + K is included in the cone of symmetric positive definite matrices.

By the classical theory of strictly hyperbolic equations, [START_REF] Bergh | Interpolation Spaces, an Introduction[END_REF] we can solve such equation with Cauchy data (u 0 , u 1 ) in the space Ḣs (R d )× Ḣs-1 (R d ) for s > d 2 +1. Notwithstanding, it is important to think about the scale invariance of such equation. It is immediate to check that if u is a solution of equation (E), then the function u λ defined by u λ (t, x) = u(λt, λx) is also a solution of (E). A large number of works have been concerned to solving nonlinear wave equations, by trying to decrease as far as possible the index of minimal regularity of the initial data towards a space of initial data invariant by the above change of scale, for instance in the space Ḣ d 2 . The goal here is to solve equation (E) for less regular Cauchy data than what is required by energy methods. This approach fits in Christodoulou-Klainerman programme for general relativity, which also includes works by Klainerman, Bourgain, Tao and their schools. To get closer to scale invariant spaces for the initial data, it is obvious that we need to use the specific properties of the wave equation, namely the dispersion effects referred to above. This necessitates the proof of Strichartz type inequalities for that equation that we can interpret as a wave equation with variable and rough coefficients. It is the alliance of geometric optics and harmonic analysis through the paradifferential calculus of Jean-Michel Bony that allows to establish these estimates, to improve the minimal regularity index, and to give an answer to a longtime open question.

As stated above, Strichartz estimates are obtained from dispersive phenomena coupled with an abstract functional argument known as T T * -argument, developed by Ginibre and Velo in [?], and generalized by Keel and Tao in [?]. As also pointed out previously, dispersive phenomena are obtained for the wave equations with constant coefficients by applying a stationary phase argument on an explicit representation of the solution. The variable coefficients case needs more attention since in this case we do not have an explicit representation, and we recur to geometric optics methods involving Hamilton-Jacobi and transport equations to approximate the solution. When the coefficients are rough, as, for example, in the quasilinear case, such approach does not work since the Hamilton-Jacobi equation produces singularities. It is the Littlewood-Paley theory that allow us to overcome this difficulty.

In fact, to perform such method in this framework requires a regularization of the coefficients. More precisely, using Bony's paradifferential calculus, we are left with the study of the part of the solution related with frequencies of size 2 j which satisfies a wave equation with regular coefficients. By a classical method, we construct a microlocal approximation of the solution to this equation, that is valid in a time interval whose size depends on the frequency and that allows to establish a microlocal Strichartz estimate. In fact, it seems impossible to construct a local approximation of the solution since the associated Hamilton-Jacobi equation generates singularities at a time related to the frequency : this is due to the fact that these regular coefficients keep memory of the original regularity of the solution. The local Strichartz estimate is obtained (with some loss) by decomposing the interval [0, T ] into intervals where the microlocal Strichartz estimate is satisfied.

The applications of the Littlewood-Paley theory, and particularly of the paradifferential calculus, are manifold and we cannot enumerate all of them here. For a wider range of perspectives, wether in the study of functional inequalities or the analysis of solutions to nonlinear partial differential equations arising in fluid mechanics or general relativity, we refer the reader to the monograph [?]. The Littlewood-Paley theory has inspired the wavelet theory which is at the origin of numerous progresses in various applied disciplines, such as signal and image processing techniques. We can illustrate wavelet theory in a simple setting by considering Haar's system introduced at the beginning of the 1920s by Alfred Haar in his PhD thesis. This system is defined by the functions

ψ j,k (x) = 2 j 2 ψ(2 j x -k), j, k ∈ Z,
where the generating wavelet

ψ = χ [0, 1 2 [ -χ [ 1 2 ,1]
is the piecewise constant function equal to 1 in [0, 1 2 [ and -1 in [1/2, 1[. This system constitutes an orthonormal basis of L 2 (R) and, thus, it is straightforward that all functions f of L 2 (R) can be decomposed as follows :

(8) f = j,k∈Z f, ψ j,k ψ j,k ,
where f, ψ j,k denotes the scalar product of f and ψ j,k in L 2 (R). In the wavelet decomposition (??), the homogeneous dyadic blocs ∆j f are replaced by the projections

P j f = k∈Z f, ψ j,k ψ j,k ,
where the index k provides an additional level of discretization.

The main drawback of Haar's system is its lack of regularity, since the mother wavelet ψ is not continuous. Other, more regular, wavelet basis have been constructed alter on, which allow to get decompositions in wavelets similar to (??), often taking into consideration the scaling of the space in question.

