P Bénard

A Lagae

P Vangorp

S Lefebvre

G Drettakis

J Thollot

A Dynamic Noise Primitive for Coherent Stylization

Keywords: Descriptors (according to ACM CCS): I, 3, 3 [Computer Graphics]: Picture/Image Generation-

We present a new solution for temporal coherence in non-photorealistic rendering (NPR) of animations. Given the conflicting goals of preserving the 2D aspect of the style and the 3D scene motion, any such solution is a tradeoff. We observe that primitive-based methods in NPR can be seen as texture-based methods when using large numbers of primitives, leading to our key insight, namely that this process is similar to sparse convolution noise in procedural texturing. Consequently, we present a new primitive for NPR based on Gabor noise, that preserves the 2D aspect of noise, conveys the 3D motion of the scene, and is temporally continuous. We can thus use standard techniques from procedural texturing to create various styles, which we show for interactive NPR applications. We also present a user study to evaluate this and existing solutions, and to provide more insight in the trade-off implied by temporal coherence. The results of the study indicate that maintaining coherent motion is important, but also that our new solution provides a good compromise between the 2D aspect of the style and 3D motion.

Introduction

All stylized animation algorithms are confronted with the temporal coherence problem: the fundamental contradiction between conveying the 2D impression of the style depicted, while simultaneously following the 3D motion of the underlying scene. In this work, we address the case of region filling, where an animation is stylized using what we call a pattern, i.e., watercolor pigments, set of strokes, paper grain or any medium an artist may want to simulate.

We define three goals, inspired by previous work [Mei96, CTP * 03, BSM * 07], that must be achieved simultaneously for successful temporal coherent stylization:

Flatness by which we mean the ability to convey the 2D aspect of a given stylization. As pointed out by Meier [Mei96], the goal is to give the impression that each image is produced on a flat medium rather than being painted on a 3D object using surface texture mapping. 2D properties of the pattern must be respected: size of the features, their 2D distribution, homogeneous contrast or density. Breslav et al. [BSM * 07] emphasized the popularity of these 2D patterns in traditional imagery because they provide greater abstraction. Coherent motion represents the high correlation between the 3D scene motion field and the pattern motion. As stated by Cunzi et al. [CTP * 03], the goal is to provide in 2D screen space a perceptual impression of motion as close as possible to the 3D displacement in object space.

In contrast, when the pattern is static in screen space, the shower door effect [Mei96] occurs: the pattern seems to slide over the scene. Temporal continuity is the quality of minimizing changes from frame to frame to ensure fluid animations. Perceptual studies [YJ84, [START_REF] Schwarz | On predicting visual popping in dynamic scenes[END_REF] have shown that human observers are very sensitive to sudden temporal incoherence such as popping. The goal is to keep the stylization as continuous as possible during the animation. This is especially important when zooming and at occlusion/disocclusion boundaries.

The conflicting nature of these goals means that any method addressing the problem of temporal coherence is necessarily a compromise. Any such method will be unable to achieve all the above goals perfectly, and thus will necessarily incur some artifacts, such as pattern deformation, sliding or popping. Previous approaches are based on primitives or textures; each solution presents a different compromise.

We present two contributions in this paper:

• First, we propose a noise primitive that provides a trade-off for the temporally coherent stylization of any kind of 3D animation including deformable objects. By merging a large number of such primitives we produce a temporally coherent noise that exhibits the benefit of both primitive-based and texture-based approaches. This noise is then used in the same way as traditional procedural texture approaches to produce a wide variety of results with styles ranging from continuous (watercolor, ink) to discrete (hatching, painterly), all in the same framework.

• Second, we present a user study which evaluates the success of various solutions according to the three goals outlined above. The design of this study is based on a careful analysis of the trade-offs of different methods. The results of the study indicate that coherent motion is important and that our solution is a good compromise, providing a strong sense of flatness, and a reasonable trade-off for the other goals.

Related work

We classify related work into two categories: primitivebased methods and texture-based methods. We also identify a third category, many-primitive methods, establishing a connection between primitive-based methods in NPR and methods for noise in procedural texturing.

Primitive-based methods

Primitive-based methods are mostly used for patterns composed of uniformly distributed features, typically painterly rendering and stippling.

Meier [Mei96] introduced the idea of attaching anchor points to the 3D surface and using them to draw overlapping primitives in 2D. Since the primitives are usually small with respect to object size, their motion remains very close to the original motion field of the 3D scene. However, no control is provided over their 2D distribution which may produce stroke cluttering. To overcome this limitation, Daniels [START_REF] Daniels | Deep canvas in disney's tarzan[END_REF] gave the control to the user who manually draws the strokes on the objects. This time-consuming process can be avoided by using dynamic distributions. Several authors generate such distributions automatically either by extending Meier's work, e.g. [START_REF]Stylized and abstract painterly rendering system using a multiscale segmented sphere hierarchy[END_REF][START_REF] Vanderhaeghe | Dynamic point distribution for stroke-based rendering[END_REF], or using the optical flow of an input video, e.g. [HE04]. Their main goal is to ensure a Poisson disk distribution in 2D, while still following the 3D scene motion. Kaplan et al. [KGC00] extended this range of styles by relaxing the distribution constraint and giving back more control to the user.

For such approaches the typical trade-off between flatness and temporal continuity involves either a dynamic distribution at the price of some popping (e.g. [START_REF] Vanderhaeghe | Dynamic point distribution for stroke-based rendering[END_REF]), or no popping but some clutter artifacts or holes (e.g. [Mei96]).

Texture-based methods

Texture-based approaches are mostly used for continuous textures (canvas, watercolor) or highly structured patterns (hatching). Textures naturally ensure spatial continuity of the pattern, which allows for simpler and faster methods than primitive-based approaches in general.

Object-space approaches compromise flatness in favor of coherent motion by relying on texture mapping. To reduce perspective distortions and scale variations, these methods try to adapt the texture to maintain a near-constant size of the pattern in screen space. A specific set of mipmaps can be used to achieve this goal, such as art maps [KLK * 00] or tonal art maps [START_REF] Praun | Real-time hatching[END_REF]. It leads to smooth transitions but global oscillations in terms of sharpness when only two mipmap levels are blended. Texture fractalization [START_REF] Bénard | Dynamic solid textures for real-time coherent stylization[END_REF] achieves a very good texture continuity, but also has a tendency to degrade the original pattern [START_REF] Bénard | Quality assessment of fractalized NPR textures: a perceptual objective metric[END_REF].

Image-space approaches compromise motion coherence in favor of flatness by directly rendering the texture in screen space and modifying it according to the scene motion. Bousseau et al. [START_REF] Bousseau | Video watercolorization using bidirectional texture advection[END_REF] locally deform the texture according to the optical flow of a video. This creates a good compromise for continuous media but introduces contrast oscillations and degrades structured patterns. Cunzi et al. [CTP * 03] approximate the scene motion by a global 2D transformation of the texture, creating a strong sliding artifact. This artifact is reduced but still visible in [CDH06, BSM * 07] where they cut the texture in a small number of patches transformed independently. Again, these methods rely on mipmaps or fractalization to obtain smooth transitions when zooming.

Many-primitive methods

Primitive-based methods in NPR typically use relatively few primitives, in which case individual primitives can be discriminated and popping is noticeable. Our key insight is that when using many primitives, individual primitives merge and form a texture, which allows to fade primitives in or out with much less noticeable popping, and these methods become highly related to sparse convolution noise.

In procedural texturing, sparse convolution noise [START_REF]Texture synthesis for digital painting[END_REF]Lew89], including spot noise [vW91] and Gabor noise [START_REF] Lagae | Procedural noise using sparse Gabor convolution[END_REF], can be classified as many-primitive methods. For an overview of procedural noise functions, see Lagae et al. [LLC * 10]. These methods allow to create a wide variety of textures using procedural texturing [EMP * 02] but do not address the problem of temporal coherence. In NPR, the only two previous methods that can be classified as many-primitive methods are the dynamic canvas of Kaplan and Cohen [KC05] and the interactive watercolor rendering of Bousseau et al. [START_REF] Bousseau | Interactive watercolor rendering with temporal coherence and abstraction[END_REF]. These methods take into account the problem of temporal coherence, but are limited to a specific medium: canvas fibers and watercolor respectively. In both cases the connection with sparse convolution noise is not made explicit, even though Bousseau et al. dresses the problem of temporal coherence and leverages procedural texturing to create a wide variety of styles. To further extend the range of patterns we can produce, we plan to construct a version of spot noise that takes into account temporal coherency. As a many-primitive method, NPR Gabor noise bridges primitive-based methods and texture based methods. As we shall show in our results and user experiment, it features good motion coherence, similar to fewprimitive methods, and a good sense of flatness, similar to image-space texture-based methods.

A noise primitive for NPR

In procedural texturing, a wide variety of textures is created using a noise primitive (typically Perlin noise) as the basic component [Per85, EMP * 02]. To fulfill the goals we determined in the introduction, we define a noise primitive for NPR that is temporally coherent. Our new noise primitive is based on Gabor noise [LLDD09, LLD09]. We chose Gabor noise because it features precise local spectral control, which allows us to maintain the 2D aspect of the noise, and to create a wide variety of different styles (see Sec. 4).

Gabor noise

Gabor noise n is a sum of randomly weighted and positioned Gabor kernels,

n(x, y) = ∑ i w i g(a i , F 0i , ω 0i ; x -x i , y -y i ), (1)
where {w i } are the random weights, g is the Gabor kernel, and {(x i , y i )} are the random positions. The Gabor kernel is parametrized by a bandwidth a, which determines the kernel radius r, a frequency F 0 and an orientation ω 0 . The random positions {(x i , y i )} are distributed according to a Poisson distribution with mean λ. We call the random positions the point distribution, and λ the point density. Depending on how a i , F 0i and ω 0i vary for different kernels, anisotropic, isotropic, or even more general kinds of noise are obtained.

The number of overlapping kernels is relatively large. The mean λ is expressed as N ′ /πr 2 , where N ′ is the average number of overlapping kernels. The choice of N ′ is a speed versus quality trade-off, since both the evaluation time and the quality of the noise are proportional to N ′ . Typical values of N ′ range from 16 to 128. Gabor noise can be seen as a primitive-based method, where the kernel is the primitive. As for other primitivebased methods in NPR, the basic principles of NPR Gabor noise follow from the goals formulated in Sec. 1:

NPR Gabor noise

• To obtain a noise with a 2D aspect, we define the noise parameters F 0 , a, ω 0 and λ in 2D screen space, and also evaluate the noise in 2D screen space. • To ensure a coherent motion of the noise, we define the point distribution used by the noise on the surface of the 3D model.

Primitive-based methods in NPR typically use splatting [KC05, BKTS06]. Noise based on kernels can be generated using splatting, as in [START_REF] Van Wijk | Spot noise texture synthesis for data visualization[END_REF][START_REF] Frisvad | Fast high-quality noise[END_REF], or procedurally, as in [Lew89, LLDD09], two conceptually opposite approaches. Procedural approaches are usually more general, while splatting is usually faster and maps better to the GPU. Since we target interactive applications, we use splatting.

Dynamic point distribution

Assuming a triangle model and a perspective camera, we first need to generate a Poisson distribution for our Gabor noise. We then need a way to guarantee temporal continuity of the noise. To achieve this while minimizing popping, we introduce an LOD mechanism which ensures a continuous and constant density of point distributions in 2D screen space throughout time. A key aspect of this LOD is the blending weight, which we determine based on the statistical properties of the noise.

Poisson distribution for NPR Gabor noise. In contrast to most few-primitive methods in NPR, which require a Poisson-disk distribution, Gabor noise only requires a Poisson distribution. We obtain this distribution per triangle using a method inspired by [KC05] and [START_REF] Lagae | Procedural noise using sparse Gabor convolution[END_REF]. We generate the points in a triangle using a pseudo-random number generator (PRNG), with a different seed for each triangle to ensure a random point distribution. We use the PRNG to generate uniformly distributed barycentric coordinates that correspond to the random kernel positions {(x i , y i )}. Since points in a Poisson distribution are independent, these positions are guaranteed to follow a Poisson distribution over edges, even if they are generated per-triangle. We always use the same seed for the same triangle, which ensures temporal continuity during the animation. If the expected number of points N on the triangles is less than 1, both generating a point or not will result in an incorrect point density. We solve this problem by generating a point and weighting it by N to account for this discretization error. Dynamic point distribution using LOD. Since the density of the point distribution is constant in 2D screen space, the number of points on a triangle in 3D object space should vary. More specifically, the required number of points N on a triangle is λA, where A is the 2D screen space area of the triangle. Note that N changes when A changes, for when the triangle moves away from the viewer. This means that NPR Gabor noise requires a dynamic point distribution, i.e., a distribution that changes over time. One way to achieve this would be to add and remove individual points in a temporally consistent manner, to maintain the required point density λ in 2D screen space, similar to [KC05, BKTS06]. Unfortunately, we have experienced that, with our noise, this fine-grained level of control can still lead to popping, since a single kernel can still appear relatively fast.

