Kévin Polisano

Marianne Clausel

Valérie Perrier

Laurent Condat

TEXTURE MODELING BY GAUSSIAN FIELDS WITH PRESCRIBED LOCAL ORIENTATION

Keywords: Prescribed orientation, anisotropic selfsimilar Gaussian fields, turning bands, oriented textures

This paper presents a new framework for oriented texture modeling. We introduce a new class of Gaussian fields, called Locally Anisotropic Fractional Brownian Fields, with prescribed local orientation at any point. These fields are a local version of a specific class of anisotropic self-similar Gaussian fields with stationary increments. The simulation of such textures is obtained using a new algorithm mixing the tangent field formulation and a turning band method, this latter method having proved its efficiency for generating stationary anisotropic textures. Numerical experiments show the ability of the method for synthesis of textures with prescribed local orientation.

INTRODUCTION

Texture modeling is a challenging issue of image processing. There is a variety of texture methods in the field of computer vision, namely structural, statistical, model-based and transformbased methods. Thus, identifying the perceived characteristics of a texture in an image (regularity, roughness, frequency, content directionality, etc.) is an important first step towards building mathematical models for textures. We are interested in textures presenting same similar patterns at different scales, as is often the case for objects appearing in the nature, like clouds or mountains. We focus on stochastic models with a property of self-similarity, characteristic of a fractal behavior. The stochastic model behind fractal analysis is the fractional Brownian field (FBF), which is a multi-dimensional extension of the famous fractional Brownian motion (FBM) introduced in 1940 by Kolmogorov [START_REF] Kolmogorov | Wienersche spiralen und einige andere interessante kurven im hilbertschen raum[END_REF] as a way to generate Gaussian "spirals" in Hilbert spaces. The systematic study of the FBM started with the seminal paper of Mandelbrot and Van Ness [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF]. The FBM has now become a standard model: it is used in many areas such as hydrology, economics, finance, physics and telecommunications (see, e.g., [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF], [START_REF] Véhel | Fractals in engineering: from theory to industrial applications[END_REF], [START_REF] Abry | Lois d'échelle, fractales et ondelettes[END_REF], and references therein for more details). The FBF has also been largely used in medical applications, with for instance the study of lesion detectability in mammogram textures [START_REF] Caldwell | Characterisation of mammographic parenchymal pattern by fractal dimension[END_REF], assessment of breast cancer risk [START_REF] Heine | Mammographic tissue, breast cancer risk, serial image analysis, and digital mammography[END_REF], and the characterization of bone architecture for the evaluation of osteoporotic fracture risk [START_REF] Benhamou | Fractal analysis of radiographic trabecular bone texture and bone mineral density: two complementary parameters related to osteoporotic fractures[END_REF].

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-13-BS03-0002-01 (ASTRES).

Nevertheless, in many cases, fractal analysis with fractional Brownian fields, which are isotropic by definition, is not completely satisfactory, in particular when the considered data display some anisotropy. Therefore, many stochastic models have been introduced in the literature to take into account these possible additional anisotropic properties. Let us cite notably fractional Brownian sheet defined in [START_REF] Kamont | On the fractional anisotropic Wiener field[END_REF][START_REF] Ayache | Drap brownien fractionnaire[END_REF] and anisotropic fractional Brownian field (AFBF) introduced by Bonami and Estrade in [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] which are two classical examples of Gaussian fields satisfying global anisotropic properties. Other models of anisotropic textures called locally parallel textures, have also been recently introduced in [START_REF] Maurel | Locally parallel texture modeling[END_REF]. The mathematical definition and computational synthesis of anisotropic textures is an important issue, since it provides flexible models enabling to test estimation procedures of the anisotropic characteristics of an image. Here we focus on anisotropic local properties of Gaussian textures and provide a new Gaussian model whose anisotropic properties are prescribed at every point. It is a first preliminary and important step in defining new statistical estimators of the local anisotropic features of a given texture.

The paper is organized as follows. Section 2 briefly reviews definitions and characterizations of a class of self-similar Gaussian fields derived from the AFBF. Section 3 is devoted to the presentation of our model, from both the theoretical and implementation point of view. Finally, we provide the synthesis of numerical textures, for several vector fields of local orientations, showing the ability of our approach.

ANISOTROPIC SELF-SIMILAR GAUSSIAN FIELDS

The fractional Brownian field

Let 0 < H < 1. The fractional Brownian field with Hurst index H, denoted by B H = {B H (x); x ∈ R 2 }, is the unique real-valued centered Gaussian field satisfying the following properties: -almost surely B H (0) = 0, -B H admits stationary increments, i.e, for every

z ∈ R 2 , B H (• + z) -B H (z) L = B H (•) -B H (0), -B H is H self-similar, i.e, ∀λ ∈ R ⋆ , B H (λ•) L = λ H B H (•), -B H is isotropic, i.e, for every rotation R in R 2 , B H • R L = B H ,
where L = denotes equality for all finite dimensional distributions. The FBM is wholly characterized by its covariance function, which is given, for every x, y ∈ R 2 by

Cov(B H (x), B H (y)) = c H ( x 2H + y 2H -x -y 2H ) ,
c H being a well-known nonnegative constant depending on H. Following [START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF], the FBM can also be defined by its harmonizable representation:

B H (x) = R 2 e ix•ξ -1 ξ H+1 d W (ξ), (1)
where d W is a complex Brownian measure and x • ξ denotes the dot-product on R 2 . The Hurst index H is a fundamental parameter of the FBF, as an indicator of the texture roughness.

The greater H is, the smoother the resulting texture is, as can be seen in Fig. 1.

General anisotropic self-similar Gaussian fields

In order to introduce anisotropy in this model, Bonami and Estrade [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] replaced the Hurst index H in (1) by a function of the direction of ξ and then derived a new class of Anisotropic Fractional Brownian Field (AFBF) by:

X(x) = R 2 e ix•ξ -1 ξ h(arg ξ)+1 d W (ξ). (2)
More generally, a larger class of anisotropic models can be defined as

X(x) = R 2 (e ix•ξ -1)f 1/2 (ξ) d W (ξ), (3)
where the spectral density f is of the form

f 1/2 (ξ) = c(arg ξ) ξ -h(arg ξ)-1 . (4)
Here, c and h are two π-periodic functions, defined on the interval (-π/2, π/2] with ranges satisfying c((-π/2, π/2]) ⊂ R + and h((-π/2, π/2]) ⊂ (0, 1). When c and h are both constant, we recover a FBF of order H ≡ h.

To define stationary anisotropic models with global orientation α 0 , one can set h ≡ H in (4) and: for some 0 < α π/2. Note that we then recover the elementary fields of [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian fields[END_REF], which are a particular case of AFBF. When α = π/2, this model corresponds to the usual isotropic FBF of Hurst index H (Fig. 1), but as soon as 0 < α < π/2, the field is no longer isotropic, since the non-zero frequency arguments are restricted between -α + α 0 and α + α 0 . Simulation algorithms for Gaussian fields use the covariance function [START_REF] Brouste | On Gaussian random fields simulation[END_REF]. But their high complexity is a real problem to produce large textures, and the covariance function is not explicitly known in general case. With respect to the AFBF, a recent fast method has been proposed in [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian fields[END_REF], called the turning band method, and used here to simulate the textures of Fig. 2, with global orientation α 0 = 0. Remark that the more the sector α decreases to 0, the more the frequencies concentrate along the horizontal axis, so the resulting texture appears vertically oriented, as a consequence of the Fourier transform properties. For small α, we obtain a strongly stationary oriented texture in the direction orthogonal to α 0 = 0 like in Fig. 2 (b).

c α0,α (arg(ξ)) = 1 [-α,α] (arg(ξ) -α 0 ), (5)

A NEW CLASS OF GAUSSIAN FIELD WITH PRESCRIBED ORIENTATION

Definition

We now define our new Gaussian model as a local version of the elementary field defined in (3) with density given by ( 4) with h ≡ H and c as in [START_REF] Abry | Lois d'échelle, fractales et ondelettes[END_REF]. More precisely, we define the following Gaussian field, that we call Locally Anisotropic Fractional Brownian Field (LAFBF):

X(x) = R 2 (e ix•ξ -1)f 1/2 (x, ξ) d W (ξ), (6)
with

f 1/2 (x, ξ) = c α0,α (x, arg ξ) ξ -H-1 , (7) c α0,α (x, arg ξ) = 1 [-α,α] (arg(ξ) -α 0 (x)), (8)
α 0 being now a differentiable function on R 2 . Our Gaussian field is derived from AFBF in a similar way than Multifractional Brownian Motion (MBM) from FBM, since we replace the orientation parameter α 0 with a function depending on the spatial location x, whereas in the case of MBM the Hurst index H of FBM was allowed to vary spatially [START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF].

V x 0 cα 0 ,α (x 0 , arg ξ)

x 0 α Texture orientation f 1/2 (x 0 , ξ) = c α0,α (x 0 , arg ξ) ξ H+1 (a) (b) (c)

Tangent fields at every point

To describe the local properties and simulate our new class of Gaussian fields, we shall use the notion of tangent fields that we now briefly review. Recall that the random field X is locally asymptotically self-similar of order H ∈ (0, 1) if for any h ∈ R 2 the random field

X(x 0 + ρh) -X(x 0 )
ρ H , admits a non-trivial limit in law Y x0 as ρ → 0 (see [START_REF] Benassi | Elliptic Gaussian random processes[END_REF], and [START_REF] Falconer | Tangent fields and the local structure of random fields[END_REF][START_REF] Falconer | The local structure of random processes[END_REF] for a more general definition). The field Y x0 is then called the tangent field of X at x 0 . Roughly speaking, the random field X admits the tangent field Y x0 at a given point x 0 if it behaves locally as Y x0 when x → x 0 . This notion has been first introduced in [START_REF] Benassi | Elliptic Gaussian random processes[END_REF] to describe the local behavior of Multifractional Brownian Motion (which behaves locally as a FBM). We can prove that the LAFBF X of (6) admits a tangent field Y x0 at any point x 0 ∈ R 2 defined as:

Y x0 (x) = R 2 (e ixξ -1)f 1/2 (x 0 , ξ) d W (ξ) . ( 9
)
We observe that the tangent field Y x0 is no more and no less than an elementary field using the terminology of [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian fields[END_REF]. This result shall be crucial when simulating this Gaussian model as detailed in the next section.

Simulation of Locally Anisotropic Fractional Brownian Field

Simulation of tangent fields. The simulation of a LAFBF will first require the simulation of a tangent field at every point x 0 . We follow below the methodology of [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian fields[END_REF] using the turning bands method.

-Discrete formulation of the tangent field By a change of variable in polar coordinates, one can derive an integral expression for the variogram of Y x0 :

v Yx 0 (x) = 1 2 R 2 |e ix•ξ -1| 2 f (x 0 , ξ)dξ = 1 2 γ(H) π/2 -π/2 c α0,α (x 0 , θ) |x • u(θ)| 2H dθ , (10)
where u(θ) = (cos θ, sin θ) and γ(H) = π HΓ(2H) sin(Hπ) . The integral ( 10) is of the form

π/2 -π/2 ṽθ (x • u(θ))dθ with ṽθ = 1 2 γ(H)c α0,α (x 0 , θ)| • | 2H .
Ignoring the factor 1 2 γ(H)c α0,α (x 0 , θ), we recognize that ṽθ is the variogram of a FBM of order H. Consequently, Y x0 can be viewed as a sum of independent FBM rotating around the origin. Discretizing θ in an ordered set (θ i ) 1 i n of n band orientations, and let be (λ i ) 1 i n the associated band weights λ i = θ i+1 -θ i , the turning band fields take the form

Y [n] x0 (x) = γ(H) 1 2 n i=1 λ i c α0,α (x 0 , θ i )B H i (x • u(θ i )), (11)
where the B H i 's are n independent FBM of order H. This discrete version is a good approximation, provided max i λ i ε for ε sufficiently small.