As in the Littlewood-Paley decompositions, we can characterize the belonging of a function to almost all classical functional spaces by conditions pertaining only to the absolute values of the coefficients of the function in a basis of unconditional normalized wavelets [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] .

For example, in the Besov space Ḃs p,p (R d ), 1 ≤ p < ∞ and s < d p , the wavelet decomposition of a function takes the form :

(9) f = λ∈∇ d λ ψ λ ,
where λ = (j, k) includes the scale index j = j(λ) and the space index k = k(λ), and

ψ λ = ψ j,k = 2 jr ψ(2 j • -k), j ∈ Z, k ∈ Z d ,
where ψ is the mother wavelet, and r = d p -s • The wavelet theory allows to characterize the belonging to Ḃs p,p (R d ) in terms of the coefficients in the above wavelet decomposition as follows : (10) f Ḃs p,p (R d ) ∼ (d λ ) λ∈∇ p . The possibility of characterizing the regularity of a function by the size of its wavelet coefficients is at the heart of the extensive applications of wavelet theory. In particular, 12. For more details consult [?, ?, ?].

we can translate the equivalence (??) by the decrease of the wavelet coefficients, with the exception of a small number of them. This property of concentration of information in a small number of coefficients, often called parsimony or sparsity, plays a crucial role in image processing. In this type of essentially nonlinear process, it is clear that the set of remaining coefficients depend on the function we are approaching. A general theory for the study of these phenomena, known as nonlinear approximation theory, was started by Ronald DeVore in the 1980s.

A first result in nonlinear approximation theory is the representation of a function by its N most significant coefficients. More precisely, given an element f of Ḃs p,p (R d ) admitting a decomposition given by (??) in the wavelet basis (ψ λ ) λ∈∇ , the goal is to keep only the nonlinear projection Q N f defined by

Q N f = λ∈E N d λ ψ λ ,
where E N = E N (f ) is the subset of ∇ with cardinal N , which corresponds to the N largest wavelet coefficients |d λ |.

Among the many applications of the nonlinear projection Q N f , we can refer to the following estimate : (11) sup

f Ḃs p,p (R d ) ≤1 f -Q N f Ḃt q,q (R d ) ≤ CN -s-t d ,
that has played a key role in [?], in the study of the lack of compactness of the critical Sobolev embedding Ḃs

p,p (R d ) → Ḃt q,q (R d ) , with 0 < 1 p - 1 q = s -t d •
In fact, given a function f of Ḃs p,p (R d ) we obtain, from (??) and using (d m ) m>0 , the decreasing rearrangment of

|d λ | f -Q N f Ḃt q,q (R d ) ∼ λ / ∈E N |d λ | q 1/q = m>N |d m | q 1/q ≤ |d N | 1-p/q m>N |d m | p 1/q ≤ N -1 N m=1 |d m | p 1/p-1/q m>N |d m | p 1/q ≤ N -(1/p-1/q) m>0 |d m | p 1/p ≤ N -s-t d (d λ ) λ∈∇ p ∼ N -s-t d f Ḃs p,p (R d ) .
The success of wavelet theory either in signal and image processing, or in the field of numerical simulations of partial differential equations is now well established. For a general survey of applications of this theory, one can consult the monograph [?] and the references therein.

The Littlewood-Paley theory is considered the simplest tool of microlocal analysis. We can see microlocal analysis as the study of functions by the decomposition of the phase space, that is the space of (x, ξ). In a general way, this process consists in localizing in physical space x then in the Fourier variable ξ, which corresponds to the localization in a ball for a metric of T R d (the cotangent space of R d ) : it is the Weyl-Hörmander calculus [START_REF] Bony | Espaces fonctionnels associés au calcul de Weyl-Hörmander[END_REF] . The interest of this type of process, introduced in the 1970s, is to allow the analysis of fine properties of functions defined in the physical space by operating in the phase space, where the number of variables has doubled. This turned out to be particularly useful in the study of nonlinear partial differential equations namely, for instance, to take into consideration certain geometric specificities.

