Tambet Treimuth
email: treimuth@recherche.enac.fr

Daniel Delahaye
email: delahaye@recherche.enac.fr

Parallel Complexity Copmputation Based on Dynamical Systems

This paper describes a parallel approach for computing an air traffic complexity metric that is based on dynamical systems. It is important in order to quantify the complexity and geometric features of traffic. Lyapunov exponents are used for understanding the geometry of current air traffic in the airspace. It is computed on different points on a spatial grid. Those values can indicate areas where air traffic controllers must pay more attention because the complexity is higher at those points. A parallel implementation of the air traffic complexity metric computation is introduced in this paper. Serial implementation of this problem is too slow for real world problems. This process is parallelizable because Lyapunov exponent values can be computed simultaneously for each grid point in the airspace.

Intdouction

The number of aircraft in the airspace that air traffic controllers have to direct is increasing but there exists a maximum number of aircraft that one air traffic controller can handle. It is not enough if we just take into account the maximum number of aircraft that are allowed at each moment in a sector of air traffic controller. There are situations when they can accept more aircraft. It depends on the geometry of the traffic. Geometric features of the traffic have to be included to quantify the complexity. This method for evaluating complexity of air traffic takes into account the intrinsic traffic disorder. ATC controllers can have different workload even if the number of aircraft present in the sector is the same. This complexity computation method is based on dynamical systems. There exists many different methods for measuring the complexity of air traffic. The first complexity indicator incorporating structural considerations along with the simple number of aircraft is the "Dynamic Density" of Laudeman et al. from NASA [START_REF] Laudeman | Dynamic density : an air traffic management metric[END_REF]. The "Dynamic Density" is a weighted sum of the traffic density (number of aircraft), the number of heading changes (> 15 degrees), the number of speed changes (>0.02 Mach), the number of altitude changes (>750 ft), the number of aircraft with 3-D Euclidean distance between 0-25 nautical miles, the number of conflicts predicted in 25-40 nautical miles. These factors are summed together using weighting factors that were determined by showing different traffic scenarios to several controllers. B.Sridhar from NASA [START_REF] Sridhar | Airspace complexity and its application in air traffic management[END_REF], has developed a model to predict the evolution of such a metric in the near future. Efforts to define "Dynamic Density" have identified the importance of a wide range of potential complexity factors, including structural considerations. However, the instantaneous position and speeds of the traffic itself does not appear to be enough to describe the total complexity associated with an airspace. A few previous studies have attempted to include structural consideration in complexity metrics, but have done so only to a restricted degree. For example, the Wyndemere Corporation proposed a metric that included a term based on the relationship between aircraft headings and dominant geometric axis in a sector [START_REF] Inc | An evaluation of air traffic control complexity[END_REF]. The importance of including structural consideration has been explicitly identified in work at Eurocontrol. In a study to identify complexity factors using judgment analysis, Airspace Design was identified as the second most important factor behind traffic volume [START_REF] Kirwan | Investigating complexity factors in u.k. air traffic management[END_REF]. Histon,et,al. [4], [START_REF] Histon | Structural consideration and cognitive complexity in air traffic control[END_REF] investigated how this structure can be used to support structure-based abstractions that controllers appear to use to simplify traffic situations. The previous models do not take into account the intrinsic traffic disorder which is related to the complexity. The first efforts related with disorder can be found in [START_REF] Delahaye | Air traffic complexity : Towards intrinsic metrics[END_REF]. This paper introduces two classes of metrics which measure the disorder of a traffic pattern.G.Aigoin has extended and refined the geometrical class by using a cluster based analysis [START_REF] Aigoin | Air traffic complexity modeling[END_REF]. All the previous metrics capture only one feature of the complexity and are not able to produce an aggregate metric which can capture all the possible situations (high-low density, how-low convergence, translation organization, round about organization).

Dynamical Systems

The first part of this algorithm consists of finding a model of dynamical system that represent current traffic situation at the airspace. Later this model can be used for computing complexity values for different points in the airspace. A dynamical system model describes the traffic situation in the airspace. This model consists of linear and non-linear part. This model is based on aircraft positions and velocities at each time step. Based on this input data we can get coefficients of model of dynamical system. The same process is repeated for each time step when the complexity metric is computed.

Linear Model

For this approach a set of aircraft that are flying in the airspace is considered as a dynamical system. A model of dynamical system is computed for each time step. A dynamical system can be described by the following equation: ˙

X = A. X + B ( 1
)
where X is the state vector of the system :

X = x y z T (2)
This equation describes relationship between aircraft velocities ˙ X to aircraft positions X. This model can be used for understanding the behaviour of the traffic in the airspace. Eigenvalues of the matrix A can be used for understanding the evolution of the traffic. See Figure 1 Those values of A and B can be computed with Least Square Minimization method. This way A and B are found that can minimize the error E. For each aircraft i, it is supposed that position X i = [x i , y i , z i ] T and speed vector V i = [vx i , vy i , vz i ] T are part of input data. An error criterion between the dynamical system

E = i=N ∑ i=1 V i -A. X i + B 2
The position matrix and the velocity matrix have following structure:

X =     x 1 ... x N y 1 ... y N z 1 ... z N 1 ... 1     V =   vx 1 ... vx N vy 1 ... vy N vz 1 ... vz N  

Non Linear Extensions in Space

A non linear dynamical system is summarized by the following equation [START_REF] Delahaye | Air traffic complexity based on dynamical systems[END_REF]:

˙ X(t) = f ( X) ( 3
)
where X is the state vector of the system ( X = [x, y, z] T ) and f : C 2 vector field, describe systems which integral curves may fit the observed trajectories. This equation associates a vector speed ˙ X to a position in the space coordinate X and then synthesis a particular vector field. Based on the observations of the aircraft (positions, speed vectors), the dynamical system has to be adjusted with the minimum error.This fitting is done with a Least Square Minimization (LMS) method for which the following criterion is used :

E 1 = i=N ∑ i=1 V i -f ( X i ) 2 (4)
where N is number of observations. If we consider criterion E 1 only, it can be shown that there is an infinite number of vector fields f which can be adjusted to the observations. In order to keep the smoothest one, another criterion is added which has to be minimized, the so-called "div-curl" criterion :

E 2 = R 3 α ∇div f ( X) 2 + β ∇curl f ( X) 2 d X (5)
with α, β positive weights controlling the smoothness of the approximation by focusing on constant divergence or constant curl. In the following, we will consider α = β = 0.5; in such case :

E 2 = ∆ f ( X) 2
where ∆ f ( X) is the Laplacian of the vector field f .

The joint minimization of E 1 and E 2 induces a unique optimum vector field:

f ( X) = N ∑ i=1 φ( X -X i ). a i + A. X + B
with a i parameter vectors (one for each observation),

φ( X -X i ) = Q( X -X i 3 )
and

Q =   γ 0 0 0 γ 0 0 0 γ   with γ = ∂ 2 xx + ∂ 2 yy + ∂ 2 zz
The resulting adjustment is done without error (⇒ minE 1 = 0).

When α = β = 0.5, the vector spline function φ has the following structure :

φ( X -X i ) = 12. X -X i (r i = X i -X ).
It must be noticed, that farthest observations has more weight in such computation. In order to compute the smoothest vector field which fit exactly the measures, all observations have to be taken into account in the computation. This velocity field is used for computing the Lyapunov exponents.They are computed at different point of this velocity field. All steps of the algorithms are described in the following section.

Algortihm

The algorithm for computing Lyapunov exponents can be divided into three parts:

• Computation of Dynamical Regression • Computation of gradients

Dynamical Regression

The first part of this method consists of finding the model of dynamical system that correspond to the current traffic in the airspace. This part was described in the previous section. The goal of this step is to find the matrix A and the vector B.That were described previously in the linear model part. Those coefficients describe a velocity field that can be used for computing Lyapunov exponents.

Gradient Computation

The Lyapunov exponents are computed at each gridpoint. Lyapunov exponents values can be computed only if gradients are known at those points. The gradient of the velocity field are computed based on the dynamical system model that was computed at the previous point. Those values can be used as an input data for the for the Runge Kutta method. Those Fig. 3. High Lyapunov exponent value with six aircraft computations for each grid point are independent and can be computed simultaneously.

Lyapunov Exponent Computation

Lyapunov exponent measures the complexity for each grid point where it is computed. Lyapunov exponent measures the divergence of two trajectories that have nearby initial points.Runge-Kutta method is a method for numerical integration.

Lyapunov Exponents

Air traffic controllers can accept more aircraft if aircraft are well structured. The first example shows the situation with six aircraft. There are many potential conflicts. It causes high workload for air traffic controller. See Figure 3 The second example has six aircraft also but all aircraft have same direction. See Figure 4. This is much easier for air traffic controller. Those differences can be analysed with Lyapunov exponents.

This metric for complexity relies on a measure of sensitivity to initial conditions of the underlying dynamical system called Lyapunov exponents. In order to figure out what Lyapunov exponents are, let consider a point and look at its evolution when transported by the dynamical system. Let x 0 be fixed (initial point) and let γ be a point trajectory of the dynamical system associated to the vector field f given by : Assume now that trajectory is disturbed by a small perturbation ε, we have :

γ(t, x 0 ) = x 0 + t 0 f (u, γ(u, x))du (6)
γ(t, x 0 + ε) = γ(t, x 0 ) + ∇ x f (γ(t, x)). ε + o( ε )
where ∇ x f (t, γ(t, x)) is the differential of the vector field f at x. Divergence to nominal trajectory with respect to time is thus γ(t, x 0 )γ(t, x) = D(t, s) (see figure). γ(t, x) being defined as a flow :

∂γ(t, x) ∂t = f (t, γ(t, x)) γ(0, x) = x
with f a smooth vector field, it is possible to show that D(t) satisfies a differential equation also. Given a nominal trajectory γ(t, x 0 ), then divergence of nearby trajectories can be found up to order one in xx 0 by solving :

∂D(t, x) ∂t = ∇ x f (t, γ(t, x)).D(t, x) D(0, x) = x -x 0
If the three space dimensions are considered (x, y, z), and since the previous equation is linear, it can be extended to the matrix form :

dM(t) dt = ∇ x f (t, γ(t, x)).M(t) M(0) = Id
Where each column of the M matrix corresponds to the divergence associated to the principal coordinate axis. This equation is called the variational equation of the flow. The variational equation describes in some sense a linear dynamical system "tangent" to the original one. Let U t (t)Σ(t)V (t) = M(t) be the singular value decomposition of M(t). The Lyapunov

κ(s) = - 1 T T 0 log(Σ ii (t))dt ∀Σ ii (t) ≤ 1
When Lyapunov exponents are high, the trajectory of a point under the action of the dynamical system is very sensitive to initial conditions (or to the parameters on which the vector field may depend), so that, situation in the future is unpredictable.On the other hand, small values of the Lyapunov exponents mean that the future is highly predictable (expected to be comfortable for a controller).The Lyapunov exponent measures the distance between two trajectories. See figure 5).

So, the Lyapunov exponent map determines the area where the underlying dynamical system is organized. It identifies the places where the relative distances between aircraft do not change with time (low real value) and the ones where such distance change a lot (high real value).

Those Lyapunov exponent values can be computed with Runge Kutta algorithm. It adds all te deviation until the last time step. This total sum is equal to the Lyapunov exponent. This value describes the potential workload to the air traffic controller.

Graphical Processing Unit (GPU)

Parallel Compuation

A parallel computing platform and programming model CUDA was used for computing complexity.

CUDA programming is used to solve problems that can be expressed as data parallel computations. Applications that must process big data sets can use this model to speed up computations. Our goal for this project was to distribute data across threads in the way that each thread working on a part of the data. A sequential implementation of algorithm was not fast enough if complexity values for larger problems. For example, if complexity values for the whole country must be computed. A parallel complexity computation algorithm is proposed instead of preceding sequential algorithm. Complexity is evaluated on points evenly spaced on a spatial grid. This velocity field is used to computed Lyapunov exponent values for each grid point. Since velocity field computation and Lyapunov exponent computation was time-consuming with sequential computation, we had to find way to parallelize this part of computation program. Velocity field values computation can be parallelized because previous implementation consisted of many nested loops. Those loops were unrolled because there are not dependencies between different iterations of loops. Velocity field values must be computed for many points of airspace simultaneously also.

Parallel Computation Patterns

There exists many patterns that can be used for parallelizing serial computation. Reduce pattern was used for improving computation time [START_REF] Mccool | Structure Parallel Programming[END_REF]. Reduce pattern can be uses if there exists a set of completely independent operation. It combines all the elements in a collection into a single element using some associative combiner operator. The usefulness of reduce is easy to understand. The serial implementation of serial addition. Four variables are added sequentially.See Figure 6 One simple example is summation, which shows up frequently in mathematical computations. In the reduce pattern, a combine function is used to combine all the elements of a collection pairwise and create a summary value. It is assumed that pair of elements can be combine in different orders.See Figure 7 Paralllel implementation The same computation must be repeated for each time step. Those gradient values must be used for Lyapunov exponent computation. It is used as input values for Runge Kutta algorithm that The airspace for the big problem could be divided into millions of evenly distributed grid points. So, we have to compute those gradient matrices for millions of times. For parallel solution nested loops can be again unrolled. There are no dependencies between gradient matrices.

Velocity Field Computation

Runge Kutta Computation

Lyapunov exponents provide a quantitative measure of the divergence or convergence of nearby trajectories for a dynamical system. Runge-Kutta algorithm can be used to compute Lyapunov exponent value for each grid point. Runge-Kutta algorithm can be executed in parallel for each point in the airspace because there are different gradient values for each grid point. The result of the Runge-Kutta function is the value that describes the complexity of the airspace at the point where it was computed. Previously described steps must be repeated for each time step because the complexity of the airspace changes frequently. Complexities for different time steps can be computed simultaneously because those velocities fields are independent. Previously described computed can be divided into four parts.

If the GPU does not have enough data the computation time could be even slower than with single threaded CPU.This is the reason why parallel computation algorithm must be designed carefully. The matrix A and the vector B that were part of the Equation 1 are computed for each time step. For this computation Intel MKL library is used. These values are computed only once at each time step. The GPU computation is not needed for this part the Intel MKL provided fast enough computation time.