-Simulation along particular bands In practice, we consider a discrete grid

r -1 Z 2 ∩ [0, 1] 2 with r = 2 k -1, k ∈ N ⋆ . Following [13], we choose θ i such that tan(θ i ) = pi qi , with p i , q i ∈ Z, then B H i (x • u(θ i )) becomes B H i k 1 r cos θ i + k 2 r sin θ i ; 0 k 1 , k 2 r L = cos θ i rq i H {B H i (k 1 q i + k 2 p i ); 0 k 1 , k 2 r} , (12)
and then can be generated using the fast algorithm of Perrin et al. [START_REF] Perrin | Fast and exact synthesis for 1-d fractional Brownian motion and fractional Gaussian noises[END_REF] on a regular grid.

-Dynamic choice of discrete bands Finally, the choice of the bands orientations (θ i ) 1 i n is gov- erned by the global computational cost of the B H i , within dynamic programming [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian fields[END_REF].

Simulation of the LAFBF. As observed in [START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF] for the MBM, a Gaussian field can be simulated from its tangent fields. The LAFBF behaving locally like its tangent fields, for every pixel x 0 , we assign X(x 0 ) = Y

[n] x0 (x 0 ). The pseudocode of the algorithm is given below, and the corresponding Matlab code is available on the webpage [START_REF] Polisano | Matlab code for the simulation of the locally anisotropic fractional Brownian field[END_REF]. A preprocessing step (instructions 1,2,3,4 in the pseudocode), which does not depend on the expected local orientations, includes the dynamic choice and sorting of discrete bands, and the simulation of the n FBM. These steps are executed once and for all. The rest of the algorithm is of complexity O(r 2 log n). Indeed, at each point (k 1 , k 2 ), a turning band θ i contributes to X(k 1 , k 2 ) if and only if c α0,α ((k 1 , k 2 ), θ i ) = 0, i.e |θ i -α 0 ((k 1 , k 2 ))| α. Thus, since the array θ i is sorted, one such index i is founded using a binary search, and then the others in its neighborhood.

Algorithm 1 Simulation of the LAFBF

Input: r = 2 k -1, H, α 0 , α, ǫ Output: X LAFBF of size (r + 1) × (r + 1) 1: (p i , q i ) 1 i n ← DynamicBandsChoice(r, ε) 2: Compute and sort angles (θ i ) 1 i n : θ i ← atan2(p i , q i ) 3: Compute width bands (λ i ) 1 i n : λ i ← θ i+1 -θ i 4: Generate n FBM : B H i ← circFBM(r(|p i | + |q i |), H) 5: Initialization : X ← 0 6: for all (k 1 , k 2 ) do 7: for i = 1 to n do 8: ω i ← λ i γ(H)c α0,α ((k 1 , k 2 ), θ i ) cos θi rqi H 9: X(k 1 , k 2 ) ← X(k 1 , k 2 ) + ω i B H i (k 1 q i + k 2 p i ) 10:
end for 11: end for Fig. 5. Texture resulting from the vector field V 3 (x,y) .

Oriented texture synthesis

The parameters used in simulations are r = 255, H = 0.2, α = 10 -1 , and ε = 10 -2 . To avoid numerical artifacts due to the discrete formula [START_REF] Bonami | Anisotropic analysis of some Gaussian models[END_REF] we consider a regularized version cα0,α of the indicator function c α0,α , typically a Gaussian. For α 0 constant, we recover the results of [START_REF] Biermé | A turning-band method for the simulation of anisotropic fractional Brownian fields[END_REF] (see Fig. 2). We present now realizations of textures with prescribed local orientation at each point x 0 , given by a vector field V x0 = u(α 0 (x 0 )). Fig. 3(a) displays a texture resulting from the vector field V 1 (x,y) = (cos(-π/2 + y), sin(-π/2 + y)). A zoom around a point x 0 (in red, Fig. 3(b)) shows that locally a LAFBF behaves as an elementary field. Fig. 3(c) sketchs the local density function at x 0 and the different parameters. We then consider two others types of vector fields, V 2 (x,y) = (cos(cos(36xy)), sin(cos(36xy))) and V 3 (x,y) = ∇F (x, y) with F (x, y) = (4x -2)e -(4x-2) 2 -(4y-2) 2

, with the resulting textures in Figs. 4,5. As expected, the textures obtained with our approach present local anisotropic behavior, with a direction orthogonal to the vector field. Moreover, the simulation of a 256 × 256 texture takes only a few seconds.

CONCLUSION

We introduced a new stochastic model defined as a local version of an anisotropic fractional Brownian field. We took advantage of tangent field formulation and the turning bands method to provide an efficient algorithm to simulate textures with prescribed local orientations. We are currently improving the method by removing numerical artifacts which appear for greater values of the Hurst index H. Future extensions of our model include Gaussian fields whose Hurst index and local orientation may vary spatially. A forthcoming work will focus on its application to natural texture characterization and classification, as well as cartoontexture image decomposition [START_REF] Ono | Cartoon-texture image decomposition using blockwise low-rank texture characterization[END_REF].

Fig. 1 .

 1 Fig. 1. Examples of FBF with (a) H = 0.3, (b) H = 0.7.

Fig. 2 .

 2 Fig. 2. AFBF with H = 0.5, α 0 = 0 and (a) α = π/6, (b) α = π/24.

Fig. 3 .

 3 Fig. 3. (a) Texture resulting from the vector field orientation V 1 (x,y) in yellow, (b) zoom around the red point x 0 = (x, y) that shows locally an oriented elementary field, (c) diagram illustrating each parameter of the LAFBF model.

Fig. 4 .

 4 Fig. 4. Texture resulting from the vector field V 2 (x,y) .


Eduard Feireisl

Antonín Novotný

Yongzhong Sun

A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system

Keywords: Navier-Stokes-Fourier system, weak solution, regularity

We show that any weak solution to the full Navier-Stokes-Fourier system emanating from the data belonging to the Sobolev space W 3,2 remains regular as long as the velocity gradient is bounded. The proof is based on the weak-strong uniqueness property and parabolic a priori estimates for the local strong solutions.

Introduction

In continuum mechanics, the motion of a general compressible, viscous, and heat conducting fluid is described by the thermostatic variables -the mass density ̺ = ̺(t, x) and the absolute temperature ϑ(t, x), and the velocity field u = u(t, x) evaluated at the time t and the spatial position x belonging to the reference physical domain Ω ⊂ R 3 . The time evolution of the fluid, emanating from the initial data

̺(0, •) = ̺ 0 , ϑ(0, •) = ϑ 0 , u(0, •) = u 0 in Ω, ( 1.1)
is governed by the Navier-Stokes-Fourier system of partial differential equations:

Equation of continuity:

∂ t ̺ + div x (̺u) = 0; (1.2)
Momentum equation:

∂ t (̺u) + div x (̺u ⊗ u) + ∇ x p(̺, ϑ) = div x S(ϑ, ∇ x u); (1.3)
Total energy balance:

∂ t 1 2 ̺|u| 2 + ̺e(̺, ϑ) + div x 1 2 ̺|u| 2 + ̺e(̺, ϑ) u + div x (p(̺, ϑ)u) (1.4)
= div x (S(ϑ, ∇ x u)u) -div x q(ϑ, ∇ x ϑ).

The symbols p = p(̺, ϑ) and e = e(̺, ϑ) stand for the pressure and the (specific) internal energy, respectively. Furthermore, S = S(ϑ, ∇ x u) denotes the viscous stress given by Newton's rheological law:

S(ϑ, ∇ x u) = µ(ϑ) ∇ x u + ∇ t x u - 2 3 div x uI , (1.5)
while q = q(ϑ, ∇ x ϑ) is the heat flux determined by Fourier's law:

q(ϑ, ∇ x u) = -κ(ϑ)∇ x ϑ. (1.6)
We restrict ourselves to bounded regular domains Ω ⊂ R 3 , with energetically insulated boundary. More specifically, we consider the standard No slip boundary condition:

u| ∂Ω = 0, (1.7)
together with

No-flux condition:

q • n| ∂Ω = 0, (1.8)
where n is the (outer) normal vector to ∂Ω. Under these circumstance, the total mass as well as the total energy of the system are constants of motion, specifically, integrating equations (1.2), (1.4) over Ω we obtain

Ω ̺(t, •) dx = Ω ̺ 0 dx = M 0 , (1.9
)

Ω 1 2 ̺|u| 2 + ̺e(̺, ϑ) (t, •) dx = Ω 1 2 ̺ 0 |u 0 | 2 + ̺ 0 e(̺ 0 , ϑ 0 ) dx = E 0 . (1.10)
From the mathematical viewpoint, the Navier-Stokes-Fourier system suffers the same deficiency as most of its counterparts in continuum mechanics -the lack of sufficiently strong a priori bounds. It is known that the problem (1.1 -1.8) is well posed, locally in time, in the framework of classical solutions, see Tani [START_REF] Tani | On the first initial-boundary value problem of compressible viscous fluid motion[END_REF], or in slightly more general energy spaces of Sobolev type, see Valli [START_REF] Valli | A correction to the paper: "An existence theorem for compressible viscous fluids[END_REF], [START_REF] Valli | An existence theorem for compressible viscous fluids[END_REF], Valli and Zajaczkowski, [START_REF] Valli | Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case[END_REF], among others. Besides these local existence results, Matsumura and Nishida established in a series of papers [START_REF] Matsumura | Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with the first order dissipation[END_REF], [START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat-conductive gases[END_REF], [START_REF] Matsumura | The initial value problem for the equations of motion of compressible and heat conductive fluids[END_REF] global-in-time existence of strong/classical solutions provided the initial data are sufficiently close to an equilibrium solution. Similarly to the well known incompressible Navier-Stokes system, the existence of global-in-time smooth solutions for any physically admissible and large initial data remains an outstanding open problem.

The system (1.1 -1.8) has been also studied in the framework of weak solutions. Hoff and Jenssen [START_REF] Hoff | Symmetric nonbarotropic flows with large data and forces[END_REF] established global existence for radially symmetric data in R 3 . They also identified one of the main stumbling blocks in the analysis of the Navier-Stokes-Fourier system, namely the (hypothetical) appearance of vacuum zones, where the density vanishes and the classical understanding of the equations breaks down. More recently, Bresch and Desjardins [START_REF] Bresch | Stabilité de solutions faibles globales pour les équations de Navier-Stokes compressibles avec température[END_REF], [START_REF] Bresch | On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids[END_REF] discovered a new a priori bound on the density gradient leading to global-in-time existence in the truly 3D-setting conditioned, unfortunately, by a very specific relation satisfied by the density dependent viscosity coefficients and a rather unrealistic formula for the pressure that has to be infinite (negative) for ̺ → 0.

