Nishant Kumar
email: nishant.kumar@cnrs.pprime.fr

Franck Kerhervé

Laurent Cordier

Laurent Cordier Data

Data Assimilation applied to a numerical 2D cylinder wake flow

Keywords: data assimilation, Dual Ensemble Kalman Filter, POD Reduced-Order Model, cylinder wake flow

Introduction

Despite decades of intensive research in shape optimization, aerodynamic mechanisms such as separation and mixing still represent an important source of energy expenditure in transport vehicles. The manufacturers have consequently developed, over the years, a range of strategies to improve the aerodynamic performance of their vehicles. One of these strategies consists in using active devices that can be actuated on and off based on the change of the inflow conditions. To drive such actuators, command laws and more globally valid controllers are required. To design these controllers, different control strategies can be employed, one of which being the model-based approach. This one requires the knowledge of a dynamical model of the flow that one wants to control in real-time. In reality, the identification of such a model is challenging because it is generally obtained from limited spatial and time discrete measurements.

This problem of state estimation is well-known in the field of meteorology where data is collected at several locations, with spatial and temporal scales varying in orders of magnitude, to estimate the evolution of the weather [START_REF] Talagrand | Variational Assimilation of Meteorological Observations with the Adjoint Vorticity Equation. I: Theory[END_REF]. Classical methodologies involve the use of data assimilation (DA) techniques where data is assimilated, when available, with a dynamical model to correct the state estimate obtained by the model, and eventually, to correct the model itself for further, more accurate predictions. In the specific context of aerodynamic-related problems, the complexity resides essentially in the large number of degrees of freedom of the flow dynamics, in the broad range of scales to examine and in non-linearities driving the flow dynamics. To design a physical model that can be managed in real-time applications, the Navier-Stokes equations are traditionally projected onto an appropriate reduced basis leading to a reduced-order model of the flow [START_REF] Rowley | Model Reduction for Flow Analysis and Control[END_REF]. Proper Orthogonal Decomposition (POD, [START_REF] Lumley | Atmospheric Turbulence and Wave Propagation. the structure of inhomogeneous turbulence[END_REF]) is typically used to design the orthonormal basis which captures the flow features which are believed essential to represent the dynamics. In general, this basis is then used in a Galerkin approach to derive a POD Reduced-Order Model (POD ROM) of the flow by projecting the Navier-Stokes equations onto the POD modes. Unfortunately, this dynamical system is sometimes not sufficiently accurate for application in flow control, and identification methods are then used to improve the prediction ability of the POD ROM [START_REF] Cordier | Calibration of POD Reduced-Order Models using Tikhonov Regularization[END_REF]). An alternative is to use the ROM in conjunction with DA techniques to design dynamic observers able to predict the flow state from limited information [START_REF] Papadakis | Assimilation de données images : application au suivi de courbes et de champs de vecteurs[END_REF].

In this paper, the stochastic data assimilation method known as Dual Ensemble Kalman Filter (Dual EnKF) is used on a numerical dataset of a 2D-cylinder wake flow considered at low Reynolds number (Re = 100). In the DA setting, the POD temporal modes serve as observations for the assimilation procedure which is used to test the predictability of the DA method. In § 2, an overview of snapshot POD ( § 2.1), POD ROM ( § 2.2), and Dual EnKF ( § 2.3) is first given. POD Reduced-Order model is then applied to the cylinder wake flow in § 3. The use of data assimilation to predict the temporal POD coefficients for the numerical flow is discussed in § 4. Section 5 offers concluding remarks.

Model reduction and ensemble Kalman filter

Snapshot POD

Proper Orthogonal Decomposition is a reduced-order model approach which was introduced to the fluid dynamics community as a mathematical tool to extract coherent structures from turbulent flow fields [START_REF] Lumley | Atmospheric Turbulence and Wave Propagation. the structure of inhomogeneous turbulence[END_REF]. It provides an objective algorithm to decompose a set of data into a minimal number of basis functions or modes to capture as much as possible the original dynamics. It is a data-driven method and therefore requires only the flow field data for modal decomposition and does not necessarily require the knowledge of the dynamical equations. POD extracts the modes based on optimizing the mean square of the field variable under observation [START_REF] Cordier | Model Reduction, POD and Data Assimilation[END_REF].

Let χ be the spatial coordinates and t represents the scalar time, we assume that the unsteady component of the vector field u(χ, t) can be decomposed as

u(χ, t) -u(χ) = i a POD i (t)φ POD i (χ), ( 1
)
where u is the temporal mean of u. In (1), φ POD i (χ) and a POD i (t) represent the spatial modes and the time-dependent projection coefficients, respectively. In the POD framework, we seek the optimal set of basis functions for a given flow field data.

The POD method requires an ensemble of snapshots of any scalar (e.g. pressure, temperature) or vectorial (e.g. velocity, vorticity) fields defined at discrete spatial points χ

i (i = 1, • • • , N χ ) and discrete time instants t k (k = 1, • • • , N t ).
From this data set, we prepare snapshots of the flow field stacked in terms of a collection of column vectors u (χ i , t k ). We then consider a collection of finite-dimensional data vectors that represent the flow field fluctuation

u (χ i , t k ) = u(χ i , t k ) -u(χ i ) ∈ R Ns , i = 1, 2, . . . , N χ , k = 1, 2, . . . , N t (2)
where N s is the number of the spatial degrees of freedom of the data and N t is the number of snapshots.

For a fluid flow data, N s is equal to the number of grid points (N χ ) times the number of components (N c ) to be considered in the data. The snapshot vector can be stacked in matrix form as

X = u (χ i , t 1 ) u (χ i , t 2 ) . . . u (χ i , t Nt ) ∈ R Ns×Nt . (3)
To obtain the POD modes, we employ the method of snapshots. The objective is to find the optimal basis vectors that can best represent the flow data in an average sense. In other words, we seek the vectors φ POD i (χ) in (1) that can represent u (χ i , t k ) in an optimal manner and with the least number of modes. The solution to this problem can be determined by finding the eigenvectors ψ i and eigenvalues λ i from

X T Xψ i = λ i ψ i , ψ i ∈ R Nt (4)
where X T X is the temporal correlation matrix1 of size N t × N t .

In most numerical cases, spatial size of the data is very large compared to the temporal size, N s N t . The method of snapshots thus uses a smaller correlation matrix than the spatial correlation matrix XX T of size N s × N s used in classical spatial POD method. It has been shown in literature that the temporal correlation matrix will yield the same dominant spatial modes while giving rise to a much smaller and computationally more tractable eigenvalue problem [START_REF] Rowley | Model Reduction for Flow Analysis and Control[END_REF]. The number of snapshots N t in this case should be chosen such that the important fluctuations in the flow field are well resolved in time.

After obtaining the eigenvectors ψ i from the smaller eigenvalue problem (4), the (spatial) POD modes can be recovered as

φ i = 1 √ λ i Xψ i ∈ R Ns , i = 1, 2, . . . , N t , (5)
which can be also written as

Φ = XΨΛ -1/2 , (6)
where

Φ = [φ 1 φ 2 . . . φ Nt ] ∈ R Ns×Nt and Ψ = [ψ 1 ψ 2 . . . ψ Nt ] ∈ R Nt×Nt .
Note that these matrices are orthonormal, i.e. Φ T Φ = ΦΦ T = I and Ψ T Ψ = ΨΨ T = I. The temporal coefficients are determined accordingly as

a i (t) = u (χ, t), φ i (χ) , i = 1, 2, . . . , N t , (7)
where • represents the spatial inner product defined as

v I (χ, t), v II (χ, t) = Ω v I (χ, t) • v II (χ, t) dχ (8)
for two given vector fields v I and v II defined in a spatial domain Ω.

POD Reduced-Order Model (POD ROM)

A POD ROM may be derived by the standard Galerkin projection of the Navier-Stokes equations onto the spatial POD modes φ i . After some algebraic manipulation [START_REF] Cordier | Calibration of POD Reduced-Order Models using Tikhonov Regularization[END_REF], the following expression is obtained for the POD ROM

ȧROM i (t) = da ROM i (t) dt = C i + N Gal j=1 L ij a ROM j (t) + N Gal j=1 N Gal k=j Q ijk a ROM j (t)a ROM k (t), (9)
where the superscript ROM is used to separately identify the temporal coefficients obtained from the POD ROM as compared to the ones (a POD i (t)) which are directly obtained from snapshot POD. The parameters C i , L ij and Q ijk are the constant, linear and quadratic coefficients which depend exclusively on the spatial POD modes φ i and mean field u. As such, their values can be directly determined (see [START_REF] Cordier | Calibration of POD Reduced-Order Models using Tikhonov Regularization[END_REF]. For the initial conditions, we have

a ROM i (0) = a POD i (0) = u(χ, 0) -u(χ), φ i (χ) . ( 10
)
The number of model parameters

C i , L ij and Q ijk is given as N b = N 0 + N 1 + N 2 , where -N 0 = 1 for the constant terms C i ,
-N 1 = N Gal for the linear terms L ij , and

-N 2 = N Gal (N Gal + 1)/2 for the quadratic terms Q ijk .
From the knowledge of N t samples of the quantities ȧROM i (t k ) and a ROM i (t k ) at discrete times t k , the dynamical parameters can be determined by a least mean-square estimation procedure. This problem leads to the minimisation of an error function ε = Aθ -B 2 [START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF], with

-A ∈ R Nt×N b as the matrix of the terms 1, a ROM j (t k ) and a ROM j (t k )a ROM k (t k ), -θ ∈ R N b ×1
as the vector of unknown parameters, and

-B ∈ R Nt×1 containing the N t values of ȧROM i (t k ).
The estimation problem is solved by forming a linear system (A T A)θ = A T B and using SVD of the matrix A to find the pseudo-inverse of A T A and solve for the parameters in θ.

However, it is well known that for various reasons (structural instability of the Galerkin projection, truncation of the POD basis, inaccurate treatment of the boundary and pressure terms), the dynamical system does not represent sufficiently well the correct dynamics. This problem is now tackled by data assimilation.

Dual Ensemble Kalman Filter (Dual EnKF)

As the POD ROM determined by Galerkin projection is not sufficient in general to replicate the correct dynamics, we seek data assimilation methods to correctly identify the model parameters. In the following, we briefly present the specific DA method known as Dual Ensemble Kalman Filter which has been used in the current study and then discuss its applicability to the POD ROM.

Data assimilation is a generic methodology which combines heterogeneous observations with the underlying dynamical principles governing the system under consideration to estimate at best the states and/or physical quantities parameterizing the dynamics. Starting from a background solution and incoming imperfect information, an optimal estimation of the unknowns of the system is determined which takes into account the respective statistical confidences of the different observations. We consider a non-linear, time-discrete dynamic system given as

x t k+1 = M k,k+1 (x t k , θ k ) + η k , (11)
where the superscript t stands for true. In ( 11), k = 0, . . . , N -1 is the time index and N is the total number of time steps. In this system, the propagation of the true state x t k ∈ R Ns (where N s is the dimension of the system of the state after spatial discretisation) from time t k to t k+1 is performed by a non-linear function M k,k+1 . The propagator depends on the vector of model parameters θ k at time t k . The model error η k is considered to be Gaussian distributed with zero mean and covariance matrix

Q k i.e. η k ∼ N (0, Q k ). ( 12
)
The observation vector y o k+1 ∈ R No (where N o is the dimension of the observations) at time t k is given by

y o k+1 = H k+1 (x t k+1 ) + k+1 , (13)
where the observation function H k+1 is also non-linear. The observation noise vector k+1 is constructed by taking all the sources of observation error into consideration. It is approximated as a Gaussian distribution with zero mean and covariance R k+1 i.e.

k+1 ∼ N (0, R k+1 ). ( 14
)
If the model parameters are known to be true, a DA method is specifically used for the state estimation.

On the other hand, in practice, we frequently encounter cases where the parameters are unknown or imprecise.

One of the approaches to solve this dual estimation problem is to use an iterative method where the filtering is alternatively applied to estimate the state and parameters in the course of each iteration.

More precisely, the model parameters are obtained from the corrected state at the previous time state and the new state is evaluated from the current analysed parameters. This method is formally known as Dual EnKF [START_REF] Moradkhani | Dual state-parameter estimation of hydrological models using ensemble Kalman filter[END_REF].

The mechanism of evolution of the Kalman filter is based on a prediction-correction scheme. First, an ensemble of N e forecasted parameters is created using a forced random walk of the parameters set up according to a Gaussian law ξ k with zero mean and covariance matrix C k i.e.

ξ k ∼ N (0, C k ). ( 15
)
During the prediction step, the ensemble of forecasted parameters is constructed as

θ f,(n) k+1 = θ a,(n) k + ξ (n) k , (16)
where the superscript (n) represents the n-th member of the ensemble of parameters. In ( 16), the su-perscripts f and a denote the forecasted and assimilated values of the parameters, respectively. The predicted ensemble of parameters are then updated using the available observations. For this, the predicted state at the instant t k+1 is built starting from the predicted parameters at the instant t k+1 and the corrected state at the instant t k i.e.

x f,(n) k+1 = M k,k+1 (x a,(n) k , θ f,(n) k+1 ). (17)
The predicted ensemble of observations is then obtained as

y f,(n) k+1 = H k+1 (x f,(n) k+1 ). (18)
Next, the random observations are built to be centered around the actual observation y o k+1 following

y o,(n) k+1 = y o k+1 + o,(n) k+1 , (19)
where the covariance matrix of the observation error at the instant t k+1 is given as

R e k+1 = 1 N e -1 Ne n=1 ( o,(n) k+1 )( o,(n) k+1 ) T . ( 20
)
As the number of Monte Carlo elements tends to infinity, the empirical estimator R e k+1 tends to the covariance matrix of the observation error R k+1 .