Those A and B describe the traffic situation at that time step. Those values can be used for computed velocity field valued at each point in the airspace. If only linear model is used then those values are not precise enough. All the situations cannot be described with A and B. That is reason why the non linear extension is used. This gives more precise results. This computation is function of the number of aircraft square.

.

Results

Air traffic data from the French airsapce was used for the complexity computation. A GeForce GTX 660 GPU Ti with Compute Capability 3.0 was used for computation. The radar data was used for computing Lyapunov Exponents. The radar data describes trajectories after the conflicts between aircraft were solved. It does not give real complexity values. The complexity values are much lower. The trajectory projections were used for more precise complexity results. See Figure 8. The parallel computation of Lyapunov exponents provided remarkable speed up of computation. See Table I. First column describes number of grid points where Lyapunov exponents were computed. Second column gives information about Lyapunov exponents computation with CPU. The third column describes the computation time with GPU.If the complexity is analysed for the French airspace then the complexity is computed at 200000 points. The complexity of air traffic can be used for measuring the workload. Red color indicates the higher complexity of air traffic and higher workload for air traffic controllers. See figure 9. If the complexity of the air traffic is not so high then air traffic controller could accept more aircraft. See Figure 10. There is not red color on the complexity map that represent high complexity of air traffic. Lyapunov exponent values can be divided into two groups: high and low Lyapunov expoenent values. High Lyapunov exponents represent high sensitivity to initial conditions:

• The future is highly unpredictable

• The traffic has low organization • Represent high rate of change of relative distances Low Lyapunov exponents represent low sensitivity to initial conditions:

• The traffic is highly predictable future • The traffic is organized

• Relative distances has low rate of change The air traffic complexity was computed also for the ATCC in Reims. The computed complexity was compared with number of air traffic controllers who work at ATCC in Reims. The red line describes normalized complexity of traffic. Correlation between two lines can be seen. When the complexity of the ATCC is increasing then the number of Air Traffic Controllers who work is increasing also. IF the complexity is not so high then the number of controllers who work is also decreasing.

Conclusion

This paper presented a method for dynamical systems based complexity computation. The number of aircraft that are in the airspace will increase in the future. We have evaluate how many aircraft can be controlled by one controller. If the complexity is estimated only as a number of aircraft in the sector then it may not describe the situation precisely. Air Traffic Controllers may sometimes accept more aircraft because the structure of traffic is different. So, if we know geometric features of traffic then Air Traffic Controllers may accept more aircraft. This metric can be used for analysing the pattern of air traffic. This complexity is computed at each grid point. The computation is implemented on GPU to reduce the computation time The complexity computation is implemented in CUDA language on the GPU. It improved the computation time because the algorithm consists of similar computation pattern for million of times. This way this complexity metric can be used for real air traffic problems.

Fig. 1 .

 1 Fig. 1. Eigenvalues describe the behaviour of dynamical systems

Fig. 2 .

 2 Fig. 2. Three main steps of complexity computation

Fig. 4 .

 4 Fig. 4. Low Lyapunov Exponent value with six aircraft

Fig. 5 .

 5 Fig. 5. Time evolution of a reference trajectory and a perturbed trajectory

Fig. 6 .

 6 Fig. 6. Serial implementation


David Gérard-Varet
email: gerard-varet@math.jussieu.fr

Aneta Wróblewska-Kamińska

Boundary layer for a non-Newtonian flow over a rough surface

Introduction

The general concern of this paper is the effect of rough walls on fluids. This effect is important at various scales. For instance, in the area of microfluidics, recent experimental works have emphasized the role of hydrophobic rough walls in the improvement of slipping properties of microchannels. Also, in geophysics, as far as large scale motions are concerned, topography or shore variations can be assimilated to roughness. For high Reynolds number flows, an important issue is to understand how localized roughness triggers instabilities, and transition to turbulence. For laminar flows, the point is rather to understand how distributed roughness may have a macroscopic impact on the dynamics. More precisely, the hope is to be able to encode an averaged effect through an effective boundary condition at a smoothened wall. Such boundary condition, called a wall law, will avoid to simulate the small-scale dynamics that takes place in a boundary layer in the vicinity of the rough surface.

The derivation of wall laws for laminar Newtonian flows has been much studied, since the pioneering works of Achdou, Pironneau and Valentin [START_REF] Achdou | Domain decomposition & wall laws[END_REF][START_REF] Achdou | Effective boundary conditions for laminar flows over periodic rough boundaries[END_REF], or Jäger and Mikelić [START_REF] Jäger | On the Roughness-Induced Effective Boundary Condition for an Incompressible Viscous Flow[END_REF][START_REF] Jäger | Couette flows over a rough boundary and drag reduction[END_REF]. See also [START_REF] Luchini | Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces[END_REF][START_REF] Amirat | Effect of rugosity on a flow governed by stationary Navier-Stokes equations[END_REF][START_REF] Gérard-Varet | Highly rotating fluids in rough domains[END_REF][START_REF] Bresch | High order multi-scale wall-laws, part I: the periodic case Q[END_REF][START_REF] Mikelić | Effective slip law for general viscous flows over an oscillating surface[END_REF]. A natural mathematical approach of this problem is by homogenization techniques, the roughness being modeled by a small amplitude/high frequency oscillation. Typically, one considers a Navier-Stokes flow in a channel Ω ε with a rough bottom:

Ω ε = Ω ∪ Σ 0 ∪ R ε .
Precisely:

• Ω = (0, 1) 2 is the flat portion of the channel.

• R ε is the rough portion of the channel: it reads R ε = {x = (x 1 , x 2 ), x 1 ∈ (0, 1), 0 > x 2 > εγ(x 1 /ε)} with a bottom surface Γ ε := {x 2 = εγ(x 1 /ε)} parametrized by ε ≪ 1. Function γ = γ(y 1 ) is the roughness pattern.

• Eventually, Σ 0 := (0, 1) × {0} is the interface between the rough and flat part. It is the artificial boundary at which the wall law is set.

Of course, within such model, the goal is to understand the asymptotic behavior of the Navier-Stokes solution u ε as ε → 0. Therefore, the starting point is a formal approximation of u ε under the form u ε app (x) = u 0 (x) + εu 1 (x) + • • • + u 0 bl (x/ε) + εu 1 bl (x/ε) + . . . .

In this expansion, the u i = u i (x) describe the large-scale part of the flow, whereas the u i bl = u i bl (y) describe the boundary layer. The typical variable y = x/ε matches the small-scale variations induced by the roughness. In the case of homogeneous Dirichlet conditions at Γ ε , one can check formally that:

• u 0 is the solution of the Navier-Stokes equation in Ω, with Dirichlet condition at Σ 0 .

• u 0 bl = 0, whereas u 1 bl satisfies a Stokes equation in variable y in the boundary layer domain Ω bl := {y = (y 1 , y 2 ), y 1 ∈ R, y 2 > γ(y 1 )}.

The next step is to solve this boundary layer system, and show convergence of u 1 bl as y 2 → +∞ to a constant field u ∞ = (U ∞ , 0). This in turn determines the appropriate boundary condition for the large scale correction u 1 . From there, considering the large scale part u 0 + εu 1 , one can show that:

• The limit wall law is a homogeneous Dirichlet condition. Let us point out that this feature persists even starting from a microscopic pure slip condition, under some non-degeneracy of the roughness: [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system with rough boundaries[END_REF][START_REF] Bonnivard | The uniform rugosity effect[END_REF][START_REF] Bonnnivard | Computation of the Effective Slip of Rough Hydrophobic Surfaces via Homogenization Math[END_REF].

• The O(ε) correction to this wall law is a slip condition of Navier type, with O(ε) slip length.

All these steps were completed in aforementioned articles, in the case of periodic roughness pattern γ : γ(y 1 + 1) = γ(y 1 ). Over the last years, the first author has extended this analysis to general patterns of roughness, with ergodicity properties (random stationary distribution of roughness, etc). We refer to [START_REF] Basson | Wall Laws for Fluid Flows at a Boundary with Random Roughness[END_REF][START_REF] Gérard-Varet | The Navier wall law at a boundary with random roughness Comm[END_REF][START_REF] Gérard-Varet | Relevance of the Slip Condition for Fluid Flow Near an Irregular Boundary[END_REF]. See also [START_REF] Dalibard | Effective boundary condition at a rough surface starting from a slip condition[END_REF] for some recent work on the same topic.

The purpose of the present paper is to extend the former analysis to non-Newtonian flows. This may have various sources of interest. One can think of engineering applications, for instance lubricants to which polymeric additives confer a shear thinning behavior. One can also think of glaciology: as the interaction of glaciers with the underlying rocks is unavailable, wall laws can help. From a mathematical point of view, such examples may be described by a power-law model. Hence, we consider a system of the following form:

     -div S(Du) + ∇p = e 1 in Ω ε , div u = 0 in Ω ε , u| Γ ε = 0, u| x2=1 = 0, u 1-periodic in x 1 .
(2)

As usual, u = u(x) ∈ R 2 is the velocity field, p = p(x) ∈ R is the pressure. The source term e 1 at the right-hand side of the first equation corresponds to a constant pressure gradient e 1 = (1, 0) t throughout the channel. Eventually, the left-hand side involves the stress tensor of the fluid. As mentioned above, it is taken of power-law type: S : R 2×2 sym → R 2×2 sym is given by

S : R 2×2 sym → R 2×2 sym , S(A) = ν|A| p-2 A, ν > 0, 1 < p < +∞, (3)
where |A| = ( i,j a 2 i,j ) 1/2 is the usual euclidean norm of the matrix A. For simplicity, we shall take ν = 1. Hence, S(Du) = |Du| p-2 Du, where we recall that Du = 1 2 (∇u+(∇u) t ) is the symmetric part of the jacobian. Following classical terminology, the case p < 2 resp. p > 2 corresponds to shear thinning fluids, resp. shear thickening fluids. The limit case p = 2 describes a Newtonian flow. Note that we complete the equation in system (2) by a standard no-slip condition at the top and bottom boundary of the channel. For the sake of simplicity, we assume periodicity in the large scale horizontal variable x 1 . Finally, we also make a simplifying periodicity assumption on the roughness pattern γ: γ is C 2,α for some α > 0, has values in (-1, 0), and is 1-periodic in y 1 .

(4)

For every ε > 0 and any value of p, the generalized Stokes system (2) has a unique solution

(u ε , p ε ) ∈ W 1,p (Ω ε ) × L p ′ (Ω ε )/R.
The main point is to know about the asymptotic behavior of u ε , precisely to build some good approximate solution. With regards to the Newtonian case, we anticipate that this approximation will take a form close to (1). Our plan is then:

• to derive the equations satisfied by the first terms of expansion (1).

• to solve these equations, and show convergence of the boundary layer term to a constant field away from the boundary.

• to obtain error estimates for the difference between u ε and u ε app . • to derive from there appropriate wall laws. This program will be more difficult to achieve for non-Newtonian fluids, in particular for the shear thinning case p < 2, notably as regards the study of the boundary layer equations on u bl := u 1 bl . In short, these equations will be seen to read -div (S(A + Du bl )) + ∇p = 0, div u = 0, y ∈ Ω bl for some explicit matrix A, together with periodicity condition in y 1 and a homogeneous Dirichlet condition at the bottom of Ω bl . Due to the nonlinearity of these equations and the fact that A = 0, the analysis will be much more difficult than in the Newtonian case, notably the proof of the so-called Saint-Venant estimates. We refer to section 2.2 for all details.

Let us conclude this introduction by giving some references on related problems. In [START_REF] Marušsić-Paloka | Steady flow of non-Newtonian fluid in unbounded channels and pipes[END_REF]; E. Marušić-Paloka considers power-law fluids with convective terms in infinite channels and pipes (the non-Newtonian analogue of the celebrated Leray's problem). After an appropriate change of unknown, the system studied in [START_REF] Marušsić-Paloka | Steady flow of non-Newtonian fluid in unbounded channels and pipes[END_REF] bears some strong similarity to our boundary layer system. However, it is different at two levels : first, the analysis is restricted to the case p > 2. Second, our lateral periodicity condition in y 1 is replaced by a no-slip condition. This allows to use Poincaré's inequality in the transverse variable, and control zero order terms (in velocity u) by ∇u, and then by Du through the Korn inequality. It simplifies in this way the derivation of exponential convergence of the boundary layer solution (Saint-Venant estimates). The same simplification holds in the context of paper [START_REF] Bourgeat | Mathematical modelling and numerical simulations of a non-Newtonian viscous flows through a thin filter[END_REF], where the behaviour of a Carreau flow through a thin filter is analysed. The corrector describing the behaviour of the fluid near the filter is governed by a kind of boundary layer type system, in a slab that is infinite vertically in both directions. In this setting, one has A = 0, and the authors refer to [START_REF] Marušsić-Paloka | Steady flow of non-Newtonian fluid in unbounded channels and pipes[END_REF] for well-posedness and qualitative behaviour. We also refer to the recent article [START_REF] Suárez-Grau | Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary Nonlinear Analysis[END_REF] dedicated to power-law fluids in thin domains, with Navier condition and anisotropic roughness (with a wavelength that is larger than the amplitude). In this setting, no boundary layer analysis is needed, and the author succeeds to describe the limit asymptotics by the unfolding method. Finally, we point out the very recent paper [START_REF] Chupin | Viscoelastic flows in a rough channel: a multiscale analysis[END_REF], where an Oldroyd fluid is considered in a rough channel. In this setting, no nonlinearity is associated to the boundary layer, which satisfies a Stokes problem.

Boundary layer analysis

From the Newtonian case, we expect the solution (u ε , p ε ) of (2) to be approximated by

u ε ≈ u 0 (x) + εu bl (x/ε), p ε ≈ p 0 (x) + p bl (x/ε),
where • (u 0 , p 0 ) describes the flow away from the boundary layer. We shall take u 0 = 0 and p 0 = 0 in the rough part R ε of the channel.