Instead, we halve or double the number of points, and weight the second half of the points to account for this operation. More specifically, rather than using N points, we use 2M points, where:

M = 2 ⌊log 2 N⌋
(2)

and we weigh the first M points with a weight of 1, and second M points with a weight of α. Note that M and 2M are the powers of 2 that bracket N. This coarse-grained level of control further reduces popping, since kernels now appear at a slower rate.

Choice of blending weight using statistical properties.

Since Gabor noise is a summation, we can express the noise n 2M based on 2M points as a linear interpolation between two noises n M and n ′ M based on M points:

n 2M (x, y) = n M (x, y) + αn ′ M (x, y). ( 3
)
it might seem as if we just trade popping artifacts for blending artifacts, in a manner similar to texture-based methods in NPR. However, this is not the case, since we do not affect the appearance of the noise. Our approach preserves the statistical properties of Gabor noise because we blend between two noises with the same parameters. In contrast, most existing approaches in NPR that blend between two noises blend noises with different parameters, for example two octaves of Perlin noise, which have a different principal frequency.

We derive a weighting function α that preserves the statistical properties of the noise n M , based on the power spectrum and the variance. In the Appendix, we derive the relation between the power spectrum and variance of noise n 2M and n N . From Eqn. 6 and Eqn. 8 we see that n 2M preserves the variance and the power spectrum of n N when (1 + α 2 )M/N equals 1. We obtain the weighting function α by solving this equation for α and substituting Eqn. 2:

α(N) = N 2 ⌊log 2 N⌋ -1.
(4)

Implementation

Since we target interactive applications, we implement NPR Gabor noise efficiently on the GPU, using geometry and fragment shaders.

We use a geometry shader to generate point sprites onthe-fly from incoming triangles. It is important for performance to discard invisible point primitives before rasterization. Therefore, we do not generate primitives for backfacing and invisible triangles. We resolve visibility using a depth test, or using a per-object ID buffer and a per-triangle ID buffer. In order to avoid popping (see Sec. 6), we use a fuzzy depth test, such as a blurred depth test [LD06] or a partial visibility test [START_REF] Finkelstein | Partial visibility for stylized lines[END_REF]. The number of point primitives that can be emitted by the geometry shader for each triangle is limited (in our setup to 64). We work around this limitation by subdividing large triangles, currently in a manual preprocess. Future GPU's with relaxed hardware limitations and the programmable tessellation will most likely remove this limitation.

We use a fragment shader to perform per-fragment visibility tests and to evaluate the Gabor kernel. The visibility tests include an object ID test, a triangle ID test, and a potentially blurred depth test. Each test affects the appearance of the noise, and can be enabled or disabled by the user. The object ID test, triangle ID test and depth test reject fragments based on object ID, triangle ID and depth, trimming the noise to object boundaries, triangle boundaries and depth edges.

Our GPU implementation runs at interactive rates, ranging from 8 to 50 fps for models of moderate complexity (4k to 50k tris), using off-the-shelf hardware (Intel Core 2 Duo 2.4GHz CPU and GeForce GTX 260 graphics card). Our GPU implementation scales well with scene complexity. Although every triangle is processed by the geometry processor, the total number of point sprites is approximately constant. Our GPU implementation supports anti-aliasing using super-sampling for higher image quality at the expense of a lower frame rate.

Styles

Our approach inherits all the advantages of noise primitives, while addressing the problem of temporal coherence. This means that we can use standard techniques from procedural texturing and modeling (see e.g., [EMP * 02]) to generate a variety of patterns that are well suited for NPR. We provide a "cookbook" with a precise description of the styles we designed in the additional material.

We model binary styles by thresholding a noise using a smooth step function (see Fig. 2), in the same spirit as Durand et al. [DOM * 01]. This produces hatches in the case of an anisotropic noise, and stipples in the case of an isotropic noise (see Fig. 3(c)). The position of the step controls the intensity of the binary pattern, and the smoothness of the step controls the smoothness of the edges of the hatches and stipples. The frequency of the noise controls the average distance between stipples or hatches, and the bandwidth of the noise controls the regularity of the pattern. We achieve local control by linking these parameters to scene attributes: examples include noise orientation linked to geometric curvature (see Fig. 3(a,e)) or noise frequency linked to shading, allowing for various hatches or stipple sizes (see Fig. 3(b,c)). We obtain cross-hatching by multiplying two thresholded anisotropic noises, and graphite by modulating hatching with a high-frequency noise layer (see Fig. 2).

We produce color styles by compositing the noise with the scene color in various ways. We produce a watercolor style by modeling pigments and paper textures using noise, similar to [START_REF] Bousseau | Interactive watercolor rendering with temporal coherence and abstraction[END_REF] (see Fig. 3(d)). This approach uses scene color in an overlay blending mode that we also use to produce felt-tip patterns (see Fig. 3

(g)).

A painterly style is produced using large hatches overlaid with an high frequency anisotropic noise that simulates brush fibers. We add a bump mapping effect to emphasize the fibers (see Fig. 3(a)).

User study

The goals we set out in the introduction are contradictory, and therefore no perfect solution exists. As a consequence, evaluating the success of any given trade-off is very complex. Previous approaches only relied on speculative visual inspection. However, each goal has specific artifacts which a user can observe. A user study can thus provide significant insight into how well each solution performs for each goal.

It also provides an indication of overall success as well as relative importance of the different criteria involved.

Note that in this user study we are only evaluating the tradeoff of different methods with respect to temporal coherence. Other aspects, such as the range of patterns and the interactivity, will be discussed in Sec. 6.

Procedure techniques.

Tab. 1 summarizes the relative merits of the compromises presented by each method, as discussed in Sec. 2. Among them, the extreme cases of shower door (SD) and texture mapping (TM) are logical choices for reference comparisons. We include the following state-ofthe-art texture-based methods in our study:

• Dynamic Solid Textures (DST) [START_REF] Bénard | Dynamic solid textures for real-time coherent stylization[END_REF], an object-space approach, • Bidirectional Texture Advection (Adv) [START_REF] Bousseau | Video watercolorization using bidirectional texture advection[END_REF], a local image-space approach, • Dynamic 2D Patterns (D2D) [BSM * 07], a global imagespace approach.

For many-primitive approaches, we only used our method, since [KC05, BKTS06] can be seen as special cases of our solution applied to canvas and watercolor. We did not include few-primitive approaches in our comparisons for several reasons: (i) there is no way to produce the same pattern in a texture-based and a few-primitive approach; (ii) the severe popping produced by these methods is a highly disturbing, and thus obvious, artifact which probably does not require a user test. We show an example from [START_REF] Vanderhaeghe | Dynamic point distribution for stroke-based rendering[END_REF] in our accompanying video (used with permission).

Setup. The study was split into two parts involving simple and complex stimuli respectively. A total number of 15 volunteers participated in both. Participants were asked to rank the stimuli, presented as still images or video loops, according to several criteria. Ranking was performed by dragging and dropping them on a dual monitor setup in a controlled environment as shown in Fig. 4(a). A detailed description of the experimental procedure and setup are provided in the additional materials. A sequence of a participant performing the study is also presented in the video.

Analysis methodology. We first determine appropriate assessment criteria for each goal defined in Sec. 1. We use them as guidelines to choose the stimuli and to formulate the questions we pose to the participants of the study. In the following sections we describe the details and results of the separate ranking tasks, in the order they were presented to the participants. The statistical significance of observed trends is confirmed by the Wilcoxon rank-sum hypothesis test [Wil45] and is indicated by the p-value. Fig. 5 provides the ordinal scales (interval scales are provided in the additional materials). and the similarity groups at the 95% significance level.

Simple stimuli

Simple stimuli were created to selectively test the separate goals in controlled conditions. Each stimulus consisted of a single object rendered with a black-and-white hatching pattern (see Fig. 4), chosen to best reveal the differand possible artifacts of each selected method. This style corresponds to an extreme case; we thus believe that our conclusions generalize more readily to other styles. Lighting is disabled to avoid additional shape or motion cues [START_REF] Landy | Measurement and modeling of depth cue combination: in defense of weak fusion[END_REF]WFGS07].

Flatness. The stimuli consist of still images depicting a sphere in front of a planar background, both rendered with the same pattern (Fig. 4(c,d)). Participants are asked to rank the images according to how flat they appear. Since lighting is disabled, the only remaining shape cues are the silhouette of the sphere and the perspective distortion of the pat- In practice, the sphere completely blends in with the flat background for methods that do not provide these shape cues in the pattern.

The results show a significant amount of variance, indicating the complexity of question. People are not accustomed to relying on shape-from-texture cues in isolation. Additionally, several methods produce similar flatness by design: Shower Door and Dynamic 2D Patterns both produce perfectly flat images without perspective distortion, while Dynamic Solid Textures and Texture Mapping both produce the correct perspective. Of the hybrid 2D/3D methods, ours seems slightly closer to the 2D methods than Advection.

Coherent motion. The stimulus videos show a pitcher on a white background (Fig. 4(e)). We chose a pitcher as a simple, recognizable, everyday object, to make it easier for the observer to concentrate on the pattern. Nevertheless, the pitcher is sufficiently to exhibit some self-occlusion, which is handled differently by the compared methods. Three separate basic motions were ranked, in random order: translation parallel to the image plane, rotation, and zoom. We need to evaluate the correlation of the motion of the 3D scene and the motion of the pattern after stylization. Insufficient correlation creates the impression that the pattern is sliding over the depicted scene. Participants were asked to rank the videos according to how coherently the pattern moves with the object. As expected, object-space methods such as Texture Mapping and Dynamic Solid Textures are significantly more coherent than the other techniques (p < 0.05), while Shower Door is consistently the least coherent (p < 0.05). The three other approaches which compromise flatness for motion coherence are perceived almost equally. Depending on the motion, the ranking varies slightly: Advection performs better for rotation (p < 0.05), whereas Dynamic 2D Patterns and our method perform better for the zoom (p < 0.05).

Temporal continuity. Given a stylized version of an anwe want to estimate the fluidity or continuity of the animation. Temporal discontinuities such as popping strongly distract the attention of human observers [YJ84, SS09], and thus are easily identified. Temporally continuous artifacts, which are more subtle and consequently harder to notice, include contrast or intensity oscillations, as well as local residual motion of the patterns. For the same pitcher stimuli, participants were asked to rank the videos according to how much the pattern changes over time, regardless of the coherence of the motion of the object and the pattern, which they had already evaluated in the coherent motion task. The results are not as clear-cut as for the previous question. This may be due to the complexity of the question. For example Shower Door, for which the pattern is constant over time, clearly shows that many participants had difficulty to both ignore the principal translation and to concentrate on residual motions.

Overall, Advection introduces the largest changes to the pattern in the form of a residual motion called "swimming" in [BNTS07, Sec. 5], which are made more apparent by subsequently applying the binary threshold. For this kind of pattern, our approach exhibits similar effects during rotations and zoom.

We constrained the orientation and scale of the pattern in 2D but chose to attach the primitives to the 3D surface. Accommodating these conflicting constraints in an animation can result in artifacts such as popping or residual motion. This may be why our method performs poorly on the temporal continuity scale for rotation and zoom. Advection, Dynamic 2D Patterns, and our method are the only ones that actually try to satisfy all these constraints, whereas the other methods do not support some 2D constraints (TM and DST) or 3D constraints (SD).

Complex stimuli

We created complex stimuli to study the behavior of the different approaches for a more aesthetically pleasing experience in a complex walkthrough. The viewer moves inside an indoor scene which contains a mix of large planar surfaces, small details, grazing angles, and depth discontinuities. To make the different objects and surfaces more apparent, lighting is enabled, black outlines are drawn, and the scene is rendered with colored cross hatching patterns on a white paper background (Fig. 4(b)).

We kept the same questions as in the first stage. Flatness is evaluated on still images; coherent motion and temporal continuity are evaluated on a complex motion path. To assess the overall aesthetic appreciation of the different approaches, participants were additionally asked to rank the videos according to how pleasant they find them in the context of cartoon animation. Once again, the variance of the answers while estimating the flatness of the scene shows the complexity of this question. This is more pronounced in the complex scene, due to the additional shape and depth cues.

