Léo Agélas
email: leo.agelas@ifpen.fr

Beyond the BKM's criterion for the 2D resistive magnetohydrodynamic equations

Keywords: MHD; Navier-Stokes, Euler, BKM's criterion

The question of whether the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only magnetic diffusion can develop a finite time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion less restrictive than the Beale-Kato-Madja (BKM) regularity criterion type, namely any so-

dt < +∞. This regularity criterion may stand as a great improvement over the usual BKM regularity criterion which states that if

 for their L ∞ ill-posedness.

Introduction

Magnetohydrodynamics equations (MHD) describes the evolution of an electrically conducting fluid. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes. The field of MHD was initiated by Hannes Alfvén [START_REF] Alfvén | Existence of electromagnetic-hydrodynamic waves[END_REF], for which he received the Nobel Prize in Physics in 1970. It addresses laboratory as well as astrophysical plasmas and therefore is extensively used in very different contexts. In astrophysics, its applications range from solar wind [START_REF] Marsch | Non-Gaussian probability distributions of solar wind fluctuations[END_REF], to the Sun [START_REF] Priest | Solar Magnetohydrodynamics[END_REF][START_REF] Priest | Magnetic reconnection, MHD theory and Applications[END_REF], to the interstellar medium [START_REF] Ng | Anisotropic fluid turbulence in the interstellar medium and solar wind[END_REF] and beyond [START_REF] Zweibel | Magnetic Fields in Galaxies and Beyond[END_REF]. At the same time, MHD is also relevant to large-scale motion in nuclear fusion devices such as tokamaks [START_REF] Strauss | Nonlinear three dimensional dynamics of noncircular tokamaks[END_REF]. A tokamak is a toroidal device in which hydrogen isotopes in the form of a plasma reaching a temperature of the order of the hundred of millions of Kelvins is confined thanks to a very strong applied magnetic field. Tokamaks are used to study controlled fusion and are considered as one of the most promising concepts to produce fusion energy in the near future. However the main problem with this approach of confinement is that hydrodynamic instabilities arise. Numerical simulations using the MHD models are therefore of uttermost importance. Nevertheless, without the existence of a unique solution there is no hope of guaranteeing that a numerical approximation is really an approximation in any meaningful sense, since it is not clear what is being approximated. Due to their prominent roles in modeling many phenomena in astrophysics, geophysics and plasma physics, the MHD equations have been studied extensively mathematically. Furthermore, while the differences in behaviour between the two-dimensional (2D) and three-dimensional (3D) hydrodynamical turbulence of neutral fluids are accepted to be important, those of the MHD system in both cases are conventionally believed to be non-significant [START_REF] Biskamp | On two-dimensional magnetohydrodynamic turbulence[END_REF]. Strong statements were made by some authors that 2D simulations can be safely used to model 3D situations because the properties of the 2D and the 3D MHD turbulence are essentially the same [START_REF] Biskamp | Nonlinear Magnetohydrodynamics[END_REF][START_REF] Biskamp | On two-dimensional magnetohydrodynamic turbulence[END_REF]. Hence, the mathematical studies on the MHD equations in the two-dimensional case appear highly relevant. However up to now, the question of spontaneous apparition of singularity from a local classical solution of the partially viscous 2D MHD [START_REF] Alfvén | Existence of electromagnetic-hydrodynamic waves[END_REF] or 2D inviscid MHD ((2) without the Laplacian term) remains a challenging open problem in the mathematical fluid mechanics. Thus, in the absence of a well-posedness theory, the development of blowup/non-blowup theory is of major importance for both theoretical and practical purposes. Indeed, for a mathematical or numerical test of the actual finite time blow-up of a given solution, it is important to have a good blow-up criterion. Thus, there have been many computational attempts to find finite-time singularities of the 2D MHD equations (see [START_REF] Brachet | Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries[END_REF][START_REF] Kerr | Evidence for a Singularity in Ideal Magnetohydrodynamics: Implications for Fast Reconnection[END_REF][START_REF] Tran | Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number[END_REF]). Moreover, recent works on the 2D MHD equations developed regularity criteria in terms of the velocity field and deals with the MHD equations with dissipation and magnetic diffusion given by general Fourier multiplier operators such as the fractional Laplacian operators (see [START_REF] Wu | Generalized MHD equations[END_REF][START_REF] Wu | Regularity criteria for the generalized MHD equations[END_REF][START_REF] Wu | Global regularity for a class of generalized magnetohydrodynamic equations[END_REF]13,[START_REF] Tran | On global regularity of 2D generalized magnetodydrodynamics equations[END_REF][START_REF] Jiu | A remark on global regularity of 2D generalized magnetohydrodynamic equations[END_REF][START_REF] Cao | The 2D Incompressible Magnetohydrodynamics Equations with only Magnetic Diffusion[END_REF][START_REF] Yamazaki | Remarks on the global regularity of two-dimensional magnetohydrodynamics system with zero dissipation[END_REF][START_REF] Yamazaki | On the global regularity of two-dimensional generalized magnetohydrodynamics system[END_REF]). Among all the regularity criteria, one of particular interest is the Beale-Kato-Majda's criterion well known for Euler equations, extended in [START_REF] Caflisch | Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD[END_REF] to the inviscid MHD equations, under the assumption on both velocity field and magnetic field: [START_REF] Gala | An improved blow-up criterion for smooth solutions of the two-dimensional MHD equations[END_REF][START_REF] Tran | On global regularity of 2D generalized magnetodydrodynamics equations[END_REF][START_REF] Jiu | A remark on global regularity of 2D generalized magnetohydrodynamic equations[END_REF][START_REF] Yamazaki | Remarks on the global regularity of two-dimensional magnetohydrodynamics system with zero dissipation[END_REF][START_REF] Jiu | Global regularity of 2D generalized MHD equations with magnetic diffusion[END_REF][START_REF] Agélas | Global regularity for logarithmically critical 2D MHD equations with zero viscosity[END_REF][START_REF] Yamazaki | On the global regularity of two-dimensional generalized magnetohydrodynamics system[END_REF][START_REF] Ye | Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system[END_REF][START_REF] Fan | Global Cauchy problem of 2D generalized MHD equations[END_REF]), the global regularity issue of 2D MHD equations [START_REF] Alfvén | Existence of electromagnetic-hydrodynamic waves[END_REF] remains a challenging open problem up to date. The main reason for the unavailability of a proof of global regularity for the system of equations ( 2) is due to the quadratic coupling between u and b which invalidates the vorticity conservation. Indeed, the structure of the vorticity is instantaneously altered due to the effects of the magnetic fields. This fact is the source of the main difficulty connected to the global existence of classical solutions, where no strong global a priori estimates are known till now. This difficulty is revealed through the equations of the 2D inviscid MHD governing the vorticity ω