• (u bl , p bl ) = (u bl , p bl )(y) is a boundary layer corrector defined on the slab

Ω bl := {y = (y 1 , y 2 ), y 1 ∈ T, y 2 > γ(y 1 )},
where T is the torus R/Z. This torus corresponds implicitly to a periodic boundary condition in y 1 , which is inherited from the periodicity of the roughness pattern γ. We denote Ω ± bl := Ω bl ∩ {±y 2 > 0} its upper and lower parts, and Γ bl := {y = (y 1 , y 2 ), y 1 ∈ T, y 2 = γ(y 1 )} its bottom boundary. As the boundary layer corrector is supposed to be localized, we expect that ∇u bl → 0 as y 2 → +∞.

With this constraint in mind, we take (u 0 , p 0 ) to be the solution of

     -div S(Du 0 ) + ∇p 0 = e 1 in Ω, div u 0 = 0 in Ω, u 0 | Σ0 = 0, u 0 | x2=1 = 0, u 0 1-periodic in x 1 . ( 5
)
The solution is explicit and generalizes the Poiseuille flow. A simple calculation yields: for all x ∈ Ω,

p 0 (x) = 0, u 0 (x) = (U (x 2 ), 0), U (x 2 ) = p -1 p √ 2 -p (p-1) - √ 2 p (p-1) x 2 - 1 2 p p-1 . (6)
We extend this solution to the whole rough channel by taking: u 0 = 0, p 0 = 0 in R ε . This zero order approximation is clearly continuous across the interface Σ 0 , but the associated stress is not: denoting

A := D(u 0 )| y2=0 + = 1 2 0 U ′ (0) U ′ (0) 0 , with U ′ (0) = √ 2 p-2 p-1 (7)
we obtain [S(Du 0 )n -

p 0 n]| Σ0 = |A| p-2 An = - √ 2 -p U ′ (0) p-1 0 = -1 2 0 with n = -e 2 = -(0, 1) t and [f ] := f | x2=0 + -f | x2=0 -.
This jump should be corrected by u bl , so that the total approximation u 0 (x) + εu bl (x/ε) has no jump. This explains the amplitude ε of the boundary layer term, as its gradient will then be O [START_REF] Achdou | Domain decomposition & wall laws[END_REF]. By Taylor expansion U (x 2 ) = U (εy 2 ) = U (0) + εU ′ (0)y 2 + . . . we get formally D(u 0 + εu bl (•/ε)) ≈ A + Du bl , where the last symmetric gradient is with respect to variable y. We then derive the following boundary layer system:

               -div S(A + Du bl ) + ∇p bl = 0 in Ω + bl , -div S(Du bl ) + ∇p bl = 0 in Ω - bl , div u bl = 0 in Ω + bl ∪ Ω - bl , u bl | Γ bl = 0, u bl | y2=0 + -u bl | y2=0 -= 0, (8)
together with the jump condition

(S(A + Du bl )n -p bl n) | y2=0 + -(S(Du bl )n -p bl n) | y2=0 -= 0, n = (0, -1) t . (9)
Let us recall that the periodic boundary condition in y 1 is encoded in the definition of the boundary layer domain. The rest of this section will be devoted to the well-posedness and qualitative properties of ( 8)-( 9). We shall give detailed proofs only for the more difficult case p < 2, and comment briefly on the case p ≥ 2 at the end of the section. Our main results will be the following:

Theorem 2.1 (Well-posedness)
For all 1 < p < 2, ( 8)-( 9) has a unique solution

(u bl , p bl ) ∈ W 1,p loc (Ω bl ) × L p ′ loc (Ω bl )/R satisfying for any M > |A|: Du bl 1 {|Du bl |≤M} ∈ L 2 (Ω bl ), Du bl 1 {|Du bl |≥M} ∈ L p (Ω bl ).
For all p ≥ 2, ( 8)-( 9) has a unique solution

(u bl , p bl ) ∈ W 1,p loc (Ω bl )×L p ′ loc (Ω bl )/R s.t. Du bl ∈ L p (Ω bl )∩L 2 (Ω bl ).
Theorem 2.2 (Exponential convergence) For any 1 < p < +∞, the solution given by the previous theorem converges exponentially, in the sense that for some C, δ > 0

|u bl (y) -u ∞ | ≤ Ce -δy2 ∀ y ∈ Ω + bl , where u ∞ = (U ∞ , 0
) is some constant horizontal vector field.

Well-posedness

A priori estimates

We focus on the case 1 < p < 2, and provide the a priori estimates on which the well-posedness is based. The easier case p ≥ 2 is discussed at the end of the paragraph. As A is a constant matrix, we have from ( 8):

-div S(A + Du bl ) + div S(A) + ∇p bl = 0 in Ω + bl , -div S(Du bl ) + ∇p bl = 0 in Ω - bl .
We multiply the two equations by Du bl and integrate over Ω + bl and Ω - bl respectively. After integrations by parts, accounting for the jump conditions at y 2 = 0, we get

Ω + bl (S(A + Du bl ) -S(A)) : Du bl dy + Ω - bl S(Du bl ) : Du bl dy = - y2=0 S(A)n • u bl dS. ( 10
)
The right-hand side is controlled using successively Poincaré and Korn inequalities (for the Korn inequality, see the appendix):

| y2=0 S(A)n • u bl dy| ≤ C u bl L p ({y2=0}) ≤ C ′ ∇u bl L p (Ω - bl ) ≤ C ′′ Du bl L p (Ω - bl ) . (11)
As regards the left-hand side, we rely on the following vector inequality, established in [22, p74, eq. (VII)]: for all 1 < p ≤ 2, for all vectors a, b

(|b| p-2 b -|a| p-2 a | b -a) ≥ (p -1)|b -a| 2 1 0 |a + t(b -a)| p-2 dt. (12)
In particular, for any M > 0, if |b -a| ≤ M , one has

(|b| p-2 b -|a| p-2 a | b -a) ≥ p -1 (|a| + M ) 2-p |b -a| 2 , ( 13
)
whereas if |b -a| > M > |a|, we get (|b| p-2 b -|a| p-2 a | b -a) ≥ (p -1)|b -a| 2 1 |a| |b-a| (2t|b -a|) p-2 dt ≥ 2 p-3 1 -(|a|/M ) p-1 |b -a| p . ( 14
)
We then apply such inequalities to [START_REF] Casado-Diaz | Why viscous fluids adhere to rugose walls: a mathematical explanation[END_REF], taking a = A, b = A + Du bl . For M > |A|, there exists c dependent on M such that

Ω + bl (S(A + Du bl ) -S(A)) : Du bl dy ≥ c Ω + bl 1l {|Du bl |≤M} |Du bl | 2 dy + Ω + bl 1l {|Du bl |>M} |Du bl | p dy, so that for some C dependent on M Ω + bl |1l {|Du bl |≤M} Du bl | 2 dy + Ω + bl |1l {|Du bl |>M} Du bl | p dy + Ω - bl |Du bl | p dy ≤ C Du bl L p (Ω - bl ) .
Hence, still for some C dependent on M :

Ω + bl 1 {|Du bl |≤M} |Du bl | 2 dy + Ω + bl 1 {|Du bl |>M} |Du bl | p dy + Ω - bl |Du bl | p dy ≤ C. (15)
This is the a priori estimate on which Theorem 2.1 can be established (for p ∈]1, 2]). Note that this inequality implies that for any height h, Du bl L p (Ω bl ∩{y2≤h}) ≤ C h (bounding the L p norm by the L 2 norm on a bounded set). Combining with Poincaré and Korn inequalities, we obtain that u bl belongs to W 1,p loc (Ω bl ). In the case p ≥ 2, we can directly use the following inequality, which holds for all a, b ∈ R n :

|a -b| p 2 2-p ≤ 2 -1 |b| p-2 + |a| p-2 |b -a| 2 ≤ |a| p-2 a -|b| p-2 |b|, a -b . ( 16
)
It provides both an L 2 and L p control of the symmetric gradient of the solution. Indeed, taking a = A+D y u bl , b = A and using [START_REF] Casado-Diaz | Why viscous fluids adhere to rugose walls: a mathematical explanation[END_REF] we get the following a'priori estimates for p ≥ 2

2 2-p Ω + bl |Du bl | p dy + Ω - bl |Du bl | p dy + 2 -1 |A| p-2 Ω + bl |Du bl | 2 dy ≤ Ω + bl (S(A + Du bl ) -S(A)) : Du bl dy + Ω - bl S(Du bl ) : Du bl dy = - Σ0 S(A)n • u bl dS ≤ c(α) S(A) p ′ L p ′ (Σ0) + α u bl p L p (Σ0) ≤ c(α) S(A) p ′ L p ′ (Σ0) + αC Γ ∇u bl p L p (Ω - bl ) ≤ c(α) S(A) p ′ L p ′ (Σ0) + αC Γ C K Du bl p L p (Ω - bl ) , (17)
where the trace theorem, the Poincaré inequality and the Korn inequality were employed. By choosing the coefficient α small enough, and by the imbedding of L p (Ω - bl ) in L 2 (Ω - bl ), [START_REF] Gérard-Varet | Relevance of the Slip Condition for Fluid Flow Near an Irregular Boundary[END_REF] provides

Ω bl |Du bl | p + |Du bl | 2 dy ≤ C S(A) p ′ L p ′ (Σ0) < ∞. (18)
Eventually, by Korn and Poincaré inequalities:

u bl ∈ W 1,p (Ω bl ) for 2 ≤ p < ∞.

Construction scheme for the solution

We briefly explain how to construct a solution satisfying the above estimates. We restrict to the most difficult case p ∈]1, 2]. There are two steps:

Step 1: we solve the same equations, but in the bounded domain Ω bl,n = Ω bl ∩ {y 2 < n}, with a Dirichlet boundary condition at the top. As Ω bl,n is bounded, the imbedding of W 1,p in L p is compact, so that a solution u bl,n can be built in a standard way. Namely, one can construct a sequence of Galerkin approximations u bl,n,m by Schauder's fixed point theorem. Then, as the estimate (15) holds for u bl,n,m uniformly in m and n, the sequence Du bl,n,m is bounded in L p (Ω bl,n ) uniformly in m. Sending m to infinity yields a solution u bl,n , the convergence of the nonlinear stress tensor follows from Minty's trick. Note that one can then perform on u bl,n the manipulations of the previous paragraph, so that it satisfies [START_REF] Gérard-Varet | Highly rotating fluids in rough domains[END_REF] uniformly in n.

Step 2: we let n go to infinity. We first extend u bl,n by 0 for y 2 > n, and fix M > |A|. From the uniform estimate [START_REF] Gérard-Varet | Highly rotating fluids in rough domains[END_REF], we get easily the following convergences (up to a subsequence):

u bl,n → u bl weakly in W 1,p loc (Ω bl ), Du bl,n → Du bl weakly in L p (Ω - bl ), Du bl,n 1l |Du bl,n |<M → V 1 weakly in L 2 (Ω + bl ), weakly-* in L ∞ (Ω + bl ), Du bl,n 1l |Du bl,n |≥M → V 2 weakly in L p (Ω + bl ). ( 19
)
Of course,

Du bl = V 1 + V 2 in Ω + bl .
A key point is that

S(A + Du bl,n ) -S(A) is bounded uniformly in n in (L p (Ω + bl )) ′ = L p ′ (Ω + bl ) and in L 2 (Ω + bl ) ∩ L ∞ (Ω + bl ) ′ .
and converges weakly-* to some S + in that space. To establish this uniform bound, we treat separately

S n,1 := (S(A + Du bl,n ) -S(A))1l |Du bl,n |<M , S n,2 := (S(A + Du bl,n ) -S(A))1l |Du bl,n |≥M .
• For S n,1 , we use the inequality [START_REF] Suárez-Grau | Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary Nonlinear Analysis[END_REF]

(L 2 ∩ L ∞ ) ′ (p ∈]1, 2]).
From there, standard manipulations give

Ω + bl (S(A + Du bl,n ) -S(A)) : Du bl,n → Ω + bl S + : (V 1 + V 2 ) = Ω + bl S + : Du bl
One has even more directly 9) in a weak sense. Finally, one can perform on u bl the manipulations of the previous paragraph, so that it satisfies [START_REF] Gérard-Varet | Highly rotating fluids in rough domains[END_REF].

Uniqueness

Let u 1 bl and u 2 bl be weak solutions of ( 8)- [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system with rough boundaries[END_REF], that is satisfying the variational formulation

Ω + bl S(A + Du i bl ) : Dϕ + Ω - bl S(Du i bl ) : Dϕ = - y2=0 S(A)n • ϕ dS, i = 1, 2 (20)
for all smooth divergence free fields ϕ ∈ C ∞ c (Ω bl ). The point is then to replace ϕ by u 1 bl -u 2 bl , to obtain

Ω + bl S(A + Du 1 bl ) -S(A + Du 2 bl ) : D(u 1 bl -u 2 bl ) + Ω - bl (S(Du 1 bl ) -S(Du 2 bl ) : D(u 1 bl -u 2 bl ) = 0. ( 21
)
Rigorously, one constructs by convolution a sequence ϕ n such that Dϕ n converges appropriately to Du 1 bl -Du 2 bl . In the case p < 2, the convergence holds in (L

2 (Ω + bl ) ∩ L ∞ (Ω + bl )) + L p (Ω + bl )
, respectively in L p (Ω - bl ). One can pass to the limit as n goes to infinity because

S(A + Du 1 bl ) -S(A + Du 2 bl ) = S(A + Du 1 bl ) -S(A) + S(A) -S(A + Du 2 bl ) ,
respectively S(Du 1 bl ) -S(Du 2 bl ), belongs to the dual space: see the arguments of the construction scheme of section 2.1. Eventually, by strict convexity of M → |M | p (p > 1), [START_REF] Kaplický | C 1,α -solutions to a class of nonlinear fluids in the 2D stationary Dirichlet problem[END_REF] implies that Du 1 bl = Du 2 bl . This implies that u 1 bl -u 2 bl is a constant (dimension is 2), and due to the zero boundary condition at ∂Ω bl , we get u 1 bl = u 2 bl .