Given the random observations from ( 19) and the empirical estimator (20), the update of the predicted ensemble of parameters is realised by the Kalman analysis, given as

θ a,(n) k+1 = θ f,(n) k+1 + K θ,e k+1 (y o,(n) k+1 -y f,(n) k+1 ). (21)
The Kalman gain allowing the correction of the parameter trajectories is expressed using the ensemble predictions as K θ,e k+1 = P f,e θy,k+1 (P f,e yy,k+1 + R e k+1 ) -1 .

In this expression, the terms P f,e θy,k+1 and P f,e yy,k+1 designate at the instant t k+1 the unbiased empirical estimators of the cross-correlation matrix between the parameters and the observations, and the correlations between the observations, respectively.

Once the predicted parameters have been updated, the second filter is implemented to estimate the state of the system at the instant t k+1 by assuming the parameters as known. In other words, the parameters just corrected by the first filter are trusted. The following prediction-correction mechanism is identical to the classical ensemble Kalman filter. In the first step, the predicted ensemble of states at the instant t k+1 is built starting from the ensemble of assimilated states at the instant t k and the corrected ensemble of parameters at the instant t k+1 i.e.

x f,(n) k+1 = M k,k+1 (x a,(n) k , θ a,(n) k+1 ) + η (n) k . ( 23
)
For the second time, an ensemble of observations is calculated as

y f,(n) k+1 = H k+1 (x f,(n) k+1 ). ( 24
)
At this stage, the observations are again considered as random variables and the covariance matrix of the observation error involved in calculating the Kalman gain is also empirically estimated. Next, the update of the predicted ensemble of the state is realised by Kalman analysis, given as

x a,(n) k+1 = x f,(n) k+1 + K e k+1 (y o,(n) k+1 -y f,(n) k+1 ). ( 25
)
The use of a Kalman gain permits the correction of the state trajectories as K e k+1 = P f,e xy,k+1 (P f,e yy,k+1 + R e k+1 ) -1 .

(26)

In this expression, P f,e xy,k+1 and P f,e yy,k+1 designate at the instant t k+1 the unbiased empirical estimators of the cross-correlation matrix between the state and the observations, and the correlation matrix between the observations, respectively. The prediction-correction step is repeated by using the analysed state at t k+1 from (25) and new parameter forecast at t k+2 from ( 16) to build the new state forecast in (17). The phase of prediction-correction of the system state can precede the operation on the parameters without consequence on the quality of the estimation. Now, we can return to the POD ROM and present the use of the Dual EnKF method to correctly identify the model parameters that will help to replicate the correct dynamics. In the DA problem, the temporal coefficients a ROM i (t k ) can be considered to form the state vector x k . The POD ROM in (10) serves as the dynamical model M k,k+1 and the POD coefficients a POD i (t k ) form the observations vector y o k . Finally, the coefficients C i , L ij and Q ijk form the parameters θ. The initial background conditions x b 0 and θ b are determined by using the temporal coefficients directly obtained from POD at the initial condition, and from the parameter values identified in § 2.2 for the POD ROM.

POD ROM of a 2D-cylinder wake flow

In order to test the data assimilation framework in a simple numerical configuration, we choose in this paper to estimate the dynamics of a 2D-cylinder wake flow. The snapshots necessary to determine the POD modes (see § 2.1) are obtained from a numerical simulation performed using the finite-element solver FreeFem++ [START_REF] Hecht | New development in FreeFem++[END_REF]. A POD reduced-order model is then built using the procedure described in § 2.2. This POD ROM serves as the dynamical model used for the application of the Dual EnKF method presented in § 2.3. The computational domain along with the dimensions and boundary conditions is shown in Fig. 1. All the dimensions have been parameterised with the diameter of the cylinder D, here set to one. For the purpose of mesh size control, we divide the domain into sub-domains and use the automatic mesh generation offered by FreeFem++'s buildmesh command [START_REF] Hecht | New development in FreeFem++[END_REF]. The mesh has N vert = 10263 vertices and N tri = 20360 triangles.

To solve the space discretised Navier-Stokes equations, an optimised Newton method is used. This approach is a variant of the classical Newton method for which the non-linear term is discretised semiimplicitly. The finite element space for velocity and pressure variables are discretised using P 2 and P 1 elements, respectively. After convergence of the iterative procedure, the pressure field is such that the resulting velocity is divergence free.

The Reynolds number of the numerical simulation based on the cylinder diameter D is set to Re = 100. The POD modes are then calculated for the fluctuating velocity fields obtained from the snapshot database.

The energy content of the modes is plotted in Fig. 3. We observe that the first four modes in total represent 99% of the energy. Therefore, for the further evaluation of the ROM, we reduce the degrees of freedom of the problem to N Gal = 4, where N Gal N t .

The four most dominant temporal and spatial modes are plotted in Fig. 4 Finally, knowing the time evolution of the temporal POD modes, we can apply the identification method described in § 2.2. For the DA application (see § 4), the parameters of the ROM is identified using the first 50 s of the time range [150,200], the remaining time range (200,250] s being used for verification of the procedure. As we retained N Gal = 4 modes in the current case, we have N b = 15 unknown parameters in the model for each mode. Moreover, as we sample the time span [150,200] s at a sampling frequency of 10 Hz, we have N t = 501 for the identification. The parameters of the POD ROM obtained from the subsequent L2-minimisation are given in Table 1.

Table 1: Initial conditions and identified coefficients of the POD ROM for the 2D-cylinder wake flow at Re = 100 (i = 1, 2, 3, 4).

Results of Dual EnKF

As already mentioned in § 2.2, the POD ROM is in general not sufficiently accurate to describe the original dynamics followed by the POD modes. This is illustrated in Fig. 6 where the ROM built using the parameter values in Table 1 is employed to determine the time evolution of the POD coefficients. We observe that the true dynamics are not exactly captured by the identified ROM. This was the main motivation to use a DA method like the Dual EnKF to correct the values of the ROM parameters.

Before proceeding with the DA, we emphasize the fact that most of the physical systems have only a few non-linear terms in the dynamics, making the right hand side of the equations ( 9) sparse in a high-dimensional nonlinear function space. Hereafter, we exploit this sparsity in the non-linear model to balance the model complexity with accuracy. To introduce sparsity in our solution, we follow the procedure presented in [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF] and introduce the concept of sparsification knob. The role of this quantity is to set the maximum value allowed for any parameter. By first running the data assimilation with no sparsification, a value of 0.5 has been found empirically for the sparsification knob. In this section, the covariances are set to diagonal matrices equal to Q = 10 -4 I, C = 10 -4 I and R = 10 -5 I. The parameters obtained from the Dual EnKF DA are given in Table 2. As anticipated, the parameter space is sparse as only four significant parameters are retained to build the ROM.

The estimation and forecast results of the Dual EnKF assimilation are shown in Fig. 7. The data is

Conclusion

The Dual EnKF data assimilation is a well-established method in meteorology. It combines different inhomogeneous sources of information (data, dynamical model) for the dual estimation of the true state and model parameters. This allows the prediction of the dynamical evolution with time. In this paper, this approach has been applied in a fluid mechanics context to a POD ROM of a 2D-cylinder wake flow at low Reynolds number (Re = 100). We observe that the initial POD ROM obtained from identification is not sufficient to predict the long-term dynamics. Therefore, we apply the Dual EnKF algorithm to correct the model parameters and subsequently apply sparsification owing to the non-complex nature of the dynamics. We observe that the original dynamics was well reproduced and a good predictability of the analysed dynamical model was found. In the future, this DA method needs to be expanded with the inclusion of parameters related to control and applied to experimental flow cases. It is a work in progress.

Figure 1 :Figure 2 :

 12 Figure 1: Computational domain and boundary conditions for the 2D-cylinder wake flow.

Figure 3 :Figure 4 :

 34 Figure 3: Energy content of the POD modes obtained for the fluctuating velocity fields of the 2Dcylinder wake flow at Re = 100. The first four modes are the most dominant and account for 99% of the total energy.

(d) φ POD 4 Figure 5 :

 45 Figure 5: Spatial POD modes φ POD i corresponding to the four most energetic modes for the 2D-cylinder wake flow at Re = 100.

Figure 6 :

 6 Figure 6: True a POD i

  time window[150, 200] s and the parameters of the ROM are estimated. The ROM with the assimilated parameters is then used to forecast the evolution of the temporal coefficients in the time window t = (200, 250] s. The results are compared with the reference POD coefficients. A good agreement between the forecast and the reference values is observed as a result of the corrected parameters of the POD ROM.

Figure 7 :

 7 Figure 7: Time evolution of the four most energetic temporal POD coefficients for the 2D-cylinder wake flow at Re = 100. Comparison between the values obtained by POD and the ones obtained by the POD ROM with the parameters determined by Dual EnKF. The true values a POD i are used for t ≤ 200 s as observation in the DA. The dotted line at t = 200 s indicates the end of the DA window and beginning of the forecast window.

For simplicity, we have considered the field data to be placed on a uniform grid. In general, the cell volume needs to be included in the formulation to represent the inner product. The covariance matrix should therefore be written as X T W X where W holds the spatial weights.


Omar Lazar

PARALINEARIZATION OF THE MUSKAT EQUATION AND APPLICATION TO THE CAUCHY PROBLEM

We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spaces Ḣ1 (R) ∩ Ḣs (R) with s > 3/2. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.

Introduction

The Muskat equation is a fundamental equation for incompressible fluids in porous media. It describes the evolution of a time-dependent free surface Σ(t) separating two fluid domains Ω 1 (t) and Ω 2 (t). A common assumption in this theory is that the motion is in two dimensions so that the interface is a curve. In this introduction, for the sake of simplicity, we assume that the interface is a graph (the analysis is done later on for a general interface). On the supposition that the fluids extend indefinitely in horizontal directions, it results that Ω 1 (t) = {(x, y) ∈ R × R ; y > h(t, x)} , Ω 2 (t) = {(x, y) ∈ R × R ; y < h(t, x)} , Σ(t) = ∂Ω 1 (t) = ∂Ω 2 (t) = {y = h(t, x)}.

Introduce the density ρ i , the velocity v i and the pressure P i in the domain Ω i (i = 1, 2). One assumes that the velocities v 1 and v 2 obey Darcy's law. Then, the equations by which the motion is to be determined are

v i = ∇(P i + ρ i gy)
in Ω i ,

div v i = 0 in Ω i , P 1 = P 2 on Σ, v 1 • n = v 2 • n on Σ,
where g is the gravity and n is the outward unit normal to Ω 2 on Σ,

n = 1 1 + (∂ x h) 2 -∂ x h 1 .
The first two equations express the classical Darcy's law and the last two equations impose the continuity of the pressure and the normal velocities at the interface. This system is supplemented with an equation for the evolution of the free surface:

∂ t h = 1 + (∂ x h) 2 v 2 • n.
The previous system has been introduced by Muskat in [START_REF] Muskat | Two fluid systems in porous media. The encroachment of water into an oil sand[END_REF] whose main application was in petroleum engineering (see [START_REF] Muskat | Physical Principles of Oil Production[END_REF][START_REF] Narasimhan | Hydraulic characterization of aquifers, reservoir rocks, and soils: A history of ideas[END_REF] for many historical comments).

In [START_REF] Córdoba | Contour dynamics of incompressible 3-D fluids in a porous medium with different densities[END_REF], Córdoba and Gancedo discovered a formulation of the previous system based on contour integral, which applies whether the interface is a graph or not. The latter work opened the door to the solution of many important problems concerning the Cauchy problem or blow-up solutions (see [START_REF]Interface evolution: the Hele-Shaw and Muskat problems[END_REF][START_REF] Castro | Breakdown of smoothness for the Muskat problem[END_REF][START_REF] Castro | Splash singularities for the one-phase Muskat problem in stable regimes[END_REF][START_REF] Castro | Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves[END_REF], more references are given below as well as in the survey papers [START_REF] Gancedo | A survey for the Muskat problem and a new estimate[END_REF][START_REF] Granero-Belinchón | Growth in the Muskat problem[END_REF]). This formulation is a compact equation where the unknown is the parametrization of the free surface, namely a function f = f (t, x) depending on time t ∈ R + and x ∈ R, satisfying

∂ t f = ρ 2π ∂ x arctan (∆ α f ) dα, (1.1)
where ρ = ρ 2 -ρ 1 is the difference of the densities, the integral is understood in the principal value sense and ∆ α f is the slope, namely

∆ α f (t, x) = f (x, t) -f (x -α, t) α •
The beauty of equation (1.1) lies in its apparent simplicity, which should be compared with the complexity of the equations written in Eulerian formulation. This might suggest that (1.1) is the simplest version of the Muskat equation one may hope for. However, since the equation is highly nonlocal (this means that the nonlinearity enters in the nonlocal terms), even with this formulation the study of the Cauchy problem for (1.1) is a very delicate problem. We refer the reader to the above mentioned papers for the description of the main difficulties one has to cope with.

Our goal in this paper is to continue this line of research. We want to simplify further the study of the Muskat problem by transforming the equation (1.1) into the simplest possible form. We shall prove that one can derive from the formulation (1.1) an explicit parabolic evolution. In particular, we shall see that one can decouple the nonlinear and nonlocal aspects. There are many possible applications that one could work out of this explicit parabolic formulation. Here we shall study the Cauchy problem in homogeneous Sobolev spaces.