Weak and dissipative solutions

We adopt the approach to the Navier-Stokes-Fourier system originated in [START_REF] Feireisl | Stability of flows of real monoatomic gases[END_REF] and further developed and detailed in [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF]. Suppose that p and e are interrelated through Gibbs' equation:

ϑDs(̺, ϑ) = De(̺, ϑ) + p(̺, ϑ)D 1 ̺ , (1.11)
where s = s(̺, ϑ) is a new thermodynamic function called (specific) entropy. If ̺, ϑ are smooth and bounded below away from zero and if u is smooth, then the total energy balance (1.4) can be replaced by Thermal energy balance:

̺ ∂e(̺, ϑ) ∂ϑ (∂ t ϑ + u • ∇ x ϑ) -div x (κ(ϑ)∇ x ϑ) = S(ϑ, ∇ x u) : ∇ x u -ϑ ∂p(̺, ϑ) ∂ϑ div x u. (1.12)
Furthermore, dividing (1.12) on ϑ we arrive at Entropy (production) equation:

∂ t (̺s(̺, ϑ)) + div x (̺s(̺, ϑ)u) + div x q(ϑ, ∇ x ϑ) ϑ = σ, (1.13)
with Entropy production rate: Another crucial observation, which is the platform of the approach developed in [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF], is that we can relax (1.14) to

σ = 1 ϑ S(ϑ, ∇ x u) : ∇ x u - q • ∇ x ϑ ̺ . ( 1
σ ≥ 1 ϑ S(ϑ, ∇ x u) : ∇ x u - q • ∇ x ϑ ̺ (1.15)
provided the system is supplemented by the integrated total energy balance (1.10 • Weak-strong uniqueness. Any weak solution coincides with a classical solution emanating from the same initial data provided the latter exists, see [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF].

Relative entropy and dissipative solutions

The property of weak-strong uniqueness mentioned above is closely related to the relative entropy inequality introduced in [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF]. We start by introducing another thermodynamic function: Ballistic free energy:

H Θ (̺, ϑ) = ̺e(̺, ϑ) -Θ̺s(̺, ϑ), (1.16)
see [START_REF] Ericksen | Introduction to the thermodynamics of solids[END_REF], together with Relative entropy functional:

E ̺, ϑ, u r, Θ, U = Ω 1 2 ̺|u -U| 2 + H Θ (̺, ϑ) - ∂H Θ (r, Θ) ∂̺ (̺ -r) -H Θ (r, Θ) dx,
(1.17) where {̺, ϑ, u} is a weak solution of the Navier-Stokes-Fourier system and {r, Θ, u} is an arbitrary trio of smooth functions satisfying r > 0, Θ > 0, U| ∂Ω = 0.

(1.18)

As observed in [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF], the weak solutions of the Navier-Stokes-Fourier system satisfy Relative entropy inequality:

E ̺, ϑ, u r, Θ, U t=τ t=0 + τ 0 Ω Θ ϑ S(ϑ, ∇ x u) : ∇ x u - q(ϑ, ∇ x ϑ) • ∇ x ϑ ϑ dx dt (1.19) ≤ τ 0 Ω ̺(U -u) • ∂ t U + ̺(U -u) ⊗ u : ∇ x U -p(̺, ϑ)div x U dx dt + τ 0 Ω S(ϑ, ∇ x u) : ∇ x U dx dt - τ 0 Ω ̺ s(̺, ϑ) -s(r, Θ) ∂ t Θ + ̺ s(̺, ϑ) -s(r, Θ) u • ∇ x Θ dx dt - τ 0 Ω q(̺, ∇ x ϑ) ϑ • ∇ x Θ dx dt + τ 0 Ω 1 - ̺ r ∂ t p(r, Θ) - ̺ r u • ∇ x p(r, Θ) dx dt
for any trio {r, Θ, U} satisfying (1.18). Taking {r, Θ, U} the strong solution of the same system emanating from the same initial data gives rise to a Gronwall type inequality for E yielding ̺ = r, ϑ = Θ, u = U, see [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF].

As the proof of the weak-strong uniqueness principle uses only the integral inequality (1.19) without any reference to the original system of equations, we may introduce a new class of dissipative solutions satisfying solely the relative entropy inequality (1.19) for any trio of admissible test functions {r, Θ, U}. Note that a similar concept was introduced by DiPerna and Lions [START_REF] Lions | Mathematical topics in fluid dynamics[END_REF] in the context of the Euler system.

Regularity criteria

A conditional regularity criterion is a condition which, if satisfied by a weak solution, implies that the latter is regular. Similarly, such a condition may be applied to guarantee that a local (strong) solution can be extended to a given time interval. The most celebrated conditional regularity criteria are due to Prodi and Serrin [START_REF] Prodi | Un teorema di unicità per le equazioni di Navier-Stokes[END_REF], [START_REF] Serrin | On the interior regularity of weak solutions of the Navier-Stokes equations[END_REF] and Beal, Kato and Majda [START_REF] Beale | Remarks on the breakdown of smooth solutions for the 3-D Euler equations[END_REF], Constantin and Fefferman [START_REF] Constantin | Direction of vorticity and the problem of global regularity for the Navier-Stokes equations[END_REF] for solutions to the incompressible Navier-Stokes and Euler systems. Recently, similar conditions were obtained also in the context of compressible barotropic fluids and the full Navier-Stokes-Fourier system, see Fan, Jiang and Ou [START_REF] Fan | A blow-up criterion for compressible viscous heatconductive flows[END_REF], Sun et al. [START_REF] Sun | A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navies-Stokes equations[END_REF], and the references cited therein.

In view of the results of Hoff et al. [START_REF] Hoff | Dynamics of singularity surfaces for compressible viscous flows in two space dimensions[END_REF], [START_REF] Hoff | Lagrangean structure and propagation of singularities in multidimensional compressible flow[END_REF], certain discontinuities imposed through the initial data in the compressible Navier-Stokes system propagate in time. In other words, unlike its incompressible counterpart, the hyperbolic-parabolic compressible Navier-Stokes system does not enjoy the smoothing property typical for purely parabolic equations. Analogously, a solution of the full Navier-Stokes-Fourier system can be regular only if regularity is enforced by a proper choice of the initial data.

Our approach to conditional regularity is based on the concept of weak (dissipative) solutions satisfying the relative entropy inequality (1.19):

• In Section 2, we introduce the structural restrictions imposed on the thermodynamic functions p, e, s and the transport coefficients µ, κ so that the Navier-Stokes-Fourier system may possess a global in time weak (dissipative) solution for any finite energy initial data.

• Now, suppose that the initial data are more regular, specifically,

0 < ̺ ≤ ̺ 0 , 0 < ϑ ≤ ϑ 0 , ̺ 0 , ϑ 0 ∈ W 3,2 (Ω), u 0 ∈ W 3,2 (Ω; R 3 ), (1.20)
satisfying, in addition, the necessary compatibility conditions

u 0 | ∂Ω = 0, ∇ x ϑ 0 • n| ∂Ω = 0, ∇ x p(̺ 0 , ϑ 0 )| ∂Ω = div x S(ϑ 0 , u 0 )| ∂Ω . (1.21)
Under these circumstance, the Navier-Stokes-Fouries system admits a local-in-time strong solution {̺, θ, ũ} constructed by Valli [START_REF] Valli | An existence theorem for compressible viscous fluids[END_REF] such that

̺, θ ∈ C([0, T ]; W 3,2 (Ω)), θ ∈ L 2 (0, T ; W 4,2 (Ω)), ∂ t θ ∈ L 2 (0, T ; W 2,2 (Ω)), (1.22) ũ ∈ C([0, T ]; W 3,2 (Ω; R 3 )) ∩ L 2 (0, T ; W 4,2 (Ω; R 3 )), ∂ t ũ ∈ L 2 (0, T ; W 2,2 (Ω; R 3 )).
(1.23)

• Modifying slightly the arguments of [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF] to accommodate the strong solutions belonging to the regularity class (1.22) we show in Section 3 that the associated (globalin-time) weak solution {̺, ϑ, u} emanating from the initial data coincides with the strong solution {̺, θ, ũ} on its existence interval [0, T ].

• We show that if lim sup t→ T - ̺(t, •) W 3,2 (Ω) + ϑ(t, •) W 3,2 (Ω) + u(t, •) W 3,2 (Ω;R 3 ) = ∞, (1.24)
then, necessarily, lim sup

t→ T - ∇ x u(t, •) L ∞ (Ω;R 3×3 ) = ∞, (1.25)
see Section 4.

• Finally, in Section 5, we observe that the solutions constructed by Valli are classical (all necessary derivatives are continuous) for any t > 0. We conclude by the following conditional regularity result for the weak solutions of the Navier-Stokes-Fourier system:

Suppose that {̺, ϑ, u} is a weak (dissipative) solution of the Navier-Stokes-Fourier system in (0, T ) × Ω, emanating from the initial data belonging to the regularity class specified through (1.20), (1.21), and satisfying ess sup

t∈(0,T ) ∇ x u(t, •) L ∞ (Ω;R 3×3 ) < ∞.
Then u is a classical solution of the Navier-Stokes-Fourier system, unique in the class of weak (dissipative) solutions, see Theorem 2.1 below.

Preliminaries, main results

We start with a list of structural restrictions imposed on the functions p, e, s, µ, and κ. The interested reader may consult [11, Chapter 1] for the physical background and possible relaxations. We suppose that the pressure is given by the formula

p(̺, ϑ) = ϑ 5/2 P ̺ ϑ 3/2 + a 3 ϑ 4 , a > 0, (2.1)
with

P ∈ C 1 [0, ∞) ∩ C 3 (0, ∞) satisfying P (0) = 0, P ′ (Z) > 0, 0 < 5 3 P (Z) -P ′ (Z)Z Z < c for all Z > 0, (2.2) lim Z→∞ P (Z) Z 5/3 > 0. (2.3)
Accordingly, in agreement with Gibbs' equation (1.11), e(̺, ϑ) = 3 2

ϑ 5/2 ̺ P ̺ ϑ 3/2 + a ̺ ϑ 4 , (2.4)
and

s(̺, ϑ) = S ̺ ϑ 3/2 + 4a 3 ϑ 3 ̺ , (2.5)
where

S ′ (Z) = - 3 2 5 3 P (Z) -P ′ (Z)Z Z 2 , lim Z→∞ S(Z) = 0. (2.6)
Finally, the transport coefficients are continuously differentiable functions of the temperature satisfying

µ(1 + ϑ Λ ) ≤ µ(ϑ) ≤ µ(1 + ϑ Λ ), |µ ′ (ϑ)| < c for all ϑ ∈ [0, ∞), 2 5 < Λ ≤ 1, (2.7
)

κ(1 + ϑ 3 ) ≤ κ(ϑ) ≤ κ(1 + ϑ 3 ) for all ϑ ∈ [0, ∞). ( 2
, ϑ 0 ∈ L ∞ (Ω), ̺ 0 , ϑ 0 > 0 a.a. in Ω, u 0 ∈ L 2 (Ω; R 3 ). (2.9)
Moreover, the weak solution {̺, ϑ, u} satisfies the relative entropy inequality (1.19), see [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF].

Accordingly, a trio {̺, ϑ, u} will be called dissipative solution of the Navier-Stokes-Fourier system (1.1 -1.8) provided it obeys the relative entropy (1.19) for any choice of smooth functions {r, Θ, U} satisfying (1.18).