T 0 ( ω(t) L ∞ + j(t) L ∞ ) dt < ∞,
= ∂ 1 u 2 -∂ 2 u 1 and the current density j = ∂ 1 b 2 -∂ 2 b 1 , ∂ t ω + u • ∇ω = b • ∇j ∂ t j + u • ∇j = b • ∇ω + T (∇u, ∇b), (1)
where,

T (∇u, ∇b) = 2∂ 1 b 1 (∂ 2 u 1 + ∂ 1 u 2 ) + 2∂ 2 u 2 (∂ 2 b 1 + ∂ 1 b 2 ).
We observe that the magnetic field contributes in the last nonlinear part of the second equation with the quadratic term T (∇u, ∇b). By virtue of this difficulty, no a priori uniform bound for ω L ∞ (R 2 ×[0,T ]) is known for the 2D MHD equations with only magnetic diffusion [START_REF] Alfvén | Existence of electromagnetic-hydrodynamic waves[END_REF]. Further in [START_REF] Fan | Global Cauchy problem of 2D generalized MHD equations[END_REF][START_REF] Jiu | Global regularity of 2D generalized MHD equations with magnetic diffusion[END_REF][START_REF] Agélas | Global regularity for logarithmically critical 2D MHD equations with zero viscosity[END_REF], by considering Fourier multiplier operators magnetic diffusion slightly stronger than the Laplacian magnetic diffusion, the authors were able to obtain an uniform bound of ∇j L 1 ([0,T ];L ∞ (R 2 )) and then from the first equation of (1) obtain an uniform bound of ω L ∞ (R 2 ×[0,T ]) deriving from estimates for transport equations (see for instance Lemma 4.1 in [START_REF] Kato | Commutator Estimates and the Euler and Navier-Stokes Equations[END_REF]). However, the approach used in [START_REF] Fan | Global Cauchy problem of 2D generalized MHD equations[END_REF][START_REF] Jiu | Global regularity of 2D generalized MHD equations with magnetic diffusion[END_REF][START_REF] Agélas | Global regularity for logarithmically critical 2D MHD equations with zero viscosity[END_REF] and based on the properties of heat equation by using singular integral representations of equations (2) fails in the case where we have only a laplacian magnetic diffusion. Then, in this paper, we consider the initial-value problem for the 2D incompressible magneto-hydrodynamics equations with Laplacian magnetic diffusion,