The well-posedness of the Cauchy problem was first proved in [START_REF] Córdoba | Contour dynamics of incompressible 3-D fluids in a porous medium with different densities[END_REF] by Córdoba and Gancedo for initial data in H 3 (R) in the stable regime ρ 2 > ρ 1 (they also proved that the problem is ill-posed in Sobolev spaces when ρ 2 < ρ 1 ). Several extensions of their results have been obtained by different proofs. In [START_REF] Arthur Cheng | Well-posedness of the Muskat problem with H 2 initial data[END_REF], Cheng, Granero-Belinchón, Shkoller proved the well-posedness of the Cauchy problem in H 2 (R) (introducing a Lagrangian point of view which can be used in a broad setting, see [START_REF] Granero-Belinchón | Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability[END_REF]) and Constantin, Gancedo, Shvydkoy and Vicol ( [START_REF] Constantin | Global regularity for 2D Muskat equations with finite slope[END_REF]) considered rough initial data which are in W 2,p (R) for some p > 1, as well they obtained a regularity criteria for the Muskat problem. We refer also to the recent work [START_REF] Granero-Belinchón | Growth in the Muskat problem[END_REF] where a regularity criteria is obtained in terms of a control of some critical quantities. Many recent results are motivated by the fact that, loosely speaking, the Muskat equation has to do with the slope more than with the curvature of the fluid interface. Indeed, one scale invariant norm is the Lipschitz norm sup x∈R |∂ x f (t, x)|. We refer the reader to the work [START_REF] Constantin | On the Muskat problem: global in time results in 2D and 3D[END_REF] of Constantin, Córdoba, Gancedo, Rodríguez-Piazza and Strain for global well-posedness results assuming that the Lipschitz semi-norm is smaller than 1 (see also [START_REF] Patel | Robert Strain Large time decay estimates for the Muskat equation[END_REF] where time decay of those solutions is proved). In [START_REF] Deng | On the Two-Dimensional Muskat Problem with Monotone Large Initial Data[END_REF], Deng, Lei and Lin proved the existence of global in time solutions with large slopes, assuming some monotonicity assumption on the data. In [START_REF] Cameron | Global well-posedness for the 2d Muskat problem with slope less than 1[END_REF], Cameron was able to prove a global existence result assuming that some critical quantity, namely the product of the maximal and minimal slopes, is smaller than 1. His result allows to consider arbitrary large slopes. By using a new formulation of the Muskat equation involving oscillatory integrals, Córdoba and the second author in [START_REF] Córdoba | Global well-posedness for the 2d stable Muskat problem in H 3 2[END_REF] proved that the Muskat equation is globally well-posed for sufficiently smooth data provided the critical Sobolev norm Ḣ 3 2 (R) is small enough. The latter is a global existence result of a unique strong solution having arbitrarily large slopes.

These observations suggest to study the local in time well-posedness of the Cauchy problem without assuming that any L p -norm of the curvature is finite. The wellposedness of the Cauchy problem in this case was obtained by Matioc [START_REF] Vasile | Viscous displacement in porous media: the Muskat problem in 2D Trans[END_REF][START_REF] Vasile | The Muskat problem in 2D: equivalence of formulations, well-posedness, and regularity results Anal[END_REF]. Using tools from functional analysis, Matioc proved that the Cauchy problem is locally in time well-posed for initial data in Sobolev spaces H s (R) with s > 3/2, without smallness assumption. We shall give a simpler proof which generalizes the latter result to homogeneous Sobolev spaces Ḣs (R). Eventually, let us mention that many recent results focus on different rough solutions, which are important for instance in the unstable regime ρ 1 > ρ 2 (see e.g. the existence mixing zones in [START_REF] Castro | Mixing solutions for the Muskat problem[END_REF][START_REF] Castro | Degraded mixing solutions for the Muskat problem Calc[END_REF][START_REF] Otto | Viscous fingering: an optimal bound on the growth rate of the mixing zone[END_REF] or the dynamic between the two different regimes [START_REF] Córdoba | A note on stability shifting for the Muskat problem[END_REF][START_REF] Córdoba | A note on stability shifting for the Muskat problem II: Stable to Unstable and back to Stable[END_REF]). We refer also to [START_REF] Córdoba | Lack of uniqueness for weak solutions of the incompressible porous media equation[END_REF][START_REF] Székelyhidi | Relaxation of the incompressible porous media equation[END_REF] where uniqueness issues have been studied using the convex integration scheme.

In this paper we assume that the difference between the densities in the two fluids satisfies ρ > 0, so, by rescaling in time, we can assume without loss of generality that ρ = 2.

A fundamental difference with the above mentioned results is that we shall determine the full structure of the nonlinearity instead of performing energy estimates. To explain this, we begin by identifying the nonlinear terms. Since ρ = 2, one can rewrite equation (1.1) as

∂ t f = 1 π ∂ x ∆ α f 1 + (∆ α f ) 2 dα
(in this introduction some computations are formal, but we shall rigorously justify them later). Consequently, the linearized Muskat equation reads

∂ t u = 1 π pv ∂ x ∆ α u dα. (1.2)
Consider the singular integral operators

Hu = - 1 π pv ∆ α u dα and Λ = H∂ x . (1.3)
Then H is the Hilbert transform (the Fourier multiplier with symbol -i sgn(ξ)) and Λ is the square root of -∂ xx . With the latter notation, the linearized Muskat equation (1.2) reads

∂ t u + Λu = 0.
With this notation, the Muskat equation (1.1) can be written under the form

∂ t f + Λf = T (f )f, (1.4)
where T (f ) is the operator defined by

T (f )g = - 1 π ∂ x ∆ α g (∆ α f ) 2 1 + (∆ α f ) 2 dα.
Our first main result will provide a thorough study of this nonlinear operator. Before going any further, let us fix some notations. Definition 1.1. i) Given a real number σ, we denote by Λ σ the Fourier multiplier with symbol |ξ| σ and by Ḣσ (R) the homogeneous Sobolev space of tempered distributions whose Fourier transform û belongs to L 1 loc (R) and satisfies

u 2 Ḣσ = Λ σ u 2 L 2 = 1 2π R |ξ| 2σ |û(ξ)| 2 dξ < +∞.
ii) We denote by H σ (R) the nonhomogeneous Sobolev space L 2 (R) ∩ Ḣσ (R). We set H ∞ (R) := ∩ σ≥0 H σ (R) and introduce X := ∩ σ≥1 Ḣσ (R), the set of tempered distributions whose Fourier transform û belongs to L 1 loc (R) and whose derivative belongs to H ∞ (R).

iii) Given 0 < s < 1, the homogeneous Besov space Ḃs 2,1 (R) consists of those tempered distributions f whose Fourier transform is integrable near the origin and such that

f Ḃs 2,1 = |f (x) -f (x -α)| 2 |α| 2s dx 1 2 dα |α| < +∞.
iv) We use the notation

• E∩F = • E + • F .
Theorem 1.2. i) (Low frequency estimate) There exists a constant C such that, for all f in Ḣ1 (R) and all g in

Ḣ 3 2 (R), T (f )g belongs to L 2 (R) and T (f )g L 2 ≤ C f Ḣ1 g Ḣ 3 2 . Moreover, f → T (f )f is locally Lipschitz from Ḣ1 (R) ∩ Ḣ 3 2 (R) to L 2 (R). ii) (High frequency estimate) For all 0 < ν < < 1/2, there exists a positive constant C > 0 such that, for all functions f, g in X = ∩ σ≥1 Ḣσ (R), T (f )g = γ(f )Λg + V (f )∂ x g + R(f, g) (1.5)
where

γ(f ) := f 2 x 1 + f 2 x ,
and R(f, g) and V (f ) satisfy

R(f, g) L 2 ≤ C f Ḣ 3 2 + g Ḃ1- 2,1 , (1.6)
and

V (f ) C 0,ν := V (f ) L ∞ + sup y∈R |V (f )(x + y) -V (f )(x)| |y| ν ≤ C f 2 Ḣ1 ∩ Ḣ 3 2 + . (1.7) iii) Let 0 < < 1/2.
There exists a non-decreasing function F : R + → R + such that, for all functions f, g in X,

Λ 1+ T (f )g -T (f )Λ 1+ g L 2 ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + g Ḣ 3 2 + ∩ Ḣ2 . (1.
8) The proof of the first statement follows directly from the definition of fractional Sobolev spaces in terms of finite differences, see Section 2. The proof of the second statement is the most delicate part of the proof, which requires to uncover some symmetries in the nonlinearity, see Section 4. The last statement is proved in Section 3 by using sharp variants of the usual nonlinear estimates in Sobolev spaces. Namely we used for the later proof a version of the classical Kato-Ponce estimate proved recently by Li and also a refinement of the composition rule in Sobolev spaces proved in Section 2.

We deduce from the previous result a paralinearization formula for the nonlinearity. We do not consider paradifferential operators as introduced by Bony ( [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF][START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]).

Instead, following Shnirelman [START_REF] Shnirelman | Microglobal analysis of the Euler equations[END_REF], we consider a simpler version of these operators which is convenient for the analysis of the Muskat equation for rough solutions.

Corollary 1.3. Consider 0 < < 1/2 and, given a bounded function a = a(x), denote by Ta : Ḣ1+ (R) → L 2 (R) ∩ Ḣ1+ (R) the paraproduct operator defined by Ta g = (I + Λ 1+ ) -1 (aΛ 1+ g).

Then, there exists a function F : R → R such that, for all f ∈ X,

T (f )f = Tγ(f) Λf + TV (f ) ∂ x f + R (f ), (1.9)
where

R (f ) H 1+ ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ2+ 2 .
(1.10)

Proof. Writing

T (f )f = (I + Λ 1+ ) -1 T (f )Λ 1+ f + (I + Λ 1+ ) -1 Λ 1+ , T (f ) f,
and using the formula (1.5) we find that (1.9) holds with

R (f ) = (I + Λ 1+ ) -1 R f, Λ 1+ f + (I + Λ 1+ ) -1 Λ 1+ , T (f ) f.
Then, it follows from (1.6) and (1.8) that

R (f ) H 1+ ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + f Ḣ 3 2 + ∩ Ḣ2 + Λ 1+ f Ḃ1- 2,1
.

So (1.10) follows from Ḣ1-3 2 (R) ∩ Ḣ1-2 (R) → Ḃ1- 2,1 (R) (see Lemma 2.
2). We now consider the Cauchy problem for the Muskat equation. Substituting the above identity for T (f ) in the equation (1.4) and simplifying, we find

∂ t -V (f )∂ x + 1 1 + f 2 x Λ Λ 1+ f = Λ 1+ R (f ).
Now, the key point is that the estimates (1.6) and (1.10) mean that the remainder term R (f ) and the operator V ∂ x contribute as operators of order stricly less than 1 (namely 1-/2 and 1-ν) to an energy estimate, and so they are sub-principal terms for the analysis of the Cauchy problem. We also observe that the Muskat equation is parabolic as long as one controls the L ∞ -norm of f x only. This observation is related to our second goal, which is to solve the Cauchy problem in homogeneous Sobolev spaces instead of nonhomogeneous spaces. This is a natural result since the Muskat equation is invariant by the transformation f → f + C. This allows us to make an assumption only on the L ∞ -norm of the slope of the initial data, allowing initial data which are not bounded or not square integrable.

Theorem 1.4. Consider s ∈ (3/2, 2) and an initial data f 0 in Ḣ1 (R)∩ Ḣs (R). Then, there exists a positive time T such that the Cauchy problem for (1.1) with initial data f 0 has a unique solution f satisfying f (t, x) = f 0 (x) + u(t, x) with u(0, x) = 0 and

u ∈ C 0 [0, T ]; H s (R) ∩ C 1 [0, T ]; H s-1 (R) ∩ L 2 0, T ; H s+ 1 2 (R) ,
where H σ (R) denotes the nonhomogeneous Sobolev space L 2 (R) ∩ Ḣσ (R).

The latter result is proved in the last section. We conclude this introduction by fixing some notations. Notation 1.5. i) We denote by f x the spatial derivative of f .

ii) A B means that there is C > 0, depending only on fixed quantities, such that A ≤ CB. iii) Given g = g(α, x) and Y a space of functions depending only on x, the notation g Y is a compact notation for α → g(α, •) Y .

Preliminaries

In this section, we recall or prove various results about Besov spaces which we will need throughout the article. We use the definition of these spaces originally given by Besov in [START_REF] Besov | Investigation of a class of function spaces in connection with imbedding and extension theorems[END_REF], using integrability properties of finite differences.

Given a real number α, the finite difference operators δ α and s α are defined by:

δ α f (x) = f (x) -f (x -α), s α f (x) = 2f (x) -f (x -α) -f (x + α).
Definition 2.1. Consider three real numbers (p, q, s) in [1, ∞] 2 × (0, 2). The homogeneous Besov space Ḃs p,q (R) consists of those tempered distributions f whose Fourier transform is integrable near the origin and such that the following quantity f Ḃs p,q is finite:

f Ḃs p,q = δ α f L p (R,dx) |α| s L q (R,|α| -1 dα)
for s ∈ (0, 1), (2.1)

f Ḃs p,q = s α f L p (R;dx) |α| s L q (R,|α| -1 dα) for s ∈ [1, 2). (2.2)
We refer the reader to the book of Peetre [45, chapter 8] for the equivalence between these definitions and the one in terms of Littlewood-Paley decomposition (see also [START_REF] Bourdaud | Le calcul fonctionnel sous-linéaire dans les espaces de Besov homogènes[END_REF]Prop. 9] or [4, Theorems 2.36, 2.37] for the case s ∈ (0, 1]).