Summarizing the results of [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF]Chapter 3] and [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF] we obtain:

Proposition 2.1 Let Ω ⊂ R 3 be a bounded domain of class C 2+ν
. Suppose that the thermodynamic functions p, e, s and the transport coefficients µ, κ obey the structural hypotheses (2.1 -2.8). Finally, let the initial data belong to the class specified in (2.9). Then the Navier-Stokes-Fourier system possesses a dissipative solution {̺, ϑ, u} on an arbitrary time interval (0, T ) that enjoys the following regularity properties:

̺ ≥ 0 a.a. in (0, T ) × Ω, ̺ ∈ C([0, T ]; L 1 (Ω)) ∩ L ∞ (0, T ; L 5/3 (Ω)) ∩ L β ((0, T ) × Ω) (2.10)
for a certain β > 5 3 ;

ϑ > 0 a.a. in (0, T ) × Ω, ϑ ∈ L ∞ (0, T ; L 4 (Ω)) ∩ L 2 (0, T ; W 1,2 (Ω)), (2.11)
ϑ 3 , log(ϑ) ∈ L 2 (0, T ; W 1,2 (Ω)); (2.12) u ∈ L 2 (0, T ; W α 0 (Ω; R 3 )), α = 8 5 -Λ , ̺u ∈ C weak (0, T ; L 5/4 (Ω; R 3 )).
(2.13)

Main result

In accordance with the programme delineated in the introductory part, our main goal is to show that for regular data the weak solution remains regular as long as we can control the gradient of the velocity field. More specifically, we will show the following result. Suppose, in addition, that ess sup t∈(0,T )

∇ x u(t, •) L ∞ (Ω;R 3×3 ) < ∞. (2.14)
Then {̺, ϑ, u} is a classical solution of the Navier-Stokes-Fourier system satisfying (1.1 -1.8) in (0, T ) × Ω.

Remark: For technical reasons, we have omitted the effect of the bulk viscosity in the viscous stress S(ϑ, ∇ x u). The possibility to extend the result to more general forms of the viscous stress is discussed in Section 5.

Here, classical solution means that all functions and all derivatives appearing in the equations (1.2 -1.4) are continuous in (0, T ) × Ω, the functions ̺, ϑ, u, together with their first order derivatives, are continuous in [0, T ] × Ω and satisfy the initial conditions (1.1) as well as the boundary conditions (1.7), (1.8).

Although the rest of the paper is essentially devoted to the proof of Theorem 2.1, we will obtain a few other results that may be of independent interest.

Local existence and weak-strong uniqueness

To begin, it is convenient to introduce a scaled temperature

Ξ = K(ϑ), where K(ϑ) = ϑ 0 κ(z) dz.
Accordingly, the thermal energy balance (1.12) reads

̺ κ(ϑ) ∂e(̺, ϑ) ∂ϑ (∂ t Ξ + u • ∇ x Ξ) -∆Ξ (3.1) = 1 K(ϑ) S(ϑ, ∇ x u) : ∇ x u Ξ - ϑ K(ϑ) ∂p(̺, ϑ) ∂ϑ div x u Ξ.

First a priori bounds

Our goal is to derive a lower and upper bounds on the temperature. We first rewrite (3.1) in the form

(∂ t Ξ + u • ∇ x Ξ) -D∆Ξ = DAΞ -Bdiv x u Ξ,
where

D = ̺ κ(ϑ) ∂e(̺, ϑ) ∂ϑ -1 , A = 1 K(ϑ) S(ϑ, ∇ x u) : ∇ x u, B = ϑκ(ϑ) K(ϑ) ∂p(̺, ϑ) ∂ϑ ̺ ∂e(̺, ϑ) ∂ϑ -1
. Now, in view of the hypotheses (2.1), (2.4), and (2.8), it is easy to check that the exist constants B, B such that 0 < B ≤ B(t, x) ≤ B for all t, x.

Applying the standard comparison argument, we therefore deduce that

Ξ(τ, •) ≥ inf x∈Ω Ξ(0, x) exp -B τ 0 div x u(t, •) L ∞ (Ω) dt , τ ≥ 0. (3.2)
In order to obtain an upper bound, we need a similar estimate for the density, namely

inf x∈Ω ̺ 0 (x) exp - τ 0 div x u(t, •) L ∞ (Ω) dt (3.3) ≤ ̺(τ, •) ≤ sup x∈Ω ̺ 0 (x) exp τ 0 div x u(t, •) L ∞ (Ω) dt
that follows easily from the equation of continuity (1.2). Now, we may use hypotheses (2.7 -2.8) to observe that

A(t, •) ≤ A ∇ x u(t, •) 2 L ∞ (Ω;R 3×3 ) ; whence Ξ(τ, •) ≤ (3.4) sup x∈Ω Ξ(0, x) exp D A τ 0 ∇ x u(t, •) 2 L ∞ (Ω;R 3×3 ) dt + exp B τ 0 div x u(t, •) L ∞ (Ω) dt for τ ∈ [0, T ],

Local existence of strong solutions

Rewriting the system of equations (1.2), ( Then there exists a positive time T such that the Navier-Stokes-Fourier system (1.1 -1.8) admits a unique strong solution {̺, θ, ũ} in the class

̺, θ ∈ C([0, T ]; W 3,2 (Ω)), ũ ∈ C([0, T ]; W 3,2 (Ω; R 3 )), (3.5) θ ∈ L 2 (0, T ; W 4,2 (Ω)), ∂ t θ ∈ L 2 (0, T ; W 2,2 (Ω)), (3.6) ũ ∈ L 2 (0, T ; W 4,2 (Ω; R 3 )), ∂ t ũ ∈ L 2 (0, T ; W 2,2 (Ω; R 3 )). (3.7)
Moreover, there exist scalar functions ̺, ̺, ϑ, ϑ, depending solely on

inf Ω ̺ 0 , sup Ω ̺ 0 , inf Ω ϑ 0 , sup Ω ϑ 0 , and on T 0 ∇ x ũ 2 L ∞ (Ω;R 3×3 ) dt, such that 0 < ̺(τ ) ≤ ̺(τ, •) ≤ ̺(τ ), 0 < ϑ(τ ) ≤ θ(τ, •) ≤ ϑ(τ ) (3.8)
for any τ ∈ [0, T ].

Weak strong uniqueness

We claim that in view of the result [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF] and its generalizations obtained in [START_REF] Feireisl | Relative entropies in thermodynamics of complete fluid systems[END_REF], the dissipative solution obtained in Proposition 2.1 coincides with the strong solution of Proposition 3.1 on the time interval [0, T ] provided they start from the same initial data (1.20), (1.21). This deserves some comments since both [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF] and [START_REF] Feireisl | Relative entropies in thermodynamics of complete fluid systems[END_REF] deal with classical solutions having all relevant derivatives continuous and bounded in (0, T ) × Ω.

Given the integrability properties of the dissipative solutions stated in (2.10 -2.13) and the regularity of the strong solutions (3.5 -3.7), we can easily check that the trio {r = ̺, Θ = θ, U = ũ} can be taken as test functions in the relative entropy inequality (1.19). Now, following step by step the arguments of [10, Section 6, formula (79)] we deduce from (1.19) the estimate

E ̺, ϑ, u ̺, θ, ũ t=τ t=0 + c τ 0 u -ũ 2 W 1,α (Ω;R 3 ) + ϑ -θ 2 W 1,2 (Ω) dt (3.9) ≤ τ 0 χ 1 (t)E ̺, ϑ, u ̺, θ, ũ dt + τ 0 χ 2 (t) Ω 1 + ̺ + ̺|s(̺, ϑ)| res 1 + |u -ũ| res + ϑ -θ res dxdt,
where, similarly to [START_REF] Feireisl | Relative entropies in thermodynamics of complete fluid systems[END_REF], we have denoted

h = [h] ess + [h] res = h -[h] ess , [h] ess = Φ(̺, ϑ)h, Φ ∈ C ∞ c ((0, ∞) 2 ), 0 ≤ Φ ≤ 1, Φ = 1 in an open neighborhood of a compact K ⊂ (0, ∞) 2
where K is chosen to contain the range of [̺, θ], specifically,

[̺(t, x), θ(t, x)] ∈ K for all x ∈ Ω, t ∈ [0, T ].
The functions χ i , i = 1, 2 are of the form

χ i (t) = b i (t) 1 + ∂ t θ L ∞ ((0, T )×Ω) + ∇ 2 θ L ∞ ((0, T )×Ω) + ∂ t ũ L ∞ ((0, T )×Ω) + ∇ 2 ũ L ∞ ((0, T )×Ω)
, where b i are bounded positive functions determined in terms of the amplitude of ̺, θ, ũ and their spatial gradients in [0, T ] × Ω. Focusing on the most difficult term, we have

χ 2 (t) Ω ̺|s(̺, ϑ)|| [u -ũ] res | dx ≤ χ 2 (t) [̺s(̺, ϑ)] res L 4/3 (Ω) u -ũ L 4 (Ω;R 3 ) ≤ ε u -ũ 2 W 1,α (Ω;R 3 ) + c(ε)χ 2 2 (t) [̺s(̺, ϑ)] res 2 L 4/3 (Ω) ≤ ε u -ũ 2 W 1,α (Ω;R 3 ) + c(ε)χ 2 2 (t)E 3/2 ̺
, ϑ, u ̺, θ, ũ for any ε > 0, where, exactly as in [10, Section 6.1], we have used the structural properties of s and the embedding

W 1,α (Ω) ֒→ L 4 , α = 8 5 -Λ , 2 5 < Λ ≤ 1.
Going back to (3.9) we may infer that

E ̺, ϑ, u ̺, θ, ũ t=τ t=0 + c τ 0 u -ũ 2 W 1,α (Ω;R 3 ) + ϑ -θ 2 W 1,2 (Ω) dt (3.10) ≤ τ 0 χ(t)E ̺, ϑ, u ̺, θ, ũ dt,
where, by virtue of (3.6), (3.7), χ ∈ L 1 (0, T ).

Applying Gronwall's lemma we obtain the following conclusion:

Conditional regularity

Our goal is to show that the energy norm

̺(t, •) W 3,2 (Ω) + θ(t, •) W 3,2 (Ω) + ũ(t, •) W 3,2 (Ω;R 3 )
associated to a strong solution of the Navier-Stokes-Fourier system remains bounded in [0, T ] as long as sup t∈(0, T )

∇ x ũ(t, •) L ∞ (Ω;R 3×3 ) ≤ G < ∞. (4.1)
We remark that we already know that

0 < ̺(τ ) ≤ ̺(τ, •) ≤ ̺(τ ), 0 < ϑ ≤ θ(τ, •) ≤ ϑ (4.2)
for any τ ∈ [0, T ], see (3.8), where the bounds depend only on G, the initial data, and the length of the time interval.