   ∂ t u + (u • ∇)u = -∇p + (b • ∇)b ∂ t b + (u • ∇)b -∆b = (b • ∇)u ∇ • u = 0, ∇ • b = 0 (2) with initial conditions u(x, 0) = u 0 (x) for a.e x ∈ R 2 b(x, 0) = b 0 (x) for a.e x ∈ R 2 (3)
which models many significant phenomena such as the magnetic reconnection in astrophysics and geomagnetic dynamo in geophysics (see [START_REF] Priest | Magnetic reconnection, MHD theory and Applications[END_REF]). The problem of global well-posedness of the 2D MHD equations with partial dissipation and magnetic diffusion has generated considerable interest recently [START_REF] Cao | Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion[END_REF][START_REF] Chae | Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity[END_REF][START_REF] Jiu | Mathematical results related to a two-dimensional magneto-hydrodynamic equations[END_REF][START_REF] Lei | BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity[END_REF][START_REF] Zhou | A regularity criterion for the 2D MHD system with zero magnetic diffusivity[END_REF][START_REF] Cao | Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion[END_REF][START_REF] Jiu | Global regularity of 2D generalized MHD equations with magnetic diffusion[END_REF]. However up-today, the problem of uniqueness and global regularity of 2D MHD system (2) remains widely open.

Let us take a new look at the main obstruction. We start by noting that we can rewrite the equation [START_REF] Fan | Global Cauchy problem of 2D generalized MHD equations[END_REF] satisfied by ω the vorticity of u as follows:

∂ t ω + (u • ∇)ω = F -b 1 b • ∇u 2 + b 2 b • ∇u 1 , (4)
where

F = b 1 (∆b 2 + b • ∇u 2 ) -b 2 (∆b 1 + b • ∇u 1 )
. Furthermore, it was shown recently in [START_REF] Yuan | Global Regularity of 2D almost resistive MHD Equations[END_REF] an uniform bound of ∆b+(b•∇)u L ∞ (R 2 ×[0,T ]) (in section 4 we give a sketch of the proof). Moreover, we get an uniform bound of b L ∞ (R 2 ×[0,T ]) deriving from some estimates for linear Stokes system (see [START_REF] Giga | Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains[END_REF]), hence we deduce an uniform bound for

F L ∞ (R 2 ×[0,T ])
. Then, we notice that our equation ( 4) fits with the study made in [START_REF] Elgindi | L ∞ Ill-posedness for a class of equations arising in hydrodynamics[END_REF] about L ∞ ill-posedness for a class of equations arising in hydrodynamics. Thus, by virtue of ∇u = R(ω Id), where R is the Riesz transform on 2 × 2 matrix-valued functions (see ( 19)), we understand that the main obstruction comes from the fact that Riesz transforms does not maps L ∞ into itself.

Let us specify the way in which the obstruction is characterized. By using the following logarithmic Sobolev inequality proved in [START_REF] Kozono | Bilinear estimates and critical Sobolev inequality in BMO, with applications to the Navier-Stokes and the Euler equations[END_REF]:

∇f L ∞ (R 2 ) 1 + ∇ × f L ∞ (R 2 ) 1 + log + f W s,p (R 2 ) with p > 1, s > 1 + 2 p , (5) where
∇ • f = 0, ∇ × f = -∂ 2 f 1 + ∂ 1 f 2 is
the vorticity of f and log + x = max(0, log x) for any x > 0, we infer that for all t ∈ [0, T [,

∇u(t) ∞ r 1 + ω(t) ∞ 1 + log + u(t) H r (6)
and also

(∇u, ∇b)(t) ∞ r 1 + (ω, j)(t) ∞ 1 + log + (u, b)(t) H r . (7)
Then thanks to (6) and by using estimates for transport equations (see for instance Lemma 4.1 in [START_REF] Kato | Commutator Estimates and the Euler and Navier-Stokes Equations[END_REF]), from (4) we infer that for all t ∈ [0, T [

ω(t) ∞ r ω 0 ∞ + t 0 ( F (s) ∞ + b(s) 2 ∞ ) ds + t 0 b(s) 2 ∞ ω(s) ∞ 1 + log + u(s) H r ds, (8)
where r > 2. As a consequence of Gronwall Lemma, we deduce

ω(t) ∞ ≤ c r ω 0 ∞ + t 0 ( F (s) ∞ + b(s) 2 ∞ ) ds e cr R t 0 b(s) 2 ∞ (1+log + u(s) H r )ds , (9)
where c r > 0 is a real depending only on r. Thus, the main obstruction to get global regularity comes from the term in logarithm which appears in [START_REF] Caflisch | Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD[END_REF], namely log + u(s) H r . Nevertheless, thanks to (6), [START_REF] Chemin | Perfect Incompressible Fluids[END_REF] and the following estimate in the Hilbert space H r

(u, b)(t) H r ≤ (u 0 , b 0 ) H r e κr R t 0 (∇u,∇b)(τ ) ∞dτ , (10)
T 0 ∇ × u(t) ∞ dt < +∞ then the solution (u, b) of the 2D MHD equations (2)
remains smooth up to time T . Indeed, in virtue of ∇u = R((∇ × u) Id) with R Riesz transform on matrix-valued functions, we get

∇u BMO(R 2 ) ∇ × u BMO(R 2 ) ,
and for any 1

< q < ∞ ∇u L q (R 2 ) q ∇ × u L q (R 2 ) .
Then, we expect that the blow-up rate at a time T of ∇u(t) ∞ behaves like the one of ∇ × u(t) ∞ (log(e + ∇ × u(t) ∞ )) γ for a given γ ≥ 0. Thus, in this case, due to the exponent 1 2 in our regularity criterion, we get a great improvement over the usual BKM regularity criterion.