In this paper, we use only Besov spaces of the form

Ḃs 2,2 (R), Ḃs ∞,2 (R), Ḃs 2,1 (
R). We will make extensive use of the fact that • Ḣs and • Ḃs 2,2 are equivalent for s ∈ (0, 2). Moreover, for s ∈ (0, 1),

u 2 Ḣs = 1 4πc(s) u 2 Ḃs 2,2 with c(s) = R 1 -cos(t) |t| 1+2s dt. (2.3)
We will also make extensive use of the fact that, for all s in (0, 2),

Ḣs+ 1 2 (R) → Ḃs ∞,2 (R). (2.4)
We will also use the following Lemma 2.2. For any s ∈ (0, 1) and any δ > 0 such that [s -δ, s + δ] ⊂ (0, 1),

Ḣs-δ (R) ∩ Ḣs+δ (R) → Ḃs 2,1 (R). Proof. We have |α|≤1 δ α f L 2 (R;dx) |α| s dα |α| = |α|≤1 |α| δ δ α f L 2 (R;dx) |α| s+δ dα |α| ≤ |α|≤1 |α| 2δ dα |α| 1 2 R δ α f 2 L 2 (R;dx) |α| 2(s+δ) dα |α| 1 2 ≤ C(δ) f Ḃs 2,2 = C(δ, s) f Ḣs+δ ,
and similarly

|α|≥1 δ α f L 2 (R;dx) |α| s dα |α| ≤ C (δ, s) f Ḣs-δ ,
which gives the result.

As an example of properties which are very simple to prove using the definition of Besov semi-norms in terms of finite differences, let us prove the first point in Theorem 1.2. Recall that, by notation,

T (f )g = - 1 π ∆ α g x (∆ α f ) 2 1 + (∆ α f ) 2 dα,
where g x := ∂ x g. Proposition 2.3. i) For all f in Ḣ1 (R) and all g in Ḣ 3 2 (R), the function

α → ∆ α g x (∆ α f ) 2 1 + (∆ α f ) 2 belongs to L 1 α (R; L 2 x (R)). Consequently, T (f )g belongs to L 2 (R). Moreover, there is a constant C such that T (f )g L 2 ≤ C f Ḣ1 g Ḣ 3 2 . (2.5)
ii) For all δ ∈ [0, 1/2), there exists a constant C > 0 such that, for all functions

f 1 , f 2 in Ḣ1-δ (R) ∩ Ḣ 3 2 +δ (R), (T (f 1 ) -T (f 2 ))f 2 L 2 ≤ C f 1 -f 2 Ḣ1-δ f 2 Ḣ 3 2 +δ . iii) The map f → T (f )f is locally Lipschitz from Ḣ1 (R) ∩ Ḣ 3 2 (R) to L 2 (R). Proof. i) Since ∆ α g x (∆ α f ) 2 1 + (∆ α f ) 2 L 2 ≤ ∆ α g x L 2 ∆ α f L ∞ = δ α g x L 2 |α| δ α f L ∞ |α| ,
by using the Cauchy-Schwarz inequality and the definition (2.1) of the Besov seminorms one finds that

T (f )g L 2 ≤ 1 π δ α g x L 2 |α| δ α f L ∞ |α| dα ≤ 1 π δ α g x L 2 |α| 1/2 δ α f L ∞ |α| 1/2 dα |α| ≤ 1 π δ α g x 2 L 2 |α| dα |α| 1 2 δ α f 2 L ∞ |α| dα |α| 1 2 ≤ 1 π g x Ḃ 1 2 2,2 f Ḃ 1 2 ∞,2
.

Recalling that

• Ḃ 1 2 2,2
and • Ḣ 1 2 are equivalent semi-norms, and using the Sobolev embedding (2.4), we have

g x Ḃ 1 2 2,2 g Ḣ 3 2 , f Ḃ 1 2 ∞,2 f Ḣ1 ,
and hence we obtain the wanted inequality (2.5).

ii) Write that

(T (f 1 ) -T (f 2 ))f 2 = - 1 π ∆ α f 2x ∆ α (f 1 -f 2 )M (α, x) dα where M (α, x) = (∆ α f 1 ) + ∆ α f 2 (1 + (∆ α f 1 ) 2 )(1 + (∆ α f 2 ) 2 )
.

Since |M (α, x)| ≤ 1, by repeating similar arguments to those used in the first part (balancing the powers of α in a different way), we get

(T (f 1 ) -T (f 2 ))f 2 L 2 ≤ 1 π δ α f 2x L 2 |α| δ α (f 1 -f 2 ) L ∞ |α| dα ≤ 1 π δ α f 2x L 2 |α| 1/2+δ δ α (f 1 -f 2 ) L ∞ |α| 1/2-δ dα |α| ≤ 1 π f 2x Ḃ 1 2 +δ 2,2 f 1 -f 2 Ḃ 1 2 -δ ∞,2
.

which implies

(T (f 1 ) -T (f 2 ))f 2 L 2 ≤ C f 1 -f 2 Ḣ1-δ f 2 Ḣ 3 2 +δ . iii) Consider f 1 and f 2 in Ḣ1 (R) ∩ Ḣ 3 2 (R). Then T (f 1 )f 1 -T (f 2 )f 2 = T (f 1 )(f 1 -f 2 ) + (T (f 1 ) -T (f 2 ))f 2 .
Then (2.5) implies that the L 2 -norm of the first term is bounded by

C f 1 Ḣ1 f 1 -f 2 Ḣ 3 2 .
We estimate the second term by using ii) applied with δ = 0. It follows that

T (f 1 )f 1 -T (f 2 )f 2 L 2 f 1 Ḣ1 ∩ Ḣ 3 2 + f 2 Ḣ1 ∩ Ḣ 3 2 f 1 -f 2 Ḣ1 ∩ Ḣ 3
2 , which completes the proof.

We gather in the following proposition the nonlinear estimates which will be needed.

Proposition 2.4. i) Let s ∈ (0, 1), then L ∞ (R) ∩ Ḣs (R) is an algebra. Moreover, for all u, v in L ∞ (R) ∩ Ḣs (R), uv Ḣs ≤ 2 u L ∞ v Ḣs + 2 v L ∞ u Ḣs . (2.6) ii) Consider a C ∞ function F : R → R satisfying ∀(x, y) ∈ R 2 , |F (x) -F (y)| ≤ K |x -y| .
Then, for all s ∈ (0, 1) and all u ∈ Ḣs (R), one has F (u) ∈ Ḣs (R) together with the estimate

F (u) Ḣs ≤ K u Ḣs . (2.7)
iii) Consider a C ∞ function F : R → R and a real number σ in (1, 2). Then, there exists a non-decreasing function

F : R → R such that, for all u ∈ Ḣσ-1 (R) ∩ Ḣσ (R) one has F (u) ∈ Ḣσ (R) together with the estimate F (u) Ḣσ ≤ F( u L ∞ ) u Ḣσ-1 + u Ḣσ . (2.8) Remark 2.5. i) The inequality (2.6) is the classical Kato-Ponce estimate ([29]
). We will use it only when 0 < s < 1, for which one has a straightforward proof (see below).

ii) Statement ii) is also elementary and classical (see [START_REF] Bourdaud | Le calcul fonctionnel sous-linéaire dans les espaces de Besov homogènes[END_REF]). Notice that (2.8) is a sub-linear estimate, which means that the constant K depends only on F and not on u (which is false in general for s > 1).

iii) The usual estimate for composition implies that

F (u) Ḣσ ≤ F( u L ∞ ) u L 2 ∩ Ḣσ .
The bound (2.8) improves the latter estimate in that one requires less control of the low frequency component. This will play a role in the proof of Lemmas 3.2 and 3.3.

Proof. i) Since δ α (uv) = uδ α v + (τ α v)δ α u where τ α v(x) = v(x -α), we have δ α (uv) L 2 ≤ u L ∞ δ α v L 2 + v L ∞ δ α u L 2 .
Directly from the definition (2.1), we deduce that

uv Ḃs 2,2 ≤ 2 u L ∞ v Ḃs 2,2 + 2 v L ∞ u Ḃs 2,2
.

This implies (2.6) by virtue of the identity (2.3) on the equivalence of • Ḣs and

• Ḃs 2,2 .
ii) Similarly, the inequality (2.7) follows directly from the fact that

δ α F (u) L 2 ≤ K δ α u L 2 .
iii) We adapt the classical proof of the composition rule in nonhomogeneous Sobolev spaces, which is based on the Littlewood-Paley decomposition. Namely,

choose a function Φ ∈ C ∞ 0 ({ξ; |ξ| < 1}) which is equal to 1 when |ξ| ≤ 1/2 and set φ(ξ) = Φ(ξ/2) -Φ(ξ) which is supported in the annulus {ξ; 1/2 ≤ |ξ| ≤ 2}. Then, for all ξ ∈ R, one has Φ(ξ) + j∈N φ(2 -j ξ) = 1, which one can use to decompose tempered distribution. For u ∈ S (R), we set ∆ -1 u = F -1 (Φ(ξ) u) and ∆ j u = F -1 (φ(2 -j ξ) u) for j ∈ N. We also use the notation S j u = -1≤p≤j-1 ∆ p u for j ≥ 0 (so that S 0 u = ∆ -1 u = Φ(D x )u).
The classical proof (see [START_REF] Alinhac | Pseudo-differential operators and the Nash-Moser theorem[END_REF][START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF][START_REF]The Navier-Stokes problem in the 21st century[END_REF]) of the composition rule consists in splitting F (u) as

F (u) = F (S 0 u) + F (S 1 u) -F (S 0 u) + • • • + F (S j+1 u) -F (S j u) + • • • = F (S 0 u) + j∈N m j ∆ j u with m j = 1 0 F (S j u + y∆ j u) dy = F (S 0 u) + j∈N m j ∆ j ũ with ũ = u -Φ(2D x )u,
where we used ∆ j • Φ(2D x ) = 0 for j ≥ 0. Then, the Meyer's multiplier lemma (see [START_REF] Meyer | Régularité des solutions des équations aux dérivées partielles non linéaires (d'après Bony)[END_REF]Theorem 2] or

[3, Lemma 2.2]) implies that j≥0 m j ∆ j ũ H σ ≤ F( u L ∞ ) ũ H σ ,
where, to clarify notations, we insist on the fact that above H σ is the nonhomogeneous Sobolev space. Since ũ H σ ≤ u Ḣσ , we see that the contribution of m j ∆ j ũ is bounded by the right-hand side of (2.8). This shows that the only difficulty is to estimate the low frequency component F (S 0 u). We claim that

F (S 0 u) Ḣσ ≤ F( u L ∞ ) u Ḣσ-1 .
(2.9)

To see this, we start with

F (S 0 u) Ḣσ = ∂ x F (S 0 u) Ḣσ-1 = F (S 0 u)∂ x S 0 u Ḣσ-1 ,
and then use the product rule (2.6) with s = σ -1 ∈ (0, 1),

F (S 0 u)∂ x S 0 u Ḣσ-1 ≤ 2 F (S 0 u) L ∞ ∂ x S 0 u Ḣσ-1 + 2 F (S 0 u) Ḣσ-1 ∂ x S 0 u L ∞ .
Since |ξΦ(ξ)| ≤ 1 one has the obvious inequality

∂ x S 0 u Ḣσ-1 ≤ u Ḣσ-1 .
On the other hand, since the support of the Fourier transform of S 0 u is included in the ball of center 0 and radius 1, it follows from the Bernstein's inequality that

S 0 u L ∞ ≤ C 1 u L ∞ , ∂ x S 0 u L ∞ ≤ C 2 u L ∞ .
The first estimate above also implies that

F (S 0 u) L ∞ ≤ F 1 ( S 0 u L ∞ ) ≤ F 2 ( u L ∞ )
where

F 1 (r) = sup y∈[-r,r] |F (y)| and F 2 (r) = F 1 (C 1 r).
It thus remains only to estimate F (S 0 u) Ḣσ-1 . Notice that we may apply the composition rule given in statement ii) since the index σ -1 belongs to (0, 1) and since F is Lipschitz on an open set containing S 0 u(R). The composition rule (2.7) implies that

F (S 0 u) Ḣσ-1 ≤ K S 0 u Ḣσ-1 ≤ K u Ḣσ-1 with K = sup [-2 S 0 u L ∞ ,2 S 0 u L ∞ ] F ≤ F 3 ( u L ∞ ).
This proves that the Ḣσ -norm of F (S 0 u) satisfies (2.9) and hence it is bounded by the right-hand side of (2.8), which completes the proof of statement iii).

For later purposes, we prove the following commutator estimate with the Hilbert transform.

Lemma 2.6. Let 0 < θ < ν < 1. There exists a constant K such that for all f ∈ C 0,ν (R), and all u in the nonhomogeneous space H -θ (R),

H(f u) -f Hu L 2 ≤ K f C 0,ν u H -θ .
(2.10)

Proof. We establish this estimate by using the para-differential calculus of Bony [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF].