The whole issue of the Weyl-Hörmander calculus consists in the use of reasonable metrics (the so called Hörmander metrics) in order to localize in phase space. As an example, the procedure of localizing in the variable x in an Euclidean ball with size α, and afterward in the Fourier variable in a ball of radius α(1 + |ξ 0 | 2 ) 1 2 is equivalent to localize in a ball for the following metric, the so called (1, 0) metric :

g (x,ξ) (dx 2 , dξ 2 ) = dx 2 + dξ 2 1 + |ξ| 2 •
The so called Weyl-Hörmander calculus, which achieved its present day formalism at the end of the 1970s in the works of L. Hörmander, generalizes this metric. In fact, it consists in the description of reasonable ways to decompose the phase space. These decompositions are chosen according to the nature and the geometry of the problem under consideration. The admissible decompositions are those whose construction is based on Hörmander's metrics, which are functions g of T R d with its standard sympletic structure in the set of positive definite quadratic forms in T R d satisfying :

-a so called slowness assumption stating that the metric does not change much on its own balls, and this in a uniform way ; -an uncertainty principle hypothesis that prevents too much localization. In particular, the uncertainty principle imposes that the volume of a g X ball of radius 1 is larger than or equal to the volume of the Euclidean ball of radius 1 ; -and finally, a so called temperance hypothesis that reflects the fact that we can estimate the ratio of metrics in arbitrary points by the dual metric.

8 .

 8 For more details see [?, ?] . 9. Profile decompositions originate in the work of Brézis-Coron [?].

10 .

 10 For an introduction to Orlicz spaces see [?, ?].

  11. See, for instance, chapter 4 of [?].


T Kim

J.-P Devolder

Adam

R Soucaille
email: remy.soucaille@gmail.com

F Garcia-Sanchez

J.-V Kim

T Devolder

J.-P Adam

Nonreciprocal flexural dynamics of Dzyaloshinskii domain walls

Nonreciprocal flexural dynamics of Dzyaloshinskii domain walls

I. INTRODUCTION

The dynamics of ferromagnet domain walls encompasses a wide range of nonlinear phenomena. A prime example is Walker breakdown, which represents the transition between the steady state motion of a wall under applied fields or currents and the precessional regime in which the wall motion is oscillatory. Despite the complexity of the magnetic interactions and dynamics governing individual moments in a given material system, much of the salient features of domain wall motion can be captured by a one-dimensional model in which the wall center, u(t), and internal wall angle, φ(t), are the only variables that describe the dynamics [1,2]. For example, the magnetic field dependence of the wall velocity predicted by the model has been observed in in-plane magnetized wires [3].

Despite the utility of the one-dimensional model, a large number of experimental observations cannot be accounted for by this description. For example, magnetic disorder can prevent the Walker transition to be attained or even identified clearly [4], and leads to a low-field creep regime where thermally-activated processes are dominant [5]. Beyond the creep regime, other deviations from the one-dimensional picture have been observed in perpendicularly-magnetized ultrathin ferromagnets. Instabilities in the wall structure lead to plateaus in the velocity versus field curves [6,7], which are largely driven by incoherent magnetization precession at the wall center [8]. Due to this incoherent precession, Néel or Bloch lines are created within and move along these domain walls, leading to periodic annihilation events resulting in spin wave bursts [9]. A number of previous works has addressed such shortcomings in different ways. For the dynamics of vortex walls in in-plane magnetized systems, an extended model has been developed that also account for the internal dynamics of the vortex in addition to the usual wall variables [10,11]. This approach is based on the method of collective coordinates, which provides a framework to incorporate internal degrees of freedom (such as spin waves) into the dynamics of the underlying spin texture [12][13][14]. Other work have sought to account for flexural modes of the domain wall, which can be excited during propagation and can lead to clear deviations from the one-dimensional behavior [15]. Evidence of wall flexing has been obtained in low moment ferromagnets [16].

In ultrathin ferromagnets, the proximity of a strong spinorbit coupling material can give rise to an additional chiral interaction of the Dzyaloshinskii-Moriya form [17][18][19]. Besides favoring Néel-type domain walls at equilibrium [20][21][22][23][24][25], this interfacial Dzyaloshinskii-Moriya interaction (iDMI) results in the asymmetric nucleation [26] and growth of magnetic domains in perpendicularly-magnetized ferromagnetic films under in-plane applied fields, where domain wall propagation is affected in the creep, steady state, and precessional regimes [21,[27][28][29][30][31][32][33][34][35][36][37][38]. With respect to internal modes, it has also been shown that spin wave channeling by domain walls can acquire a nonreciprocal character [39,40], which is similar to behavior seen for magnetostatic spin wave modes induced by dipolar effects [41]. Some theoretical work has been undertaken to explore how the iDMI affects the wall motion in the creep regime [42], but there remain open questions on its role of the dynamics of flexural modes.