Saint-Venant estimate

We focus in this paragraph on the asymptotic behaviour of u bl as y 2 goes to infinity. The point is to show exponential convergence of u bl to a constant field. At first, we can use interior regularity results for the generalized Stokes equation in two dimensions. In particular, pondering on the results of [START_REF] Wolf | Interior C 1,α regularity of weak solutions to the equations of stationary motion to certain non-Newtonian fluids in two dimensions[END_REF] for p < 2, and [START_REF] Kaplický | C 1,α -solutions to a class of nonlinear fluids in the 2D stationary Dirichlet problem[END_REF] for p ≥ 2, we have :

Lemma 2.1
The solution built in Theorem 2.1 satisfies: u bl has C 1,α regularity over Ω bl ∩ {y 2 > 1} for some 0 < α < 1. In particular, ∇u bl is bounded uniformly over Ω bl ∩ {y 2 > 1}.

Proof. Let 0 ≤ t < s. We define Ω t,s bl := Ω bl ∩ {t < y 2 ≤ s}. Note that Ω bl ∩ {y 2 > 1} = ∪ t∈N * Ω t,t+1 bl . Moreover, from the a priori estimate [START_REF] Gérard-Varet | Highly rotating fluids in rough domains[END_REF] or [START_REF] Glowinski | Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problémes de Dirichlet non linéaires[END_REF], we deduce easily that

Du bl L p (Ω t,t+2 bl ) ≤ C (22)
for all t > 0, for a constant C that does not depend on t. We then introduce:

v t := 2A 0, y 2 -(t + 1 2 ) + u bl - 1 2 Ω t-1 2 ,t+ 3 2 bl u bl dy, y ∈ Ω t-1 2 ,t+ 3 2 bl , ∀t ∈ N * .
From (8):

-div (S(Dv t )) + ∇p bl = 0, div v t = 0 in Ω t-1/2,t+3/2 bl , ∀t ∈ N * .
Moreover, we get for some C independent of t:

v t W 1,p (Ω t-1/2,t+3/2 bl ) ≤ C ∀t ∈ N * . (23)
Note that this W 1,p control follows from ( 22): indeed, one can apply the Poincaré inequality for functions with zero mean, and then the Korn inequality. One can then ponder on the interior regularity results of articles [START_REF] Wolf | Interior C 1,α regularity of weak solutions to the equations of stationary motion to certain non-Newtonian fluids in two dimensions[END_REF] and [START_REF] Kaplický | C 1,α -solutions to a class of nonlinear fluids in the 2D stationary Dirichlet problem[END_REF], depending on the value of p: v t has C 1,α regularity over Ω t,t+1 bl for some α ∈ (0, 1) (independent of t): for some C ′ ,

v t C 1,α (Ω t,t+1 bl ) ≤ C ′ , and in particular ∇v t L ∞ (Ω t,t+1 bl ) ≤ C ′ ∀t ∈ N * .
Going back to u bl concludes the proof of the lemma.

We are now ready to establish a keypoint in the proof of Theorem 2.2, called a Saint-Venant estimate: namely, we show that the energy of the solution located above y 2 = t decays exponentially with t. In our context, a good energy is

E(t) := {y2>t} |∇u bl | 2 dy
for t > 1. Indeed, from Lemma 2.1, there exists M such that |Du bl | ≤ M for all y with y 2 > 1. In particular, in the case p < 2, when localized above y 2 = 1, the energy functional that appears at the left hand-side of (15) only involves the L 2 norm of the symmetric gradient (or of the gradient by the homogeneous Korn inequality, cf the appendix). Hence, ∇u bl ∈ L 2 (Ω bl ∩ {y 2 > 1}). The same holds for p ≥ 2, thanks to (18). Proposition 2.1 There exists C, δ > 0, such that E(t) ≤ C exp(-δt). In particular, E(t) goes to zero as t → +∞ and the point is to quantify the speed of convergence. By the use of inequality [START_REF] Dibenedetto | On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems[END_REF] (with a = A, b = A + Du bl ), we find

Proof. Let

E(t) ≤ C Ω t bl |Du bl | 2 dy ≤ C ′ Ω t bl |A + Du bl | p-2 (A + Du bl ) -|A| p-2 A : Du bl dy ≤ C ′ lim n→∞ Ω bl |A + Du bl | p-2 (A + Du bl ) -|A| p-2 A : Du bl χ n (y 2 ) dy (24)
for a smooth χ n with values in [0, 1] such that χ n = 1 over [t, t + n], χ n = 0 outside [t -1, t + n + 1], and |χ ′ n | ≤ 2. Then, we integrate by parts the right-hand side, taking into account the first equation in [START_REF] Bresch | High order multi-scale wall-laws, part I: the periodic case Q[END_REF]. We write

Ω bl |A + Du bl | p-2 (A + Du bl ) -|A| p-2 A : Du bl χ n (y 2 ) dy = - Ω bl ∇p bl • u bl χ n (y 2 ) dy - Ω bl (S(A + Du bl ) -S(A)) 0 χ ′ n • u bl dy = Ω bl S(A) -S(A + Du bl )) 0 χ ′ n • u bl dy + Ω bl p bl χ ′ n u bl,2 dy := I 1 + I 2 . ( 25
)
To estimate I 1 and I 2 , we shall make use of simple vector inequalities. Namely:

for all p ∈]1, 2], for all vectors a, b, a = 0, , ||b| p-2 b -|a| p-2 a| ≤ C p,a |b -a|, (26)
whereas for all p > 2, for all vectors a, b,

|b| ≤ M , ||b| p-2 b -|a| p-2 a| ≤ C p,a,M |b -a|. ( 27
)
The latter is a simple application of the finite increment inequality. As regards the former, we distinguish between two cases:

• If |b -a| < |a| 2
, it follows from the finite increments inequality.

• If |b -a| ≥ |a| 2 , we simply write

||b| p-2 b -|a| p-2 a| ≤ |b| p-1 + |a| p-1 ≤ (3 p-1 + 2 p-1 )|b -a| p-1 ≤ (3 p-1 + 2 p-1 )( |a| 2 ) p-2 |b -a| using that 2|b-a| |a| p-1 ≤ 2|b-a| |a| for 1 < p ≤ 2.
We shall also make use of the following:

Lemma 2.2 For any height t > 0 i) {y2=t} u bl,2 = 0. ii) {y2=t} (S(A + Du bl ) -S(A)) ( 0 1 ) • ( 1 0 ) = 0. Proof of the lemma.
i) The integration of the divergence-free condition over Ω 0,t bl leads to

0 = Ω 0,t bl div u bl = {y2=t} u bl,2 - {y2=0 + } u bl,2 = {y2=t} u bl,2 - {y2=0 -} u bl,2 = {y2=t} u bl,2 - Ω bl - div u bl + Γ bl u bl • n = {y2=t} u bl,2 ,
where the second and fourth inequalities come respectively from the no-jump condition of u bl at y 2 = 0 and the Dirichlet condition at Γ bl . ii) By integration of the first equation in [START_REF] Bresch | High order multi-scale wall-laws, part I: the periodic case Q[END_REF] over Ω 0,t bl we get:

y2=t (S(A + Du bl ) -S(A) -p bl Id) ( 0 1 ) = y2=0 + (S(A + Du bl ) -S(A) -p bl Id) ( 0 1 ) .
In particular, the quantity

I := y2=t (S(A + Du bl ) -S(A) -p bl Id) ( 0 1 ) • ( 1 0 ) = y2=t (S(A + Du bl ) -S(A)) ( 0 1 ) • ( 1 0 )
is independent of the variable t. To show that it is zero, we apply inequality ( 26) or ( 27) with a = A and b = A + Du bl , so that

I 2 ≤ C {y2=t} |Du bl | 2 ≤ C ′ {y2=t} |Du bl | 2
(C is bounded by Lemma 2.1). As Du bl belongs to L 2 (Ω 1 bl ), there exists a sequence t n such that {y2=tn} |Du bl | 2 → 0 as n → +∞. It follows that I = 0. This concludes the proof of the Lemma.

We can now come back to the treatment of I 1 and I 2 .

• Treatment of I 1 . We note that χ ′ n is supported in [t -1, t] ∪ [t + n, t + n + 1]
. By Lemma 2.2 we can write

I 1 = Ω t-1,t bl S(A) -S(A + Du bl )) 0 χ ′ n • (u bl -c) (28)
+

Ω t+n,t+n+1 bl S(A) -S(A + Du bl )) 0 χ ′ n • (u bl -c n ) := I 1,1 + I 1,2 , (29)
where

c := - Ω t-1,t bl u bl = -
Again, we apply inequality [START_REF] Suárez-Grau | Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary Nonlinear Analysis[END_REF] or [START_REF] Sohr | The Navier-Stokes equations: an elementary functional analytic approach[END_REF] to find

I 1,1 ≤ C Ω t-1,t bl |Du bl | |u bl -c|
and by the Poincaré inequality for functions with zero mean, we easily deduce that

I 1,1 ≤ C ′ Ω t-1,t bl |∇u bl | 2 = C ′ (E(t -1) -E(t)) .
An upper bound on I 1,2 can be derived in the same way:

I 1,2 ≤ C ′ (E(t + n) -E(t + n + 1)),
where the right-hand side going to zero as n → +∞ since E(t ′ ) → 0 as t ′ → ∞. Eventually:

lim sup n→+∞ I 1 ≤ C (E(t -1) -E(t)) . ( 31
)
• Treatment of I 2 .

We can again use the decomposition

I 2 = Ω t-
) ≤ C χ ′ n (y 2 )u bl,2 (y) L 2 (Ω t-1,t bl ) ≤ C ′ u bl,2 L 2 (Ω t-1,t bl
) , for constants C, C ′ that do not depend on t. As w is zero at the boundary:

I 1,2 = Ω t-1,t bl p bl div w = - Ω t-1,t bl ∇p bl • w = Ω t-1,t bl (S(A + Du bl ) -S(A)) • ∇w
where the last equality comes from [START_REF] Bresch | High order multi-scale wall-laws, part I: the periodic case Q[END_REF]. We find as before (cf ( 26) or ( 27)):

|I 1,2 | ≤ C Ω t-1,t bl |Du bl ||∇w| ≤ C Du bl L 2 (Ω t-1,t bl ) ∇w L 2 (Ω t-1,t bl ) ≤ C ′ Du bl L 2 (Ω t-1,t bl ) u bl,2 L 2 (Ω t-1,t bl ) ≤ C ′′ ∇u bl 2 L 2 (Ω t-1,t bl )
where we have controlled the L 2 norm of u bl,2 by the L 2 norm of its gradient (we recall that u bl,2 has zero mean). A similar treatment can be performed with I 2,2 , so that

I 2,1 ≤ C(E(t -1) -E(t)), I 2,2 ≤ C(E(t + n) -E(t + n + 1)) and lim sup n→+∞ I 2 ≤ C (E(t -1) -E(t)) . (33)
Finally, combining ( 24), ( 25), ( 31) and (33), we get

E(t) ≤ C(E(t -1) -E(t))
for some C > 0. It is well-known that this kind of differential inequality implies the exponential decay of Proposition 2.1 (see the appendix). The proof of the Proposition is therefore complete. We have now all the ingredients to show Theorem 2.2.

Proof of Theorem 2.2. Thanks to the regularity Lemma 2.1, we know that ∇u bl is uniformly bounded over Ω 1 bl , and belongs to L 2 (Ω 1 bl ). Of course, this implies that ∇u bl belongs to L q (Ω 1 bl ) for all q ∈ [2, +∞]. More precisely, combining the L ∞ bound with the L 2 exponential decay of Proposition 2.1, we have that

∇u bl L q (Ω t bl ) ≤ C exp(-δt) (34)
(for some C and δ depending on q). This exponential decay extends straightforwardly to all 1 ≤ q < +∞.

Let us now fix q > 2. To understand the behavior of u itself, we write the Sobolev inequality: for all y and y ′ ∈ B(y, r),

|u(y ′ ) -u(y)| ≤ Cr 1-2 q B(y,2r) |∇u(z)| q dz 1/q . ( 35
)
We deduce from there that: for all y 2 > 2, for all s ≥ 0,

|u bl (y 1 , y 2 + s) -u bl (y 1 , y 2 )| ≤ |u bl (y 1 , y 2 + s) -u bl (y 1 , y 2 + ⌊s⌋)| + ⌊s⌋-1 k=0 |u bl (y 1 , y 2 + k + 1) -u bl (y 1 , y 2 + k)| ≤ C   ∇u bl L q (B((y1,y2+s) t ,1)) + ⌊s⌋-1 k=0 ∇u bl L q (B((y1,y2+k) t ,1))   ≤ C ′   e -δ(y2+s) + ⌊s⌋-1 k=0 e -δ(y2+k)  
where the last inequality comes from (35). This implies that u bl satisfies the Cauchy criterion uniformly in y 1 , and thus converges uniformly in y 1 to some u ∞ = u ∞ (y 1 ) as y 2 → +∞. To show that u ∞ is a constant field, we rely again on (35), which yields for all |y 1 -y ′ 1 | ≤ 1:

|u bl (y 1 , y 2 ) -u bl (y ′ 1 , y 2 )| ≤ C|y 1 -y ′ 1 | 1-2 q ∇u bl L q (B((y1,y2) t ,1)) ≤ C ′ e -δy2 .
Sending y 2 to infinity gives: u ∞ (y 1 ) = u ∞ (y ′ 1 ). Finally, the fact that u ∞ is a horizontal vector field follows from Lemma 2.2, point i). This concludes the proof of the Theorem 2.2.

Eventually, for later purposes, we state Corollary 2.1 (higher order exponential decay)

• There exists α ∈ (0, 1), such that for all s ∈ [0, α), for all 1 ≤ q < +∞, one can find C and δ > 0 with

u bl -u ∞ W s+1,q (Ω t bl ) ≤ C exp(-δt), ∀t ≥ 1.
• There exists α ∈ (0, 1), such that for all s ∈ [0, α), for all 1 ≤ q < +∞, one can find C and δ > 0 with p bl -p t W s,q (Ω t,t+1 bl ) ≤ C exp(-δt), ∀t ≥ 1, for some constant p t .