We use the Littlewood-Paley decomposition (see the proof of Proposition 2.4) and denote by T f the operator of para-multiplication by f , so that

T f u = j≥1 S j-1 (f )∆ j u.
Denote by f the multiplication operator u → f u and introduce H 0 , the Fourier multiplier with symbol -i(1 -Φ(ξ))ξ/ |ξ| where Φ ∈ C ∞ 0 (R) is such that Φ(ξ) = 1 on a neighborhood of the origin. With these notations, one can rewrite the commutator H, f as

H, f = [H, T f ] + H(f -T f ) -(f -T f )H = [H 0 , T f ] + (H -H 0 )T f -T f (H -H 0 ) + H(f -T f ) -(f -T f )H. (2.11)
Notice that H -H 0 is a smoothing operator (that is an operator bounded from H σ to H σ+t for any real numbers σ, t ∈ R). We then use two classical estimates for paradifferential operators (see [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF][START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]). Firstly,

∀σ ∈ R, T f H σ →H σ ≤ c(σ) f L ∞ , so (H -H 0 )T f H -θ →L 2 ≤ H -H 0 H -θ →L 2 T f H -θ →H -θ f L ∞ , T f (H -H 0 ) H -θ →L 2 ≤ T f L 2 →L 2 H -H 0 H -θ →L 2 f L ∞ .
Secondly, since H 0 is a Fourier multiplier whose symbol is a smooth function of order 0 (which means that its kth derivative is bounded by

C k (1 + |ξ|) -k ), one has ∀σ ∈ R, [H 0 , T f ] H σ →H σ+ν ≤ c(ν, σ) f C 0,ν . In particular, [H 0 , T f ] H -ν →L 2 f C 0,ν .
It remains only to estimate the last two terms in the right-hand side of (2.11). We claim that

H(f -T f ) H -θ →L 2 + (f -T f )H H -θ →L 2 f C 0,ν .
Since H is bounded from H σ to itself for any σ ∈ R, it is enough to prove that

f -T f H -θ →L 2 f C 0,ν .
To do so, observe that

f g -T f g = j,p≥-1 (∆ j f )(∆ p g) - -1≤j≤p-2 (∆ j f )(∆ p g) = j≥-1 (S j+2 g)∆ j f.
Then, using the Bernstein's inequality and the characterization of Hölder spaces in terms of Littlewood-Paley decomposition, it follows from the assumption θ < ν that the series 2 j(θ-ν) converges, so

f g -T f g L 2 ≤ S j+2 g L 2 ∆ j f L ∞ 2 θ(j+2) g H -θ 2 -jν f C 0,ν g H -θ f C 0,ν .
By combining the previous estimates, we have H, f H -θ →L 2 f C 0,ν , which gives the result.

Commutator estimate

In this section we prove statement iii) in Theorem 1.2. Namely, we prove the following proposition. Proposition 3.1. Let 0 < < 1/2. There is a non-decreasing function F : R + → R + such that, for all functions f, g in ∩ σ≥1

Ḣσ (R),

Λ 1+ T (f )g -T (f )Λ 1+ g L 2 ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + g Ḣ 3 2 + ∩ Ḣ2 . (3.1) Proof. Recall that T (f )g = - 1 π (∆ α f ) 2 1 + (∆ α f ) 2 ∆ α g x dα. Since Λ 1+ ∆ α g x = ∆ α Λ 1+ g x , we have Λ 1+ T (f )g = T (f )Λ 1+ g + R 1 (f )g + R 2 (f )g, where R 1 (f )g = - 1 π ∆ α g x Λ 1+ (∆ α f ) 2 1 + (∆ α f ) 2 dα, (3.2)
and

R 2 (f )g = - 1 π Λ 1+ u α v α -u α Λ 1+ v α -v α Λ 1+ u α dα with u α = ∆ α g x , v α = (∆ α f ) 2 1 + (∆ α f ) 2 .
We shall estimate these two terms separately. Classical results from paradifferential calculus (see [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF][START_REF] Coifman | Au delà des opérateurs pseudo-différentiels[END_REF][START_REF] Métivier | Para-differential calculus and applications to the Cauchy problem for nonlinear systems[END_REF]) would allow us to estimate them provided that we work in nonhomogeneous Sobolev spaces. In the homogeneous spaces we are considering, we shall see that one can derive similar results by using only elementary nonlinear estimates.

We begin with the study of R 1 (f )g.

Lemma 3.2.

There exists a non-decreasing function

F : R + → R + such that R 1 (f )g L 2 ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + g Ḣ 3 2 + ∩ Ḣ2 . Proof. By definition R 1 (f )g = - 1 π ∆ α g x Λ 1+ (∆ α f ) 2 1 + (∆ α f ) 2 dα, so R 1 (f )g L 2 ∆ α g x L ∞ (∆ α f ) 2 1 + (∆ α f ) 2 Ḣ1+
dα.

(3.

3)

The Sobolev embedding

L 2 (R) ∩ Ḣ 1 2 + (R) → L ∞ (R) implies that, for all α in R, sup x∈R |∆ α f (x)| ≤ sup x∈R |f x (x)| f x L 2 ∩ Ḣ 1 2 + = f Ḣ1 ∩ Ḣ 3 2 + , (3.4)
so that the composition rule (2.8) implies that

(∆ α f ) 2 1 + (∆ α f ) 2 Ḣ1+ ≤ F f Ḣ1 ∩ Ḣ 3 2 + ∆ α f Ḣ + ∆ α f Ḣ1+ . (3.5)
We claim that we have the two following inequalities

∆ α g x L ∞ ∆ α f Ḣ dα g Ḣ 3 2 + f Ḣ1 , (3.6)
∆ α g x L ∞ ∆ α f Ḣ1+ dα g Ḣ2 f Ḣ 3 2 + . (3.7)
Let us prove (3.6). Directly from the definition of ∆ α , we have

∆ α g x L ∞ ∆ α f Ḣ dα = δ α g x L ∞ |α| δ α Λ f L 2 |α| dα = δ α g x L ∞ |α| δ α Λ f L 2 |α| 1- dα |α| .
So, using the Cauchy-Schwarz inequality,

∆ α g x L ∞ ∆ α f Ḣ dα ≤ δ α g x 2 L ∞ |α| 2 dα |α| 1 2 δ α Λ f 2 L 2 |α| 2(1-) dα |α| 1 2
and hence, using the definition of Besov semi-norms (see (2.1)),

∆ α g x L ∞ ∆ α f Ḣ dα g x Ḃ ∞,2 Λ f Ḃ1- 2,2
.

By using (2.3) and (2.4), we obtain that

∆ α g x L ∞ ∆ α f Ḣ dα g Ḣ 3 2 + f Ḣ1 ,
which is the first claim (3.6). To prove the second claim (3.7), we repeat the same arguments except that we balance the powers of α in a different way:

∆ α g x L ∞ ∆ α f Ḣ1+ dα = δ α g x L ∞ |α| 1/2 δ α Λ 1+ f L 2 |α| 1/2 dα |α| ≤ δ α g x 2 L ∞ |α| dα |α| 1 2 δ α Λ 1+ f 2 L 2 |α| dα |α| 1 2 ≤ g x Ḃ 1 2 ∞,2 Λ 1+ f Ḃ 1 2 2,2 g Ḣ2 f Ḣ 3 2 + ,
R 1 (f )g L 2 ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + g Ḣ 3 2 + ∩ Ḣ2
, which is the desired result.

We now move to the second remainder term R 2 (f )g.

Lemma 3.3. There exists a non-decreasing function

F : R + → R + such that R 2 (f )g L 2 ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + g Ḣ 3 2 + ∩ Ḣ2 . Proof. We use the classical Kenig-Ponce-Vega commutator estimate Λ s (uv) -uΛ s v -vΛ s u L r ≤ C Λ s 1 u L p 1 Λ s 2 v L p 2 (3.8)
where s = s 1 +s 2 and 1/r = 1/p 1 +1/p 2 . Kenig, Ponce and Vega considered the case s < 1. Since, for our purpose we need s > 1, we will use the recent improvement by Li [START_REF] Li | On Kato-Ponce and fractional Leibniz[END_REF] (see also D'Ancona [START_REF] Piero | A short proof of commutator estimates[END_REF]) showing that (3.8) holds under the assumptions

s = s 1 + s 2 ∈ (0, 2), s j ∈ (0, 1), 1 r = 1 p 1 + 1 p 2 , 2 ≤ p j < ∞.
With

p 1 = 4, p 2 = 4, r = 2, s = 1 + , s 1 = 3 2 , s 2 = 1 -2 , this implies that R 2 (f )g L 2 Λ 3 2 ∆ α g x L 4 Λ 1-2 (∆ α f ) 2 1 + (∆ α f ) 2 L 4 dα.
We now use the Sobolev inequality

u L 4 Λ 1 4 u L 2 , to obtain R 2 (f )g L 2 Λ 1 4 + 3 2 ∆ α g x L 2 Λ 5 4 -2 (∆ α f ) 2 1 + (∆ α f ) 2 L 2 dα.
By combining the composition rule (2.8) (applied with σ = 5 4 -2 ∈ (1, 2)) and (3.4), we obtain that

R 2 (f )g L 2 F(M ) Λ 1 4 + 3 2 ∆ α g x L 2 Λ 1 4 -2 ∆ α f L 2 + Λ 5 4 -2 ∆ α f L 2 dα where M = f Ḣ1 ∩ Ḣ 3
Besov semi-norms (2.1) and the Sobolev embedding to obtain that Λ

1 4 + 3 2 ∆ α g x L 2 Λ 1 4 -2 ∆ α f L 2 dα = Λ 1 4 + 3 2 δ α g x L 2 |α| Λ 1 4 -2 δ α f L 2 |α| dα = Λ 1 4 + 3 2 δ α g x L 2 |α| 1/4-/2 Λ 1 4 -2 δ α f L 2 |α| 3/4+ /2 dα |α| ≤ Λ 1 4 + 3 2 g x Ḃ 1 4 -/2 2,2 Λ 1 4 -2 f Ḃ 3 4 + /2 2,2 g Ḣ 3 2 + f Ḣ1 .
One estimates the second term in a similar way. We begin by writing that Λ

1 4 + 3 2 ∆ α g x L 2 Λ 5 4 -2 ∆ α f L 2 dα = Λ 1 4 + 3 2 δ α g x L 2 |α| Λ 5 4 -2 δ α f L 2 |α| dα = Λ 1 4 + 3 2 δ α g x L 2 |α| 3/4-3 /2 Λ 5 4 -2 δ α f L 2 |α| 1/4+3 /2 dα |α| .
Since belongs to (0, 1/2) we have 3/4 -3 /2 > 0 and 1/4 + 3 /2 < 1. Therefore one can use the definition (2.1) of the Besov semi-norms to deduce that Λ

1 4 + 3 2 ∆ α g x L 2 Λ 5 4 -2 ∆ α f L 2 dα ≤ Λ 1 4 + 3 2 g x Ḃ 3 4 -3 2 2,2 Λ 5 4 -2 f Ḃ 1 4 + 3 2 2,2 g Ḣ2 f Ḣ 3 2 +
. By combining the above inequalities, we have proved that

R 2 (f )g L 2 ≤ F f Ḣ1 ∩ Ḣ 3 2 + f Ḣ1 ∩ Ḣ 3 2 + g Ḣ 3 2 + ∩ Ḣ2 , which concludes the proof.
This completes the proof of the proposition.

High frequency estimate

We now prove the second point of Theorem 1.2 whose statement is recalled in the next proposition. Proposition 4.1. For all 0 < ν < < 1/2, there exists a positive constant C > 0 such that, for all functions f, g in ∩ σ≥1

Ḣσ (R),

T (f )g = f 2 x 1 + f 2 x Λg + V (f )∂ x g + R(f, g) where R(f, g) L 2 ≤ C f Ḣ 3 2 + g Ḃ1- 2,1 , V (f ) C 0,ν ≤ C f 2 Ḣ1 ∩ Ḣ 3
2 + . We shall prove this proposition in this section by using a symmetrization argument which consists in replacing the finite differences δ α f (x) = f (x) -f (x -α) by the symmetric finite differences 2f (x)-f (x-α)-f (x+α). To do so, it will be convenient to introduce a few notations. Notation 4.2. Given a function f = f (x) and a real number α, we define the functions δα f , ∆α f , s α f , S α f and D α f by: δα

f (x) = f (x) -f (x + α), s α f (x) = δ α f (x) + δα f (x) = 2f (x) -f (x -α) -f (x + α),
and

∆α f (x) = f (x) -f (x + α) α , S α f (x) = ∆ α f (x) + ∆α f (x) = s α f (x) α = 2f (x) -f (x + α) -f (x -α) α , D α f (x) = ∆ α f (x) -∆α f (x) = f (x + α) -f (x -α) α .
Lemma 4.3. One has

D α f = 2f x - 1 α α 0 s η f x dη, (4.1)
where

s η f x (x) = 2f x (x) -f x (x + η) -f x (x -η). Furthermore, ∂ α (D α f ) = -S α f x + 1 α 2 α 0 s η f x dη, (4.2)
and

∂ α (S α f ) = ∆α f x -∆ α f x - S α f α . (4.3)
Proof. The formula (4.1) can be verified by two direct calculations: one is

1 α α 0 2f x (x) dη = 2f x (x),
and the other is

1 α α 0 (f x (x -η) + f x (x + η)) dη = 1 α α 0 ∂ η (f (x + η) -f (x -η)) dη = 1 α (f (x + α) -f (x -α)).
Now, the value for ∂ α (D α f ) in (4.2) follows by differentiating (4.1). The formula for ∂ α (S α f ) follows from the definition of S α f and the chain rule.