The responses concerning motion coherence are consistently in favor of object-space approaches. Our noise ranks with the best of the image-space methods for this goal.

The results for motion coherence and pleasantness (Fig. 5(i) and (k)) are strongly correlated (Kendall rank correlation τ = 0.58 [START_REF] Kendall | A new measure of rank correlation[END_REF]). This indicates that motion co-herence is probably the most important quality to preserve in the overall temporal coherence compromise. Both Dynamic Solid Textures and our method perform well on the motion coherence scale, and trade off better temporal continuity (DST) versus better flatness (ours).

Discussion

Popping

The GPU implementation of our noise primitive based on splatting is subject to residual popping artifacts, which may be reinforced by the stylization. This popping is caused by aliasing in the per-point-sprite visibility tests, due to the mismatch between the continuous point sprite position and the discretized visibility. This problem is typically alleviated using fuzzy visibility tests, such as the blurred depth test [LD06]. We use a partial visibility test based on the work of Cole and Finkelstein [START_REF] Finkelstein | Partial visibility for stylized lines[END_REF]. It works relatively well in practice, although some popping may remain according to the mesh and style. Moreover, partial visibility avoids binary visibility decisions at (dis)occlusion boundaries, which reduces the residual motion of the pattern near silhouettes.

We believe it might be possible to completely remove the remaining popping artifacts by using a procedural approach, as in [START_REF] Lagae | Procedural noise using sparse Gabor convolution[END_REF]. However, initial experiments were not enin terms of performance. Because we target interactive rates, which is especially important for the artistic design process, we believe that a splatting approach is currently more suited for NPR Gabor noise It would be interesting to develop such a high quality offline approach which could use the current method as a previsualization tool. Appropriate guarantees on similarity of visual results must of course be provided.

Styles

Our goal is not to precisely reproduce a specific style, which explains why our results do not always look like their realworld equivalent (e.g., hatching and stippling). Instead, we provide a noise primitive that is temporally coherent, such that the artist can focus on style creation rather than on the dynamic behavior of the stylization.

Our noise primitive does not offer direct control at the stroke level. Consequently, we cannot create styles with clearly independent primitives. This limitation is shared with other texture-based and many-primitives methods. However, in contrast to most other texture-based approaches, our noise primitive does offer local control, for example by varying the noise parameters according to scene attributes.

Our current method for style design is not entirely trivial. Nevertheless, we were able to create a wide variety of styles, with reasonable effort, and without an advanced user interface (see supplemental material and the video for a stylecreation sequence). Of course, more intuitive methods for artistic control would significantly facilitate this process. In this context, it is important to note that procedural texturing and noise, the concepts we use for style design, have been used by artist for decades.

User study

Our user study is necessarily limited by the choices we have made. For example, the choice of hatching as a stimulus, which makes the fractalization artifacts introduced by dynamic solid textures hard to see, or the choice for naive users, which could respond differently than expert users such professional artists, and fundamentally the choice of the goals we evaluated. Of course, a more elaborate user study would be very interesting, particularly to refine the relative importance of these goals.

Our user study provides a comparison of several methods, but has to be interpreted with care: First, as explained in the introduction, the problem of temporal coherence in NPR involves intrinsically contradictory constraints. Therefore, any possible solution to this problem will be a trade-off, and no method can have the best performance for all of the criteria simultaneously. We believe a good trade-off should provide a satisfactory compromise for as many criteria as possible, and should not completely break down for one of them. Overall, our new method achieves this goal well.

Second, the user study only covers some aspects of the methods. Our approach has several advantages over the others that are not or only partially revealed by the user study. Compared to dynamic 2D Patterns [BSM * 07], our method is not subject to sliding, and is better at handling complex scenes. Compared to dynamic solid textures [START_REF] Bénard | Dynamic solid textures for real-time coherent stylization[END_REF], our method is better in flatness, and does not degrade the pattern. Compared to bidirectional texture advection [START_REF] Bousseau | Video watercolorization using bidirectional texture advection[END_REF], our method runs at interactive rates, does not require knowledge of the entire animation, and preserves the pattern better.

Conclusion

We have presented a new dynamic noise primitive for coherent stylization, and a user study investigating temporal coherence. Our new noise primitive allows the use of standard techniques from procedural texturing to create various temporally coherent styles for interactive NPR applications. Our user study provides more insight into the problem of temporal coherence, for example, that motion coherence is one of the most important qualities to preserve in the overall temporal coherence compromise, and that our new solution provides a good compromise between the 2D aspect of the style and 3D motion.

Our noise primitive for NPR makes the connection between primitive-based methods in NPR and methods for sparse convolution noise in procedural texturing more explicit. This places existing work in a different perspective. For example, the watercolor of Bousseau et al. [START_REF] Bousseau | Interactive watercolor rendering with temporal coherence and abstraction[END_REF] is actually a sparse convolution noise [Lew89] with a Gaussian-like kernel, and the dynamic canvas of Kaplan and Cohen [KC05] is actually a spot noise [vW91] with a fiber as the spot shape. This connection is likely to be productive. For example, to further extend the range of patterns we can produce, we plan to construct a version of spot noise [vW91] that takes into account temporal coherency, using the ideas of NPR Gabor noise.

Our user study is a first attempt to better understand the trade-offs involved in temporal coherence, and a first step towards a more formal evaluation procedure for this problem. We plan to do more user studies, taking into account what we have learned so far, but also to explore objective measures to quantify these trade-offs, for example, statistical texture measures to quantify flatness, in the spirit of [START_REF] Bénard | Quality assessment of fractalized NPR textures: a perceptual objective metric[END_REF], and flow to quantify motion coherence and temporal continuity. This is likely to lead to deeper insight in the evaluation of solutions for the temporal coherence problem.

  use Perlin noise [Per85]. Our dynamic noise primitive for coherent stylization adc 2011 The Author(s) Journal compilation c 2011 The Eurographics Association and Blackwell Publishing Ltd.

  None of the existing kinds of Gabor noise addresses the problem of temporal coherence for NPR. 2D Gabor noise (see Fig.1(a)) is inherently 2D, and therefore behaves like the shower door effect, while surface Gabor noise (see Fig.1(b)) and solid Gabor noise are inherently 3D, and therefore behave like regular texture mapping. We present a new kind of Gabor noise, which we call NPR Gabor noise (see Fig.1(c)), that takes into account the problem of temporal coherence.

Figure 1 :

 1 Figure 1: NPR Gabor noise. (a) 2D Gabor noise. (b) Surface Gabor noise. (c) NPR Gabor noise. NPR Gabor noise has a 2D aspect, similar to 2D Gabor noise, and moves coherently with the object (see video), similar to surface Gabor noise.

  c 2011 The Author(s) Journal compilation c 2011 The Eurographics Association and Blackwell Publishing Ltd.

Figure 2 :

 2 Figure 2: Style design. (a, b) Two anisotropic noises are designed. (c) The thresholded results are multiplied to produce a crosshatching. The inset shows the x-toon texture used for thresholding. (d) Adding a third high-frequency noise layer allows to produce a graphite style.

  Figure 3: A variety of styles produced with our method. See the accompanying video to see these styles in motion.

  c 2011 The Author(s) Journal compilation c 2011 The Eurographics Association and Blackwell Publishing Ltd.

  inria-00487986, version 3 -2 Aug 2011

  Figure 4: Some examples of the simple stimuli (c,d) from the flatness task and (e) from the coherent motion and temporal continuity tasks, and (b) a complex stimulus. tern [Pal99, Ch. 5].In practice, the sphere completely blends in with the flat background for methods that do not provide these shape cues in the pattern.

Figure 5 :

 5 Figure5: Ordinal scales indicating the relative merits of each method. The methods are classified into similarity groups based on the Wilcoxon rank-sum hypothesis test at 95% significance level. Within a group all methods perform similarly. Conversely, methods that are not grouped together are significantly different. A method can belong to multiple groups, shown here by overlapping rectangles.

  c 2011 The Author(s) Journal compilation c 2011 The Eurographics Association and Blackwell Publishing Ltd.

  inria-00487986, version 3 -2 Aug 2011 P. Bénard, A. Lagae, P. Vangorp, S. Lefebvre, G. Drettakis & J. Thollot / A Dynamic Noise Primitive for Coherent Stylization

  c 2011 The Author(s) Journal compilation c 2011 The Eurographics Association and Blackwell Publishing Ltd. inria-00487986, version 3 -2 Aug 2011 P. Bénard, A. Lagae, P. Vangorp, S. Lefebvre, G. Drettakis & J. Thollot / A Dynamic Noise Primitive for Coherent Stylization

Table 1 :

 1 P.Bénard, A. Lagae, P. Vangorp, S. Lefebvre, G. Drettakis & J. Thollot / A Dynamic Noise Primitive for Coherent Stylization Summary of the trade-offs made by various solutions for the temporal coherency problem.

			Flatness	Coherent motion		Temporal continuity
	Naive	Shower door Texture mapping	++ --	--++	++ ++
	Few-primitive	[Mei96] [VBTS07]	-/+ ++	+ +	--	popping
	Object-space textures	[PHWF01] [BBT09]	--	++ ++	-/+ mipmap + fractalization
		[BNTS07]	++	++	-/+ regeneration
	Image-space texture	[CDH06, BSM * 07]	+	-	-/+ mipmap
		[CTP * 03]	++	--	+	fractalization
	Many-primitive	[KC05, BKTS06] Ours	++ ++	+ +	--/+ secondary motion, residual popping flickering, oscillations

Acknowledgments

We would like to thank the anonymous reviewers, Kartic Subr and Adrien Bousseau. We are especially grateful to Laurence Boissieux for 3D modeling and for producing the advection sequences, to Marcio Cabral for producing the video, and to the participants of the user study. Ares Lagae is a Postdoctoral Fellow of the Research Foundation -Flanders (FWO), and acknowledges K.U.Leuven CREA funding (CREA/08/017).

Appendix

To determine the weight α in Eqn. 4, we need to determine the power spectrum and variance of n 2M in terms of n N .

Power spectrum. First, we take the magnitude of the Fourier transform of both sides of Eqn. 3. Then we simplify, noting that the Fourier transform is a linear operator, and that n M and n ′ M have the same expected Fourier transform,

Note that with expected Fourier transform we mean the Fourier transform of the stochastic processes corresponding to n M and n ′ M rather than the Fourier transform of specific instances of n M and n ′ M . Finally, we apply equation 10 of [START_REF] Lagae | Procedural noise using sparse Gabor convolution[END_REF], noting that λ

This establishes a relation between the power spectrum of the noises n N and n 2M .

Variance. First, we take the variance of both sides of Eqn. 3. Then we simplify, noting that n M and n ′ M are uncorrelated and have the same expected variance,

Finally, we apply Eqn. 10 of [START_REF] Lagae | Procedural noise using sparse Gabor convolution[END_REF],

This establishes a relation between the variance of the noises n N and n 2M .


Hayk Nersisyan
email: hayk.nersisyan@u-cergy.fr

Stabilization of the 2D incompressible Euler system in an infinite strip

The paper is devoted to the study of a stabilization problem for the 2D incompressible Euler system in an infinite strip with boundary controls. We show that for any stationary solution (c, 0) of the Euler system there is a control which is supported in a given bounded part of the boundary of the strip and stabilizes the system to (c, 0).

Introduction

We consider the incompressible two-dimensional Euler system u + u, ∇ u + ∇p = 0, div u = 0, (1.1) where u = (u 1 , u 2 ) and p are unknown velocity field and pressure of the fluid, and

u, ∇ v = 3 i=1 u i (t, x) ∂ ∂x i v.
The space variable x = (x 1 , x 2 ) belongs to the strip D defined by (1.

D := {(x 1 , x 2 ) : x 1 ∈ R, x 2 ∈ (-1, 1)}. ( 1
3)

The aim of this paper is the study of stabilization of (1.1) with boundary controls supported by Γ 0 . System (1.1) is completed with the boundary and initial conditions

u • n = 0 on Γ \ Γ 0 , (1.4)
u(x, 0) = u 0 (x), (

where Γ := ∂D and n is the outward unit normal vector on Γ. In particular, (1.4) is equivalent to u 2 = 0 on Γ \ Γ 0 . For any integer s ≥ 0 we denote by H s (D) the space of vector functions u = (u 1 , u 2 ) whose components belong to the Sobolev space of order s and by

• s,D the corresponding norm. If there is no confusion, we drop the index D. In the case s = 0, we write • := • 0 . For any integer s > 0 we define H s (D) as the space of distributions u in D with ∇u ∈ H s-1 (D). We equip H s (D) with the semi-norm u H s (D) := ∇u s-1 .