Here, we introduce an intermediate model between a full micromagnetic description of a domain wall and the 1D model [21]. We explicitly allow for a nonuniform propagation of the domain wall, where the spatial dependence of the wall center is taken into account. This model allows us to have a more complete description of the domain wall dynamics. We will focus on the bending motion of a straight domain wall with iDMI and in-plane field. To describe the dynamics of the domain wall, especially at long wavelength we model the evolution of the domain wall dynamics in system with perpendicular magnetic anisotropy and in the presence of iDMI. We also examine the effect of a small in-plane magnetic field and pinning potentials.

The article is organized as follows. In Section II, we compute the dynamics of the domain wall via a Lagrangian description of the magnetic texture. Our model is a direct extension of the 1D model where the magnetization is supposed rigid. In Section III, we compare the obtained result for the flexural motion with micromagnetic simulations. In Section IV, we discuss in more details the non-reciprocity of the flexural modes. Some concluding remarks are provided in Section V.

K u1 K u K u1 φ(y,t) H x x y z u(y,t)
FIG. 1. Geometry for the domain wall dynamics. The domain wall is set at the center (dotted line). An anisotropy well where K u < K u,1 is used to model the effect of the pinning . The two function φ (y, t) and u (y, t) parametrize respectively the in-plane angle and the out-ofplane component of the magnetization. The red line is a schematic of the domain wall position during an excitation.

II. DOMAIN WALL ENERGY AND DYNAMICS

We describe the energy and dynamics of a magnetic domain wall in an ultrathin film in this section. Within the micromagnetic description, the magnetization unit vector m = M/M s can be described by the two spherical angles θ and φ, m (θ, φ) = (sin θ cos φ, sin θ sin φ, cos θ) .

(

) 1
We consider a domain wall running along the y direction that separates two magnetic domains along the x direction. The magnetization within each domain is taken to point along the z direction, perpendicular to the film plane. A schematic illustration of this geometry is given in Fig. 1. We assume that the component of the magnetization perpendicular to the plane, parametrized by the angle polar angle θ, can be described by the usual profile,

θ(x, y, t) = 2 tan -1 exp x -u (y, t) ∆ , (2)
where u (y, t) represents the domain wall center displacement and ∆ = √ A/K eff is the domain wall width, which remains constant. A is the exchange constant and K eff is the perpendicular anisotropy constant. Both θ and φ are assumed to be uniform across the film thickness along z. We also assume that φ is uniform along the wall in the x direction, φ = φ (y, t) and that the domain wall remains close to a straight configuration (||∂u/∂y|| 1).

We now describe the different contributions to the magnetic energy densities with this ansatz. The effective perpendicular uniaxial anisotropy is given by

E anis = K eff d sin 2 θ dx = 1 2 σ 0 d, (3)
where d is the film thickness and σ 0 = 4 √ AK eff is the Bloch wall energy. This energy is independent of the displacement u and the angle φ. The exchange interaction energy density is given by

E ex = Ad (∇θ) 2 + sin 2 θ (∇φ) 2 dx, = 1 2 σ 0 d        1 + ∂u ∂y 2 + ∆ 2 ∂φ ∂y 2        . (4)
The (positive) squared derivative terms are a reminder that the exchange energy favors a uniform configuration along the transverse (i.e., y) direction. The sum of these two terms give the usual domain wall energy, σ 0 , with additional contributions to the wall elastic energy, proportional to (∂ y u) 2 and (∂ y φ) 2 , which arise from small deformations from the straight wall profile. In perpendicularly magnetized films described by these two magnetic energies only, domain walls are of the Bloch type which minimize volume dipolar charges. Deviations from this profile can appear when other interactions are present. First, proximity of a strong spin-orbit coupling material to the ferromagnetic film induces an iDMI, which can be described by [21,43]

E D = Dd (m z (∇ • m) -(m • ∇) m z ) dx, = -πDd cos φ - ∂u ∂y sin φ . (5)
One can rewrite this equation as a function of φ+tan -1 (∂u/∂y) and the length of the domain wall which shows that the iDMI effective field [START_REF] Thiaville | Spin dynamics in confined magnetic structures III[END_REF] is always normal to the domain wall. Second, applied magnetic fields can also modify the wall profile. The effect of a magnetic field H can be separated into two parts, an in-plane (//) and an out-of-plane (⊥) component.