Recall that

T (f )g = - 1 π (∆ α f ) 2 1 + (∆ α f ) 2 ∆ α g x dα.
The idea is to decompose the factor

(∆ α f ) 2 1 + (∆ α f ) 2
into its even and odd components with respect to the variable α. We define

E(α, •) = 1 2 (∆ α f ) 2 1 + (∆ α f ) 2 + 1 2 ( ∆α f ) 2 1 + ( ∆α f ) 2 , (4.4) O(α, •) = 1 2 (∆ α f ) 2 1 + (∆ α f ) 2 - 1 2 ( ∆α f ) 2 1 + ( ∆α f ) 2 , ( 4.5)
where the dots in the notations E(α, •) and O(α, •) are placeholders for the variable x (notice that ( ∆α f ) 2 = (∆ -α f ) 2 and ∆α f = -∆ -α f ). Then,

T (f )g = - 1 π ∆ α g x E(α, •) dα - 1 π ∆ α g x O(α, •) dα,
and hence, since α → E(α, •) is even, this yields T (f )g = T e (f )g + T o (f )g with

T e (f )g = - 1 2π ∆ α g x -∆α g x E(α, •) dα, T o (f )g = - 1 π ∆ α g x O(α, •) dα.
The following result is the key point of the proof where we identify the main contribution of the nonlinearity.

Proposition 4.4. There exists a constant C such that

T e (f )g - f 2 x 1 + f 2 x Λg L 2 ≤ C f Ḣ 3 2 + g Ḃ1- 2,1
,

T o (f )g -V ∂ x g L 2 ≤ C f Ḣ 3 2 + g Ḃ1- 2,1 , (4.6)
where

V (x) = - 1 π R O(α, x) α dα. (4.7)
Proof. i) The main difficulty is to extract the elliptic component from T e (f )g. To uncover it, we shall perform an integration by parts in α. The first key point is that

∆ α g x -∆α g x = g x (• + α) -g x (• -α) α = ∂ α (g(• + α) + g(• -α) -2g(•)) α = - ∂ α (s α g) α .
Consequently, directly from the definition of T e (f )g, by integrating by parts in α, we obtain that

T e (f )g = 1 2π ∂ α (s α g) α E(α, •) dα = 1 2π s α g α 2 E(α, •) dα - 1 2π s α g α ∂ α E(α, •) dα. ( 4.8)
We now have to estimate the coefficients E(α, •) and ∂ α E(α, •). Lemma 4.5. i) We have

E(α, x) = f x (x) 2 1 + f x (x) 2 + Q(α, x) (4.9)
for some function Q satisfying

|Q(α, x)| |s α f (x)| |α| + 1 α α 0 s η f x (x) dη . (4.10)
ii) Furthermore,

|∂ α E(α, x)| ≤ C δα f x (x) |α| + |δ α f x (x)| |α| + |s α f (x)| |α| 2 + 1 α 2 α 0 s η f x (x) dη (4.11)
for some fixed constant C.

Proof. i) We introduce the function

F (a) = a 2 1 + a 2 .
Then we have the identity (4.9) with

Q := 1 2 F (∆ α f ) + F ( ∆α f ) -F (f x ). (4.12)
Since F is bounded, the Taylor formula implies that, for all (a, b)

∈ R 2 , 1 2 (F (a) + F (b)) -F a + b 2 ≤ F L ∞ 8 |a -b| 2 .
On the other hand, since F is bounded, one has the obvious inequality

1 2 (F (a) + F (b)) -F a + b 2 ≤ 2 F L ∞ .
By combining these two inequalities, we find that

1 2 (F (a) + F (b)) -F a + b 2 |a -b| .
Since F is even we have

F (b) = F (-b) and hence 1 2 (F (a) + F (b)) -F a -b 2 |a + b| .
We now apply this inequality with a = ∆ α f and b = ∆α f . Since, by definition,

D α f = ∆ α f -∆α f , S α f = ∆ α f + ∆α f , we conclude that 1 2 (F (∆ α f ) + F ( ∆α f )) -F 1 2 D α f |S α f | . (4.13)
We now use the fact that F is Lipschitz to infer from (4.1) that

F 1 2 D α f -F (f x ) 1 α α 0 s η f x dη . (4.14)
In light of (4.12), by using the triangle inequality, it follows from (4.13) and (4.14) that

|Q| |S α f | + 1 α α 0 s η f x dη ,
which gives the result (4.10).

ii) Since

E(α, •) = 1 2 F (∆ α f ) + 1 2 F ( ∆α f ),
and since F is bounded, the chain rule implies that

|∂ α E(α, •)| |∂ α ∆ α f | + ∂ α ∆α f .
By combining this estimate with the identities It follows directly from (4.8) and (4.9) that T e (f )g = 1 2π

2∆ α f = S α f + D α f, 2 ∆α f = S α f -D α f, we deduce that |∂ α E(α, •)| |∂ α S α f | + |∂ α D α f |.
f 2 x 1 + f 2 x s α g α 2 dα + 1 2π s α g α Q(α, •) α -∂ α E(α, •) dα. (4.15)
Observe that

s α g α 2 dα = -s α g∂ α 1 α dα = ∂ α s α g α dα = g x (x -α) -g x (x + α) α dα = -2 ∆ α g x dα = 2πΛg,
where we used (1.3). So, the first term in the right-hand side of (4.15) is the wanted elliptic component

f 2 x 1 + f 2 x Λg.
To conclude the proof of the first statement in (4.6), it remains only to prove that the second term in the right-hand side of (4.15) is a remainder term. Putting for shortness

I = s α g α Q(α, •) α -∂ α E(α, •) dα L 2
, we will prove that

I f Ḣ 3 2 + g Ḃ1- 2,1 . (4.16) The L ∞ -norm of Q(α,•) α -∂ α E(α, •) is controlled from (4.10
) and (4.11). We have

I I 1 + I 2 + I 3 + I 4
with

I 1 = s α g L 2 |α| 1- δα f x L ∞ |α| dα |α| , I 2 = s α g L 2 |α| 1- δ α f x L ∞ |α| dα |α| , I 3 = s α g L 2 |α| 1- s α f L ∞ |α| 1+ dα |α| , I 4 = s α g L 2 |α| 1- 1 |α| 1+ α 0 s η f x L ∞ dη dα |α| , (4.17)
where, as above, we have distributed the powers of |α| in a balanced way. Using the Cauchy-Schwarz inequality and the definition (2.1) of Besov semi-norms, it follows that

I 1 + I 2 ≤ g Ḃ1- 2,2 f x Ḃ ∞,2
.

and, similarly, it results from (2.2) that

I 3 ≤ g Ḃ1- 2,2 f Ḃ1+ ∞,2
.

Consequently, the Sobolev embeddings

Ḃ1- 2,1 → Ḃ1- 2,2 , Ḣ 3 2 + (R) → Ḃ ∞,2 (R), imply that I 1 + I 2 + I 3 g Ḃ1- 2,1 f Ḣ 3 2 + .
To estimate I 4 , the key point consists in using the Cauchy-Schwarz inequality to verify that

1 |α| 1+ α 0 s η f x L ∞ dη ∞ 0 s µ f x 2 L ∞ µ 2 dµ µ 1 2
(notice that the variable η above could be negative, while µ here is always positive). It follows from (2.1) that

∞ 0 s µ f x 2 L ∞ µ 2 dµ µ 1 2 f x Ḃ ∞,2 f Ḣ 3
2 + , we obtain, using again (2.1) with (p, q, s) = (2, 1, 1 -),

I 4 f Ḣ 3 2 + s α g L 2 |α| 1- dα |α| f Ḣ 3 2 + g Ḃ1- 2,1 . (4.18)
This completes the proof of (4.16) and hence the proof of the desired result (4.6).

ii) It remains to study T o (f )g. Recall that

T o (f )g = - 1 π ∆ α g x O(α, •) dα,
where O(α, •) is given by (4.5). By splitting the factor ∆ α g x into two parts

∆ α g x (x) = g x (x) α - g x (x -α) α ,
we obtain at once that

T o (f )g = V ∂ x g + B
where V is given by (4.7) and where the remainder B is given by

B(x) = 1 π 1 α g x (x -α)O(α, x) dα.
The analysis of B is based on the observation that

g x (x -α) = ∂ α (g(x) -g(x -α)) = ∂ α (δ α g),
which allows to integrate by parts in α, to obtain

B = 1 π δ α g α 1 α O(α, •) -∂ α O(α, •) dα.
Consequently, by writing

B L 2 δ α g L 2 |α| 1 α O(α, •) -∂ α O(α, •) L ∞ dα |α| ,
we are back to the situation already treated in the first step. The estimate for ∂ α O is proved by repeating the arguments used to prove the estimate (4.11) for ∂ α E.

To bound α -1 O(α, x), remembering the expression of O(α, x) given by (4.5), it is sufficient to notice that

O(α, •) α = 1 2 |α| (∆ α f ) 2 1 + (∆ α f ) 2 - ( ∆α f ) 2 1 + ( ∆α f ) 2 ≤ 1 2 |α| ∆ α f -∆α f (1 + (∆ α f ) 2 )(1 + ( ∆α f ) 2 ∆ α f + ∆α f ≤ |S α f | |α| • (4.19) This gives that |O(α, x)| |s α f (x)| / |α| 2 . Therefore, we obtain that the L ∞ -norm of O(α,•) α -∂ α O(α, •
) is estimated by the right-hand side of (4.11). Then we may repeat the arguments used in the proof of the first step to estimate I. We call the attention to the fact that, previously, in (4.17), the expressions involved the more favorable symmetric differences s α g instead of δ α g. However, this is not important for our purpose since, to estimate I 1 , . . . , I 4 , we used only the characterization of Besov norms valid for 0 < s < 1, which involves only the finite differences δ α f . This proves that B L 2 is controlled by the right-hand side of (4.16), which implies that B ∼ 0. Lemma 4.6. Let 0 < ν < < 1/2. There exists a positive constant C > 0 such that

V C 0,ν = V L ∞ + sup y∈R |V (x + y) -V (x)| |y| ν ≤ C f 2 Ḣ1 ∩ Ḣ 3 2 + .
Proof. As already seen, we have

V (x) = - 1 π R O(α, x) α dα,
where

O(α, •) = M α (f )S α f with M α (f ) = 1 2 ∆ α f -∆α f (1 + ( ∆α f ) 2 )(1 + ( ∆α f ) 2 . Since |M α (f )| ≤ |∆ α f | + ∆α f , we obtain that V L ∞ ≤ 2 ∆ α f L ∞ + ∆α f L ∞ S α f L ∞ dα |α| ≤ δ α f L ∞ + δα f L ∞ |α| 1- s α f L ∞ |α| 1+ dα |α| ≤ f Ḃ1- ∞,2 f Ḃ1+ ∞,2 f 2 Ḣ1 ∩ Ḣ 3 2 +
, where we used the Cauchy-Schwarz inequality, the definitions (2.1) and (2.2) of the Besov semi-norms, and the Sobolev embedding.

We now have to estimate the Hölder-modulus of continuity of V . Given y ∈ R and a function u = u(x), we introduce the function [u] y defined by

[u] y (x) = u(x + y) -u(x) |y| ν • We want to estimate the L ∞ -norm of [V ] y uniformly in y ∈ R. Notice that [O(α, •)] y = M α (f )S α ([f ] y ) + [M α (f )] y τ y (S α f )
where τ y u(x) = u(x + y). The contribution of the first term is estimated as above: by setting δ = -ν > 0, we have

M α (f )S α ([f ] y ) L ∞ dα ≤ ∆ α f L ∞ + ∆α f L ∞ S α [f ] y L ∞ dα |α| ≤ δ α f L ∞ + δα f L ∞ |α| 1-δ s α [f ] y L ∞ |α| 1+δ dα |α| ≤ f Ḃ1-δ ∞,2 [f ] y Ḃ1+δ ∞,2
. Now, using Plancherel theorem and the inequality e iyξ -1 ≤ |yξ| ν , we have

[f ] y Ḃ1+δ ∞,2 [f ] y Ḣ 3 2 +δ f Ḣ 3 2 +δ+ν = f Ḣ 3 2 + , since δ + ν = . On the other hand, since |[M α (f )] y | |∆ α [f ] y | + ∆α [f ] y ,
by repeating the previous arguments, we get

[M α (f )] y τ y (S α f ) L ∞ dα ≤ δ α [f ] y L ∞ + δα [f ] y L ∞ |α| 1-ν s α f L ∞ |α| 1+ν dα |α| ≤ [f ] y Ḃ1-ν ∞,2 f Ḃ1+ν ∞,2 f Ḣ 3 2 f Ḣ 3 2 +ν . This concludes the proof of Lemma 4.6
This completes the proof of Theorem 1.2.

Cauchy problem

In this section we prove Theorem 1.4 about the Cauchy problem. We prove the uniqueness by estimating the difference of two solutions. With regards to the existence, we construct solutions to the Muskat equation as limits of solutions to a sequence of approximate nonlinear systems, following here [START_REF] Alazard | On the water waves equations with surface tension[END_REF][START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF][START_REF] Lannes | Well-posedness of the water waves equations[END_REF][START_REF] Lannes | Water waves: mathematical analysis and asymptotics[END_REF]. We split the analysis in three parts.