Energy bounds, temperature

Multiplying equation (3.1) on ̺ κ( θ)

∂e(̺, θ) ∂ϑ -1 ∆ Ξ, Ξ = K( θ),
we obtain, after a routine manipulation, ess sup

t∈(0, T ) Ξ(t, •) W 1,2 (Ω) + ∂ t Ξ 2 L 2 (0,T ;L 2 (Ω)) + Ξ 2 L 2 (0,T ;W 2,2 (Ω)) ≤ c(B, data). (4.3)
Since θ is already known to be bounded, the estimates (4.3) transfer to θ. Indeed note that

D 2 x θ = K -1 ( Ξ) -1 D 2 x Ξ + K -1 ( Ξ) |D x Ξ| 2 ,
where, since Ξ satisfies the homogeneous Neumann boundary condition, the term ∇ x Ξ may be estimates in terms of the Gagliardo-Nirenberg inequality

∇ x Ξ 2 L 4 (Ω;R 3 ) ≤ Ξ L ∞ (Ω) ∆Ξ L 2 (Ω) . (4.4)
Consequently, relation (4.3) implies ess sup t∈(0, T )

θ(t, •) W 1,2 (Ω) + ∂ t θ 2 L 2 (0,T ;L 2 (Ω)) + θ 2 L 2 (0,T ;W 2,2 (Ω))
≤ c(B, data). (4.5)

Energy bounds, velocity

Our next goal is to deduce similar bounds for the velocity field. To this end, we write the momentum equation in the form

∂ t ũ - 1 ̺ div x S( θ, ∇ x ũ) = -ũ • ∇ x ũ - 1 ̺ ∂p(̺, θ) ∂ϑ ∇ x θ + 1 ̺ ∂p(̺, θ) ∂̺ ∇ x ̺ (4.6) = h 1 + 1 ̺ ∂p(̺, θ) ∂̺ ∇ x ̺,
where, in accordance with the previous estimates sup

t∈[0, T ] h 1 (t, •) L 2 (Ω;R 3 ) ≤ c(B, data).
Taking the scalar product of (4.6) with -div x S( θ, ∇ x ũ) we obtain Ω S( θ, ũ) :

∂ t ∇ x ũ dx + Ω 1 ̺ div x S( θ, ∇ x ũ) 2 dx = - Ω h 1 + 1 ̺ ∂p(̺, θ) ∂̺ ∇ x ̺ • div x S( θ, ∇ x ũ) dx,
where, furthermore,

Ω S( θ, ũ) : ∂ t ∇ x ũ dx = d dt Ω µ( θ) 4 ∇ x ũ + ∇ t x ũ - 2 3 div x ũI 2 dx - Ω µ ′ ( θ) 4 ∂ t θ ∇ x ũ + ∇ t x ũ - 2 3 div x ũI 2 dx.
In view of the estimate (4.5) we conclude that

d dt Ω µ( θ) 4 ∇ x ũ + ∇ t x ũ - 2 3 div x ũI 2 dx + Ω 1 ̺ div x S( θ, ∇ x ũ) 2 dx (4.7)
= -

Ω h 1 + 1 ̺ ∂p(̺, θ) ∂̺ ∇ x ̺ • div x S( θ, ∇ x ũ) dx + Ω h 2 dx,
where h 2 L 2 (0, T ;L 2 (Ω) ≤ c(B, data).

Elliptic estimates

In order to exploit (4.7), we have to show that the L 2 -norm of div x S( θ, ∇ x ũ) is "equivalent" to the L 2 -norm of ∇ 2 ũ. We have

H ≡ div x S( θ, ∇ x ũ) = µ( θ) ∆ũ + 1 3 ∇ x div x ũ + h 3 ,
where, in accordance with (4.5),

h 3 L ∞ (0, T ;L 2 (Ω;R 3 ) ≤ c(B, data).
Consequently, we may integrate (4.7) with respect to t and use the standard elliptic estimates to obtain

τ 0 ũ 2 W 2,2 (Ω;R 3 ) dt ≤ c(B, data) 1 + τ 0 ∇ x ̺ 2 L 2 (Ω;R 3 ) , τ ∈ [0, T ]. (4.8)
Now, we differentiate the equation of continuity with respect to x to obtain

∂ t (∂ x i ̺) + ũ • ∇ x (∂ x i ̺) = -∂ x i (ũ) • ∇ x ̺ -(∂ x i ̺) div x ũ -̺∂ x i div x ũ; (4.9) whence d dt Ω |∇ x ̺| 2 dx ≤ c(B, data) Ω |∇ x ̺| 2 + |∇ x ̺||∇ x div x ũ| dx. (4.10)
Combining (4.8), (4.10) with the standard Gronwall type argument we get the following estimates: sup

t∈[0, T ] ∇ x ̺(t, •) L 2 (Ω;R 3 ) ≤ c(B, data). (4.11)
Next, due to (4.6),

∂ t ũ L 2 (0, T ;L 2 (Ω;R 3 )) + ũ L 2 (0, T ;W 2,2 (Ω;R 3 )) ≤ c(B, data), (4.12)
and, finally, by means of the equation of continuity,

∂ t ̺ L 2 (0, T ;L 2 (Ω)) ≤ c(B, data). (4.13)

Energy estimates for the time derivatives

Differentiating the momentum equation (1.3) with respect to t and setting Ṽ = ∂ t ũ we obtain

̺ ∂ t Ṽ + ũ • ∇ x Ṽ -div x S( θ, ∇ x Ṽ) = -∂ t ̺ Ṽ + ̺ Ṽ • ∇ x ũ + h 1 , (4.14)
with

h 1 = -∂ t ̺ũ • ∇ x ũ + div x µ ′ ( θ)∂ t θ ∇ x ũ + ∇ t x ũ - 2 3 div x ũI -∇ x ∂ t p(̺, θ) ,
where, in accordance with the previous estimates,

h 1 L 2 (0, T ;W -1,2 (Ω;R 3 )) ≤ c(G, data). (4.15)
Seeing that

-∂ t ̺ Ṽ + ̺ Ṽ • ∇ x ũ • V = | Ṽ| 2 ̺div x ũ
we may take the scalar product of (4.14) with Ṽ and, integrating over Ω, we deduce the energy estimates for Ṽ = ∂ t ũ:

sup t∈[0, T ] ∂ t ũ L 2 (Ω;R 3 ) + ∂ t ũ L 2 (0, T ;W 1,2 (Ω;R 3 )) ≤ c(G, data). (4.16)
and, according to Sobolev's embedding theorem, Going back to (1.3) we compute

∂ t ũ L 2 (0, T ;L 6 (Ω;R 3 )) ≤ c(G, data). (4.
µ( θ) ∆ũ + 1 3 ∇ x div x ũ (4.18) = ̺ (∂ t ũ + ũ • ∇ x ũ) -µ ′ ( θ) ∇ x ũ + ∇ x ũt - 2 3 div x ũI ∇ x θ + ∇ x p(̺,
ũ(t, •) 2 W 2,6 (Ω;R 3 ) dt ≤ c 1 + τ 0 ∇ x ̺(t, •) 2 L 6 (Ω;R 3 ) dt . (4.20)
Furthermore, multiplying (4.9) on |∇ x ̺| 4 ∇ x ̺ yields

d dt ∇ x ̺ 6 L 6 (Ω;R 3 ) ≤ c ∇ x ̺ 6 L 6 (Ω;R 3 ) + ũ W 2,6 (Ω;R 3 ) ∇ x ̺ 5 L 6 (Ω;R 3 ) . ( 4
∇ x ̺(t, •) L 6 (Ω;R 3 ) + ∂ t ̺(t, •) L 6 (Ω;R 3 ) + ũ L 2 (0, T ;W 2,6 (Ω;R 3 )) ≤ c(G, data). (4.22)
4.5 L p -L q estimates for the temperature Our next goal is to apply the technique of L p -L q estimates to the parabolic equation (1.12). For this purpose, we first need Hölder continuity for θ. To this end, let us write equation (1.12) in terms of ẽ(̺, Ξ) = e(̺, K -1 ( Ξ)).

̺∂ t ẽ(̺, Ξ) + ̺ũ • ∇ x ẽ(̺, Ξ) -∆ x Ξ = S( θ, ∇ x ũ) : ∇ x ũ -p(̺, θ)div x ũ := h. Note that ∆ x Ξ = div x ∇ x e e Θ (̺, Ξ) -div x ∂ ̺ ẽ(̺, Ξ) ∂ Θ ẽ(̺, Ξ) ∇ x ̺ .
We obtain

̺∂ t ẽ(̺, Ξ) + ̺ũ • ∇ x ẽ(̺, Ξ) -div x ∇ x ẽ ∂ Θ ẽ(̺, Ξ) = h + div x ∂ ̺ ẽ( ̺, Ξ) ∂ Θ ẽ(̺, Ξ) ∇ x ̺ ,
which, after dividing on both sides by ̺, yields Now we write (1.12) in the following form: being Hölder continuous. The L p -L q theory for parabolic equations is now applicable yielding

∂ t ẽ(̺, Ξ) + ũ - ∇ x ̺ ̺2 ∂ Θ ẽ(̺, Ξ) • ∇ x ẽ(̺, Ξ) -div x ∇ x e ̺∂ Θ ẽ(̺, Ξ) = h ̺ + ∂ ̺ ẽ(̺, Ξ) ̺2 ∂ Θ ẽ(̺, Ξ) |∇ x ̺| 2 + div x ∂ ̺ ẽ(̺, Ξ) ̺∂ Θ ẽ(̺, Ξ) ∇ x ̺ . (4.23) Note that ∇ x ̺ ∈ L ∞ (0, T ; L 6 (Ω, R 3
∂ t Ξ + ũ • ∇ x Ξ -D∆ x Ξ = Dh. ( 4
θ ∈ L p (0, T ; W 2,6 (Ω)), ∂ t θ ∈ L p (0, T ; L 6 (Ω)) for any 1 < p < ∞. (4.26)
see Amann [START_REF] Amann | Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems[END_REF], [START_REF] Amann | Linear and quasilinear parabolic problems[END_REF], Krylov [START_REF] Krylov | Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms[END_REF]. Note that the integrability in the spatial variable is limited by the integrability of the initial data.

Remark: Applying similar arguments to the momentum equation (1.3), we could obtain analogous estimates for the velocity field:

ũ ∈ L p (0, T ; W 2,6 (Ω; R 3 )), ∂ t ũ ∈ L p (0, T ; L 6 (Ω; R 3 )) for any 1 < p < ∞, (4.27)
however, we do not need this refinement in the future analysis.

Full regularity

Differentiating (4.25) with respect to time yields As for the final estimates

∂ t Ξt -D∆ x Ξt = (Dh) t + D t ∆ x Ξ -(ũ • ∇ x Ξ) t := h. ( 4
(Ω)) + Ξt L 2 (0, T ;W 1,2 (Ω)) ≤ c(G, data), (4.30) Ξ L ∞ (0, T ;W 2,2 (Ω)) + Ξ L 2 (0, T ;W 3,2 (Ω)) ≤ c(G, data), (4.31) as well as Ξt L ∞ (0, T ;W 1,2 (Ω)) + Ξtt L 2 (0, T ;L 2 (Ω)) ≤ c(G, data), (4.32) Ξ L ∞ (0, T ;W 3,2 (Ω)) + Ξt L 2 (0, T ;W 2,2 (Ω)) ≤ c(G, data), (4.33) Ξ L 2 (0, T ;W 4,2 (Ω)) ≤ c(G, data). ( 4
ũ L 2 (0, T ;W 4,2 (Ω,R 3 )) + ̺ L ∞ (0, T ;W 3,2 (Ω)) ≤ c(G, data), (4.37)
one needs to combine with the transport equation. The treatment is similar to Section 4.3 and we omit the details. Summarizing the previous considerations, we are allowed to state the following result. In particular, if T is the maximal existence time, then T = T .