The out-of-plane component results in domain wall displacement and contributes to the total energy density through the Zeeman interaction as,

E Z,⊥ = -2µ 0 M s H z ud, (6)
In contrast, the in-plane component leads to changes in the internal structure φ of the domain wall. In the limit of a small applied in-plane magnetic field, H 2 x + H 2 y H K , where H K = 2K eff /µ 0 M s is the effective anisotropy field, the associated Zeeman term is

E Z,// = -πµ 0 M s ∆d H x cos φ -H y sin φ . (7)
Note that as compared to the one dimensional model [START_REF] Thiaville | Spin dynamics in confined magnetic structures III[END_REF] we have an additional term proportional to ∂ y u in the iDMI energy, which is absent in the Zeeman energy. While part of the iDMI energy can still be assimilated to an effective magnetic field along the x direction, we note that the y component of the effective field is linked to gradients in the domain wall displacement u, ∂ y u. This result may provide a theoretical basis for asymmetric domain growth that has been observed experimentally in the presence of in-plane fields [27,29,32,34,42,[START_REF] Balk | [END_REF].

We approximate the dipolar interaction with a transverse anisotropy term, E ⊥ . We assume that the domain wall profile varies slowly compared to the domain wall width and consider flexural modes in the long wavelength limit, k∆ 1. Under this approximation, we can write

E ⊥ = 2K ⊥ ∆d cos φ + tan -1 ∂u ∂y 2 , (8)
where K ⊥ = (ln 2) µ 0 M 2 s d/π∆ accounts for the difference in dipolar energy between the Néel and Bloch wall profiles [46]. We note that the dipolar interaction, like the iDMI, leads to a coupling between the internal angle φ and deformations in the wall, ∂ y u. The dipolar interaction favors the Bloch profile, where the magnetization at the wall center is tangent to the domain wall, while the interfacial iDMI favors the Néel profile, where the magnetization at the wall center is normal to the domain wall. Finally, domain wall pinning due to material inhomogeneities need to be accounted for realistic systems. This can be introduced by assuming a quadratic potential well of the form,

E pin ≈ 1 2 κu 2 d, (9)
where κ characterizes the strength of the pinning potential.

For systems with uniform properties along the y axis, such as the line anisotropy defect shown in Fig. 1, we can assume that the equilibrium domain wall profile is also uniform along this direction. As such, φ(y) = φ 0 and all spatial derivatives in the domain wall position are vanishing, ∂u/∂y = 0. In this case, the equilibrium domain wall angle φ 0 can be found by minimizing the total energy. The equilibrium angle φ 0 can be determined by minimizing the total energy. In the case where the in-plane field is applied along the x direction, φ 0 is given by [46]

φ 0 =              π µ 0 πM s H x < πD/∆ -2K ⊥ 0 µ 0 πM s H x > πD/∆ + 2K ⊥ 2 tan -1 2K ⊥ +πD/∆+µ 0 πM s H x 2K ⊥ -πD/∆-µ 0 πM s H x otherwise (10)
Depending on the in-plane field the domain wall structure transforms from a right handed Néel wall to a left handed Néel passing through a mixed Néel/Bloch wall. We now discuss the dynamics of the domain wall as an elastic line using Lagrangian formalism to derive the equa-tions of motion of the domain wall. For spin dynamics, the Lagrangian density for the spherical angles θ and φ can be written as [START_REF] Kim | Solid State Physics[END_REF][START_REF] Braun | [END_REF],

L = M s d γ φ (1 -cos θ) -E θ, φ dx, (11)
where the first term on the right hand side is the Berry phase term and E (θ, φ) is the total energy density of the domain wall, which is the sum of the different contributions given in Eqs.

(3) to (8). By using the domain wall ansatz in Eq. ( 2), integrating over x and removing terms which are not relevant for the dynamics of the system, we find

L = -2 M s d γ φu -E u, φ , (12)
Gilbert damping can be accounted for through the dissipation function

W G = α M s d 2γ θ2 + φ2 sin 2 θ dx, (13)
which, with the same ansatz, leads to the density

W G = αM s ∆d γ u2 ∆ 2 + φ2 . (14)
The equations of motion correspond to the usual Euler-Lagrange equations,

d dt ∂L ∂(∂ t u) + d dy ∂L ∂(∂ y u) - ∂L ∂u + ∂W G ∂(∂ t u) = 0, ( 15
)
d dt ∂L ∂(∂ t φ) + d dy ∂L ∂(∂ y φ) - ∂L ∂φ + ∂W G ∂(∂ t φ) = 0. ( 16
)
Explicitly, this leads to the set of coupled nonlinear differential equations,

2M s γ ∂φ ∂t + M s α γ∆ ∂u ∂t -σ 0 ∂ 2 u ∂y 2 -πD ∂φ ∂y cos φ -2µ 0 M s πH z + κu- K ⊥ ∂χ ∂y sin 2 (χ + φ) sin 2χ -2 cos 2 (χ + φ) ∂φ ∂y + ∂χ ∂y cos 2 χ = 0, ( 17
)
- 2M s γ ∂u ∂t + M s α γ ∆ ∂φ ∂t -4A∆ ∂ 2 φ ∂y 2 + πD ∂u ∂y cos φ + sin φ -K ⊥ ∆ sin (2χ + 2φ) + µ 0 M s π∆ H x sin φ -H y cos φ = 0, ( 18
)
where χ ≡ tan -1 (∂u/∂y). Note that we recover the usual onedimensional domain wall model [21] when the spatial derivatives in u and φ are set to zero.