We denote by Ḣs (D) the quotient space H s (D)/R. The following theorem is our main result.

Main result. Let s ≥ 4 be an integer. Then for any constant c ∈ R and initial function u 0 ∈ H s (D) that decays fast at infinity and satisfies the relations

div u 0 = 0, u 0 • n = 0 on Γ \ Γ 0
there exists a solution (u, p) ∈ C(R + , C(D) ∩ Ḣs (D)) × C(R + , Ḣs (D)) of (1.1), (1.4) and (1.5) such that lim t→∞ ( u(•, t) -(c, 0) L ∞ + ∇u s-1 + ∇p s-1 ) = 0.

For the exact statement see Theorem 3.1. In this formulation the control is not given explicitly, but we can assume that control acts on the system as a boundary condition on Γ 0 .

Before turning to the ideas of the proof, let us describe in a few words some previous results on the controllability of Euler and Navier-Stokes systems. Coron [START_REF] Coron | On the controllability of 2-D incompressible perfect fluids[END_REF] introduced the return method to show exact boundary controllability of 2D incompressible Euler system in a bounded domain. Glass [START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF] generalized this result for 3D Euler system. Chapouly [START_REF] Chapouly | On the global null controllability of a Navier-Stokes system with Navier slip boundary conditions[END_REF] using return method proved the global null controllability of the Navier-Stokes system in rectangle. Recently, Glass and Rosier [START_REF] Glass | On the control of the motion of a boat[END_REF] proved the controllability of the motion of a rigid body, which is surrounded by an incompressible fluid. Controllability of Euler and Navier-Stokes systems with distributed controls is studied in [START_REF] Agrachev | Navier-Stokes equations controllability by means of low modes forcing[END_REF][START_REF] Fursikov | Exact controllability of the Navier-Stokes and Boussinesq equations[END_REF][START_REF] Nersisyan | Controllability of 3D incompressible Euler equations by a finite-dimensional external force[END_REF][START_REF] Shirikyan | Approximate controllability of three-dimensional Navier-Stokes equations[END_REF]; see also the book [START_REF] Coron | Control and nonlinearity[END_REF] for further references.

Notice that the above papers concern the problem of controllability of the fluid in a bounded domain. In this paper, we develop Coron's return method to get the controllability of velocity of 2D Euler system in an unbounded strip. This method consists in reducing the controllability of nonlinear system to the linear one. To this end, one constructs a particular solution (u, p) of Euler system and a sequence of balls {B i } covering D, such that (P ) Any ball B i driven by the flow of u leaves D through Γ 0 at some time.

Then the linearized system around u is controllable. In our case, since the domain D is unbounded, the number of balls B i is infinite, thus we cannot construct a bounded function u, whose flow moves all balls outside D in a finite time. However, we can find a particular solution u such that property (P ) holds in infinite time. This proves the stabilization of linearized system in infinite time.

To show that controllability of linearized system implies that of the nonlinear system, we need to prove that (P ) also holds for any ũ sufficiently close to u. This is obvious in the case of bounded domain. In our case, to prove this, we need some additional properties for u. In particular, we need to construct a solution u, which decays at infinity faster than 1/x 2 1 . As our particular solution u is a combination of the Green functions of the Laplacian with Neumann boundary condition, we need to prove that Green functions decay at infinity. This property is a consequence of elliptic regularity and some explicit formulas for solutions of the Laplace equation in a strip.

The paper is organized as follows. In Section 2, we give preliminaries on Poisson and Euler equations in an unbounded strip. The main results of the paper are presented in Section 3. In Section 4, we construct the particular solution u. In the Appendix, we prove an auxiliary result used in Section 2.

Acknowledgments. The author would like to express deep gratitude to Armen Shirikyan for drawing his attention to this problem and for many fruitful suggestions and also to Nikolay Tzvetkov for useful remarks on the Euler system.

Notation.

Let J T := [0, T ). The space of continuous functions u : J T → X is denoted by C(J T , X). For any integer s ≥ 0 or s = ∞, we denote

C s b (D) = {u ∈ C s (D) : u L ∞ (D) < ∞}. We set Ḣ∞ (D) := ∩ ∞ s=0 Ḣs (D). Define S(D) := {u ∈ L 2 (D) : x α 1 ∂ β u(x 1 , x 2 ) ∈ L 2 (D) for any α ∈ R + , β ∈ Z 2 + }.
For a vector field u = (u 1 , u 2 ) we set

curl u = ∂ 1 u 2 -∂ 2 u 1 .
The interior of a set K is denoted by int(K). Let B(x 0 , r) be the closed ball in R 2 of radius r centred at x 0 . We denote by C a universal constant whose value may change from line to line.

Preliminaries

In this section, we present some auxiliary results on Poisson and Euler equations in an unbounded strip. The methods used in their proofs are well known and in many cases we confine ourselves to a brief description of the main ideas.

Poisson equations in an unbounded strip

First, let us describe the spaces Ḣs (D).

Proposition 2.1. For any integer s ≥ 1 we have (i) The space Ḣs (D) is complete.

(ii) H s (D) = {u ∈ H s loc (D) : ∇u ∈ H s-1 }. (iii) If s ≥ 3, then for any u ∈ H s (D) there is a constant C depending on u such that |u(x 1 , x 2 )| ≤ C|x 1 | + C
holds for all x ∈ D.

Proof. Let {u n } ⊂ Ḣs (D) be a Cauchy sequence. Then there is v ∈ H s-1 (D) such that ∇u n →v in H s-1 (D) as n→∞, and for any ϕ ∈ C ∞ 0 (D) such that div ϕ = 0, we have

0 = lim n→∞ (∇u n , ϕ) L 2 = (v, ϕ) L 2 .
Hence, v = ∇z, where z ∈ Ḣs (D). This proves that Ḣs (D) is complete. Now let us prove assertion (ii). Clearly the space in the right-hand side is contained in Ḣs (D). Let us take a function u ∈ Ḣs (D), a compact set K ⊂ D and let us show that u ∈ H s (K). Take two functions χ, χ 1 ∈ C ∞ 0 (D) and a compact set K 1 ⊂ D with int(K 1 ) ⊃ K such that χ = 1 in K 1 and χ 1 = 1 in K1 := supp χ. Then there exists r ∈ N such that χ 1 u ∈ H -r (D). This implies that u ∈ H -r ( K1 ), hence ∆(χu) = 2∇χ∇u + χ∆u + u∆χ ∈ H min(-r;s-2) ( K1 ).

The elliptic regularity implies χu ∈ H min(-r+2;s) (D), thus u ∈ H min(-r+2;s) (K 1 ).

Repeating this argument for a compact set

K 2 ⊂ K 1 with int(K 2 ) ⊃ K we can show that u ∈ H min(-r+4;s) (K 2 )
. Iterating this, we get u ∈ H s (K). This completes the proof of assertion (ii).

It is easy to see that (ii) implies (iii). Indeed, from (ii) we get

u(x 1 , x 2 ) = x1 0 ∂ 1 u(y, x 2 )dy + u(0, x 2 ).
The Sobolev inequality yields (iii).

D := {(x 1 , x 2 ) : x 1 ∈ R, x 2 ∈ (-1 -γ(x 1 -d), 1 + γ(x 1 ))} (2.1)
(see figure 1).

∆u = f in D ′ , (2.2) u = 0 on Γ ′ , (2.3)
where

Γ ′ = ∂D ′ and f ∈ L 2 (D ′ ). We say that u ∈ H 1 0 (D ′ ) is a solution of (2.2), (2.3) if D ′ ∇u∇θdx = - D ′ f θdx for any θ ∈ H 1 0 (D ′ ).
We have the following result for the well-posedness of this problem.

Proposition 2.2. For any integer s ≥ 0 and for any f ∈ H s (D ′ ) problem (2.2), (2.3) has a unique solution u ∈ H s+2 (D ′ ). Moreover,

u s+2 ≤ C f s , (2.4)
where C depends only on s.

Proof. The existence of the solution u ∈ H 1 0 (D ′ ) is a consequence of the Riesz representation theorem. Clearly, we have

∇u 2 ≤ C f u .
(2.5)

The Poincaré inequality applied to u(x 1 , •) gives

u ≤ C ∂ 2 u .
Combining this with (2.5), we obtain

u 1 ≤ C f . (2.6)
To show the regularity of the solution and estimate (2.4), we need the following lemma.

Lemma 2.3. For any integer s ≥ 1 we have

H s (D ′ ) = {z ∈ L 2 (D ′ ) : curl z ∈ H s-1 (D ′ ), div z ∈ H s-1 (D ′ ), z •n ∈ H s-1/2 (Γ ′ )},
where n is the outward unit normal vector on Γ ′ . Moreover, any function z ∈ H s (D ′ ) satisfies the inequality

z s ≤ C z + curl z s-1 + div z s-1 + z • n s-1/2 ,
where C depends only on s.

The proof of this lemma is given in the Appendix. Let us denote z = ∇ ⊥ u := (∂ 2 u, -∂ 1 u). Then curl z = -∆u = -f , div z = 0. Notice that (2.3) implies that z •n = 0. It follows from Lemma 2.3 and inequality (2.6) that z ∈ H s+1 (D ′ ) and z s+1 ≤ C f s . Thus, we obtain u ∈ H s+2 (D ′ ) and (2.4).

Let us take g ∈ H 1 (D ′ ) and consider the Neumann problem for the Poisson equation: (ii) For any

∆u = div g in D ′ , (2.7) ∂u ∂n = g • n on Γ ′ . ( 2
x ∈ D \ {a} ∇G a (x) = - 1 2π |x -a| 2 -2(x 1 -a 1 ) 2 |x -a| 4 , -2(x 1 -a 1 )(x 2 -a 2 ) |x -a| 4 +ψ a (x), (2.12)
where ψ a ∈ H ∞ ( D).

(iii) Let a ∈ D \ D, then G a ∈ Ḣ∞ (D) and for any integers 1 ≤ i, j ≤ 2 we have

∂ i ∂ j G a (x 1 , x 2 ) ∈ S(D). (2.13)
(iv) For any fixed x ∈ D the function G a (x) is analytic in a ∈ D \ {x}.

Proof. The existence of a solution G a ∈ C ∞ ( D \ {a}) will be established when proving assertion (ii). To prove the uniqueness of the solution, we assume that there are two solutions G 1,a and G 2,a . For G = G 1,a -G 2,a we have

∆ G = 0 in D, ∂ G ∂n = 0 on ∂ D, Let χ ∈ C ∞ 0 ( D) with χ = 1 in Q. Then ∆(χ G) = h,
where h ∈ C ∞ 0 ( D). The elliptic regularity for a bounded domain implies that χ G ∈ H ∞ ( D). Since G ∈ Ḣs ( D \ Q), we get G ∈ Ḣs ( D). It follows from Proposition 2.4 that G = 0.

To prove (ii), we seek the solution in the form

G a = ∂ 1 (F a χ) + u a , (2.14)
where F a (x) = -1 2π ln |x-a| is the fundamental solution of the Laplace operator in R 2 , χ ∈ C ∞ 0 ( D), χ is 1 in a neighborhood of a. Then u a must be the solution of the problem

∆u a = -∂ 1 (2∇F a • ∇χ + F a ∆χ) := ∂ 1 f in D, ∂u a ∂n = 0 on ∂ D.
Since f ∈ C ∞ 0 ( D), applying Proposition 2.4 for g = (f, 0), we conclude that this problem has a solution u a ∈ H ∞ ( D). Property (2.12) follows from the construction of G a . Now let us show (2.13). We have that G a satisfies the following problem in D:

∆G a = 0 in D, (2.15) ∂G a ∂n = ϕ on Γ, (2.16)
where ϕ ∈ C ∞ (Γ) and supp ϕ ⊂ Γ 0 . To show that the second derivatives of the solution belong to S(D), let us apply the Fourier transform in x 1 to (2.15), (2.16). We obtain

d 2 dx 2 2 Ĝa -ξ 2 Ĝa = 0 in D, d Ĝa dx 2 (ξ, -1) = φ1 (ξ), d Ĝa dx 2 (ξ, 1) = φ2 (ξ),
where Ĝa , φ1 and φ2 are Fourier transforms of G a , ϕ(•, -1) and ϕ(•, 1), respectively. The solution of this ODE is given by Ĝa (ξ,

x 2 ) = φ2 -φ1 2ξ sinh(ξ) cosh(ξx 2 ) + φ2 + φ1 2ξ cosh(ξ) sinh(ξx 2 ).
Since ϕ 1 and ϕ 2 are compactly supported, we have

F (∂ i ∂ j G a ) ∈ S(D), 1 ≤ i, j ≤ 2, whence it follows that ∂ i ∂ j G a ∈ S(D).
This completes the proof of (iii).