Proof of the corollary. It was established above that

|u(y 1 , y 2 + s) -u(y 1 , y 2 )| ≤ C ′   e -δ ′ (y2+s) + ⌊s⌋-1 k=0 e -δ ′ (y2+k)   .
for some C ′ and δ ′ > 0. From there, after sending s to infinity, it is easily deduced that

u bl -u ∞ L q (Ω t bl ) ≤ C exp(-δt).
It then remains to control the W s,q norm of ∇u bl . This control comes from the C 0,α uniform bound on ∇u bl over Ω 1 bl , see Lemma 2.1. By Sobolev imbedding, it follows that

∇u bl W s,q (Ω t,t+1 bl ) ≤ C, ∀s ∈ [0, α), ∀1 ≤ q < +∞ uniformly in t.
Interpolating this bound with the bound ∇u bl L q (Ω t,t+1 bl ) ≤ C ′ exp(-δ ′ t) previously seen, we get ∇u bl W s,q (Ω t,t+1 bl ) ≤ C ′′ exp(-δ ′′ t), ∀s ∈ [0, α), ∀1 ≤ q < +∞. The first inequality of the Lemma follows.

The second inequality, on the pressure p bl , is derived from the one on u bl . This derivation is somehow standard, and we do not detail it for the sake of brevity.

3 Error estimates, wall Laws We now go back to our primitive system [START_REF] Achdou | Effective boundary conditions for laminar flows over periodic rough boundaries[END_REF]. A standard estimate on u ε leads to

Ω ε |Du ε | p ≤ Ω ε e 1 • u ε .
The Korn inequality implies that

Ω ε |∇u ε | p ≤ C Ω ε |Du ε | p
for a constant C independent of ε: indeed, one can extend u ε by 0 for x 2 < εγ(x 1 /ε) and apply the inequality on the square T × [-1, 1], cf the appendix. Also, by the Poincaré inequality:

| Ω ε e 1 • u ε | ≤ C u ε L p (Ω ε ) ≤ C ′ ∇u ε L p (Ω ε ) .
We find that

u ε W 1,p (Ω ε ) ≤ C. (36)
In particular, it provides strong convergence of u ε in L p by the Rellich theorem (up to a subsequence). As can be easily guessed, the limit of u 0 in Ω is the generalized Poiseuille flow u 0 . One can even obtain an error estimate by a direct energy estimate of the difference (extending u 0 and p 0 by zero in R ε ). We focus on the case 1 < p ≤ 2, and comment briefly the easier case p ≥ 2 afterwards. We write u ε = u 0 + w ε and p ε = p 0 + q ε . We find, taking into account (5):

-div x S(Du 0 + Dw ε ) + div x S(Du 0 ) + ∇q ε = 1l R ε e 1 in Ω ε \ Σ 0 , div x w ε = 0 in Ω ε , w ε = 0 on Γ ε ∪ Σ 1 , w ε is periodic in x 1 with period 1, [w ε ]| Σ0 = 0, [S(Du 0 + Dw ε )n -S(Du 0 )n -q ε n]| Σ0 = -S(Du 0 )n| x2=0 + . (37)
In particular, performing an energy estimate and distinguishing between Ω and R ε , we find

Ω S(Du 0 + Dw ε ) -S(Du 0 ) : Dw ε + R ε S(Dw ε ) : Dw ε = - Σ0 S(Du 0 )n| x2=0 + •w ε dS + R ε e 1 •w ε (38)
Relying on inequalities ( 13)-( 14), we get for any M > Du 0 L ∞ :

Dw ε p L p (Ω∩{|Dw ε |≥M}) + Dw ε 2 L 2 (Ω∩{|Dw ε |≤M}) + Dw ε p L p (R ε ) ≤ C Σ0 S(Du 0 )n • w ε dS + R ε e 1 • w ε (39)
Then by the Hölder inequality and by Proposition 4.2 in the appendix, we have that

| R ε e 1 • w ε | ≤ ε p-1 p w ε L p (R ε ) ≤ Cε 1+ p-1 p ∇w ε L p (R ε ) . (40)
Next, since Du 0 is given explicitly and uniformly bounded, the Proposition 4.2 provides

| Σ0 (S(Du 0 )n| x2=0 + • w ε dS| ≤ C w ε L p (Σ0) ≤ C ′ ε p-1 p ∇w ε L p (R ε ) . (41)
Note that, as w ε is zero at the lower boundary of the channel, we can extend it by 0 below R ε and apply Korn inequalities in a strip (see the appendix). We find

∇w ε L p (R ε ) ≤ C Dw ε L p (R ε )
for some constant C > 0 independent of ε. Summarising, we get

Dw ε p L p (Ω∩{|Dw ε |≥M}) + Dw ε 2 L 2 (Ω∩{|Dw ε |≤M}) + Dw ε p L p (R ε ) ≤ Cε p-1 p Dw ε L p (R ε )
and consequently

Dw ε p L p (Ω∩{|Dw ε |≥M}) + Dw ε 2 L 2 (Ω∩{|Dw ε |≤M}) + Dw ε p L p (R ε ) ≤ Cε (42)
In the case p ≥ 2 one needs to use ( 16) instead of ( 13)-( 14) what yields

Dw ε L p (Ω ε ) ≤ Cε 1 p , p ∈ [2, ∞). ( 43
)

Construction of a refined approximation

The aim of this section is to design a better approximation of the exact solution u ε of (2). This approximation will of course involve the boundary layer profile u bl studied in the previous section. Consequences of this approximation in terms of wall laws will be discussed in paragraph 3.4.

From the previous paragraph, we know that the Poiseuille flow u 0 is the limit of u ε in W 1,p (Ω). However, the extension of u 0 by 0 in the rough part of the channel was responsible for a jump of the stress tensor at Σ 0 . This jump was the main limitation of the error estimates (42)-( 43), and the reason for the introduction of the boundary layer term u bl . Hence, we hope to have a better approximation replacing u 0 by u 0 + εu bl (•/ε). Actually, one can still improve the approximation, accounting for the so-called boundary layer tail u ∞ . More precisely, in the Newtonian case, a good idea is to replace u 0 by the solution u 0,ε of the Couette problem:

-∆u 0,ε + ∇p 0,ε = 0, div u 0,ε = 0, u 0,ε | Σ0 = εu ∞ , u 0,ε | x2=1 = 0.
One then defines:

u ε = u 0,ε + ε(u bl (•/ε) -u ∞ ) + r ε in Ω, u ε = εu bl (•/ε) in R ε ,
where r ε is a small divergence-free remainder correcting the O(exp(-δ/ε)) trace of u bl -u ∞ at {x 2 = 1}.

However, for technical reasons, the above approximation is not so successful in our context, so that we need to modify it a little. We proceed as follows. Let N a large constant to be fixed later. We introduce:

Ω ε N := Ω ε ∩ {x 2 > N ε| ln ε|}, Ω ε 0,N = Ω ε ∩ {0 < x 2 < N ε| ln ε|}, and Σ N = Π × {x 2 = N ε| ln ε|}.
First, we introduce the solution u 0,ε of

             -div S(Du 0,ε ) + ∇p 0,ε = e 1 , x ∈ Ω ε N , div u 0,ε = 0, x ∈ Ω ε N , u 0,ε | ΣN = x → U ′ (0)x2 0 + εu ∞ | ΣN , u 0,ε | {x2=1} = 0. (44)
As for the generalized Poiseuille flow, the pressure p 0,ε is zero, and one has an explicit expression for u 0,ε = (U ε (x 2 ), 0). In particular, one can check that

U ε (x 2 ) = β(ε) - ( √ 2) p ′ p ′ 1 2 + α(ε) -x 2 p ′ , (45)
where α(ε) satisfies the equation (x 2,N := N ε| ln ε|)

- 1 p ′ ( √ 2) p ′ 1 2 + α(ε) -x 2,N p ′ - 1 2 -α(ε) p ′ = U ′ (0)x 2,N + εU ∞ (46)
and

β(ε) = ( √ 2) p ′ p ′ 1 2 -α(ε) p ′ .
By the Taylor expansion, we find that

α(ε) = - √ 2 p ′ -4 εU ∞ + O(ε 2 | ln ε| 2 ). (47)
This will be used later.

Then, we consider the Bogovski problem

     div r ε = 0 in Ω ε N , r ε | ΣN = ε(u bl (•/ε) -u ∞ )| ΣN , r ε | {x2=1} = 0. (48) Since u ∞ = (U ∞ , 0), note that ΣN ε(u bl (•/ε) -u ∞ ) • e 2 = Ω ε N ∪R ε div y u bl (•/ε) = 0.
Hence, the compatibility condition for solvability of (48) is fulfilled: there exists a solution r ε satisfying

r ε W 1,p (Ω ε N ) ≤ Cε u bl (•/ε) -u ∞ W 1-1 p ,p (ΣN )
.

Using the first estimate of Corollary 2.1, we find

r ε W 1,p (Ω ε N ) ≤ Cε 1 p exp(-δN | ln ε|). (49)
Finally, we define the approximation (u ε app , p ε app ) by the formula

u ε app (x) =        u 0,ε (x) + r ε (x) x ∈ Ω ε N , U ′ (0)x2 0 + εu bl (x/ε), x ∈ Ω ε 0,N , εu bl (x/ε), x ∈ R ε , (50) whereas p ε app (x) = 0 x ∈ Ω ε N , p bl (x/ε) x ∈ Ω ε 0,N ∪ R ε . (51)
With such a choice:

u ε app | ∂Ω ε = 0, div u ε app = 0 over Ω ε N ∪ Ω ε 0,N ∪ R ε . Moreover, u ε app has zero jump at the interfaces Σ 0 and Σ N : [u ε app ]| Σ0 = 0, [u ε app ]| ΣN = 0.
Still, the stress tensor has a jump. More precisely, we find

S(Du ε app )n -p ε app n | Σ0 = 0, S(Du ε app )n -p ε app n | ΣN = S(Du 0 ε + Dr ε )| {x2=(N ε| ln ε|) + } -S(A + Du bl (•/ε))| {x2=(N ε| ln ε|) -} e 2 -p bl (•/ε)| {x2=(N ε| ln ε|) -} e 2 . ( 52
)
The next step is to obtain error estimates on u ε -u ε app .

Error estimates

We prove here:

Theorem 3.1 (Error estimates)

• For 1 < p ≤ 2, there exists C such that

u ε -u ε app W 1,p (Ω ε ) ≤ C(ε| ln ε|) 1+ 1 p ′ .
• For p ≥ 2, there exists C such that

u ε -u ε app W 1,p (Ω ε ) ≤ C(ε| ln ε|) 1 p-1 + 1 p .
Remark 3.1 A more careful treatment would allow to get rid of the ln factor in the last estimate (p ≥ 2).

We do not detail this point here, as we prefer to provide a unified treatment. Also, we recall that the shear thinning case (1 < p ≤ 2) has a much broader range of applications. More comments will be made on the estimates in the last paragraph 3.4.

Proof of the theorem. We write v ε = u ε -u ε app , q ε = p ε -p ε app . We start from the equation

-div S(Du ε ) + div S(Du ε app ) + ∇q ε = e 1 + div S(Du ε app ) + ∇p ε app := F ε (53) satisfied in Ω ε \ (Σ 0 ∪ Σ N ). A quick computation shows that F ε = div S(Du 0,ε + Dr ε ) -S(Du 0,ε ), x ∈ Ω ε N , e 1 , x ∈ Ω ε 0,N ∪ R ε . Defining F ε , v ε := Ω ε N F ε • v ε + Ω ε 0,N F ε • v ε + R ε F ε • v ε we get: | F ε , v ε | ≤ α ε ∇v ε L p (Ω ε N ) + β ε v ε L p (ΣN ) + v ε L 1 (Ω ε \Ω ε N )
where α ε := S(Du 0,ε + Dr ε ) -S(Du 0,ε ) L p ′ (Ω ε N ) , β ε := S(Du 0,ε + Dr ε ) -S(Du 0,ε ) e 2 L p ′ (ΣN ) . We then use the inequalities

v ε L p (ΣN ) ≤ C(ε| ln ε|) 1/p ′ ∇v ε L p (Ω ε ) , v ε L 1 (Ω ε \Ω ε N ) ≤ Cε 1 p ′ v ε L p (Ω ε \Ω ε N ) ≤ Cε 1 p ′ (ε| ln ε|) ∇v ε L p (Ω ε ) (54)
(see the appendix for similar ones). We end up with

| F ε , v ε | ≤ C α ε + β ε (ε| ln ε|) 1/p ′ + ε 1 p ′ (ε| ln ε|) ∇v ε L p (Ω ε ) . (55)
Back to (53), after multiplication by v ε and integration over Ω ε , we find: for some s < α (where α ∈ (0, 1)) and any q > 1 s . Let q > max(p ′ , 2 s ). The solution r ε of (48) satisfies: r ε ∈ W 1+s,q (Ω ε N ) with r ε W 1+s,q (Ω ε N ) ≤ Cε for N large enough.

To treat α ε , we still have to pay attention to the cancellation of Du 0,ε . Indeed, from the explicit expression of u 0,ε , we know that there is some x 2 (ε) ∼ 1 2 at which Du 0,ε | x2=x2(ε) = 0. Namely, we write The first integral is bounded by

I 1 ≤ C {x∈Ω ε N , |x2-x2(ε)|≤ε 10p ′ } |Du 0,ε | p + |Dr ε | p ≤ Cε 10p ′ ,
where we have used the uniform bound satisfied by Du 0,ε and Dr ε over Ω ε N , see (58). For the second integral, we can distinguish between p < 2 and p ≥ 2. For p < 2, see [START_REF] Suárez-Grau | Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary Nonlinear Analysis[END_REF] and its proof, we get

I 2 ≤ C {x∈Ω ε N , |x2-x2(ε)|≥ε 10p ′ } |Du 0,ε | (p-2)p ′ |Dr ε | p ′ ≤ C ′ ε -M exp(-δ ′ N | ln ε|) (60)
for some M, C ′ , δ ′ > 0, see (58). In the case p ≥ 2, as Du 0,ε and Dr ε are uniformly bounded, we derive a similar inequality by [START_REF] Sohr | The Navier-Stokes equations: an elementary functional analytic approach[END_REF]. In both cases, taking N large enough, we obtain I 2 ≤ C ′′ ε 10p ′ , to end up with α ε ≤ Cε 10 .