III. DISPERSION RELATION OF THE FLEXURAL MODE

In this section we focus on the the dispersion relation for the flexural dynamics of the domain wall derived from the Lagrangian formalism in the previous section. We limit our study to the case where the slope of the wall is small, ∂u/∂y

1, for small wavelengths, k∆ 1, and for small applied in-plane fields along the normal to the domain wall, H // = H x H K eff . A first order series expansion on the set of two coupled differential equations is done over u (y, t) and φ (y, t) around φ = φ 0 and u = 0 where φ 0 is the equilibrium angle of the magnetization. We then look for the propagating solutions of theses equations to get the dispersion relation ω (k), which is found to be

ω (k) = Ω NR + Ω u Ω φ , (19)
where Ω NR is linear with respect to the wave vector and is responsible for non-reciprocal propagation,

Ω NR = ω D,k cos φ 0 -k∆ω ⊥ cos 2φ 0 . ( 20
)
Ω u describes the stiffness of the domain wall position,

Ω u = ω k + ω pin -(k∆) 2 ω ⊥ cos 2φ 0 , (21)
and Ω φ describes the stiffness of the angle φ of the magnetization,

Ω φ = ω k + ω D,k k∆ cos φ 0 + ω H,x cos φ 0 -ω ⊥ cos 2φ 0 . ( 22
)
The angular frequencies in Eqs. ( 20) to ( 22) are given by [40]

ω k = 2 γA M s k 2 , (23)
ω D,k = π 2 γD M s k, (24)
ω ⊥ = γK ⊥ M s , (25)
ω H,x = π 2 γµ 0 H x , (26)
ω pin = 1 2 γ∆ M s κ. (27)
We note that this description provides a more accurate treatment of the dipolar interaction at long wavelengths in comparison to previous work [40,49,50], since we take into account fluctuations in the wall position through the term ∂ y u. The variation of the flexural mode frequency with applied in-plane field, H x , and the wave vector, k, is presented in Fig. 2. The dispersion relation allows us to identify instabilities in the straight domain wall configuration assumed as the equilibrium profile. Instabilities occur when the mode frequency vanish, which typically corresponds to a change in the equilibrium state. We note that the uniform mode, k = 0, which corresponds to the uniform displacement of the domain wall, is not necessarily the eigenmode with the lowest energy. This can be seen in Fig. (2) where there are two field intervals over which the straight domain wall is unstable. Similarly to the case where the iDMI is strong enough to create maze domain patterns [40], this instability occurs for one propagation direction of the flexural model.

Unlike this case, here the domain wall energy is still positive and the sign of wave vector which leads to negative frequency depends on the chirality of the domain wall and on D. This instability is shown in Figure 2, where the frequency reaches negative values for wave vectors between 0 and 10 µm -1 at around -50 mT and for -5 µm -1 < k < -1 µm -1 at -20 mT. For an applied in-plane field value close to the field value corresponding to the iDMI strength a straight wall is stable. This result is surprising since one expects faceting of the wall when the in-plane field compensates the iDMI field [37,42]. We do not find such an instability here. This is linked to the initial state that we have considered, where φ 0 is constant in the initial configuration whereas for a curved domain wall φ 0 can vary along the domain wall. The two side of facets has opposite slope (∂u/∂y) and opposite angle φ. To create facets the angle φ needs to go continuously from positive to negative value. For this, the domain wall needs to overcome an energy barrier which is linked to the dipolar interaction: a straight domain wall configuration is a metastable state. The dipolar energy has two opposite effects, it broadens the region where a straight wall is metastable and it tends to increase the stability of the domain wall in this region by increasing the energy barrier. For example, if we do not consider the dipolar interaction (Eq. 8) the two regions of instability in Fig. 2 merge into one region which is centered close to the iDMI effective field.