Let Ω be any domain such that Ω ⊂ D and Ω ∩ ( D \ D) = ∅. Then for any fixed x ∈ Ω the function G a (x) is analytic in a ∈ Ω \ {x}. Indeed, let χ in (2.14) be 1 in Ω. Then the analyticity of G a (x) is consequence of the facts that F a is analytic in a and u a is a linear operator in F a . Since G a is the unique solution of (2.10), (2.11), we have the analyticity of G a (x) in D \ {x}.

Euler equations in an unbounded strip

We consider the incompressible Euler system:

u + u, ∇ u + ∇p = 0, div u = 0 in D,
(2.17)

u • n = 0 on Γ, (2.18) u(x, 0) = u 0 (x), (2.19) It is well known that if D is a bounded domain or if D = R 2 , then problem (2.17)-(2.19
) is well posed in various function spaces (e.g., see [START_REF] Kato | On classical solutions of the two-dimensional non-stationary Euler equation[END_REF][START_REF] Kato | Commutator estimates and the Euler and Navier-Stokes equations[END_REF][START_REF] Wolibner | Un théorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long[END_REF]).

In this subsection, we study the well-posedness of Euler system in D defined by (1.2). Definition 2.6. For any integer s ≥ 3 we say that (u, p) is a solution of Euler system if (u, p) ∈ C(J T , H s (D)) × C(J T , Ḣs+1 (D)) and (2.17) is satisfied in the sense of distributions.

Let us show that the Euler system is equivalent to the problem

ẇ + u, ∇ w = 0, w(x, 0) = curl u 0 (x), (2.20)
curl u = w, div u = 0, u • n| Γ = 0. (2.21)
Clearly, if (u, p) is a solution of the Euler system, then (2.20), (2.21) hold. Now let us show that to any solution

(u, w) ∈ C(J T , H s (D)) ∩ C 1 (J T , H s-1 (D)) × C(J T , H s-1 (D))
of (2.20), (2.21) there corresponds a unique solution (u, p) ∈ C(J T , H s (D)) × C(J T , Ḣs+1 (D)) of (2.17)- (2.19). From (2.20) and (2.21) it follows that curl( u + u, ∇ u) = 0.

Hence, there exists p ∈ C(J T , Ḣs (D)) such that -∇p = u + u, ∇ u. It is easy to see that

-div ∇p = div( u, ∇ u) = 2 i,j=1 ∂ i u j ∂ j u i ∈ H s-1 , curl ∇p = 0, - ∂p ∂n = ( u, ∇ u) • n = u, ∇ (u • ñ) - 2 i,j=1 u j u i ∂ j ñi = - 2 i,j=1 u j u i ∂ j ñi ∈ H s-1/2 ,
where ñ is a regular extension of n. Thus, it follows from Lemma 2.3 that ∇p ∈ C(J T , H s (D)), whence we conclude that p ∈ C(J T , Ḣs+1 (D)) .

We have the following result on the local well-posedness of Euler system. The ideas used in the proof of existence of a solution play an important role in the study of stabilization problem (see Section 3). Therefore we present a rather complete proof, even though we do not really need this result.

Theorem 2.7. Let s ≥ 4. For any u 0 ∈ H s (D) satisfying the conditions

div u 0 = 0, u 0 • n = 0 on Γ, there is T * = T * ( u 0 s ) such that system (2.17)-(2.19) has a unique solution (u, p) ∈ C(J T * , H s (D)) × C(J T * , Ḣs+1 (D)).
Proof. Uniqueness. To prove the uniqueness, we argue as in the case of bounded domain. We assume that there are two solutions u 1 and u 2 . Then

for v = u 1 -u 2 , we have v + u 1 , ∇ v + v, ∇ u 2 + ∇p = 0, (2.22) div v = 0, v • n| Γ = 0, v(x, 0) = 0.
Multiplying (2.22) by v and integrating over D, we get

∂ t v(•, t) 2 ≤ - D u 1 , ∇ v • vdx + C v(•, t) 2 - D ∇p • vdx, (2.23)
where C > 0 is a constant depending only on u 2 . Since u 1 • n = 0, the first term on the right-hand side of (2.23) is zero. Let us show that the last term is also zero. Let us denote

Ω (R) := {x ∈ D : |x 1 | < R},
and let χ ∈ C ∞ (D) such that χ(x) = 0, if x / ∈ Ω (2) , 1, if x ∈ Ω (1) .
Clearly, we have lim

R→∞ D χ( x R )∇p(x) • v(x)dx = D ∇p(x) • v(x)dx.
On the other hand, integrating by parts, we obtain

D χ( x R )∇p(x) • v(x)dx = - Ω (2R) \Ω (R) ∇χ( x R ) p(x) R • v(x)dx.
Since p ∈ Ḣs+1 , from assertion (iii) of Proposition 2.1 we have sup

x∈Ω (2R) | p(x) R | < C,
where C does not depend on R. Thus, dominated convergence theorem yields

D ∇p(x) • v(x)dx = 0.
Applying the Gronwall inequality to (2.23), we obtain v = 0. Existence. To prove the existence of the solution, we shall need the following result.

Lemma 2.8. Let ũ ∈ C(R + , H s ), ũ•n| Γ×R+ = 0, f ∈ C(R + , H s ) and w 0 ∈ H s , s ≥ 3. Then the problem ∂ t w + ũ, ∇ w = f, (2.24) w(x, 0) = w 0 , (2.25)
has a unique solution w ∈ C(R + , H s ), which satisfies the inequality

w(•, t) s ≤ w 0 s + t 0 ( f (•, τ ) s + C w(•, τ ) s ∇ũ(•, τ ) s-1 ) dτ. (2.26)
Proof. Let us denote by φ g : D × R + → D the flow associated to g, i.e., the solution of the problem

∂φ g ∂t = g(φ g , t), φ g (x, 0) = x.
Since (2.24), (2.25) is an inhomogeneous transport equation, its solution is given by

w(φ ũ(x, t), t) = w 0 (x) + t 0 f (φ ũ(x, τ ), τ )dτ.
Let us derive formally inequality (2.26). Taking the ∂ α := ∂ α ∂x α , |α| ≤ s derivative of (2.24) and multiplying the resulting equation by ∂ α w, we get

1 2 d dt ∂ α w 2 = D ∂ α f ∂ α wdx - D ∂ α (ũ • ∇) w • ∂ α wdx ≤ D (ũ • ∇)∂ α w • ∂ α wdx + f s w s + C ∇ũ s-1 w 2 s .
Integrating We now return to the proof of the theorem. The proof is based on some ideas from [START_REF] Bardos | Finite-time regularity for bounded and unbounded ideal incompressible fluids using Hölder estimates[END_REF] and [START_REF] Bona | The initial-value problem for the Korteweg-de Vries equation[END_REF].

Step 1. Let E : H k (D)→H k (R 2 ), 0 ≤ k ≤ s + 1 be an extension operator. Let ρ ∈ S(R 2 ) be the function such that ρ(ξ) = exp(-|ξ| 2 1-|ξ| 2 ) |ξ| < 1, 0 |ξ| ≥ 1.
Define J m : H s (D)→H s+1 (D) by

J m (v) := (m 2 ρ(mx) * E(v))| D . (2.31) For u 0 ∈ H s (D) we define u m 0 := J m (u 0 ). Then u m 0 →u 0 in H s (D), u m 0 s ≤ C u 0 s , u m 0 s+1 ≤ mC u 0 s , (2.32)
u m 0 -u k 0 s = o(1) and u m 0 -u k 0 1 = o( 1 m s-1 ) as m→∞, (2.33)
where (2.33) holds uniformly in k > m. Using Lemmas 2.8 and 2.9, we define the sequences u m ∈ C(R + , H s+1 ) and w m ∈ C(R + , H s ) by

u 0 = u 0 , ∂ t w m+1 + u m , ∇ w m+1 = 0, w m+1 (0) = curl u m+1 0 , curl u m+1 = w m+1 , div u m+1 = 0, u m+1 • n| Γ = 0.
Our strategy is to show that sequence u m is convergent and the limit is the solution of Euler system. From (2.26) we derive

w m (•, t) i ≤ curl u m 0 i + C 1 t 0 w m (•, τ ) i u m-1 (•, τ ) i dτ (2.34) for i = s -1, s.
Step 2. In this step, we show that there exists a time

T * = T * ( u 0 s ) such that for any t ∈ J T * w m (•, t) s-1 ≤ C u m 0 s , w m (•, t) s ≤ C u m 0 s+1 ≤ mC u 0 s . (2.35)
By induction, let us prove for i = s -1, s the inequality

w m (•, t) i ≤ y m (t), (2.36)
where C does not depend on m and y m (t) is the solution of

ẏm = C 1 y 2 m , y m (0) = curl u m 0 i . (2.37)
Clearly (2.36) holds for m = 0 for a sufficiently large C. Assume that it holds also for m -1 and let us prove it for m. From the construction of ρ we have

u m-1 0 i ≤ u m 0 i , hence y m-1 ≤ y m .
Thus, from (2.34), (2.37) and induction hypothesis, we have

w m (•, t) i -y m ≤ C 1 t 0 ( w m (•, τ ) i u m-1 (•, τ ) i -y 2 m )dτ ≤ C 1 t 0 y m ( w m (•, τ ) i -y m )dτ.
Inequality (2.36) follows from the Gronwall inequality. It is easy to see that (2.36) yields (2.35).

Step 3. Now let us show that w m converges in C(J T * , H s-1 ). In view of Lemma 2.9, sequence u m converges in C(J T * , H s ) and the limit u is the solution of Euler problem.

Notice that for m < k we have

∂ t w m -w k + u k-1 , ∇ w m -w k = u k-1 -u m-1 , ∇ w m . (2.38) Denote K m,k (t) := w m (•, t) -w k (•, t) s-1 . Lemma 2.8 implies K m,k (t) ≤ u m 0 -u k 0 s + C t 0 K m,k (τ ) u k-1 (•, τ ) s-1 + u m-1 (•, τ ) -u k-1 (•, τ ) s-1 w m (•, τ ) s dτ. (2.39)
On the other hand,

w m s ≤ Cm, u m-1 -u k-1 s-1 ≤ u m-1 -u k-1 1 s-1 1 u m-1 -u k-1 s-2 s-1 s . (2.40) Assume for a moment that U m,k := w m-1 -w k-1 ≤ o( 1 m s-1 ).
(2.41) Substituting (2.40) into (2.39) and using (2.33) and (2.41), we obtain

K m,k (t) ≤ o(1) + C t 0 K m,k (τ ) u k-1 (•, τ ) s-1 dτ.
Using the Gronwall inequality, we obtain the convergence of w m in C(J T * , H s-1 (D)).

Step 4. To complete the proof of the theorem, it remains to show (2.41). Taking the scalar product of (2.38) with w mw k in L 2 , we get

U m,k (t) ≤ C u m 0 -u k 0 1 + C t 0 U m-1,k-1 (t 1 )dt 1 .
Iterating this inequality, one deduces

U m+p,k+p (t) ≤C u m+p 0 -u k+p 0 1 + C t 0 U m+p-1,k+p-1 (t 1 )dt 1 ≤C u m+p 0 -u k+p 0 1 + C t 0 C u m+p-1 0 -u k+p-1 0 1 + C t1 0 U m+p-2,k+p-2 (t 2 ) dt 2 dt 1 ≤C u m+p 0 -u k+p 0 1 + C t 0 C u m+p-1 0 -u k+p-1 0 1 + • • • + C tp-1 0 C u m+1 0 -u k+1 0 1 + C tp 0 U m,k (t p ) dt p • • • dt 2 dt 1 .
Hence, for any t ∈ J T * we obtain

U m+p,k+p ≤ p j=1 C p-j+1 T * p-j (p -j)! u m+j 0 -u k+j 0 1 + C p+1 T * p p! max t∈[0,T * ] U m,k . (2.42) Since ∞ j=1 C j+1 T * j j! < ∞,
inequalities (2.33) and (2.42) imply (2.41).

Remark 2.10. We have the following assertions:

• Adapting the Beale-Kato-Majda criterion (see [START_REF] Beale | Remarks on the breakdown of smooth solutions for the 3-D Euler equations[END_REF]) for an unbounded strip, one can prove that the solution of (2.17)-(2. [START_REF] Wolibner | Un théorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long[END_REF]) is global in time. However, we shall not need this result.

• Let us take any non-zero function g ∈ H s-1/2 (Γ). If the homogeneous boundary condition (2.18) is replaced by u • n| Γ = g, then, to our knowledge, neither existence nor uniqueness of a solution is known to hold (even in the case of bounded domain).