(1) Firstly, we prove that the Cauchy problem for these approximate systems are well-posed locally in time by means of an ODE argument. (2) Secondly, we use Theorem 1.2 and an elementary L 2 -estimate for the paralinearized equation to prove that the solutions of the later approximate systems are bounded in C 0 ([0, T ]; Ḣ1 (R) ∩ Ḣs (R)) on a uniform time interval. (3) The third task consists in showing that these approximate solutions converge to a limit which is a solution of the Muskat equation. To do this, one cannot apply standard compactness results since the equation is non-local. Instead, we prove that the solutions form a Cauchy sequence in an appropriate space, by estimating the difference of two solutions.

Approximate systems.

To define approximate systems, we use a version of Galerkin's method based on Friedrichs mollifiers. We find convenient to use smoothing operators which are projections and consider, for n ∈ N \ {0}, the operators J n defined by

J n u(ξ) = û(ξ) for |ξ| ≤ n, J n u(ξ) = 0 for |ξ| > n.
Notice that J n is a projection, J 2 n = J n . This will allow us to simplify some technical arguments. Now we consider the following approximate Cauchy problems:

∂ t f + Λf = J n T (f )f , f | t=0 = J n f 0 . (5.1)
The following lemma states that this system has smooth local in time solutions.

Lemma 5.1. For all f 0 ∈ Ḣ1 (R), and any n ∈ N\{0}, the initial value problem (5.1) has a unique maximal solution, for some time T n > 0, of the form

f n = J n f 0 + u n where u n ∈ C 1 ([0, T n [; H ∞ (R)) is such that u n (0) = 0. Moreover, either T n = +∞ or lim sup t→Tn u n (t) L 2 = +∞. ( 5.2)
Proof. We begin by studying an auxiliary system. Consider the following Cauchy problem

∂ t f + J n Λf = J n T (J n f )J n f , f | t=0 = J n f 0 . (5.3)
Set u = f -J n f 0 . Then the Cauchy problem (5.3) has the form

∂ t u = F n (u), u| t=0 = 0, (5.4)
where

F n (u) = -ΛJ n u -ΛJ n f 0 + J n T (J n (f 0 + u))J n (f 0 + u)
(we have used J 2 n = J n to simplify the expression of F ). The operator J n is a smoothing operator: it is bounded from Ḣ1 (R) into Ḣµ (R) for any µ ≥ 1, and from L 2 (R) into H µ (R) for any µ ≥ 0. Consequently, if u belongs to L 2 (R), then J n (f 0 + u) belongs to Ḣµ (R) for any µ ≥ 1. Thus, it follows from statement i) in Proposition 2.3 and the assumption f 0 ∈ Ḣ1 (R) that F n maps L 2 (R) into itself. This shows that (5.4) is in fact an ODE with values in a Banach space for any n ∈ N\{0}.

The key point is that statement iii) in Proposition 2.3 implies that the function F n is locally Lipschitz from L 2 (R) to itself. Consequently, the Cauchy-Lipschitz theorem gives the existence of a unique maximal solution

u n in C 1 ([0, T n [; L 2 (R)). Then the function f n = J n f 0 + u n is a solution to (5.3). Since J 2 n = J n , we check that the function (I -J n )f n solves ∂ t (I -J n )f n = 0, (I -J n )f n | t=0 = 0.
This shows that (I -J n )f n = 0, so J n f n = f n . Consequently, the fact that f n solves (5.3) implies that f n is also a solution to (5.1).

The alternative (5.2) is a consequence of the usual continuation principle for ordinary differential equations. Eventually, integrating (5.4) in time and using the fact that J n is a smoothing operator, we obtain that u n belongs to

C 0 ([0, T n [; H ∞ (R)). Using again (5.4), we conclude that ∂ t u n belong to C 0 ([0, T n [; H ∞ (R)).

5.2.

A priori estimate for the approximate systems. In this paragraph we prove two a priori estimates which will play a key role to prove uniform estimates for the solutions (f n ) and also to estimate the differences between two such solutions. We begin with the following estimate in L 2 (R) ∩ Ḣs (R).

Proposition 5.2. For all real number s ∈ (3/2, 2), there exists a positive constant C > 0 and a non-decreasing function F : R → R such that, any n ∈ N \ {0}, for any T ∈ (0, T n ), the norm

M n (T ) = sup t∈[0,T ] f n (t) -f 0 2 L 2 ∩ Ḣs satisfies M n (T ) + C 1 + K 2 T 0 f n (t) 2 Ḣs+ 1 2 dt ≤ (2 + T ) 2 f 0 2 Ḣ1 ∩ Ḣs + T F sup t∈[0,T ] f n (t) 2 Ḣ1 ∩ Ḣs , (5.5)
where K := sup

(t,x)∈[0,T ]×R |∂ x f n (t, x)| . (5.6)
Proof. Set T n (f ) = J n T (f )f . We estimate the • L 2 -norm and • Ḣs -norm by different methods.

First step : low-frequency estimate. Since

∂ t f n + Λf n = J n T (f n )f n , f n | t=0 = J n f 0 , (5.7)
we have

f n (t) = exp (-tΛ) J n f 0 + t 0 exp -(t -t )Λ T n (f n )(t ) dt , so f n (t) -J n f 0 = (exp (-tΛ) -I) J n f 0 + t 0 exp -(t -t )Λ T n (f n )(t ) dt ,
where I denotes the identity operator. Using the Fourier transform and Plancherel identity, one obtains immediately that

(exp (-tΛ) -I) J n f 0 L 2 ≤ tΛJ n f 0 L 2 ≤ T f 0 Ḣ1 .
On the other hand,

exp

-(t -t )Λ T n (f n )(t ) L 2 ≤ T n (f n )(t ) L 2 .
Consequently,

f n (t) -J n f 0 L 2 ≤ T f 0 Ḣ1 + T sup t ∈[0,T ] T n (f n )(t ) L 2 .
Now we want to replace the left hand side of the above inequality by f n (t) -f 0 L 2 .

To do so, notice that, since the spectrum of

J n f 0 -f 0 is contained in {|ξ| ≥ 1}, we have J n f 0 -f 0 L 2 ≤ f 0 Ḣ1 .
By combining the above estimates, we deduce that

f n (t) -f 0 L 2 ≤ (1 + T ) f 0 Ḣ1 + T sup t ∈[0,T ] T n (f n )(t ) L 2 .
Now, we estimate the L 2 -norm of the nonlinearity T n (f n ) by means of the first statement in Theorem 1.2. We conclude that, for T < 1,

f n (t) -f 0 2 L 2 ≤ 2(1 + T ) 2 f 0 2 Ḣ1 + CT 2 sup [0,T ] f n 2 Ḣ1 sup [0,T ] f n 2 Ḣ 3 2 .
This is in turn estimated by the right side of (5.5). This concludes the first step.

Second step : High frequency estimate. Denote by (•, •) the scalar product in L 2 (R).

To estimate the Ḣs -norm of f n , we make act Λ s on the equation, and then take its scalar product with Λ s f n . We get

(∂ t Λ s f n , Λ s f n ) + (Λ s+1 f n , Λ s f n ) = Λ s T n (f n ), Λ s f n .
Since the Muskat equation is parabolic of order one, we will be able to gain one half-derivative. We exploit this parabolic regularity by writing that

(Λ s+1 f n , Λ s f n ) = f n 2 Ḣs+ 1
2 , and

Λ s T n (f n ), Λ s f n = Λ s J n T (f n )f n , Λ s f n = Λ s T (f n )f n , J n Λ s f n = Λ s T (f n )f n , Λ s f n since J n f n = f n , = Λ s-1 2 T (f n )f n , Λ s+ 1 2 f n .
Consequently, we find 1 2

d dt f n 2 Ḣs + f n 2 Ḣs+ 1 2 = Λ s-1 2 T (f n )f n , Λ s+ 1 2 f n .
We next use a variant of the paralinearization formula given by Corollary 1.3. Set = s -3 2 .

We claim that, for any function g,

Λ 1+ (T (g)g) = V (g)∂ x Λ 1+ g + g 2 x 1 + g 2 x Λ 2+ g + Λ 1+ R (g)
where V (g) and R (g) are two functions satisfying, for any fixed ν < ,

R (g) Ḣ1+ ≤ F( g Ḣ1 ∩ Ḣs ) g Ḣs+ 1 2 -2 , (5.8)
V (g) C 0,ν ≤ F( g Ḣ1 ∩ Ḣs ) g Ḣs+ 1 2 -2 .
(5.9)

where F depends only on (that is s) and ν (which will be specified later). The proof of this claim is similar to the one of (1.9). With notations as above, set

V n = V (f n ), R n = Λ 1+ R (f n ), γ n = f 2 nx 1 + f 2 nx where f nx = ∂ x f n .
Then,

1 2 d dt f n 2 Ḣs + f n 2 Ḣs+ 1 2 = γ n Λ s+ 1 2 f n , Λ s+ 1 2 f n , + V n ∂ x Λ s-1 2 f n + R n , Λ s+ 1 2 f n .
(5.10)

Now the key point is that

f n 2 Ḣs+ 1 2 -γ n Λ s+ 1 2 f n , Λ s+ 1 2 f n = Λ s+ 1 2 f n 2 1 + f 2 nx dx.
On the other hand, the Cauchy-Schwarz inequality and the estimate (5.8) imply that

R n , Λ s+ 1 2 f n ≤ R n L 2 Λ s+ 1 2 f n L 2 ≤ F( f n Ḣ1 ∩ Ḣs ) f n Ḣs+ 1 2 -2 f n Ḣs+ 1 2 .
It remains to estimate the contribution of V n to the second term in the right-hand side of (5.10). Here we use the commutator estimate given by Lemma 2.6. To do so, one uses the identity HΛ = -∂ x where H is the Hilbert transform, to write

V n ∂ x Λ s-1 2 f n = -V n Λ s+ 1 2 Hf n .
Since H is skew-symmetric, we deduce that

V n ∂ x Λ s-1 2 f n , Λ s+ 1 2 f n = 1 2 H, V n Λ s+ 1 2 f n , Λ s+ 1 2 f n .
Now we exploit the regularity result for V n given by (5.9). Fix ν = 2 /3 and θ = /2. By applying the commutator estimate in Lemma 2.6, we obtain

H, V n Λ s+ 1 2 f n L 2 V n C 0,ν Λ s+ 1 2 f n H -θ V n C 0,ν f n Ḣs+ 1 2 -2 ≤ F( f n Ḣ1 ∩ Ḣs ) f n Ḣs+ 1 2 -2 .
So, by combining the above estimates,

1 2 d dt f n 2 Ḣs + Λ s+ 1 2 f n 2 1 + f 2 nx dx ≤ F( f n Ḣ1 ∩ Ḣs ) f n Ḣs+ 1 2 -2 f n Ḣs+ 1 2 .
The end of the proof will consist in exploiting the parabolic regularity and a variant of Gronwall's lemma to absorb the right-hand side. Set

K(t) = sup x∈R |∂ x f n (t, x)|. Then 1 1 + f 2 nx ≥ 1 1 + K 2 , so 1 2 d dt f n 2 Ḣs + C 1 + K 2 f n 2 Ḣs+ 1 2 ≤ F( f n Ḣ1 ∩ Ḣs ) f n Ḣs+ 1 2 -2 f n Ḣs+ 1 2 .
(5.11) Then we observe that

F( f n Ḣ1 ∩ Ḣs ) f n Ḣs+ 1 2 -2 f n Ḣs+ 1 2 ≤ C 2(1 + K 2 ) f n 2 Ḣs+ 1 2 + 1 + K 2 2C F( f n Ḣ1 ∩ Ḣs ) 2 f n 2 Ḣs+ 1 2 -2 .
Since K f n Ḣ1 ∩ Ḣs by Sobolev embedding, up to modifying the value of the function F, by inserting the above inequality in (5.11), we get 1 2

d dt f n 2 Ḣs + C 2(1 + K 2 ) f n 2 Ḣs+ 1 2 ≤ F( f n Ḣ1 ∩ Ḣs ) f n 2 Ḣs+ 1 2 -2 .
(5.12)

To conclude, it will suffice to replace in the right side the norm

f n 2 Ḣs+ 1 2 -2 by f n 2
Ḣs . To do this, we begin by using the interpolation inequality:

f n 2 Ḣs+ 1 2 -2 ≤ f n 2θ Ḣs f n 2-2θ Ḣs+ 1 2 , (5.13)
for some θ ∈ (0, 1). Next, because of the Young's inequality

xy ≤ 1 p x p + 1 p y p with 1 p + 1 p = 1, (5.14)
applied with p = 2/(2 -2θ), we infer that 1 2

d dt f n 2 Ḣs + C 4(1 + K 2 ) f n 2 Ḣs+ 1 2 ≤ F( f n Ḣ1 ∩ Ḣs ) f n 2 Ḣs , (5.15)
where as above we modified the value of the function F. From this, it is now an easy matter to obtain the conclusion of the proposition. Firstly, integration of the above estimate gives

1 2 f n (t) 2 Ḣs + C 4(1 + K 2 ) t 0 f n (t ) 2 Ḣs+ 1 2 dt ≤ 1 2 f n (0) 2 Ḣs + t sup t ∈[0,t] F( f n (t ) Ḣ1 ∩ Ḣs ) f n (t ) 2
Ḣs .