We compared the analytical model with full micromagnetic simulations using Mumax3 [51]. The simulation consists of a domain wall pinned in an anisotropy well where the uniaxial anisotropy is increased by 10% outside the well. The width of the anisotropy well is taken to be 80 nm. The dynamics are induced by a pulsed magnetic field, with a time dependence given by a sinc function, along the z direction localized at the center of the frame. The simulation geometry is given by a rectangular window with dimensions of 8 µm by 125 nm, which is discretized with 2048 × 64 finite difference cells. The layer thickness is 1 nm and periodic boundary conditions are applied along the y direction (along the domain wall). The saturation magnetization is taken to be M s = 788 kA/m, the exchange stiffness A = 22.5 pJ/m, the uniaxial anisotropy K 0 = 641.5 kJ/m 3 , the iDMI D = 0.28 mJ/m 2 and the Gilbert damping constant α = 0.015. These values represent the material system W(3 nm)/CoFeB(1 nm)/MgO, as discussed in Ref. 35.

A comparison is given in Fig. 3 for the finite frequency gap for the uniform k = 0 mode, which appears due to the uniform pinning potential as illustrated in Fig. 1. An expression for this gap is given by

ω gap = ω pin ω D,k k∆ cos φ 0 -ω ⊥ cos 2φ 0 . (28)
Besides setting the pinning potential to zero (i.e. K u = K u1 ), the frequency gap can also vanish when the applied in-plane field leads to changes in the domain wall profile. This can be seen in Fig. 3 at around µ 0 H x = -20 mT and -50 mT, where transitions toward the two different chiral Néel domain wall states occur. The figure shows that good agreement between the analytical model and the micromagnetics simulations is obtained, both for the critical fields and the gap frequencies.

In the micromagnetic simulations pinning also occurs due to the presence of edges close to the domain wall. In systems with iDMI the boundary conditions are modified [39,52] which can contribute to pinning. In order to estimate the anisotropy well contribution to the pinning potential, we analyze the wall displacement with an out-of-plane field. We focus on the quadratic pinning potential, Eq. ( 9), and the Zeeman interaction, Eq. ( 6). We minimize these two energy terms. The pinning potential is then related to the out-of-plane field and the wall position by, The pinning potential is then obtained by a linear fit to the positive field branch, as shown in the inset of Figure 4. The pinning exhibits a variation that is similar to the domain wall width [46]. When the applied field is opposed to the wall chirality the wall width is slightly reduced [46] in our simulation, as the width of the anisotropy well is larger than the wall width the pinning is reduced. From the pinning strength the resonant frequency can be estimated.

κ = 2µ 0 M s H z u . (29)
In order to compute the full dispersion relation we extract the domain wall position, u (y, t), and the angle of the magnetization at the domain wall center, φ (y, t). Linear extrapolation is used to obtain values of the displacement smaller than the cell size. The dispersion relation is then computed from a two-dimensional Fourier transform of the complex-valued function u + i∆φ, as shown in Fig. 5. The remaining branch in the frequency wave vector space leads to negative frequency when instabilities occur. At this applied field the energy difference between a straight domain wall and the relaxed state is small. In this section we have seen that an in-plane field allows to modify the resonant frequency. To properly describe the dynamic of flexural motion of the domain wall in domain walls at relatively low frequencies, i.e. in the sub-GHz range, we have to consider the effect of pinning in the flexural modes.

IV. NONRECIPROCITY OF THE FLEXURAL MODE

In this section we discuss the nonreciprocal behavior of the flexural mode. In the absence of dipolar interactions [40], the frequency difference between two counterpropagating spin waves is proportional to the iDMI constant d. However, as seen in Eq. (20), dipolar interactions also lead to an additional nonreciprocity. While the contribution from the iDMI is periodic with respect to φ, the nonreciprocity related to the dipolar interaction is π-periodic [see Eq. ( 19)] and vanishes when FIG. 6. Illustration of the magnetization state for two opposite propagation directions. The "+" and "-" represent the excess of virtual magnetic charges which appear for k < 0. φ = π/4. This is related to the fact that the dipolar contribution does not depend on the domain wall chirality but rather on the presence of volume dipolar charges (i.e., whether the wall is of the Bloch or Néel type), whereas the iDMI is sensitive to the domain wall chirality. If we consider a Bloch wall and only the in-plane component of the magnetization, the two counterpropagating waves lead to two different configurations as represented in Figure 6.