Main result

Let D and Γ 0 be defined by (1.2) and (1.3). Consider the Euler system:

u + u, ∇ u + ∇p = 0 in D × (0, ∞), (3.1) div u = 0, (3.2)
u • n = 0 on Γ \ Γ 0 × R + , (3.3)
u(x, 0) = u 0 (x). (3.4)
For any integer s we denote

X s (D) = C(R + , C b (D) ∩ Ḣs (D)),
and

x 1 := (1 + x 2 1 ) 1/2 .
The following theorem is our main result.

Theorem 3.1. For any constants α, β > 0, c ∈ R and integer s ≥ 4, for any initial data u 0 ∈ H s (D) such that

div u 0 = 0, (3.5)
u 0 • n = 0 on Γ \ Γ 0 , (3.6)
exp(α

x 1 2+β ) curl u 0 (x 1 , x 2 ) s-1 < ∞ (3.7)
there is a solution (u, p) ∈ X s (D) × C(R + , Ḣs (D)) of (3.1)-(3.4) with

lim t→∞ ( u(•, t) -(c, 0) L ∞ (D) + ∇u(•, t) s-1 + ∇p(•, t) s-1 ) = 0. (3.8)
As explained in Introduction, in this formulation the control is not given explicitly, but we can assume that control acts on the system as a boundary condition on Γ 0 . So we show that there exists control η such that there is a solution of our system with u • n| Γ0 = η verifying (3.8). As we mentioned in Remark 2.10, we are not able to show that this solution is unique.

Using a standard scaling argument for Euler system, we can reduce this theorem to a small neighborhood of the origin.

( u M (•, t) -( c M , 0) L ∞ (D) + ∇u M (•, t) s-1 + ∇p M (•, t) s-1 ) = 0. Then (u, p) = (M u M (x, M t), M 2 p M (x, M t)
) is a solution of our system with u(0) = u 0 and it satisfies (3.8).

Proof of Theorem 3.2. The proof of this theorem is based on generalization of the Coron return method to the case of an unbounded strip. It consists in construction of a particular solution (u, p) of (3.1)-( 3.3) such that the solution of linearized system around (u, p) verifies property (3.8). Then, in the small neighborhood of u, we construct a solution u of Euler system satisfying (3.8).

Step 1. In this step, we construct a particular solution (u, p) of (3.1)-( 3.3) such that any point of strip D, driven by the flow of u, leaves D at some time. Let D ⊂ R 2 be the strip

D := {(x 1 , x 2 ) : x 1 ∈ R, x 2 ∈ (-2, 2)}.
Let us admit the proposition below, which is proved in Section 4.1.

Proposition 3.3. There are scalar functions θ i ∈ C 1 ( D×R + ) with ∇θ i ∈ X s ( D), open balls B i , a sequence τ i ⊂ R + , constants M, λ and an integer N ∈ N such that the following properties are true.

1. Covering. For any integer k ≥ 0, we have

[k, k + 1] × [-1, 1] ⊂ N j=1 B 2kN +j , (3.9)
[-k -1, -k] × [-1, 1] ⊂ N j=1 B (2k+1)N +j . (3.10)
In particular, the union of balls B i covers D and any square

[k, k + 1] × [-1, 1] is covered by N balls.
2. Support.

supp θ i ⊂ D × (0, τ i ). (3.11)
3. Vector field. The time dependent vector field ∇θ i is divergence-free in D and tangent to Γ \ Γ 0 and ∂ D:

∆θ i = 0 in D × [0, τ i ], (3.12
)

∂θ i ∂n = 0 on (Γ \ Γ 0 ) ∪ ∂ D × [0, τ i ]. (3.13)
4. Time decay. For any i ≥ 1 we have

||∇θ i (•, t)|| X s ( D) ≤ 1 i for any t ∈ [0, τ i ], (3.14)
τ i ≤ M i. (3.15)
5. Flow. For any i ≥ 1 and c ∈ R with |c| < λ the flow associated with ∇θ i + (c, 0) is such that

φ ∇θ i +(c,0) (B i , τ i ) ⊂ D \ D. (3.16)
Moreover, there are two closed balls B1 , B2 ⊂ D \ D such that

∪ ∞ i=1 φ ∇θ i +(c,0) (B i , τ i ) ⊂ B1 ∪ B2 . (3.17)
Let us set t 0 = 0,

t i = 2 i j=1 τ j , t i+1/2 = t i + t i+1 2 , i ≥ 1. (3.18)
We define θ in the following way:

θ(x, t) = θ i (x, t -t i-1 ) for t ∈ [t i-1 , t i-1/2 ], (3.19) θ(x, t) = -θ i (x, t i -t) for t ∈ [t i-1/2 , t i ]. (3.20)
Notice that from the construction of t i we have t it i-1/2 = τ i . Thus (3.11) shows that θ ∈ C 1 ( D × R + ) and ∇θ ∈ X s ( D). We define

u : = ∇θ + (c, 0), p : = -∂ t θ - |∇θ| 2 2 -c∂ 1 θ.
Then (u, p) is a solution of (3.1)-(3.3). Indeed, by construction, (u, p) satisfies (3.1). Properties (3.12) and (3.13) imply (3.2) and (3.3), respectively. Moreover, it follows from (3.14), (3.16) that for any i ∈ N, we have

φ u (B i , t i-1/2 ) ⊂ D, lim t→∞ ( u(•, t) -(c, 0) L ∞ (D) + ∇u(•, t) s-1 ) = 0. (3.21)
We deduce from (3.19) and (3.20) that

φ u (x, t i ) = x (3.22)
for any i ≥ 1 and x ∈ D. We shall need the following result, which is proved in Section 4.1.

Proposition 3.4. There is a constant ν > 0 such that the functions θ i in Proposition 3.3 can be chosen in a way that, for any u ∈ X s ( D) satisfying the inequality

∞ 0 u(t) -u(t) s, D dt ≤ ν, we have φ u (B i , t i-1/2 ) ⊂ D \ D for any i ≥ 1.
From now on, we assume that functions θ i verify this proposition.

Step 2. In this step, we construct an application F u0 such that its fixed point is a solution of our stabilization problem. First, for any constant ν > 0 let us introduce the set

Y ν (u 0 ) := {u ∈ X s (D) : div u = 0, ∞ 0 u(t) -u(t) s,D dt ≤ ν, u(x, t) • n(x) = (u 0 (x)µ(t) + u(x, t)) • n(x) on Γ × R + }, where µ ∈ C ∞ 0 ([0, ∞)) is a non-negative function such that µ(0) = 1, ∞ 0 µ(t)dt < 1. Let D 1 := R×(-3 2 , 3
2 ) and π : H s (D) → H s ( D) be any linear bounded extension operator such that supp πu ⊂ D 1 for any u ∈ H s (D). Let κ i ∈ C ∞ 0 ( D) be a partition of unity subordinate to B i , i.e.,

supp κ i ⊂ B i , ∞ i=1 κ i = 1 in D.
Take any u ∈ Y ν (u 0 ) and let w l ∈ C(R + , H s-1 ( D)) be the solution of the linear problem For any t ∈ R + we define the function

ẇl + ũ, ∇ w l = -(div ũ)w l in D × R + , (3.23) w l (0) = κ l curl(πu 0 ), ( 3
w(•, t) = ∞ l=i+1 w l (•, t), when t ∈ [t i-1/2 ,
w l (t) s-1, D ≤ C( κ l curl(πu 0 ) s-1, D + t 0 ∇ũ(τ ) s-1, D w l (τ ) s-1, D dτ ).
It follows from the Gronwall inequality and relation (3.25) that

w l (t) s-1, D ≤ C κ l curl (πu 0 ) s-1, D exp C t 0 ∇ũ (τ ) s-1, Ddτ ≤ C κ l curl (πu 0 ) s-1, D exp C t 0 ∇u (τ ) s-1, D + u (τ ) -ũ (τ ) s, D dτ .
Using the fact that u ∈ X s ( D), we get

w l (t) s-1, D ≤ C κ l curl(πu 0 ) s-1, D exp(C(t i+1/2 + ν)) for any t ∈ [t i-1/2 , t i+1/2 ]. Thus ∞ l=i w l (t) s-1, D ≤ C exp(Ct i+1/2 ) ∞ l=i κ l curl(πu 0 ) s-1, D.
(3.28) Using (3.7) and assertion 1 of Proposition 3.3, we derive that the right-hand side of (3.28) is finite. Hence, w ∈ C([t i-1/2 , t i+1/2 ], H s-1 ( D)) for any i ≥ 0. Moreover, assertion (3.26) yields that w is continuous at t i-1/2 , thus w ∈ C(R + , H s-1 (D)) (we emphasize that, in general, this is not true for D). Furthermore, we have

ẇ + ũ, ∇ w = -(div ũ)w in D × [t i-1/2 , t i+1/2 ], w(0) = ∞ l=1 κ l curl πu 0 in D.
In Step 3, we prove that for this w there exists a v ∈ Y ν (u 0 ) such that curl v = w.

(3.29)

For any u ∈ Y ν (u 0 ), let F u0 (u) := v. In Step 4, we show that the mapping F u0 : Y ν (u 0 )→Y ν (u 0 ) has a fixed point. We shall prove that this fixed point is a solution of our stabilization problem.

Step 3. In this step, we prove the existence of the solution v ∈ Y ν (u 0 ) of (3.29). By Lemma 2.9, there is a function

z ∈ C(R + , H s (D)) such that curl z = w, div z = 0, z • n = 0, z(•, t) s,D ≤ C w(•, t) s-1,D .
(3.30)

Let us take the solution of the following problem

∆ϕ = 0 in D, ∂ϕ ∂n = (u 0 µ) • n on Γ.
From Proposition 2.4 we have ϕ ∈ C(R + , Ḣs+1 (D)) and

ϕ(•, t) Ḣs+1 (D) ≤ C u 0 µ(t) s,D . Denote v = z + ∇ϕ + u. Let us show that v ∈ Y ν (u 0 ) and (3.29) is verified. Clearly curl v = curl z = w, div v = div z + ∆ϕ = 0, v • n = (u 0 (x)µ + u) • n on Γ × R + .
Hence, to show v ∈ Y ν (u 0 ), it suffices to prove for sufficiently small u 0 that

∞ 0 v(t) -u(t) s,D dt ≤ ν. (3.31) It follows from the construction of v that v(•, t) -u(t) s,D ≤ ϕ(•, t) Ḣs+1 (D) + z(•, t) s,D .
Proposition 2.4 and (3.30) imply

∞ 0 v(t) -u(t) s,D dt ≤ u 0 s,D ∞ 0 µ(t)dt + C ∞ 0 w(•, t) s-1,D dt. From (3.27) we have ∞ 0 w(•, t) s-1,D dt = ∞ i=0 t i+1/2 ti-1/2 ∞ l=i+1 w l (•, t) s-1,D dt. Applying Lemma 2.8 to ∞ l=i+1 w l , we obtain ∞ l=i+1 w l (x, t) s-1,D ≤ C exp C t 0 ∇ũ (•, τ ) s-1,D dτ ∞ l=i+1 κ l curl u 0 s-1,D . Thus ∞ 0 w (•, t) s-1,D dt ≤ C ∞ i=0 t i+1/2 ti-1/2 ∞ l=i+1 κ l curl u 0 s-1,D × × exp C t 0 ∇ũ (•, τ ) s-1,D dτ dt ≤ C 1 ∞ i=0 t i+1/2 ti-1/2 exp Ct i+1/2 curl u 0 s-1,D\∪ i l=1 B l dt.
Combining (3.7), (3.15), (3.18) and assertion 1 of Proposition 3.3, we get

(t i+1/2 -t i-1/2 ) exp(Ct i+1/2 ) curl u 0 s-1,D\∪ i l=1 B l ≤ C 2 1
i 2 for any i > 0, where C 2 does not depend on i. Let K be a constant such that

C 1 C 2 ∞ i=K 1 i 2 < ν 2 .
Taking u 0 sufficiently small such that

u 0 s,D + K i=1 t i+1/2 ti-1/2 ∞ l=i+1 κ l curl u 0 s-1,D exp t 0 ∇ũ (•, τ ) s-1,D dτ dt ≤ ν 2 ,
we get (3.31).