Bound on γ ε . We have

where the last inequality can be deduced from (57). It is again O(ε 10 ) for N large enough. For the second term of the right-hand side, we rely on the explicit expression of u 0,ε . On the basis of ( 45

  bl,n ) : Du bl,n → Ω - bl S -: Du bl and one concludes by Minty's trick that S + = S(A+Du bl )-S(A), S -= S(Du bl ). It follows that u bl satisfies (8)-(

3. 1

 1 Approximation by the Poiseuille flow.

1 p

 1 ε ) -S(Du ε app ) : ∇v ε ≤ C α ε + β ε (ε| ln ε|) 1/p ′ + ε ′ (ε| ln ε|) ∇v ε L p (Ω ε ) + ΣN [S(Du ε app )e 2 ]| ΣN • v ε -[p ε app ]| ΣN v ε 2 .

1 qN ) ≤ Cε 1 q 1 q

 111 -s exp(-N δ| ln ε|) so that by Sobolev imbeddingDr ε L ∞ (ΣN ) + Dr ε L q (ΣN ) + Dr ε L ∞ (Ω ε N ) ≤ C Dr ε W s,q (Ω ε -s exp(-N δ| ln ε|)(58)This last inequality allows to evaluate β ε . Indeed, for x ∈ Σ N , C ≥ |Du 0,ε (x)| ≥ c > 0 uniformly in x. We can then use the upper bound (26) for p < 2, or[START_REF] Sohr | The Navier-Stokes equations: an elementary functional analytic approach[END_REF] for p ≥ 2, to obtainβ ε ≤ C Dr ε L p ′ (ΣN ) ≤ C Dr ε L q (ΣN ) ≤ C ′ ε -s exp(-N δ| ln ε|) ≤ C ′ ε 10(59)

  0,ε + Dr ε ) -S(Du 0,ε )| p ′ = {x∈Ω ε N , |x2-x2(ε)|≤ε 10p ′ } |S(Du 0,ε + Dr ε ) -S(Du 0,ε )| p ′ + {x∈Ω ε N , |x2-x2(ε)|≥ε 10p ′ } |S(Du 0,ε + Dr ε ) -S(Du 0,ε )| p ′ := I 1 + I 2 .

  γ ε ≤ S(Du 0,ε + Dr ε ) -S(Du 0,ε ) e 2 L p ′ (ΣN ) + S(Du 0,ε ) -S(A) e 2 L p ′ (ΣN ) + (S(A) -S(A + Du bl (•/ε)))e 2 L p ′ (ΣN ) + p bl (•/ε) -p ε,N L p ′ (ΣN ) .The first term is β ε , so O(ε 10 ) by previous calculations. The third term can be treated similarly to β ε . As A = 0,[START_REF] Suárez-Grau | Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary Nonlinear Analysis[END_REF] implies that(S(A) -S(A + Du bl (•/ε)))e 2 L p ′ (ΣN ) ≤ C Du bl (•/ε) L p ′ (ΣN ) ≤ C ′ exp(-δ ′ N ln ε),

  )-(47), we find that D(u 0,ε )| ΣN = A + O(ε| ln ε|)Proof. Let φ(y) = ϕ(εy), where y ∈ S k = S + (k, -1) (a rescaled single cell of rough layer). Then φ ∈ W 1,p (S k ) for all k ∈ N, and ϕ = 0 on Γ. By the trace theorem and the Poincaré inequality: for all p ∈ [1, ∞)S k ∩{y2=0} | φ(ȳ, 0)| p dȳ ≤ C S k |∇ y φ| p dy.A change of variables providesεS k ∩{x2=0} |ϕ(x, 0)| p ε -1 dx ≤ C εS k ε p |∇ x φ(x)| p ε -2 dx.Summing over k we obtain Σ |ϕ(x, 0)| p dx is proved. The inequality (70) is proved in the same way, as a consequence of the (one-dimensional) Poincaré inequality.

  . It gives |S n,1 | ≤ C|Du bl,n |1l |Du bl,n |<M , which provides a uniform bound in L 2 ∩ L ∞ , and so in particular in L p ′ and in L 2 . • For S n,2 , we use first that |S n,2 | ≤ C|Du bl,n | p-1 1l |Du bl,n |≥M , so that it is uniformly bounded in L p ′ . We use then (26), so that |S n,2 | ≤ C|Du bl,n |1l |Du bl,n |≥M , which yields a uniform bound in L p , in particular in

  t > 1, Ω t bl := Ω bl ∩ {y 2 > t}. Let M such that |Du bl | is bounded by M over Ω 1 bl , which exists due to Lemma 2.1. As explained just above, one has Ω 1 bl |Du bl | 2 < +∞, and by Korn inequality E(1) is finite.

Acknowledgements:

The work of AWK is partially supported by Grant of National Science Center Sonata, No 2013/09/D/ST1/03692.

Let p ε,N be a constant to be fixed later. As v ε is divergence-free and zero at Γ ε , its flux through Σ N is zero: ΣN v ε 2 = 0. Hence, we can add p ε,N to the pressure jump [p ε app ]| ΣN without changing the surface integral. We get:

where

. Note that we used again the first bound in (54) to go from the third to the fourth inequality. Lemma 3.1 For N large enough, and a good choice of p ε,N there exists C = C(N ) such that

Let us temporarily admit this lemma. Then, we can conclude the proof of the error estimates:

• In the case 1 ≤ p ≤ 2, we rely on the inequality established in [START_REF] Glowinski | Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problémes de Dirichlet non linéaires[END_REF]Proposition 5.2]: for all p ∈]1, 2],

there exists c such that for all u, u ′ ∈ W 1,p 0 (Ω ε )

We use this inequality with u = u ε , u ′ = u ε app . With the estimate (36) and the Korn inequality in mind, we obtain

Combining this lower bound with the upper bounds on α ε , β ε , γ ε given by the lemma, we deduce from (56) the first error estimate in Theorem 3.1.

• In the case 2 ≤ p, we use the easier inequality

The second error estimate from Theorem 3.1 follows.

The final step is to establish the bounds of Lemma 3.1.

Bound on α ε and β ε . From Corollary 2.1 and the trace theorem, we deduce that

Finally, to handle the pressure term, we use the second term of Corollary 2.1, which implies p bl -p t L q ({y2=t}) ≤ C exp(-δt) for some constant p t . We take t = N ln ε and p ε,N = p t to get

Taking N large enough, we can make this term neglectible, say O(ε 10 ). Gathering all contributions, we obtain γ ε ≤ Cε| ln ε| as stated.

Comment on possible wall laws

On the basis of the previous error estimates, we can now discuss the appropriate wall laws for a non-Newtonian flow above a rough wall. We focus here again on the shear thinning case (1 < p ≤ 2).

We first notice that the field u ε app (see ( 50)) involves in a crucial way the solution u 0,ε of (44). Indeed, we know from (49) that the contribution of r ε in W 1,p (Ω ε N ) is very small for N large enough. Hence, the error estimate of Theorem 3.1 implies that

In other words, away from the boundary layer, u ε is well approximated by u 0,ε , with a power of ε strictly bigger than 1. Although such estimate is unlikely to be optimal, it is enough to emphasize the role of the boundary layer tail u ∞ . Namely, the addition of the term εu ∞ in the Dirichlet condition for u 0,ε (see the third line of (44)) allows to go beyond a O(ε) error estimate. A contrario, the generalized Poiseuille flow u 0 leads to a O(ε) error only (away from the boundary layer). Notably,

where the lower bound for u 0,ε -u 0 is obtained using the explicit expressions.

Let us further notice that instead of considering u 0,ε , we could consider the solution

It reads u 0 ε = (U ε , 0) with

We can compare directly these expressions to (45)-( 46) and deduce that

Hence, in view of (62) and (64), we distinguish between two approximations (outside the boundary layer):

• A crude approximation, involving the generalized Poiseuille flow u 0 .

• A refined approximation, involving u 0 ε .

The first choice corresponds to the Dirichlet wall law u| Σ0 = 0, and neglects the role of the roughness. The second choice takes it into account through the inhomogeneous Dirichlet condition: u| Σ0 = εu ∞ = ε(U ∞ , 0). Note that this last boundary condition can be expressed as a wall law, although slightly abstract. Indeed, U ∞ can be seen as a function of the tangential shear (D(u

Denoting by F this application, we write

whereas (u 0 ε ) n = 0. This provides the following refined wall law :

This wall law generalizes the Navier wall law derived in the Newtonian case, where F is simply linear. Of course, it is not very explicit as it involves the nonlinear system ( 8)-( 9). More studies will be necessary to obtain qualitative properties of the function F , leading to a more effective boundary condition.

4 Appendix : A few functional inequalities Proposition 4.1 (Korn inequality) Let S a := T × (a, a + 1), a ∈ R. For all 1 < p < +∞, there exists C > 0 such that: for all a ∈ R, for all u ∈ W 1,p (S a ),

Proof. Without loss of generality, we can show the inequality for a = 0: the independence of the constant C with respect to a follows from invariance by translation. Let us point out that the keypoint of the proposition is that the inequality is homogeneous. Indeed, it is well-known that the inhomogeneous Korn inequality

holds. To prove the homogeneous one, we use reductio at absurdum : if (65) is wrong, there exists a sequence

Up to replace u n by u ′ n := (u n -S0 u n )/ u n L p , we can further assume that

Combining (66) and (67), we deduce that 1 ≥ n-C ′ C ′ D(u n ) L p which shows that D(u n ) converges to zero in L p . Using again (66), we infer that (u n ) is bounded in W 1,p , so that up to a subsequence it converges weakly to some u ∈ W 1,p , with strong convergence in L p by Rellich Theorem. We have in particular

Moreover, as D(u n ) goes to zero, we get D(u) = 0. This implies that u must be a constant (dimension is 2), which makes the two statements of (68) contradictory. Proof. From the previous inequality, we get for all n ∈ N:

The result follows by summing over n.

Corollary 4.2 Let 1 < p < +∞. There exists C > 0, such that for all u ∈ W 1,p (Ω - bl ), resp. u ∈ W 1,p (Ω bl ), satisfying u| Γ bl = 0, one has ∇u L p (Ω - bl ) ≤ C Du L p (Ω - bl ) , resp. ∇u L p (Ω bl ) ≤ C Du L p (Ω bl ) . Proof. One can extend u by 0 for all y with -1 < y 2 < γ(y 1 ), and apply the previous inequality on S -1 , resp. H -1 . Proposition 4.2 (Rescaled trace and Poincaré inequalities) Let ϕ ∈ W 1,p (R ε ). We have

ϕ L p (Rε) ≤ Cε ∇ x ϕ L p (Rε) .

(70)


Lingmin Liao
email: lingmin.liao@u-pec.fr

Micha L Rams
email: rams@impan.pl

SUBEXPONENTIALLY INCREASING SUMS OF PARTIAL QUOTIENTS IN CONTINUED FRACTION EXPANSIONS

Keywords:

We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients Sn(x) = n j=1 aj(x), where x = [a1(x), a2(x), • • • ] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function ϕ : N → N, one is interested in the Hausdorff dimension of the sets

Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case exp(n γ ), γ ∈ [1/2, 1). We show that when γ ∈ [1/2, 1), Eϕ has Hausdorff dimension 1/2. Thus, surprisingly, the dimension has a jump from 1 to 1/2 at ϕ(n) = exp(n 1/2 ). In a similar way, the distribution of the largest partial quotient is also studied.

Introduction

Each irrational number x ∈ [0, 1) admits a unique infinite continued fraction expansion of the form x = 1 a 1 (x) + 1 a 2 (x) + 1 a 3 (x) + . . . , (1.1) where the positive integers a n (x) are called the partial quotients of x. Usually, (1.1) is written as x = [a 1 , a 2 , • • • ] for simplicity. The n-th finite truncation of (1.1): p n (x)/q n (x) = [a 1 , • • • , a n ] is called the n-th convergent of x. The continued fraction expansions can be induced by the Gauss transformation T : [0, 1) → [0, 1) defined by T (0) := 0, and T (x) := 1 x (mod 1), for x ∈ (0, 1).

It is well known that a 1 (x) = x -1 ( • stands for the integer part) and a n (x) = a 1 (T n-1 (x)) for n ≥ 2.

For any n ≥ 1, we denote by S n (x) = n j=1 a j (x) the sum of the n first partial quotients. It was proved by Khintchine [START_REF] Ya | Metrische Kettenbruchprobleme[END_REF] in 1935 that S n (x)/(n log n) converges in measure (Lebesgue measure) to the constant 1/ log 2. In 1988, Philipp [START_REF] Philipp | Limit theorems for sums of partial quotients of continued fractions[END_REF] showed that there is no reasonable normalizing sequence ϕ(n) such that a strong law of large numbers is satisfied, i.e., S n (x)/ϕ(n) will never converge to a positive constant almost surely.

From the point of view of multifractal analysis, one considers the Hausdorff dimension of the sets

E ϕ = x ∈ (0, 1) : lim n→∞ S n (x) ϕ(n) = 1 .
where ϕ : N → N is an increasing function.