In one case the in-plane magnetization is tangent to the domain wall position, k > 0, while this is not the case for the opposite direction, k < 0. This leads to a difference in the dipolar interaction for this two cases; when the magnetization is tangent to the domain wall the dipolar energy is reduced. To describe accurately this nonreciprocity we cannot neglect the dipolar interaction if the iDMI is small. In this case the dipolar energy and the iDMI have a similar magnitude,

π 2 D ∼ 2 ln 2µ 0 M 2 s d. (30)
For k∆ 1, the function ∆ω = ω(k) -ω(-k) is linear with respect to the wave vector due to both the dipolar and the iDMI contributions. As seen in Figure 7 the non-reciprocity does not follow the same linear trend for larger wave vectors. For k∆ 1, the dipolar contribution to the nonreciprocity saturates and the slope of the function ∆ω is dominated by the contribution from the iDMI. In order to have a closer look to the contributions of the different energy terms to the nonreciprocity we have considered the group velocity defined by v g (k) = ∂ω/∂k as a function to the in-plane magnetic field. To match our assumption, we consider only small wave vector for the group velocity and we finally plot the mean group velocity for two opposite direction (i.e. (v g (0 + ) + v g (0 -))/2) in Figure 8 versus the applied inplane field. The two components of the nonreciprocity shown (dotted lines) depends on the in-plane field H x , which provide a means to modify the nonreciprocity of the domain wall flexural modes. Micromagnetic simulations are in good agreement with the model in the long wavelength limit. As the non-reciprocity is mainly related to the domain wall structure, as soon as the domain wall saturates in one configuration a change in the applied in-plane field does not longer modify the frequency non-reciprocity. Our analytical model allows an accurate description of the non-reciprocity in this case. This long wavelength regime might be probed in Brillouin light scattering experiments [53] where the wavelength probed is in the order of few hundred nanometers.

V. CONCLUSION

We have developed an analytical model for domain wall dynamics beyond the one-dimensional model by allowing for inhomogeneous displacements of the domain wall under the influence of a magnetic field. In our model we have supposed that the domain wall structure remains rigid while the domain wall profile is allowed to bend. This description of a flexible domain wall allows for a quantitative analytical description of the flexural mode propagation in the domain wall at long wavelengths. We show that both the interfacial Dzyaloshinskii-Moriya and dipolar interactions contribute to the nonreciprocity of the flexural mode. We have studied the influence of an in-plane field on the domain wall dynamics, which allows the band gap induced by domain wall pinning to be tuned. This field can also induce instabilities in the flexural mode propagation, which take place at the Bloch-Néel transition. This change in the internal wall structure also modifies the direction and the amplitude of the nonreciprocity of the flexural modes.

FIG. 2 .

 2 FIG. 2. Dispersion relation of the flexural motion of the domain wallwith respect to the in-plane field µ 0 H x normal to the domain wall and the wave vector k y which propagates along the domain wall [from Eq. (19)]. The wall is pinned in an anisotropy well with a width of 80 nm. Contour lines are separated by 100 MHz and the black areas close to -20 mT and -50 mT represent the (µ 0 H x , k y ) space range in which the frequencies are negative, indicating straight domain wall instabilities.

FIG. 3 .

 3 FIG. 3. Influence of the in-plane field on the frequency gap at k = 0. The solid line is derived from the model and circles are result of micromagnetic simulations. The vertical lines show the different internal angles φ of the domain wall at equilibrium.

2 FIG. 4 .

 24 FIG. 4. Variation of the pinning constant κ as a function of the inplane applied field. The vertical lines delimit the different domain wall configurations for the angle φ. The inset shows one example of the fitting between the out-of-plane field with respect to the domain wall position. We deduce the position pinning from this linear fit and with the equation 29.

FIG. 5 .

 5 FIG.5. Detail of the dispersion relation for an applied field µ 0 H x = -10 mT. It corresponds to the 2D Fourier transformation of the function u + i∆φ with respect to the time and the "y" variable. The blue dotted line corresponds to the dispersion relation given by Eq. (19).

FIG. 7 .

 7 FIG.7. Evolution of the non-reciprocity with respect to the wave vector with applied in-plane field. Circle are results from micromagnetic simulation, the red line corresponds to the approximation where the demagnetizing field is taken locally.

FIG. 8 .

 8 FIG.8. Evolution of the group velocity for small wavevector with respect to in-plane field µ 0 H x along the x axis. Circle are results from micromagnetic simulation, the black line correspond to the model. The two dotted lines correspond to the iDMI and to the dipolar field contribution to the wall excitation non-reciprocity.

ACKNOWLEDGMENTS

The authors acknowledge fruitful discussions with Yves Henry and Matthieu Bailleul. This work was partially supported by Agence National de la Recherche (France) under contract No. ANR-16-CE24-0027 (SWANGATE).