Conclusion

To begin, the standard parabolic regularity theory implies that the solutions belonging to the class specified in Proposition 3.1 are, in fact, classical solutions, meaning all relevant derivatives appearing in the system (1.2 -1.4) are continuous in the open set (0, T ) × Ω, the fields θ, ̺ are continuously differentiable up to the boundary ∂Ω, and the boundary conditions (1.7), (1.8) hold, see Matsumura and Nishida [START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heat-conductive gases[END_REF], [START_REF] Matsumura | The initial value problem for the equations of motion of compressible and heat conductive fluids[END_REF]. Consequently, combining Proposition 3.1 with Proposition 4.1 we obtain the conclusion of Theorem 2.1. We conclude the paper by comments concerning the hypotheses of Theorem 2.1. In the light of the results obtained by Fan et al. [START_REF] Fan | A blow-up criterion for compressible viscous heatconductive flows[END_REF], the "optimal" regularity criterion is expected to be the bound

T 0 ∇ x u L ∞ (Ω;R 3×3 ) dt < ∞.
Note, however, that, unlike [START_REF] Fan | A blow-up criterion for compressible viscous heatconductive flows[END_REF], we have to handle the transport coefficients that depend effectively on the temperature. The advantage of this approach is that, again unlike [START_REF] Fan | A blow-up criterion for compressible viscous heatconductive flows[END_REF], the Navier-Stokes-Fourier system is known to possess a global in time dissipative solution.

The regularity of the strong solutions seems optimal, at least in view of the result [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF]. Indeed the time derivative ∂ t ũ of the strong solutions should be at least of class L 2 (0, T ; L ∞ (Ω; R 3 )) for the technique of [START_REF] Feireisl | Weak-strong uniqueness property for the full Navier-Stokes-Fourier system[END_REF] to be applicable.

The next remark concerns the possibility to include the more general class of Newtonian stress, namely S(ϑ, ∇ x u) = µ(ϑ) ∇ x u + ∇ t x u -2 3 div x uI + η(ϑ)div x uI,

where we have added the bulk viscosity, with the coefficient η = η(ϑ). Clearly, we can show the same result if η(ϑ) µ(ϑ) = A -a constant.

Moreover, our method still applies if η(ϑ) µ(ϑ) ≈ "small", where small is determined in terms of the optimal constant λ in the elliptic estimate ∆v + 1 3 div x vI L 2 (Ω;R 3×3 ) ≥ λ v W 2,2 (Ω;R 3 ) , v| ∂Ω = 0.

The general case may be handled in the following way:

1. After establishing the uniform bounds in Section 3.1, apply the theory of Krylov and Safonov [START_REF] Krylov | A certain property of solutions of parabolic equations with measurable coefficients[END_REF] for parabolic equation in non-divergence form to equation (3.1) to obtain uniform bounds on θ in the Hölderian norm.

2. Use the regularity theory for elliptic systems with Hölder coefficients in Section 4.3.

Finally, we remark that the methods of the present paper could be extended to other types of boundary conditions of the Navier type for the velocity and to more general classes of domains Ω including certain unbounded domains like exterior domains or a half-space.

1

 1 Weak and dissipative solutions . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Relative entropy and dissipative solutions . . . . . . . . . . . . . . 6 1.2 Regularity criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Theorem 2 . 1

 21 Under the hypotheses of Proposition 2.1, let {̺, ϑ, u} be a dissipative solution of the Navier-Stokes-Fourier system on the time interval (0, T ) belonging to the regularity class (2.10 -2.13), with the (regular) initial data satisfying (1.20), together with the compatibility conditions (1.21).

  where D depends only on inf x∈Ω ̺ 0 (x), inf x∈Ω Ξ(0, x), and T 0 div x u L ∞ (Ω) dt.

Proposition 3 . 1

 31 , (1.12) in terms of the new unknowns {̺, Ξ, u} and taking the a priori bounds (3.2), (3.4) into account, we can apply the local existence result of Valli [28, Theorem A and Remark 3.3]. Going back to the original variables, we obtain: Under the hypotheses of Proposition 2.1, suppose that the initial data {̺ 0 , ϑ 0 , u 0 } satisfy (1.20), and the compatibility conditions (1.21).

Proposition 3 . 2

 32 Under the hypotheses of Proposition 3.1, let {̺, θ, ũ} be the local strong solution specified in Proposition 3.1 and {̺, ϑ, u} a dissipative solution obtained in Proposition 2.1, emanating from the same initial data. Then ̺ = ̺, ϑ = θ, u = ũ in [0, T ] × Ω.

17

 17

  )

  )) according to(4.22), we can apply the standard theory of parabolic equations with bounded measurable coefficients to deduce that ẽ(̺, Ξ) is Hölder continuous in [0, T ] × Ω.See Ladyzhenskaya et al.[START_REF] Ladyzhenskaya | Linear and qusilinear equations of parabolic type[END_REF]. Since ̺ is already Hölder continuous in the set [0, T ] × Ω (cf. the estimates (4.22)), we find Ξ(hence θ) is Hölder continuous in [0, T ] × Ω.(4.24)

. 25 )

 25 with the diffusion coefficient D = ̺∂e(̺, θ) ∂ϑ -1

  .34) Similar estimates for θ hold. With these estimates in hand, we can go back to equation (4.14) for Ṽ = ∂ t ũ and in a similar way to conclude ũt L ∞ (0, T ;W 1,2 (Ω,R 3 )) + ũtt L 2 (0, T ;L 2 (Ω,R 3 )) ≤ c(G, data), (4.35)ũ L ∞ (0, T ;W 3,2 (Ω,R 3 )) + ũt L 2 (0, T ;W 2,2 (Ω,R 3 )) ≤ c(G, data). (4.36)

Proposition 4 . 1

 41 Let T > 0 be given. Suppose that {̺, θ, ũ} is a strong solution of the Navier-Stokes-Fourier on the time interval [0, T ], T ≤ T , the existence of which is claimed in Proposition 3.1. Assume, in addition, thatsup t∈[0, T ] ∇ x ũ(t, •) L ∞ (Ω;R 3 ) ≤ G. (4.38) Then sup t∈[0, T ] ̺(t, •) W 3,2 (Ω) + θ(t, •) W 3,2 (Ω) + ũ(t, •) W 3,2(Ω;R 3 ) ≤ c(G, data, T ). (4.39)

programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. † Yongzhong Sun is supported by NSF of China(Grants No. 11171145 and 10931007) and the PAPD of Jiangsu Higher Education Institutions.


Chantal Enguehard
email: chantal.enguehard@univ-nantes.fr

Mathieu Mangeot
email: mathieu.mangeot@imag.fr

Computerization of African languages-French dictionaries

Keywords: DiLAF, dictionary, Jibiki

This paper relates work done during the DiLAF project. It consists in converting 5 bilingual African language-French dictionaries originally in Word format into XML following the LMF model. The languages processed are Bambara, Hausa, Kanuri, Tamajaq and Songhai-zarma, still considered as under-resourced languages concerning Natural Language Processing tools. Once converted, the dictionaries are available online on the Jibiki platform for lookup and modification. The DiLAF project is first presented. A description of each dictionary follows. Then, the conversion methodology from .doc format to XML files is presented. A specific point on the usage of Unicode follows. Then, each step of the conversion into XML and LMF is detailed. The last part presents the Jibiki lexical resources management platform used for the project.

Introduction

The work behind this paper has been done during the DiLAF project to computerize African languages-French dictionaries (Bambara, Hausa, Kanuri, Tamajaq, Zarma) in order to disseminate them widely and extend their coverage. We present a methodology for converting dictionaries from Word .doc format in a structured XML format following the Unicode character encodings and Lexical Markup Framework (LMF) standards. The natural language processing of African languages is in its infancy. It is our duty to help our colleagues from the South in this way. This requires, among other things, the publication of articles, that are a valuable resource for under-resourced languages. Many studies have been conducted in the past in this area. However, it seemed interesting to redefine a new methodology taking into account recent developments such as the Open Document Format (ODF) or LMF standards. On the other hand, we wanted to develop the simplest possible method based solely on free and open source tools so that it can be reused by many. This method can also be used for other dictionaries and by extension, any text document (language resource at large) to be converted to XML.

Presentation of the DiLAF project

If access to computers is considered as the main indicator of the digital divide in Africa, we must recognize that the availability of resources in African languages is a handicap with incalculable consequences for the de v el op m e n t of I nf o rm a t i o n Te c hn ol og y a n d Communication Technologies (ICT). Most languages in francophone West Africa area are under-resourced (πlanguage) [START_REF] Berment | Méthodes pour informatiser des langues et des groupes de langues « peu dotées[END_REF]: electronic resources are scarce, poorly distributed or absent, making use of these languages difficult when it comes to introducing them into the education system and especially develop their use in writing in the administration and daily life. Dictionaries are the cornerstone of processing natural language, be it in the mother tongue or in a foreign language. The primary function of communication is conveying meaning, yet meaning is primarily conveyed through vocabulary. As David Wilkins, a british linguist (1972) wrote "so aptly "While without grammar little can be conveyed, without vocabulary nothing can be conveyed". Thus, to help bridge this gap, we are engaged with colleagues from North and South to improve the equipment of some African languages through, among others, the computerization of printed dictionaries of African languages. The DiLAF project aims to convert published dictionaries into XML format for their sustainability and sharing [START_REF] Streiter | Implementing NLP projects for non-central languages: Instructions for funding bodies, strategies for developers[END_REF]. This international project brings together partners from Burkina Faso (CNRST), France (LIG & LINA), Mali (National Resource Centre of the Non-Formal Education) and Niger (INDRAP, Department of Education, and University of Niamey). Based on work already done by lexicographers we formed multidisciplinary teams of linguists, computer scientists and educators. Five dictionaries were converted and integrated into the Jibiki lexical resources management platform [START_REF] Mangeot | Environnements centralisés et distribués pour lexicographes et lexicologues en contexte multilingue[END_REF]. These dictionaries are therefore available on the Internet 1 under a Creative Commons license: The aim of these usage dictionaries is to popularize the written form of the daily use of African languages in the pure lexicographical tradition [START_REF] Matoré | La Méthode en lexicologie[END_REF] [START_REF] Eluerd | La Lexicologie[END_REF]. Departing from interventionist approaches of normative dictionaries [START_REF] Mortureux | La lexicologie entre langue et discours. Paris, SEDES. République du Niger[END_REF], the present descriptive dictionaries remain open to contributions and their online availability online will, hopefully, develop a sense of pride among users of these languages. Similarly, they will participate in the development of a literate environment conducive to increase the literacy whose low level undermines the achievements of progress in other sectors.

Presentation of the dictionaries

Four of the five dictionaries have been produced by the Soutéba project (program to support basic education) with funding from the German cooperation and support of the European Union. These dictionaries for basic education have a simple structure because they were designed for children of primary school class in a bilingual school (education is given there in a national language and in French). Most terms of lexicology, such as lexical labels, parts-of-speech, synonyms, antonyms, genres, dialectal variations, etc. are noted in the language in question in the dictionary, contributing to forge and disseminate a metalanguage in the local language, a specialized terminology. The entries are listed in alphabetical order, even for Tamajaq (although it is usual for this language to sort entries based on lexical roots) because the vowels are written explicitly (this mode of classification was preferred because it is well known by children).