The case ϕ(n) = γn with γ ∈ [1, ∞) was studied by Iommi and Jordan [START_REF] Iommi | Multifractal analysis of Birkhoff averages for countable Markov maps[END_REF]. It is proved that with respect to γ, the Hausdorff dimension (denoted by dim H ) of E ϕ is analytic, increasing from 0 to 1, and tends to 1 when γ goes to infinity. In [START_REF] Wu | On the distribution for sums of partial quotients in continued fraction expansions[END_REF], Wu and Xu proved that if ϕ(n) = n γ with γ ∈ (1, ∞) or ϕ(n) = exp(n γ ) with γ ∈ (0, 1/2), then dim H E ϕ = 1. Later, it was shown by Xu [START_REF] Xu | On sums of partial quotients in continued fraction expansions[END_REF], that if ϕ(n) = exp(n) then dim H E ϕ = 1/2 and if ϕ(n) = exp(γ n ) with γ > 1 then dim H E ϕ = 1/(γ + 1). The same proofs of [START_REF] Xu | On sums of partial quotients in continued fraction expansions[END_REF] also imply that for ϕ(n) = exp(n γ ) with γ ∈ (1, ∞) the Hausdorff dimension dim H E ϕ stays at 1/2. So, only the subexponentially increasing case: ϕ(n) = exp(n γ ), γ ∈ [1/2, 1) was left unknown. In this paper, we fill this gap.

Theorem 1.1. Let ϕ(n) = exp(n γ ) with γ ∈ [1/2, 1). Then dim H E ϕ = 1 2 .
We also show that there exists a jump of the Hausdorff dimension of E ϕ between ϕ(n) = exp(n 1/2 ) and slightly slower growing functions, for example

ϕ(n) = exp( √ n(log n) -1 ). Theorem 1.2. Let ϕ(n) = exp( √ n • ψ(n))
Then dim H E ϕ = 1.
We remark that the assumption (1.2) on the function ψ says that ψ decreases to 0 slower than any polynomial. We also remark that when ψ is decreasing, then the first condition of (1.2) is automatically satisfied.

Theorems 1.1 and 1.2 show that, surprisingly, there is a jump of the Hausdorff dimensions from 1 to 1/2 in the class ϕ(n) = exp(n γ ) at γ = 1/2 and that this jump cannot be easily removed by considering another class of functions. See Figure 1 for an illustration of the jump of the Hausdorff dimension.

By the same method, we also prove some similar results on the distribution of the largest partial quotient in continued fraction expansions. For x ∈ [0, 1) \ Q, define

T n (x) := max{a k (x) : 1 ≤ k ≤ n}.
One is interested in the following lower limit:

T (x) := lim inf n→∞ T n (x) log log n n . 2 L I N G M I N L I A O A N D M I C H A L RAMS 1 1 0 1 / 2 1/2 1 dim H E ϕ 1/(1+γ)
linear γn subexponential e n γ superexponential e γ n such that a strong law of large numbers is satisfied, i.e., S n (x)/ϕ(n) will never converge to a positive constant almost surely.

From the point of view of multifractal analysis, one considers the Hausdorff dimension of the sets

E ϕ = x ∈ (0, 1) : lim n→∞ S n (x) ϕ(n) = 1 .
where ϕ : N → N is an increasing function.

The case ϕ(n) = θn with θ ∈ [1, ∞) was studied by Iommi and Jordan [START_REF] Iommi | Multifractal analysis of Birkhoff averages for countable Markov maps[END_REF]. It is proved that with respect to θ, the Hausdorff dimension of E ϕ is analytic, increasing from 0 to 1, and tends to 1 when θ goes to infinity. In [START_REF] Wu | On the distribution for sums of partial quotients in continued fraction expansions[END_REF], Wu and Xu proved that if ϕ(n) = n α with α ∈ (0, ∞) or ϕ(n) = exp{n β } with β ∈ (0, 1/2), the Hausdorff dimension of E ϕ is always 1. It was shown by Xu [START_REF] Xu | On sums of partial quotients in continued fraction expansions[END_REF], that if ϕ(n) = exp{n} then the Hausdorff dimension of E ϕ is 1/2 and if ϕ(n) = exp{γ n } with γ > 1 then the Hausdorff dimension is 1/(γ +1). The same proofs of [START_REF] Xu | On sums of partial quotients in continued fraction expansions[END_REF] also imply that for ϕ(n) = exp{n β } with β ∈ (1, ∞) the Hausdorff dimension of E ϕ stays at 1/2. So, only the subexponentially increasing case: ϕ(n) = exp{n β }, β ∈ [1/2, 1) was left unknown. In this paper, we fill this gap. We also show that for increasing rates slightly slower than e √ n , for example ϕ(n) = e √ n(log n) -1 , the Hausdorff dimension will jump.

n→∞ nψ (n) ψ(n) = 0.
Then the Hausdorff dimension of E ϕ is equal to one.

Theorems 1.1 and 1.2 show that, surprisingly, there is a jump of the Hausdorff dimensions from 1 to 1/2. By the same method, we also prove some similar results on the distribution of the largest partial quotients in continued fraction expansions. For

x ∈ Figure 1. dim H E ϕ for different ϕ.
It was conjectured by Erdös that almost surely T (x) = 1. However, it was proved by Philipp [START_REF] Philipp | A conjecture of Erdös on continued fractions[END_REF] that for almost all x, one has T (x) = 1/ log 2. Recently, Wu and Xu [START_REF] Wu | The distribution of the largest digit in continued fraction expansions[END_REF] showed that

∀α ≥ 0, dim H x ∈ [0, 1) \ Q : lim n→∞ T n (x) log log n n = α = 1.
They also proved that if the denominator n is replaced by any polynomial the same result holds. In this paper, we show the following theorem.

Theorem 1.3. For all α > 0,

F (γ, α) = x ∈ [0, 1) \ Q : lim n→∞ T n (x)/exp(n γ ) = α satisfies dim H F (γ, α) = 1 if γ ∈ (0, 1/2) 1 2 if γ ∈ (1/2, ∞).
We do not know what happens in the case γ = 1/2.

Preliminaries

For any

a 1 , a 2 , • • • , a n ∈ N, call I n (a 1 , • • • , a n ) := {x ∈ [0, 1) : a 1 (x) = a 1 , • • • , a n (x) = a n }
a rank-n basic interval. Denote by I n (x) the rank-n basic interval containing x. Write |I| for the length of an interval I. The length of the basic interval

I n (a 1 , a 2 , • • • , a n ) satisfies n k=1 (a k + 1) -2 ≤ I n (a 1 , • • • , a n ) ≤ n k=1 a -2 k . (2.1) Let A(m, n) := (i 1 , . . . , i n ) ∈ {1, . . . , m} n : n k=1 i k = m . Let ζ(•) be the Riemann zeta function.
Lemma 2.1. For any s ∈ (1/2, 1), for all n ≥ 1 and for all m ≥ n, we have

(i 1 ,...,in)∈A(m,n) n k=1 i -2s k ≤ 9 2 2 + ζ(2s) n m -2s .
Proof. The proof goes by induction. First consider the case n = 2. For m = 2 the assertion holds, assume that m > 2. We will estimate the sum

m-1 i=1 i -2s (m -i) -2s . For any u ∈ [1, m/2] we have m-1 i=1 i -2s (m -i) -2s = 2 u-1 i=1 i -2s (m -i) -2s + m-u i=u i -2s (m -i) -2s ≤ 2 u-1 i=1 i -2s (m -u) -2s + (m -2u + 1)u -2s (m -u) -2s ≤ 2ζ(2s)(m -u) -2s + (m -2u + 1)u -2s (m -u) -2s .
Take u = m/3 . Then one has

(m -2u + 1)u -2s = (m + 1)u -2s -2u 1-2s ≤ (m + 1) m 3 -2s -2 ≤ 4.
Hence, the above sum is bounded from above by

(4 + 2ζ(2s)) • 2m 3 -2s ≤ 9 2 (2 + ζ(2s)) • m -2s .
Suppose now that the assertion holds for n ∈ {2, n 0 }. Then for n = n 0 +1, we have

(i 1 ,...,i n 0 +1 )∈{1,...,m} n 0 +1 , i k =m n 0 +1 k=1 i -2s k = m-1 i=1 i -2s (i 1 ,...,in 0 )∈{1,...,m} n 0 , i k =m-i n 0 k=1 i -2s k ≤ m-1 i=1 i -2s 9 2 2 + ζ(2s) n 0 (m -i) -2s = 9 2 2 + ζ(2s) n 0 • m-1 i=1 i -2s (m -i) -2s ≤ 9 2 2 + ζ(2s) n 0 • 9 2 2 + ζ(2s) m -2s = 9 2 2 + ζ(2s) n 0 +1 m -2s . Let A(γ, c 1 , c 2 , N ) := x ∈ (0, 1) : c 1 < a n (x) e n γ < c 2 , ∀n ≥ N .
Denote by N 0 the smallest integer n such that (c

2 -c 1 ) • e n γ > 1. Then the set A(γ, c 1 , c 2 , N ) is non-empty when N ≥ N 0 .
Lemma 2.2. For any γ > 0, any N ≥ N 0 and any

0 < c 1 < c 2 , dim H A(γ, c 1 , c 2 , N ) = 1 2 .
Proof. This lemma is only a simple special case of [2, Lemma 3.2], but we will sketch the proof (based on [START_REF] Jordan | Increasing digit subsystems of infinite iterated function systems[END_REF]), needed for the next lemma. Without loss of generality, we suppose N 0 = 1 and let N = 1 (the proof for other N is almost identical).

Let a 1 , a 2 , . . . , a n satisfy c 1 < a j e -j γ < c 2 for all j. Those are exactly the possible sequences for which the basic interval I n (a 1 , . . . , a n ) has nonempty intersection with A(γ, c 1 , c 2 , 1).

There are approximately

(2.2) n j=1 (c 2 -c 1 )e j γ ≈ e n 1 j γ of such basic intervals, each of diameter (2.3) |I n (a 1 , . . . , a n )| ≈ e -2 n 1 j γ ,
(both estimations are up to a factor exponential in n). Hence, by using the intervals {I n (a 1 , . . . , a n )} as a cover, we obtain

dim H A(γ, c 1 , c 2 , 1) ≤ 1 2 .
To get the lower bound, we consider a probability measure µ uniformly distributed on A(γ, c 1 , c 2 , 1), in the following sense: given a 1 , . . . , a n-1 , the probability of a n taking any particular value between c 1 e n γ and c 2 e n γ is the same.

The basic intervals I n (a 1 , . . . , a n ) have, up to a factor c n , the length exp(-2 n 1 j γ ) and the measure exp(-n 1 j γ ). They are distributed in clusters: all I n (a 1 , . . . , a n ) contained in a single I n (a 1 , . . . , a n-1 ) form an interval of length exp(n γ ) • exp(-2 n 1 j γ ) (up to a factor c n , with c being a constant), then there is a gap, then there is another cluster. Hence, for any r ∈ (exp(-2 n 1 j γ ), exp(-2 n-1 1 j γ )) and any x ∈ A(γ, c 1 , c 2 , 1) we can estimate the measure of B(x, r):

µ(B(x, r)) ≈ r • e -n 1 j γ if r < e -2 n 1 j γ +n γ e -n-1 1 j γ if r > e -2 n 1 j γ +n γ
(up to a factor c n ). The minimum of log µ(B(x, r))/ log r is thus achieved for r = e -2 n 1 j γ +n γ , and this minimum equals

-n-1 1 j γ -2 n 1 j γ + n γ ≈ -n γ+1 /(γ + 1) -2n γ+1 /(γ + 1) -n γ = 1 2 -O(1/n).
Hence, the lower local dimension of µ equals 1/2 at each point of A(γ, c 1 , c 2 , 1), which implies

dim H A(γ, c 1 , c 2 , 1) ≥ 1 2 by the Frostman Lemma (see [1, Principle 4.2]).
Let now c 1 and c 2 not be constant but depend on n:

B(γ, c 1 , c 2 , N ) = x ∈ (0, 1) : c 1 (n) < a n (x) e n γ < c 2 (n) ∀n ≥ N .
A slight modification of the proof of Lemma 2.2 gives the following.

Lemma 2.3. Fix γ > 0. Assume 0 < c 1 (n) < c 2 (n) for all n. Assume also that lim n→∞ log(c 2 (n) -c 1 (n)) n γ = 0 and lim inf n→∞ log c 1 (n) log n > -∞ and lim sup n→∞ log c 2 (n) log n < +∞.
Then there exists an integer

N 1 such that (c 2 (n) -c 1 (n)) • e n γ > 1 for all n ≥ N 1 , and for all N ≥ N 1 , dim H B(γ, c 1 , c 2 , N ) = 1/2.
Proof. We need only to replace the constants c 1 and c 2 by c 1 (n) and c 2 (n) in the proof of Lemma 2.2. Notice that by the assumptions of Lemma 2.3, the formula (2.2) holds up to a factor exp(ε n 1 j γ ) for a sufficiently small ε > 0. While the formula (2.3) holds up to a factor exp(cn log n) for some bounded c. All these factors are much smaller than the main term exp( n 1 j γ ) which is of order exp(n 1+γ ). The rest of the proof is the same as that of Lemma 2.2.

Proofs

Proof of Theorem 1.1. Let ϕ : N → N be defined by ϕ(n) = exp(n γ ) with γ > 0. For this case, we will denote E ϕ by E γ .

Let us start from some easy observations, giving (among other things) a simple proof of dim H E γ = 1/2 for γ ≥ 1.

Consider first γ ≥ 1/2. If x ∈ E γ then for any ε > 0 and for n large enough

(1 -ε)e n γ ≤ S n (x) ≤ (1 + ε)e n γ (3.1) and (1 -ε)e (n+1) γ ≤ S n+1 (x) ≤ (1 + ε)e (n+1) γ . Hence (1 -ε)e (n+1) γ -(1 + ε)e n γ ≤ a n+1 (x) ≤ (1 + ε)e (n+1) γ -(1 -ε)e n γ .
For γ ≥ 1 this implies

E γ ⊂ N A(γ, c 1 , c 2 , N ) for some constants c 1 , c 2 . By Lemma 2.2, dim H E γ ≤ 1 2 , ∀γ ≥ 1.
Consider now any γ > 0. Set

c 1 (n) = (e n γ -e (n-1) γ )e -n γ and c 2 (n) = n + 1 n c 1 (n).
For γ ≥ 1, c 1 (n) and c 2 (n) are bounded from below. For γ < 1 and n large, we have (e n γe (n-1) γ )e -n γ ≈ γn γ-1 .