Step 4. In this step, we show that the mapping F u0 : Y ν (u 0 )→Y ν (u 0 ) admits a fixed point, which is the solution of our stabilization problem. Let us take a sequence u m 0 := J m (u 0 ), where J m is the operator defined by (2.31). We have that u m 0 ∈ H s+1 (D) verifies (2.32), (2.33). Take u 0 (x, t) = µ(t)u 0 (x) + u(x, t). For sufficiently small u 0 we have u 0 ∈ Y ν (u 0 ). Let u 1 = F u 1 0 (u 0 ) and let w 1 be defined as in (3.27) with u = u 0 and u 0 (x) = u 1 0 (x). In this way we introduce the sequences u m ∈ X s and w m ∈ C(R + , H s (D)) by the relations

u m+1 = F u m+1 0 (u m ), w m+1 defined as in (3.27) with u = u m and u 0 = u m+1 0 . Let us show the convergence of w m in C([0, t 1/2 ], H s-1 ( D)
). This will be proved by using the same arguments as in the proof of Theorem 2.7. It is easy to see

∂ t (w m -w k ) + ũk-1 , ∇ (w m -w k ) = ũk-1 -ũm-1 , ∇ w m -div ũk-1 (w m -w k ) -div ũm-1 -div ũk-1 w m .
Setting K m,k (t) := w m (•, t)w k (•, t) s-1, D and using Lemmas 2.8 and 2.9, we obtain

K m,k (t) ≤ u m 0 -u k 0 s + C t 0 K m,k (τ ) ∇ũ k-1 (•, τ ) s-1 + ũm-1 (•, τ ) -ũk-1 (•, τ ) s-1 w m (•, τ ) s + K m-1,k-1 (τ ) w m (•, τ ) s-1 dτ. (3.32) Let us show that for any m ∈ N sup t∈[0,t 1/2 ] w m (•, t) s-1, D < C u m 0 s, D, (3.33)
where C depends only on u(t) L 1 ((0,t 1/2 ), Ḣs ( D)) and does not depend on m.

From the construction of w m , we have

ẇm + ũm-1 , ∇ w m = -(div ũm-1 )w m in D × R + , w m (0) = ∞ l=1 κ l curl πu m 0 in D.
Applying Lemma 2.8, we get

w m (t) s-1, D ≤ C u m 0 s, D + t 0 w m s-1, D ∇ũ m-1 s-1, Ddt ≤ C u m 0 s, D + t 0 w m s-1, D ∇u s-1, D + u -ũm-1 s, D dt .
Using the Gronwall inequality and the fact that ũm-1 ∈ Y ν (u m 0 ), we derive

w m (t) s-1, D ≤ C( u 0 s exp( t 1/2 0 ( ∇u s-1, D + u -ũm-1 s, D)dt) ≤ C 1 ,
where C 1 does not depend on m. Thus, we obtain (3.33). The construction of u m implies boundedness of sup t∈[0,t 1/2 ] u m s, D uniformly in m. In the same way we can show that sup

t∈[0,t 1/2 ] w m (•, t) s, D ≤ C u m 0 s+1, D.
Combining this with (2.32) and (2.33), we get

ũm-1 (•, τ ) -ũk-1 (•, τ ) s-1 w m (•, τ ) s ≤ ũm-1 (•, τ ) -ũk-1 (•, τ ) 1/s × × ũm-1 (•, τ ) -ũk-1 (•, τ ) 1-1/s s w m (•, τ ) s ≤ a m,k (3.34)
for any t ∈ J t 1/2 , where sup k≥m a m,k →0 as m→∞ and a m,k is decreasing sequence in m for any fixed k > m (this properties we can obtain arguing in the same way as in Theorem 2.7). Using this with (3.32) and (3.33), for any t ∈ J t 1/2 we get

K m,k (t) ≤ C t 0 (K m-1,k-1 (t 1 ) + K m,k (t 1 ))dt 1 + a m,k .
By the Gronwall inequality, for any t ∈ [0, t 1/2 ] we have

K m+p,k+p (t) ≤ C t 0 K m+p-1,k+p-1 (σ 1 )e Ct1 dσ 1 + Ca m+p,k+p ≤ C 2 t 0 σ1 0 K m+p-2,k+p-2 (σ 2 )e Cσ1 e Cσ2 dσ 2 dσ 1 + Ce Ct 1/2 a m+p-1,k+p-1 + Ca m+p,k+p ≤ C 3 t 0 σ1 0 σ2 0 K m+p-3,k+p-3 (σ 2 )e Cσ1 e Cσ2 e Cσ3 dσ 3 dσ 2 dσ 1 + C e 2Ct 1/2 2 a m+p-2,k+p-2 + Ce Ct 1/2 a m+p-1,k+p-1 + Ca m+p,k+p ≤ C p t 0 σ1 0 • • • σp-1 0 K m,k (σ p )e Cσ1+Cσ2+•••+Cσp dσ p • • • dσ 2 dσ 1 + p-1 j=0 C (e Ct 1/2 ) j j! a m+p-j,k+p-j .
Thus, we derive

K m+p,k+p ≤ Ce pC p! max t∈[0,T ] K m,k + Ca m,k .
Hence, w m is a convergent sequence in C([0, t 1/2 ], H s-1 ( D)). In the same way we can get the convergence of w m in C([t i-1/2 , t i+1/2 ], H s-1 ( D)). Finally, the fact w m ∈ C(R + , H s-1 (D)) implies that w m converges to some w * in C(R + , H s-1 (D)). The convergence of w m implies the convergence of u m to some u * in X s (D). We have curl u * = w * , (3.35) div u * = 0, (3.36)

u * (x, t) • n(x) = (u 0 (x)µ(t) + u(x, t)n(x) on Γ × R + . (3.37)
Let us show that

w * (•, t) = ∞ l=i+1 w * l (•, t) for t ∈ [t i-1/2 , t i+1/2 ], (3.38)
where w * l is the solution of

∂ t w * l + ũ * , ∇ w * l = -(div ũ * )w * l in D × R + , (3.39)
w * l (0) = κ l curl(πu 0 ). (3.40) To this end, recall that

w m (•, t) = ∞ l=i+1 w l m (•, t), when t ∈ [t i-1/2 , t i+1/2 ],
where w l m is the solution of

ẇl m + ũm-1 , ∇ w l m = -(div ũm-1 )w l m in D × R + , w l m (0) = κ l curl(πu m+1 0
).

We have that w l m →w * l in C(R + , H s-1 ( D)) uniformly with respect to l as m→∞ (this can be proved in the same way as in the proof of the convergence of w m ). Thus we have (3.38)

∞ l=i w * l (t) s-1, D ≤ C ∞ l=i exp(Ci 2 ) κ l curl(πu 0 ) s-1, D ≤ C ∞ l=i exp(Ci 2 ) exp(-Ci 2+β ).
τ i : = i sup t∈[0,1] ∇ θi (•, t) s, D, (4.5)
θ i (x, t) : = θi (x, t τi ) τ i . (4.6)
Then B i , τ i and θ i verify (3.9)-(3.16) for i = 1, . . . , N . Moreover, there are closed balls B1 , B2 ⊂ D \ D such that

∪ N i=1 φ ∇θ i +(c,0) (B i , τ i ) ⊂ B1 ∪ B2 .
We denote B 2kN +j := B(x j , r j )+(k, 0) and B (2k+1)N +j := B(x j , r j )-(k +1, 0), j = 1, . . . , N . Then properties (3.9) and (3.10) are satisfied. Let h ∈ C ∞ ([0, 1]) be such that h(t) = 0 for any t ∈ [0, 1/4],

h(t) = 1 for any t ∈ [3/4, 1], |h(t)| ≤ 1 for any t ∈ [0, 1]. For any x = (x 1 , x 2 ) ∈ D and c ∈ R define θ2kN+j (x, t) = (-k -c)x 1 h ′ (t) for t ∈ [0, 1], θj (x, t -1) for t ∈ [1, 2]. (4.7)
It follows from the constructions of θj , j = 1, . . . , N that (3.11)-(3.13) are verified for τ i = 2. It is easy to see that for any t ∈ [0, 1] we have

φ ∇ θ2kN+j +(c,0) (x, t) = (-k -c, 0)h(t) + (c, 0)t + x. (4.8)
Thus φ ∇ θ2kN+j +(c,0) (B 2kN +j , 2) = φ ∇ θj +(c,0) (B j , 1) ⊂ D, which implies (3.16) and (3.17). Notice that ∇ θi ∈ X s ( D). In order to have also (3.14) and (3.15), we define τ i by (4.5) and

θ i (x, t) : = 2 θi (x, 2t τi ) τ i . (4.9)
This completes the proof.

Proof of Lemma 4.1. The proof is based on the ideas of [START_REF] Coron | On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions[END_REF]Lemma A.1].

Step 1. We denote by A the vector space of functions ξ ∈ Ḣs+1 ( D) with the following properties ∆ξ = 0 in D, ∂ξ ∂n = 0 on Γ \ Γ 0 , supp ξ ⊂ D. (4.10)

First, let us show that for any x 0 ∈ D we have

R 2 = {∇ξ(x 0 ) : ξ ∈ A}. (4.11)
Suppose that (4.11) does not hold. Then, there is a vector

V ∈ R 2 , V = 0 such that V • ∇ξ(x 0 ) =
(x) = 1, if x / ∈ B 2 , 0, x ∈ B 1 . Clearly π(ρG a ) ∈ A, thus V • ∇π(ρG a )(x 0 ) = 0. Since x 0 / ∈ B 2 , we have V • ∇G a (x 0 ) = 0 (4.12)
for all a ∈ D \ D. On the other hand G a is analytic in a ∈ D \ {x 0 } (see Proposition 2.5, (iii)). Thus, we have (4.12) for all a ∈ D \ {x 0 }. Using (2.12), one can find a sequence a n →x 0 such that V • ∇G an (x 0 )→∞ as n→∞, which is a contradiction to V = 0.

Step 2. Take any

x 0 ∈ D ∪ Γ 0 , x 1 ∈ D \ D and let F : [0, 1] → D be a continuous function such that F (t) = x 0 for any t ∈ [0, 1/4], F (t) = x 1 for any t ∈ [3/4, 1], F (t) / ∈ Γ \ Γ 0 for any t ∈ [0, 1].
Then for any ε > 0 we can find ξ Step 3. It remains to study the case x 0 ∈ Γ \ Γ 0 . Let y 0 ∈ Γ 0 and k ∈ R be such that x 0 = y 0 + (k, 0). Then, the function We shall need the following lemma.

i ∈ A, h i ∈ C ∞ ([0, 1]), i = 1, . . . ,
θ(x, t) = (-c -k)x 1 h ′ (t) for t ∈ [0, 1/2], 2θ y0 (x, 2(t -1/2)) for t ∈ [1/2, 1] satisfies (4.1)-(4.4), where h ∈ C ∞ ([0, 1/2]) is any function with h(0) = 0, h(1/2) = 1 and θ y0 is the function constructed in Step 2 for y 0 ∈ Γ 0 .
Lemma 4.2. The functions θ i constructed in the proof of Proposition 3.3 are such that there exist ϕ i ∈ C(R + ) with

sup x∈D |φ ∇θ i +(c,0) (x, t) -x| ≤ i 2N + M for any t ∈ [0, τ i ], (4.16
)

|∇θ i (x, t) -∇θ i (y, t)| ≤ ϕ i (t) (m + 1) 2 |x -y| for any x, y ∈ D m + or x, y ∈ D m -, (4.17)
where τi

0 ϕ i (t)dt ≤ M , [ i 2N ] is the integer part of i 2N and M ∈ R does not depend on i.
Proof. It is easy to see that (4.7) and (4.9) imply

φ ∇θ i +(c,0) (x, t) = (-k -c, 0)h( 2t τi ) + (c, 0) 2t τi + x for t ∈ [0, τ i /2], φ ∇ θj +(c,0) (x, 2t τi -1) for t ∈ [τ i /2, τ i ].
where k = i 2N and j = i -2N k. This yields (4.16) for a sufficiently large M . To prove (4.17), notice that in the proof of Lemma 4.1, the functions θ can be chosen such that

x 2 1 ∂ β θ s, D < C(x 0 ), where |β| = 2. Indeed, since Proposition 2.5 implies that the second order derivatives of G a belong to S(D), one can replace (4.11) by

R 2 = {∇ξ(x 0 ) : ξ ∈ A and x 2 1 ∂ β ξ s, D < C(x 0 ), |β| = 2}.
Hence, we can find a constant M 1 such that sup i=1,...,N,|β|=2

1 0 x 2 1 ∂ β θi (t, •) L ∞ ( D) dt < M 1 .
Combining this with (4.7) and (4.9), we get (4.17).