Hausa-French dictionary

The Hausa-French dictionary includes 7,823 entries sorted according to the following lexicographical order: a b ɓ c d ɗ e f fy g gw gy h i j k kw ky ƙ ƙw ƙy l m n o p r s sh t ts u w y ƴ z (République du Niger, 1999a). They are structured with different patterns according to the part-of-speech. All entries are typographical, followed by the pronunciation (tones are marked with diacritics placed on vowels) and part-of-speech. On the semantic level, there is a definition in Hausa, a usage example (identified by the use of italics), and the equivalent in French. For a noun, the gender, feminine, plurals and sometimes dialectal variants are noted. For verbs, it is sometimes necessary to specify the degree to calculate morphological derivatives. Morpho-phonological variants of feminine and plural adjectives derivations are also written. Example: jaki [jàakíi] s. babbar dabbar gida mai kamar doki, wadda ba ta kai tsawon doki ba amma ta fi shi dogayen kunnuwa. Ya aza wa jaki kaya za ya tafi kasuwa. Jin.: n. Sg.: jaka. Jam.: jakai, jakuna. Far.: âne

Kanuri-French dictionary

The Kanuri-French dictionary includes 5,994 entries sorted according to the following lexicographical order: a b c d e ǝ f g h i j k l m n ny o p r ɍ s sh t u w y z (République du Niger, 1999b). The orthographic form of the entry is followed by an indication of pronunciation targeting rating tones. The part-of-speech is shown in italics, followed by a definition, a usage example, a French translation and meaning in French.

Soŋay Zarma-French dictionary

The Zarma-French dictionary includes 6916 entries sorted according to the following lexicographical order: a ã b c d e e f g h i ĩ j k l m n ŋ ɲ o õ p r s t u ũ w y z (République du Niger, 1999d). Each entry has an orthographic form followed by a phonetic transcription in which the tones are rated according to the conventions already set for the Kanuri. The part-of-speech specify explicitly the transitivity or intransitivity of verbs. For some entries, antonyms, synonyms and references are indicated. A gloss in French, a definition and an example end the entry. Example:

ɲagas [ɲagas] mteeb. • brusquement (détaler) • sanniize no kaŋ ga cabe kaŋ boro na zuray sambu nda gaabi sahã-din • Za zankey di hansu-kaaro no i te ɲagas

Tamajaq-French dictionary

The Tamajaq-French dictionary includes 5,205 entries sorted according to the following lexicographical order: a â ă ǝ b c d ḍ e ê f g ǧ h i î j ǰ ɣ k l ḷ m n ŋ o ô q r s ṣ š t ṭ u û w x y z ẓ (République du Niger, 1999c)

The orthographic form of the entry is followed by the part-of-speech and a gloss in French displayed in italics.

For nouns, morphological information about the state of annexation is often included, the plural and gender are also explicitly stated. A definition and an example of usage follow. Other information may appear as variants, synonyms, etc. As Tamajaq is not a tonal language, phonetics does not appear. Example: əbeɣla sn . mulet ♦ Ag-anɣer əd tabagawt. Ibeɣlan wər tan-taha tamalaɣa. anam m el u. : fakr-ejaḍ. təmust.: yy. iget.: ibəɣlan.

Bambara-French dictionary

The Bambara-French dictionary of Father Charles Bailleul (1996 edition) includes more than 10,000 entries sorted according to the following lexicographical order: a b c d e ɛ f g h i j k l m n ɲ ŋ o ɔ p r s t u w y z. This dictionary is primarily intended for French speakers wishing to improve Bambara but it is also a resource for Bambara speakers. In the words of the author himself, the dictionary "plays the role of a working tool for literacy, education and Bambara culture." To date, it can be considered as the most comprehensive dictionary of the language. It is also used by specialist of other varieties of this language like Dyula (Burkina Faso, Côte d'Ivoire) and Malinké (Guinea, Gambia, Sierra Leone, Liberia, etc.).

General conversion methodology and tools

The main objective is to convert dictionaries from a word processor format adapted for human use into XML and explicitly mark all the information so one can use them automatically in natural language processing tasks. The constraints are, on the one hand, working with free, open and multi-platform tools and on the other hand define a simple process that can be understood and then performed independently by linguists having no computer knowledge except regular expressions.

Conversion methodology

The conversion methodology follows these steps: 1. conversion of problematic characters to Unicode; 2. conversion of OpenOffice format to XML; 3. identification and explicit tagging of each part of information (headword, part-of-speech, etc.); 4. XML validation and manual correction of errors in the data (closed lists of values, references); 5. entries structuring following the LMF standard.

Tools used

The first tool allows one to edit files in original format and then convert them into XML. For this step OpenOffice (or LibreOffice) is ideal. It is free and open source. Furthermore, besides the ISO standard Open Document Format (odt), it can open many Microsoft formats (rtf, .doc as well as .docx). Finally, the XML produced is simple, especially compared to Office Open XML Microsoft. OpenOffice has a regular expression engine for search/replace functions. This tool can be used for many steps before converting to XML. However, the search/replace may be problematic because during replacements, the boundaries of text styles can be changed (part of a word is suddenly in italics). Because we rely on styles to convert to XML, we limit the conversion to Unicode characters to keep styles intact.

Then, we need an editor to modify the files. For these operations, the XML editors are not very useful because they do not directly change the plain text with regular expressions and most are not able to edit large files like dictionaries. We recommend using a simple "raw" text editor supporting regular expressions and syntax highlighting.

For XML validation and verification steps, a web browser such as FireFox does it very nicely. It is able to detect and display the XML validation errors and can interpret CSS and XSLT style to enhance the display.

Incremental backups needed

The methodology intends to make backups at each stage and keep track of all search/replace operations done in order to go back when errors resulting from improper action are identified. Sometimes it happens that an error is noticed long after being made. If an error can not be corrected simply by a new search/replace, it is possible to go back from a previous version. Despite all precautions, sometimes errors are detected very late and it is very difficult to go back. If the error can not be detected automatically, it will require manual correction. One must keep in mind that nobody is perfect and yet others even better trained had to forget the possibility to automatically correct all the errors in the conversion process.

Use of Unicode

Characters conversion to Unicode

Although the alphabets of languages on which we have [START_REF] Enguehard | Les langues d'Afrique de l'ouest : de l'imprimante au traitement automatique des langues[END_REF]) are mainly of Latin origin, new characters needed to note specific sounds in some languages with a single character has been adopted by linguists in a series of meetings. Thus, each of the alphabets we previously presented comprises at least one of these special characters: ɓ ɗ Ǝ ɛ ɣ Ƙ ɲ ŋ ɔ ƴ. Characters composed of a Latin character and a diacritical mark were also created: âêîôûăãeĩõũḍḷṣṭẓǧǰšɍ. Although most of these characters are present for several years in the Unicode standard (based on the work of the ISO 10646 (Haralambous 2004)), dictionaries were written using old hacked fonts. A methodology has been defined to identify and replace the inadequate characters with the ones defined in the Unicode standard. It implies that all identified characters are recorded in a file so one can easily repeat this operation if necessary. Table 1 shows part of the list for Zarma. There is no automatic method that will detect these problematic characters. It is imperative to look at the data.

Origin

Unicode

Digraphs lexicographical order

Digraphs can be easily typed using two characters but their use changes the sort order which determines the lexicographic presentation of dictionary entries. Thus, for Hausa and Kanuri, the digraph 'sh' is located after the letter 's'. So, in the Hausa dictionary, the word "sha" (drink) is located after the word "suya" (fried), and, in Kanuri, the word "suwuttu" (undo) precedes the name "shadda" (basin). These subtle differences can hardly be processed by software and require that digraphs appear as a proper sign in the Unicode repertoire. Some used by other languages are already there, sometimes under their different letter cases: 'DZ' (U+01F1), 'Dz' (U+01F2

Characters with diacritics

Some characters with diacritics are included in Unicode as a unique sign, others can only be obtained by composition.

Thus, vowels with tilde 'a', 'i', 'o' and 'u' can be found in Unicode in their lowercase and uppercase forms while the 'e' with a tilde is missing and must be composed with the character 'e' or 'E' followed by the tilde accent (U+303), which can cause renderings different from other letters with tilde when viewing or printing (tilde at a different height for example).

Letter j with caron exists in Unicode as a sign ǰ (U+1F0), but its capitalized form J ̌ must be composed with the letter J and caron sign (U+30C).

The characters e, E et J ̌ should be added to the Unicode standard.

Letter case change

processors usually provide the letter case change function, but do not always realize it the correct way. Thus, we found during our work that OpenOffice Writer software (3.2.1 version) fails in transforming 'ɍ' to 'Ɍ' from lowercase to uppercase or vice versa (the character remains unchanged) while Notepad++ (5.8.6 version) fails in transforming ǰ in J ̌.

Conversion of the format towards XML

Figure 1 shows an excerpt of the Zarma-French dictionary in the original .odt format. All the following examples are based on this dictionary. Open Document Format has the great advantage of being based on XML. Instead of a conversion, we will retrieve the contents of the XML document, then transform it to get what we want. A document in ODF format is actually a zip archive containing multiple files including the text content in XML. This content is stored in the "content.xml" file in the archive. To retrieve this file, some clever manipulations must be followed. On MacOs, one has to an empty folder and then copy the .odt file inside. Then, with a terminal, the "unzip" command must be launched to unzip the file. On Windows, the .odt file extension must be changed into .zip and then the. zip archive can be opened. The file "content.xml" can now be extracted from the archive and then renamed and placed in another location. It becomes the base file on which we will continue our work. The next step consists in editing this file with a "raw" text editor.

may first think that since the source file "content.xml" is already in XML, it may be enough to write an XSLT stylesheet to convert the file into an XML dictionary, but the XML used in the source file is completely different from the XML targeted. Indeed, the source file comes from a word processor. It is designed for styling a document and not for structuring a dictionary entry. Therefore, it is finally easier to convert the XML file "by hands" with regular expressions than to write an XSLT stylesheet for automatically converting the source file.

Explicit tagging of the information

This step consists in tagging explicitly all pieces of information. Each piece of information is usually distinguished from others in the original file with a different style. Figure 2 shows a part of the "abiyanso" entry (airport) in the Zarma-French dictionary. The style used to indicate the pronunciation is "Phonetic_form". After locating the pieces of information, one must choose a set of tags to mark them. This raises the question of the choice of the language used for tags. The choice of English as the international language of research may be privileged. But in our case, English is not a language present in our dictionaries and furthermore, it is not mastered by all linguists colleagues working on the project. The use of French solves this problem since all partners master the language. However, in the case of under-resourced languages computerization projects, we believe that it is important to encourage partners to use the words of their language to define the name of the tags. This may possibly give rise to the creation of new terms that did not exist in these languages. From a political perspective, it helps to move away from a post-colonial vision of the social status of African languages and brings new value to these languages. The set of tag now defined, the next step is to replace the ODF markup by this new "homemade" tagset. Simply perform search/replace operations for each type of information. For the example, the following regular expression (perl syntax) removes the tag "T7": s/<text:span text:style-name="T7">([^<]+) <\/text:span>//g The second expression replaces the tag "Phonetic_form" with "ciiyaŋ": s/<text:span text:style-name="Phonetic_20_form"> ([^<+)<\/text:span>/<ciiyaŋ>$1<\/ciiyaŋ>/g Replacing all tags leads to the result in Figure 4.

Correction of the data

this stage, several corrections are performed on the data.

XML Validation

In order to use XML tools, our file must be well formed. The manipulations of the previous step almost always introduce XML syntax errors. FireFox includes an XML parser and is also able to indicate exactly where the errors are located in the file. Once the error is located, one has to check if it is not repeated elsewhere in the file. If this is the case, a regular expression must be written to correct the error in a systematic way instead of doing it by hand. In our case, the following regular expression can solve the problem: s/<sanniize([^<+)<\/sanniize>/<sanniize>$1<\/sanniize>/g The XML file is now well formed. It is then possible to manipulate it with XML tools.