Thus, in both cases the assumptions of Lemma 2.3 are satisfied. Checking B(γ, c 1 , c 2 , N ) ⊂ E γ , we deduce by Lemma 2.3 that

dim H E γ ≥ 1 2 , ∀γ > 0.
Therefore, we have obtained dim

H E γ = 1/2 for γ ≥ 1 and dim H E γ ≥ 1/2 for γ > 0. What is left to prove is that for γ ∈ [1/2, 1) we have dim H E γ ≤ 1/2.
Let us first assume that γ > 1/2. Remember that if x ∈ E γ , then for any ε > 0 and for n large enough we have (3.1). Take a subsequence n 0 = 1, and n k = k 1/γ (k ≥ 1). Then there exists an integer N ≥ 1 such that for all Observe that every set A(γ, k, N ) has a product structure: the conditions on a i for i ∈ (n 1 , n 1 +1 ] and for i ∈ (n 2 , n 2 +1 ] are independent from each other. Hence, for any s ∈ (1/2, 1) we can apply Lemma 2.1 together with the formula

k ≥ N , (1 -ε)e n γ k ≤ S n k (x) ≤ (1 + ε)e n γ k , and (as exp(n γ k ) = e k ) (1 -ε)e k -(1 + ε)e k-1 ≤ S n k (x) -S n k-1 (x) ≤ (1 + ε)e k -(1 -ε)e k-1 . Thus E γ ⊂ N k≥N A(γ, k, N ), with A(γ, k, N ) being the union of the intervals {I n k (a 1 , a 2 , • • • , a n k )} such that n j=n -1 +1 a j = m with m ∈ D , N ≤ ≤ k,
|I n k | s ≤ k =1 (a n -1 +1 a n -1 +2 • • • a n ) -2s
to obtain

In k ⊂A(γ,k,1) |I n k | s ≤ k =1 m∈D 9 2 2 + ζ(2s) n -n -1 m -2s .
Denote r 1 := 2ε(1e -1 ) and r 2 := (e -1εeε)/e. Then we have |D | ≤ r 1 e and any m ∈ D is not smaller than r 2 e . Thus we get

In k ⊂A(γ,k,1) |I n k | s ≤ k =1 r 1 e • 9 2 2 + ζ(2s) 1/γ -( -1) 1/γ • r 2s 2 e -2s . (3.2)
We have 1/γ -( -1) 1/γ ≈ 1/γ-1 . As γ > 1/2, we have 1/γ -1 < 1, and the main term in the above estimate is e (1-2s) . Thus for any s > 1/2, the product is uniformly bounded. Thus dim H E (1)

ϕ ≤ 1/2.
If γ = 1/2, we take n k = k 2 /L 2 with L being a constant and we repeat the same argument. Observe that now exp(n γ k ) = e k/L . Then the same estimation will lead to

In k ⊂A(γ,k,1) |I n k | s ≤ k =1 r 1 r 2s 2 • 9 2 2 + ζ(2s) 2 -( -1) 2 L 2 e (1-2s) /L . (3.3)
The main term of the right side of the above inequality should be

9 2 2 + ζ(2s) 2 /L 2 • e (1-2s) /L .
We solve the equation

9 2 2 + ζ(2s) 2/L 2 • e (1-2s)/L = 1,
which is equivalent to

(3.4) 9 2 2 + ζ(2s) = e 2s-1 2 L .
Observe that the graphs of the two sides of (3.4) (as functions of the variable s) always have a unique intersection for some s L ∈ [1/2, 1], when L is large enough. These s L are upper bounds for the Hausdorff dimension of E

ϕ . Notice that the intersecting point s L → 1/2 as L → ∞ since the zeta function ζ has a pole at 1. Thus the dimension of E

ϕ is not greater than 1/2. So, in both cases, we have obtained dim

H E γ ≤ 1/2.
Sketch proof of Theorem 1.2. The proof goes like Section 4 of [START_REF] Wu | On the distribution for sums of partial quotients in continued fraction expansions[END_REF] with the following changes. We choose ε k = ψ(k). Let n 1 be such that ϕ(n 1 ) ≥ 1 and define n k as the smallest positive integer such that

ϕ(n k ) ≥ (1 + ε k-1 )ϕ(n k-1 ). (3.5)
For a large enough integer M , set

E M (ϕ) := x ∈ [0, 1) : a n 1 (x) = (1 + ε 1 )ϕ(n 1 ) + 1, a n k (x) = (1 + ε k )ϕ(n k ) -(1 + ε k-1 )ϕ(n k-1 ) + 1 for all k ≥ 2, and 1 ≤ a i (x) ≤ M for i = n k for any k ≥ 1 . We can check that E M (ϕ) ⊂ E ϕ .
To prove dim H E ϕ = 1, for any ε > 0, we construct a (1/(1 + ε))-Lipschitz map from E M (ϕ) to E M , the set of numbers with partial quotients less than some M in its continued fraction expansion. The theorem will be proved by letting ε → 0 and M → ∞.

Such a Lipschitz map can be constructed by send a point x in E M (ϕ) to a point x by deleting all the partial quotients a n k in its continued fraction expansion. Define r(n) := min{k : n k ≤ n}. The (1/(1 + ε))-Lipschitz property will be assured if

lim n→∞ r(n) n = 0, (3.6) and lim n→∞ log(a n 1 a n 2 • • • a n r(n) ) n = 0. (3.7)
In fact, by (1.2), we can check for any δ > 0, ψ(n) ≤ n δ for n large enough. Thus by definition of n k , we can deduce that r(n) ≤ n 1/2+δ . Hence (3.6) is satisfied.

Further, we have

r(n) k=1 ε k ≈ r(n)ψ(r(n)). (3.8) By (3.5) ϕ(n) ≥ ϕ(n r(n) ) ≥ r(n)-1 k=1 (1 + ε k )ϕ(n 1 ) ≥ e r(n) k=1 ε k /2 ϕ(n 1 ). Thus (3.8) implies r(n)ψ(r(n)) √ nψ(n), (3.9)
where a n b n means that a n /b n is bounded by some constant when n → ∞. On the other hand, by (2.1) and (3.5), we have

log(a n 1 a n 2 • • • a n r(n) ) ≤ r(n) log(2ϕ(n)) + r(n) k=1 ε k .
Hence (3.8) and (3.9) give log(a

n 1 a n 2 • • • a n r(n) ) r(n) √ nψ(n) + r(n)ψ(r(n)) nψ 2 (n) ψ(r(n)) + r(n).
Finally, (3.7) follows from the assumption (1.2) and the already proved formula (3.6).

Proof of Theorem 1.3. For the case γ < 1/2, the set constructed in Section 4 of [START_REF] Wu | On the distribution for sums of partial quotients in continued fraction expansions[END_REF] (as a subset of the set of points for which S n (x) ≈ e n γ ) satisfies also T n (x) ≈ e n γ and has Hausdorff dimension one. We proceed to the case γ > 1/2. The lower bound is a corollary of Lemma 2.3. Take c 1 (n) = α(1 -1 n ) and c 2 (n) = α. Let N 1 be the smallest integer n such that α n e n γ > 1. Then the conditions of Lemma 2.3 are satisfied, and for all points x such that c 1 (n)e n γ < a n (x) < c 2 (n)e n γ , we have

T n (x)/e n γ ≥ c 1 (n) = α 1 - 1 n ,
and

T n (x)/e n γ = a k /e n γ ≤ αe k γ /e n γ ≤ α,
where k ≤ n is the position at which the sequence a 1 , . . . , a n achieves a maximum. Thus for all x ∈ B(γ, c 1 , c 2 , N 1 ) lim n→∞ T n (x)/e n γ = α.

Hence, B(γ, c 1 , c 2 , N 1 ) ⊂ F (γ, α) and the lower bound follows directly from Lemma 2.3. The upper bound is a modification of that of Theorem 1.1. We consider the case α = 1 only, since for other α > 0, the proofs are similar.

Notice that for any ε > 0, if x ∈ F (γ, 1), then for n large enough,

(1 -ε)e n γ ≤ S n (x) ≤ n(1 + ε)e n γ .
Take a subsequence

n k = k 1/γ (log k) 1/γ 2 . Then (1 -ε)e k(log k) 1/γ ≤ S n k (x) ≤ k 1/γ (log k) 1/γ 2 (1 + ε)e k(log k) 1/γ , and u k ≤ S n k (x) -S n k-1 (x) ≤ v k , with
u k := (1 -ε)e k(log k) 1/γ -(k -1) 1/γ (log(k -1)) 1/γ 2 (1 + ε)e (k-1)(log(k-1)) 1/γ , and
v k := k 1/γ (log k) 1/γ 2 (1 + ε)e k(log k) 1/γ -(1 -ε)e (k-1)(log(k-1)) 1/γ .
We remark that As in the proof of Theorem 1.1, we need only study the set B(γ, 1). For any s ∈ (1/2, 1), since

u k > 1 2 e k(log k) 1/γ , v k < 3 2 k 1/γ (log k) 1/γ 2 e k(log k) 1/γ (3.
|I n k | s ≤ k =1 (a n -1 +1 a n -1 +2 • • • a n ) -2s , by Lemma 2.1, In k ⊂B(γ,N ) |I n k | s ≤ k =1 m∈D 9 2 2 + ζ(2s) n -n -1 m -2s .
Note that by (3.10) the number of integers in D satisfies

|D | ≤ v -u ≤ v < 3 2 • 1/γ (log ) 1/γ 2 .
By (3.10), we also have

m ≥ u > 1 2 e (log )
⊂B(γ,N ) |I n k | s is less than k =1 3 2 • 1/γ (log ) 1/γ 2 e (log ) 1/γ 9 2 2 + ζ(2s) n -n -1
2 2s e -2s (log ) 1/γ .

Since nn -1 ≈ 1/γ-1+o(ε) and 1/γ -1 < 1, the main term in the above estimation is e (1-2s) (log ) 1/γ . Thus for any s > 1/2 the product is uniformly bounded and we have the Hausdorff dimension of B(γ, 1) is not greater than 1/2. Then we can conclude dim H F (γ, 1) ≤ 1/2 and the proof is completed.

Generalizations

In this section we consider after [START_REF] Jordan | Increasing digit subsystems of infinite iterated function systems[END_REF] certain infinite iterated function systems that are natural generalizations of the Gauss map. For each n ∈ N, let f n : [0, 1] → [0, 1] be C 1 maps such that (1) there exists m ∈ N and 0 < A < 1 such that for all (a 1 , ..., a m ) ∈ N m and for all x ∈ [0, 1]

0 < |(f a 1 • • • • • f am ) (x)| ≤ A < 1, (2)
for any i, j ∈ N f i ((0, 1)) ∩ f j ((0, 1)) = ∅, (3) there exists d > 1 such that for any ε > 0 there exist C 1 (ε), C 2 (ε) > 0 such that for i ∈ N there exist constants ξ i , λ i such that for all x ∈ [0, 1] ξ i ≤ |f i (x)| ≤ λ i and

C 1 i d+ε ≤ ξ i ≤ λ i ≤
C 2 i d-ε . We will call such an iterated function system a d-decaying system. It will be further called Gauss like if

∞ i=1 f i ([0, 1]) = [0, 1)
and if for all x ∈ [0, 1] we have that f i (x) < f j (x) implies i < j.

We have a natural projection Π : N N → [0, 1] defined by

Π(a) = lim n→∞ f a 1 • • • • • f an (1),
which gives for any point x ∈ [0, 1] its symbolic expansion (a 1 (x), a 2 (x), . . .). This expansion is not uniquely defined, but there are only countably many points with more than one symbolic expansions. For a d-decaying Gauss like system we consider S n (x) = n 1 a i (x). Given an increasing function ϕ : N → N we denote The proofs (both from Section 3 and from [START_REF] Wu | On the distribution for sums of partial quotients in continued fraction expansions[END_REF][START_REF] Xu | On sums of partial quotients in continued fraction expansions[END_REF]) go through without significant changes.

  be an increasing function with ψ being a C 1 positive function on R + satisfying lim x→∞ sup y≥x ψ(y)

Figure 1 .

 1 Figure1. dim H E ϕ for ϕ with different increasing rate.

Theorem 1 . 1 .

 11 Let ϕ(n) = exp{n β } with β ∈ [1/2, 1). Then the Hausdorff dimension of E ϕ is one-half.

Theorem 1 . 2 .

 12 Let ϕ(n) = e √ n•ψ(n) be an increasing function with ψ being a C 1 positive function on R + satisfying lim n→∞ ψ(n) = 0 and lim

  where D := [(1ε)e n γ -(1 + ε)e n γ -1 , (1 + ε)e n γ -(1ε)e n γ -1 ]. Now, we are going to estimate the upper bound of the Hausdorff dimension of E (1) ϕ = k A(γ, k, 1). For E (N ) ϕ = k≥N A(γ, k, N ) with N ≥ 2 we have the same bound and the proofs are almost the same.

  [START_REF] Xu | On sums of partial quotients in continued fraction expansions[END_REF] when k is large enough.Observe that F (γ, 1)⊂ N B(γ, N ),with B(γ, N ) being the union of the intervals{I n k (a 1 , a 2 , • • • , a n k )} k≥N such that n j=n -1 +1 a j = m with m ∈ D , N ≤ ≤ k,where D is the set of integers in the interval [u , v ].

E 1 . 4 . 1 .

 141 d (ϕ) = x ∈ (0, 1) : lim n→∞ S n (x) ϕ(n) =Theorem Let {f i } be a d-decaying Gauss like system. We havei) if ϕ(n) = e n γ with γ < 1/d, dim H E d (ϕ) = 1, ii) if ϕ(n) = e n γ with γ > 1/d, dim H E d (ϕ) = 1 d , iii) if ϕ(n) = e γ n with γ > 1, dim H E d (ϕ) = 1 γ + d -1.

  Similar to (3.2) and (3.3), we deduce that In k

1/γ

for any m ∈ D .

M.R. was partially supported by the MNiSW grant N201 607640 (Poland). L.L. was partially supported by 12R03191A -MUTADIS (France). 2010