Now we return to the proof of Proposition 3.4. It suffices to show that for any ε > 0 there is ν > 0 such that the inequality

sup x∈Bi |φ u (x, t) -φ u (x, t)| ≤ ε (4.18) holds for any i ≥ 1 and t ∈ [0, t i-1/2 ]. Let us denote X(t) = φ u (x, t), Y (t) = φ u (x, t),
where x ∈ B i . We shall prove (4.18) in the case when i is even. The proof when i is odd is similar. Let k := i 2N , then

B i ⊂ [k -2, k + 3] × [-2, 2]. ( 4

.19)

Step 1. First let us show that to establish (4.18) it suffices to prove that

|X(t) -Y (t)| < 1 for all t ∈ R + . (4.20)
It is easy to see that

∂ t (X(t) -Y (t)) = u(X(t), t) -u(Y (t), t) = (u(X(t), t) -u(X(t), t)) + (u(X(t), t) -u(Y (t), t)) =: I 1 (t) + I 2 (t). (4.21)
We have that where 2 , t ∈ [t j-1/2 , t j ] for j < 2N (k -3 -M ) (here we use (4.17) for m = k -3 -j 2N -M ) and Ψ(t) = ϕ j (t), t ∈ [t j-1 , t j-1/2 ], ϕ j (t jt), t ∈ [t j-1/2 t j ] for j ≥ 2N (k -3 -M ) (in this case we use (4.17) for m = 0). Thus we have

∞ 0 |I 1 (t)|dt ≤ ν. ( 4
Ψ(t) =    ϕ j (t-tj-1) (k-2-[ j 2N ]-M) 2 , t ∈ [t j-1 , t j-1/2 ], ϕ j (tj -t) (k-2-[ j 2N ]-M)
t i-1/2 0 Ψ(t)dt = t 2N (k-3-M )-1 0 Ψ(t)dt + t i-1/2 t 2N (k-3-M )-1 Ψ(t)dt ≤ 2N (k-3-M)-1 j=1 2M (k -2 -j 2N -M ) 2
+ (2N (M + 4) + 1)2M. Choosing ν such that the right-hand side of (4.25) is smaller than ε, we prove (4.18) for all i.

Step 2. To complete the proof, it remains to show (4.20). To this end, let us assume that (4.20) does not hold for some t > 0. Denote by t0 the first time such that |X( t0 ) -Y ( t0 )| = 1. Hence, we have (4.20) for all t < t0 . Since the right-hand side of (4.26) does not depend on t0 , choosing ν sufficiently small, we get (4.20).

5 Appendix: proof of Lemma 2.3

Let us consider the space

H 0 (D ′ ) = {z ∈ L 2 (D ′ ) : curl z ∈ L 2 (D ′ ), div z ∈ L 2 (D ′ ), z • n| Γ ′ = 0}
endowed with the norm z H0 = z + curl z + div z .

Here D ′ is a strip or is the domain D defined (2.1). Recall the following result (see [10, Chapter 7, Theorem 6.1]). In the case of bounded domains it is shown in [18, Appendix 1, Proposition 1.4] that H s (Ω) = {z ∈ L 2 (Ω) : curl z ∈ H s-1 (Ω), div z ∈ H s-1 (Ω), z • n ∈ H s-1/2 (∂Ω)}.

(

Let us generalize this result to the case of domain D ′ . We shall need the following lemma.

(∂ 1 z 1 )ñ 1 + (∂ 1 z 2 )ñ 2 ∈ H s-1 .
Combining this with div z ∈ H s-1 and curl z ∈ H s-1 , we obtain ñ•∇ ⊥ z i ∈ H s-1 and ñ • ∇z i ∈ H s-1 for i = 1, 2. Thus ∇z i ∈ H s-1 , which completes the proof.

. 2 )

 2 Let us take two open intervals (a, b), (a + d, b + d) ⊂ R and denote Γ 0 = (a, b) × {1} ∪ (a + d, b + d) × {-1}.

  Now we summarize some facts about Poisson equation. Let us take a nonnegative function γ ∈ C ∞ 0 (R) such that supp γ = [a, b] and γ = 0 in (a, b) and define

Figure 1 :

 1 Figure 1: Domain D

Theorem 3 . 2 .By Theorem 3 . 2 ,

 3232 There exists ε > 0 such that for any u 0 ∈ H s (D) and c ∈ R verifying (3.5)-(3.7) and u 0 s < ε, |c| < ε there is a solution (u, p) ∈ X s (D)×C(R + , Ḣs (D)) of (3.1)-(3.4) satisfying (3.8). Proof of Theorem 3.1. Let ε > 0 be the constant in Theorem 3.2. Take any u 0 ∈ H s (D) and c ∈ R verifying (3.5)-(3.7). Let M > 0 be such that there exists a solution (u M , p M ) of (3.1)-(3.3) with initial condition u M (0) = u0 M , such that lim t→∞

  .24) where ũ = u + π(uu).(3.25) Take ν such that Proposition 3.4 holds. Since supp w l (0) ⊂ B l , we obtain w l (x, t l-1/2 ) = 0 for any x ∈ D.(3.26)

Thus

4 Construction of the particular solution 4 . 1 3 3 . 4 . 1 .φ

 413341 lim t→∞ u * (t)u(t) s,D = 0. (3.41)Combining this with (3.21), we see that the first two terms on the left-hand side of (3.8) go to zero as t→∞. Recall that∆p * =div( u * , ∇ u * ) ∂p * ∂n = -( u * , ∇ u * ) • n.Thus, Proposition 2.4 implies lim t→∞ ∇p * (t) s-1 = 0. This completes the proof of Theorem 3.1. Proof of Proposition 3.We have the following simplified version of Proposition 3.Lemma For any x 0 ∈ D there exist a function θ ∈ C ∞ ([0, 1], Ḣs+1 ( D)) and a constant λ > 0 such that ∇θ+(c,0) (x 0 , 1) / ∈ D for any |c| < λ. (4.4) This lemma is proved at the end of this subsection. Proof of Proposition 3.3. It follows from Lemma 4.1 that there are functions θi ∈ C ∞ ([0, 1], Ḣs+1 ( D)) and open balls B i = B(x i , r i ) ⊂ R 2 , i = 1, . . . , N covering the rectangle [0, 1]×[-1, 1] such that properties (3.11)-(3.13) and (3.16) are verified for τ i = 1. For i = 1, . . . , N let us take

  0 for all ξ ∈ A. Let D be the domain defined in (2.1) and let D ⊂ D 1 . Take any a ∈ D \ D, and let G a be the solution of (2.10), (2.11). Let B 1 , B 2 ⊂ D \ D be two open neighborhoods of a such that B 1 ⊂ B 2 and let ρ ∈ C ∞ ( D) be such that ρ

4. 2

 2 Proof of Proposition 3.4 For any m ∈ R + , let us denote D m -:= (-∞, -m] × [-2, 2] and D m + := [m, +∞) × [-2, 2]. (4.15)

  21), using (4.22)-(4.24) and the Gronwall inequality, we obtain|X(t i-1/2 ) -Y (t i-1/2 )|

  Step 1 implies|X( t0 ) -Y ( t0 )| ≤ ν exp

Theorem 5 . 1 .

 51 The following equality holds{z ∈ H 1 (D ′ ) : z • n| Γ ′ = 0} = H 0 .

Lemma 5 . 2 .ΓCase 1 .

 521 Let g ∈ H 1/2 (Γ ′ ). Then the problem ∆uu = 0 in D ′ , (5.2) ∂u ∂n = g on Γ ′ (5.3) has a unique solution u ∈ H 2 (D ′ ), which satisfiesu 2 ≤ C g 1/2 .(5.4)Proof. Problem (5.2), (5.3) is equivalent to gθdσ for any θ ∈ H 1 (D ′ ).Since g ∈ H -1/2 (Γ ′ ), the Riesz representation theorem implies the existence of a unique solution u ∈ H 1 (D ′ ). Assume D ′ = D, and let us prove that u ∈ H 2 (D). It is easy tosee that v := ∂ 1 u is the solution of the problem ∆vv = 0 in D, ∂v ∂n = ∂ 1 g on Γ.whereh ∈ C ∞ b ( D), h| ∂ D = 0 and h(x 1 , 0) = 1 + γ ′ (x1) √ 1+γ ′ (x1) 2 . Then we have (ñ 1 , ñ2 )| ∂ D = n and |(ñ 1 , ñ2 )| > δ for sufficiently small δ > 0. Hence, ñ(x) = (ñ1,ñ2) |(ñ1,ñ2)| is an extension of n. Let us take v := ∇ ⊥ (z • ñ). Then v ∈ L 2 , div v = 0. Since v • ñ is the tangential derivative of z • ñ along Γ ′ , we have v • ñ ∈ H s-3/2 (Γ ′ ).On the other handcurl v = ∆(z • ñ) = (∆z 1 )ñ 1 + (∆z 2 )ñ 2 + ṽ,where ṽ ∈ H s-2 . It follows from the facts ∆z1 = ∂ 1 div z + ∂ 2 curl z and ∆z 2 = ∂ 2 div z -∂ 1 curl z that curl v ∈ H s-2 . Thus the induction hypothesis yields ∇ ⊥ (z • ñ) ∈ H s-1 . Hence, (∂ 2 z 1 )ñ 1 + (∂ 2 z 2 )ñ 2 ∈ H s-1 ,

  by parts, one verifies that the first integral in the right-hand side vanishes. Integrating in time, we obtain (2.26). Lemma 2.9. Let w ∈ H s , s ≥ 0. Then the problem

	curl z = w,	(2.27)
	div z = 0,	(2.28)
	z • n| Γ = 0	(2.29)
	has a unique solution z ∈ H s+1 . Moreover, there is C > 0 depending only on s such that
	z s+1 ≤ C w s .	(2.30)
	Proof. Let us consider the following Dirichlet problem for the Poisson equation:
	∆v = w in D,
	v = 0 on Γ.

By Proposition 2.2, v ∈ H s+2 and v s+2 ≤ C w s . Then for z = -∇ ⊥ v properties (2.27)-(2.30) are satisfied.

  := 0 and i ≥ 0. Let us show that for any t ∈ [t i-1/2 , t i+1/2 ] the sum in the right-hand side of (3.27) exists and belongs to C(R + , H s-1 (D)).

	t i+1/2 ],	(3.27)
	where t -1/2 Applying Lemma 2.8 to (3.23), (3.24), we obtain

  . Clearly (3.35)-(3.40) imply that u * is a solution of the Euler system (3.1)-(3.3).

	As in (3.28), using (3.35)-(3.40) for any t ∈ [t i-1/2 , t i+1/2 ] and (3.7), we can show that

Thus ∂ 1 u ∈ H 1 (D) and

Combining this with the fact that ∆u ∈ H 1 (D), we obtain u ∈ H 2 (D) and (5.4).

Case 2. Now consider the case D ′ = D. Let

where N is so large that D \ D ⊂ Ω 1 . Let us take some function χ ∈ C ∞ ( D) such that such that

Then w := χu is the solution of ∆ww = 2∇χ∇u + ∆χu =: f in Ω 2 ,

(5.5) ∂w ∂n =: g on ∂Ω 2 .

(5.6)

It is easy to see that f ∈ L 2 (Ω 2 ) and g ∈ H 1/2 (∂Ω 2 ). This implies that w ∈ H 2 (Ω 2 ) (e.g., see [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditionsns[END_REF]). Thus u ∈ H 2 (Ω 1 ). On the other hand, from the fact Γ 0 ⊂ Ω 1 we derive ∂u ∂n | Γ ∈ H 1/2 (Γ). Hence, using the result for D ′ = D, we see that u ∈ H 2 (D). This completes the proof of Lemma 5.2. Now let us prove (5.1) for Ω = D ′ . Clearly the space in the left-hand side is contained in the right-hand side of (5.1). By induction, let us show the other inclusion. Assume s = 1. Let us take some function z from the right-hand side of (5.1) and consider the problem:

By Lemma 5.2, we have p ∈ H 2 (D ′ ) and p 2 ≤ C z • n 1/2 . Let us take w = z -∇p. Clearly w ∈ H 0 , thus Theorem 5.1 implies w ∈ H 1 (D ′ ). Hence, z ∈ H 1 (D ′ ) and

Now assume that (5.1) holds for s-1 and let us prove it for s. Let ñ be a regular extension of n in D ′ such that |ñ(x)| = 1. Let us show that such an extension exists. To simplify the proof, let us assume that d = 0 in the definition of D (see (2.1)). We define ñ1 (x 1 , x 2 ) = -γ ′ (x 1 )

1 + γ ′ (x 1 ) 2 + h(x 1 , x 2 ), ñ2 (x 1 , x 2 ) = x 2

(1 + γ(x 1 )) 1 + γ ′ (x 1 ) 2 ,