Verification of closed lists of values

The stage of verification of information taking their value in a closed list is important. Some errors come from bad handling in the previous steps, while others were present in the original file before conversion. For example, a dictionary uses parts-of-speech, a termbase uses a list of domains, etc. Make a copy of the file and keep only the values to check is a systematic approach for verification. In the example of Figure 4, the part-of-speech marked by "kanandi" can be extracted with the following expression: s/^*<kanandi>([^<]+)<\/kanandi>*$/$1/ The resulting list must then be sorted alphabetically. TextWrangler and Notepad + + plugin with its TextFX have the necessary commands. If the editor does not offer this option, OpenOffice Calc spreadsheet can be used. This approach is then used to quickly detect irregularities. If a value appears only once, it is very likely that this is a mistake. In the dictionary used in the examples, we corrected "alteeb" to "alteeb.", "Dah." to "dab.", "m/tsif." to "m / tsif.", etc.

Simple corrections

A CSS style sheet can be set to view the data directly in a browser. A compact display with a different style for each type of information helps to detect structuring errors in an entry. In the example of Figure 3, we see immediately that definition (in bold) and example (in italics) are lacking for the entry "abunaadam". With an XSL stylesheet, one can modify the data before display like adding a unique identifier for each entry, then, for each reference define a hypertext link to the corresponding entry. When the linguist browses the file, s/he can click on the hyperlinks to verify that the references are also entries of the dictionary like the entry "abunaadam" in Figure 3 with a reference to the entry "adamayse".

It is essential to scrutinize the data to detect some errors, even if they can be fixed automatically thereafter with regular expressions. The data visualization step is also very important from a pedagogical point of view. It allows to show the benefits of XML encoding the data, in particular that several forms (style) can be associated with the information (data). By learning the basics of CSS, the lexicographers can modify the style sheets themselves.

Structure of the entries

The entries can now be restructured. In files coming from word processors, the data structure is usually implied. We will have to add new structural elements to move towards a more standardized structure, allowing subsequent reuse. Concerning standards, LMF [START_REF] Romary | Standards going concrete: from LMF to Morphalou[END_REF]) became an ISO standard in November 2008 [START_REF] Francopoulo | Multilingual resources for nlp in the lexical markup framework (LMF)[END_REF]. It suits ideally our goals. As it is a meta-model and not a format, we can apply the principle of the LMF model to our entry structure and keep our tags without using the LMF syntax. The core meta-model LMF is shown in Figure 5. The object "Lexical Entry" contains a "Form" and one or more "Sense" objects.

Our lexical entries must now follow this meta-model. Figures 6 and7 show an example of an entry before and after the addition of structuring tags. The "article" tag corresponds to the "Lexical Entry" object; the "blocvedette" tag correspond to the "Form" object and the "bloc-semantique" tag is the "Sense" object.

A simple XSLT stylesheet is then provided for download with each dictionary. <LexicalEntry id="abiyanso"> <Lemma> <feat att="writtenForm" val="abiyanso"/> <feat att="phoneticForm" val="abiyansoo"/> </Lemma> <feat att="partOfSpeech" val="m."/> <Sense id="1"> <Equivalent> <feat att="language" val="fra"/> <feat att="writtenForm" val="aeroport"/> </Equivalent> <Definition> <feat att="writtenForm" val="batama kaŋ ra abiyey ga zumbu"/> </Definition> <Context> <TextRepresentation> <feat att="language" val="dje"/> <feat att="writtenForm" val="Tilbeeri nda Dooso sinda abiyanso kaŋ ra abiyo beeri ga zumbu."/> </TextRepresentation> </Context> </Sense> </LexicalEntry> It converts each dictionary into the LMF syntax (see Figure 8). For more detailed information about this part, refer to [START_REF] Enguehard | LMF for a selection of African Languages. Chapter 7[END_REF]. The next step planned is to convert the resources into the Lemon format 2 and integrate them into dbnary 3 [START_REF] Sérasset | Dbnary: Wiktionary as a Lemon Based RDF Multilingual Lexical Resource[END_REF], the team database for linked data.

Web access via the Jibiki platform 10.1. Presentation of the platform

Jibiki [START_REF] Mangeot | Construction collaborative de données lexicales multilingues, le projet Papillon. (Papillon, a project for collaborative building of multilingual lexical resources). special issue of the journal TAL (Electronic dictionaries: for humans, machines or both?[END_REF][START_REF] Mangeot | Dictionary building with the Jibiki platform: the GDEF case[END_REF][START_REF] Mangeot | Dictionary Building with the Jibiki Platform. Software Demonstration[END_REF] is a generic platform for handling online lexical resources with users and groups management. It was originally developed for the Papillon Project. The platform is programmed entirely in Java based on a the "Enhydra" environment. All data is stored in XML format in a Postgres database. This website mainly offers two services: a unified interface for simultaneous access to many heterogeneous resources (monolingual or bilingual dictionaries, multilingual databases, etc.) and a specific editing interface for contributing directly to the dictionaries available on the platform. Several lexical resources construction projects used or still use this platform successfully. This is the case for the GDEF project [START_REF] Chalvin | Méthodes et outils pour la lexicographie bilingue en ligne : le cas du Grand Dictionnaire Estonien-Français[END_REF] building an Estonian-French bilingual dictionary 4 , the LexALP project about multilingual terminology on the Alpine Convention or more recently MotÀMot project on southeast Asias' languages 5 . The source code for this platform is freely available for download from the forge of the LIG laboratory 6 . An instance of the platform has been adapted specifically to DiLAF project 1 because, in addition to dictionaries, specific project information must be accessible to visitors: -presentation of the project and partners; 2 http://lemon-model.net/ 3 http://dbnary.forge.imag.fr/ 4 http://estfra.ee 5 http://jibiki.univ-savoie.fr/motamot/ 6 http://jibiki.ligforge.imag.fr -general methodology form converting published dictionaries to LMF format; -stylesheets for different tools or tasks to be performed: tutorial on regular expressions, methodology of converting a document that uses fonts not conform to the Unicode standard to a document conforming to the Unicode standard, list of software used (exclusively opensource), methodology to monitor the project; -presentation of each dictionary: original authors, principles that governed the construction of the dictionary, language, alphabet, structure of the lexical entries, etc.

-dictionaries in LMF format.

is also envisaged to localize the platform for each language of the project.

Lookup interfaces

Three different interfaces are available to the user: -the generic lookup allows the user to lookup a word or a prefix of a word in all the dictionaries available on the platform. The language of the word must be specified.

-the volume lookup allows the user to lookup a word or prefix on a specific volume. On the left part of the result window, the volume headwords are displayed, sorted in alphabetical order. An infinite scroll allows the user to browse the entire volume. On the right part of the window, the entries previously selected on the left part are displayed.

-the advanced lookup is available for complex multicriteria queries. For example, it is possible to lookup an entry with a specific part-of-speech, and created by a specific author. On the left part of the result window, the headwords of the matching entries are displayed, sorted in alphabetical order. An infinite scroll allows the user to browse all the matching entries. On the right part, the entries previously selected on the left part are displayed.

Editing process

The editor [START_REF] Mangeot | Online Generic Editing of Heterogeneous Dictionary Entries in Papillon Project[END_REF]) is based on an HTML interface model instantiated with the lexical entry to be published. The model is generated automatically from an XML schema describing the entry structure. It can then be modified to improve the rendering on the screen. Therefore, it is possible to edit any type of dictionary entry provided that it is encoded in XML. The editing process can be adapted for specific needs through levels and status. A quality level (eg: from 1 to 5 stars, an entry with 1 star is a draft and one with 5 stars is certified by a linguist) can be assigned to each contribution. Similarly, a competence level can be assigned to each contributor (1 star is a beginner and 5 stars is a certified linguist). Then, when a 3 stars level user edits a 2 stars entry, the entry level raises to 3 stars.

can also be assigned to entries and roles to users. For example, in order to produce a high quality dictionary, an entry must follow 3 steps: creation by a registered user, revision by a reviewer and validation by a validator.

<xsl:template match="article"> <LexicalEntry id="{sanniize}{sanniize/@lamba}"> <xsl:apply-templates /> </LexicalEntry> </xsl:template> <xsl:template match="bloc-vedette"> <Lemma> <xsl:apply-templates /> </Lemma> </xsl:template> <xsl:template match="sanniize"> <feat att="writtenForm" val="{.}"/> </xsl:template> <xsl:template match="ciiya"> <feat att="phoneticForm" val="{.}"/> </xsl:template> Once dictionaries are uploaded into the Jibiki server, they can be accessed via a REST API. Lookup commands are available for querying indexed information: headword, pronunciation, part-of-speech, domain, example, idiom, translation, etc. The API can also be used for editing entries. The user must be previously registered in the website.

Conclusion

We presented a methodology for dictionaries conversion from word processing files to XML format. The DiLAF project does not stop in so good way. Before distributing dictionaries, there are still manual correction steps and possibly data addition. For example, examples of the Zarma-French dictionary will be translated into French.

Once dictionaries are converted, we can then extend their coverage through a system of contribution / editing / validation that can be done online live on the Jibiki platform. The low Internet access in Africa will require us to develop alternative methods. We can then use the data as raw material to increase the computerization of these languages: morphological analysers, spell-checkers, machine translation systems, etc.

Figure 1 :

 1 Figure 1: Excerpt of the Zarma-French dictionary in original format.

Figure 4 :

 4 Figure 2: Part of an entry (prononciation) in XML ODF format

Figure 3 :

 3 Figure 3: Compact view in a browser

Figure 5 :

 5 Figure 5: LMF kernel meta-model

Figure 8 :

 8 Figure 8: Zarma entry in LMF syntax

Figure 9 :

 9 Figure 9: excerpt of the Zarma XSL stylesheet for producing LMF syntax

•

  Bambara-French dict. Charles Bailleul, 1996 edition; • Hausa-French dict. for basic cycle, 2008 Soutéba; • Kanuri-French dict. for basic cycle, 2004 Soutéba; • Tamajaq-French dict. for basic cycle, 2007 Soutéba; • Zarma-French dict. for basic cycle, 2007 Soutéba.

1 http://dilaf.org/

Kǝska tǝngǝr̵ i, kalu ngǝwua dawulan tada cakkidǝ. Kǝryende kannua

	Additional information may appear as
	variants.
	Example:
	abǝɍwa [àbǝ ɍwà] cu. nangaro,
	abǝr ̵ wa cakkiwawo. [Fa.: ananas]

Table 1 :

 1 Partial view of the Unicode correspondence table for Zarma.

	§	ã
	é	e
	$	ɲ
	ù	ŋ
	£	Ɲ

Table 2 :

 2 Hausa and Kanuri digraphs missing in Unicode.

			), 'dz' (U+01F3) are
	used in Slovak; 'NJ' (U+01CA), 'Nj' (U+01BC), 'nj'
	(U+01CC) in Croatian and for transcribing the letter " Њ "
	of the Serbian Cyrillic alphabet, etc.
	It would be necessary to complete the Unicode standard
	with digraphs of Hausa and Kanuri alphabets in their
	various letter cases.
	fy	Fy	FY
	gw	Gw	GW
	gy	Gy	GY
	ky	Ky	KY
	kw	Kw	KW
	ƙy	Ƙy	ƘY
	ƙw	Ƙw	ƘW
	sh	Sh	SH
	ts	Ts	TS

Acknowledgements

The DiLAF project is funded by the Fonds Francophone des Inforoutes of the International Organization of the Francophonie. We also thank all the linguists of the team without whom this project would not have been possible: Sumana Kane, Issouf Modi, Michel, Radji, Rakia, Mamadou Lamine Sanogo.