Modifying F(•) and C, we deduce that Ḣs follows from the triangle inequality and the fact that f n (0) Ḣs ≤ f 0 Ḣs . We will also need another energy estimate to compare two different solutions f 1 and f 2 . The main difficulty here will be to find the optimal space in which one can perform an energy estimate. The most simpler way to do so would be to estimate their difference f 1 -f 2 in the biggest possible space and to use an interpolation inequality to control the latter in a space of smoother function. This suggests to estimate f 1 -f 2 in C 0 ([0, T ]; L 2 (R)). On the other hand, by thinking of the fluid problem, we might think that it is compulsory to control the difference between the two functions parametrizing the two free surfaces in a space of smooth functions. We will see later that, somewhat unexpectedly, that it is enough to estimate f 1 -f 2 in C 0 ([0, T ]; Ḣ1/2 (R)). In this direction, we will use the following proposition. Proposition 5.3. i) For all s in (3/2, 2), there exists a non-decreasing function F : R + → R + such that, for any n ∈ N \ {0}, any T > 0, and any functions

f n (t) 2 Ḣs + C 1 + K 2 t 0 f n (t ) 2 Ḣs+ 1 2 dt ≤ f n (0) 2 Ḣs + T F sup t∈[0,T ] f n (t)
f ∈ C 0 ([0, T ]; Ḣ1 (R) ∩ Ḣs (R)), g ∈ C 1 ([0, T ]; H 1 2 (R)) with J n g = g, F ∈ C 0 ([0, T ]; L 2 (R)),
satisfying the equation

∂ t g -J n V (f )∂ x g) + J n 1 1 + f 2 x Λg = F, (5.16)
where V (f ) is as above, we have the estimate

1 2 d dt g 2 Ḣ 1 2 + (Λg) 2 1 + f 2 x dx ≤ F( f Ḣ1 ∩ Ḣs ) g Ḣ1-2 g Ḣ1 + (F, Λg) L 2 .
(5.17)

where = s -3/2 and C = F f L ∞ ([0,T ]; Ḣ1 ∩ Ḣs ) .
ii) Moreover, the same result is true when one replaces J n by the identity.

Proof. To prove (5.17) we take the L 2 -scalar product of the equation (5.16) with Λg. Since

1 2 d dt g 2 Ḣ 1 2 = (∂ t g, Λg), J n 1 1 + f 2 x Λg , Λg = (Λg) 2 1 + f 2 x dx,
(where we used J n g = g), we only have to estimate J n V (f )∂ x g), Λg . As above, writing ∂ x = -HΛ, where H is the Hilbert transform satisfying H * = -H, we obtain

J n V (f )∂ x g), Λg = 1 2 H, V (f ) Λg, Λg .
Set = s -3/2, ν = 2 /3 and θ = /2. We use Lemma 2.6 to obtain

J n V (f )∂ x g), Λg ≤ 1 2 H, V (f ) Λg L 2 Λg L 2 ≤ V (f ) C 0,ν Λg H -θ Λg L 2 ≤ F( f Ḣ1 ∩ Ḣs ) Λg H -2 Λg L 2 ≤ F( f Ḣ1 ∩ Ḣs ) g Ḣ1-2 g Ḣ1 .
This completes the proof of i) and the same arguments can be used to prove ii).

5.3.

End of the proof. In this paragraph we complete the analysis of the Cauchy problem. We begin by proving the uniqueness part in Theorem 1.2. Proof. Set

g = f -f , M = f L ∞ ([0,T ]; Ḣ1 ∩ Ḣs ) + f L ∞ ([0,T ]; Ḣ1 ∩ Ḣs ) .
We denote by C(M ) various constants depending only on M .

We want to prove that g = 0. To do so, we use the energy estimate in Ḣ1/2 (R). The key point is to write that g is a smooth function, in C

1 ([0, T ]; H 1 2 (R)), satisfying ∂ t g + Λg = T (f )g + F 1 with F 1 = T (f )f -T (f )f .
This term is estimated by means of point ii) in Proposition 2.3 with δ = = s -3/2,

F 1 L 2 = (T (f ) -T (f ))f L 2 ≤ C f -f Ḣ1-f Ḣ 3 2 + = C(M ) g Ḣ1-.
(5.18) Recall from (1.5) that

T (f )g = f 2 x 1 + f 2 x Λg + V (f )∂ x g + R(f, g) where R(f, g) satisfies (setting = s -3/2), R(f, g) L 2 ≤ C f Ḣ 3 2 + g Ḃ1- 2,1 ≤ C(M ) g Ḃ1- 2,1 . (5.19)
Therefore, g satisfies

∂ t g -V ∂ x g + 1 1 + f 2 x Λg = F
where F = F 1 + R(f )g. In view of the estimates (5.18), (5.19) and the embedding Ḣ1

-3 /2 (R) ∩ Ḣ1-/2 (R) → Ḃ1- 2,1 (see Lemma 2.2), we have |(F, Λg)| ≤ F L 2 g Ḣ1 ≤ C(M ) g Ḣ1-2 ∩ Ḣ1-3 2 g Ḣ1 . Hence, it follows from Proposition 5.3 (see point ii)) that 1 2 d dt g 2 Ḣ 1 2 + (Λg) 2 1 + f 2 x dx ≤ C(M ) g Ḣ1-2 ∩ Ḣ1-3 2 g Ḣ1 .
Next, we use interpolation inequalities as in the proof of Proposition 5.2. More precisely, by using arguments parallel to those used to deduce (5.15) from (5.13)-(5.14), we get

1 2 d dt g 2 Ḣ 1 2 + 1 4(1 + f x L ∞ t,x ) (Λg) 2 dx ≤ C(M ) g 2 Ḣ 1 2 .
This obviously implies that

1 2 d dt g 2 Ḣ 1 2 ≤ C(M ) g 2 Ḣ 1 2 .
Since g(0) = 0, the Gronwall's inequality implies that g Ḣ 1 2 = 0 so g = 0, which completes the proof.

Having proved the uniqueness of solutions, we now study their existence. The key step will be here to apply the a priori estimates proved in Proposition 5.2. This will give us uniform bounds for the solutions f n defined in §5.1.

Lemma 5.5. There exists T

0 > 0 such that T n ≥ T 0 for all n ∈ N \ {0} and such that (f n -f 0 ) n∈N is bounded in C 0 ([0, T 0 ]; H s (R)).
Proof. We use the notations of §5.1 and Proposition 5. Let us prove this claim by contradiction. Assume that for some n there exists τ n ∈ I n such that N n (τ n ) ≥ A and consider the smallest of such times (then τ n > 0 since T → N n (T ) is continuous and N n (0) < A by construction). Then, by definition, for all 0 < T ≤ τ n , one has N n (T ) ≤ A and N n (τ n ) = A. Since ∂ x f n (t) L ∞ (R)

f n (t) Ḣ1 ∩ Ḣs (see (3.4)), we have a uniform control of the L ∞ x -norm of ∂ x f n on [0, T ] in terms of A, hence we are in position to apply the a priori estimate (5.5). Now, if we add f 0 2 Ḣ1 ∩ Ḣs to both sides of (5.5) we deduce that

N n (τ n ) + C 1 + C(A) 2 τn 0 f n (t) 2 Ḣs+ 1 2 dt ≤ 3(1 + τ n ) 2 f 0 2 Ḣ1 ∩ Ḣs + τ n F(N n (τ n )).
We infer that

A = N n (τ n ) ≤ 3(1 + τ n ) 2 f 0 2 Ḣ1 ∩ Ḣs + τ n F(N n (τ n )) ≤ 3(1 + T 0 ) 2 f 0 2 Ḣ1 ∩ Ḣs + T 0 F(A) < A,
hence the contradiction. We thus have proved that, for all n ∈ N and all T ≤ min{T 0 , T n }, we have sup

t∈[0,T ] f n (t) -f 0 2 L 2 ∩ Ḣs ≤ A.
This obviously implies that sup t∈[0,T ]

f n (t) -f 0 L 2 ≤ √ A. Since u n = f n -J n f 0 = f n -f 0 + (I -J n )f 0 ,
and since (I -J n )f 0 L 2 ≤ f 0 Ḣ1 , the previous bound implies that the norm u n (t) L 2 is bounded for all t ≤ min{T 0 , T n }. The alternative (5.2) then implies that the lifespan of f n is bounded from below by T 0 . And the previous inequality shows that (f n -f 0 ) is bounded in C 0 ([0, T 0 ]; H s (R)). This completes the proof.

At that point, we have defined a sequence (f n ) of solutions to well-chosen approximate systems. The next task is to prove that this sequence converges. Here a word of caution is in order: Ḣs (R) is not a Banach space when s > 1/2. To overcome this difficulty, we use the fact that u n = f n -f 0 is bounded in C 0 ([0, T 0 ]; H s (R)), where H s (R) is the nonhomogeneous space L 2 (R) ∩ Ḣs (R), which is a Banach space. We claim that, in addition, (u n ) is a Cauchy sequence in C 0 ([0, T 0 ]; H s (R)) for any s < s. Let us assume this claim for the moment. This will imply that (u n ) converges in the latter to some limit u. Now, setting f = f 0 + u and using the continuity result for T (f )f given by iii) in Proposition 2.3, we verify immediately that f is a solution to the Cauchy problem for the Muskat equation. It would remain to prove that u is continuous in time with values in H s (R) (instead of H s (R) for any s < s). For the sake of shortness, this is the only point that we do not prove in details in this paper (referring to [START_REF] Alazard | On the Cauchy problem for gravity water waves[END_REF] for the proof of a similar result in a case with similar difficulties).

To conclude the proof of Theorem 1.4, it remains only to establish the following Lemma 5.6. For any real number s in [0, s), the sequence (u n ) is a Cauchy sequence in C 0 ([0, T 0 ]; H s (R)).

Proof. The proof is in two steps. We begin by proving that (f n ) is a Cauchy sequence in C 0 ([0, T 0 ]; Ḣs (R)) for 1/2 ≤ s < s. Then, we use this result and an elementary L 2 -estimate to infer that (u n ) is a Cauchy sequence in C 0 ([0, T 0 ]; L 2 (R)). By using estimates parallel to those used to prove Lemma 5.4, one obtains that (f n ) is a Cauchy sequence in C 0 ([0, T 0 ]; Ḣ 1 2 (R)). Now consider 1/2 < s < s. By interpolation, there exists α in (0, 1) such that

u Ḣs u α Ḣ 1 2 u 1-α Ḣs .
Consequently, since (f n ) is bounded in C 0 ([0, T 0 ]; Ḣs (R)), we deduce that (f n ) is a Cauchy sequence in C 0 ([0, T 0 ]; Ḣs (R)) for any s < s.

It remains only to prove that (u n ) is a Cauchy sequence in C 0 ([0, T 0 ]; L 2 (R)). To do so, we proceed differently. Starting from (see (5.7)),

∂ t f n + Λf n = J n T (f n )f n , we obtain that u n -u p = f n -f p -(J n -J p )f 0 satisfies ∂ t (u n -u p ) + Λ(u n -u p ) = F np + G np (5.20)
where

F np = J n (T (f n )f n -T (f p )f p ), G np = (J n -J p ) -Λf 0 + T (f p )f p .
We now use an elementary L 2 -estimate. We take the L 2 -scalar product of the equation (5.20) with u n -u p , to obtain, since u n (0) -u p (0) = 0, is arbitrarily small for n, p large enough. This completes the proof.

  which proves the claim (3.7). Now, by combining the two claims (3.6), (3.7) with (3.3) and (3.5), we obtain that

  Then the second estimate (4.11) follows from the values for ∂ α S α f and ∂ α D α f given by Lemma 4.3.

Lemma 5 . 4 .

 54 Assume that f and f are two solutions of the Muskat equation with the same initial data and satisfying the assumptions of Theorem 1.4. Then f = f .

2 . 2 L 2 ∩ 2 Ḣ1 2 Ḣ1 3 ( 1 + T 0 ) 2 f 0 2 Ḣ1∩

 22222312 Given T < T n , we defineM n (T ) = sup t∈[0,T ] f n (t) -f 0 Ḣs , N n (T ) = M n (T ) + f 0 ∩ Ḣs .Denote by F the function whose existence is the conclusion of Proposition 5.2 and set A = 10 f 0 ∩ Ḣs .We next pick 0 < T 0 ≤ 1 small enough such that Ḣs + T 0 F(A) < A, We claim the uniform bound ∀n ∈ N \ {0}, ∀T ∈ I n := [0, min{T 0 , T n }), N n (T ) < A.

. 2 )≤ 1 min(n, p) u L 2 ,

 212 sup t∈[0,T ] u n (t) -u p (t) L 2 ≤ F np L 1 ([0,T ];L 2 ) + G np L 2 ([0,T ]; Ḣ-1 2 )So it remains only to prove that F np L 1 ([0,T ];L 2 ) and G np L 2 ([0,T ]; Ḣ-1 are arbitrarily small for n, p large enough. Here we use the result proved in the first part of the proof. Namely, since (f n ) is a Cauchy sequence in C 0 ([0, T ]; Ḣ1 (R) ∩ Ḣ 3 2 (R)), we deduce from point iii) in Proposition 2.3 thatT (f n )f n -T (f p )f p is small in C 0 ([0, T ]; L 2 (R))for n, p large enough. On the other hand, using the estimate(J n -J p )u Ḣ-1 2we verify that G np L 2 ([0,T ]; Ḣ-1 2 )

+ . We now proceed as in the previous proof. More precisely, we balance the powers of α, use the Cauchy-Schwarz inequality, the definition of the

Acknowledgments

T. A. acknowledges the support of the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR). O. L. has been partially supported by the National Grant MTM2014-59488-P from the Spanish government and the ERC through the Starting Grant project H2020-EU. 1.1.-63922