Then, the paper is organized as follows:

• In section 1, we give some notations and introduce the functional spaces.

• In section 2, we deal with the local well-posedness of the Cauchy problem of the partially viscous magneto-hydrodynamic system (2).

• In section 3, we give two energy estimates and some estimates from the properties of heat equation by using singular integral representations of equations.

• In section 4, we recall and give a sketch of the proof of new estimates obtained in [START_REF] Yuan | Global Regularity of 2D almost resistive MHD Equations[END_REF] related to the term ∆b + (b • ∇)u.

• In section 5, we give a new estimate for ∇u(t) ∞ in Lemma 5.1 and from this estimate, we obtained a new regularity criterion in Theorem 5.1 less restrictive than the BKM regularity criterion.

Some notations

We use X Y to denote the estimate X ≤ CY for an absolute constant C. If we need C to depend on a parameter, we shall indicate this by subscripts, thus for instance X s Y denotes the estimate X ≤ C s Y for some C s depending on s. For any f ∈ L p (R 2 ), with 1 ≤ p ≤ ∞, we denote by f p and f L p , the L p -norm of f . We denote by BMO(R 2 ) the space of functions of bounded mean oscillation equipped with the norm f BMO def = sup

x∈R 2 ,r>0 1 |B x,r | Bx,r
|f (y)f Bx,r |dy, where B x,r is the ball of radius r centered at x, |B x,r | its measure and f Bx,r = 1 |Bx,r| Bx,r f (y) dy. We denote by Id the 2 × 2 identity matrix. Given an absolutely integrable function f ∈ L 1 (R 2 ), we define the Fourier transform f : R 2 -→ C by the formula,

f (ξ) = R 2 e -2πix•ξ f (x) dx,
and extend it to tempered distributions. We will use also the notation F(f ) for the Fourier transform of f . We define also the inverse Fourier transform f : R 2 -→ C by the formula,

f (x) = R 2 e 2πix•ξ f (ξ) dξ. For s ∈ R, we define the Sobolev norm f H s (R 2 ) of a tempered distribution f : R 2 -→ R by, f H s (R 2 ) = R 2 (1 + |ξ| 2 ) s | f (ξ)| 2 dξ 1 2
, and then we denote by H s (R 2 ) the space of tempered distributions with finite H s (R 2 ) norm, which matches when s is a non negative integer with the classical Sobolev space H k (R 2 ), k ∈ N. The Sobolev space H s (R 2 ) can be written as

H s (R 2 ) = J -s L 2 (R 2 ) where J = (1 -∆) 1 2 .
For s > -1, we also define the homogeneous Sobolev norm,

f Ḣs (R 2 ) = R 2 |ξ| 2s | f (ξ)| 2 dξ 1 2 , (11)
and then we denote by Ḣs (R 2 ) the space of tempered distributions with finite Ḣs (R 2 ) norm. We use the Fourier transform to define the fractional Laplacian operator (-∆) α , -1 < α ≤ 1. We define it as follows,

(-∆) α f (ξ) = |ξ| 2α f (ξ).
We denote by

H s σ (R 2 ) the Sobolev space H s σ (R 2 ) def = {ψ ∈ H s (R 2 ) 2 : divψ = 0}
. We denote by P the projector onto divergence free vector fields given by P = Id -∇∆ -1 div. The operator P, which acts on vector-valued functions, is a projection : P 2 = P, annihilates gradients and maps into solenoidal (divergencefree) vectors; it is a bounded operator from (vector-valued) L q to itself for all 1 < q < ∞ and commutes with translation. We can notice that the operator P can be written under the form,

P = Id -∇∆ -1 div, (12)
which yields to Helmholtz decomposition, indeed for all v ∈ L q (R 2 ) 2 , 1 < q < ∞,

v = Pv + ∇ψ, with div Pv = 0, ψ = ∆ -1 div v. (13)

Local regularity of solutions of the 2D MHD equation

This section is devoted to the local well-posedness of the 2D MHD equations. By using P the matrix Leray operator, the first equation of ( 2) can be re-written as follows, ∂u ∂t

+ P((u • ∇)u -(b • ∇)b) = 0. ( 14
)
For (u, b) solution of (2), let us introduce

ω = ∇ × u = -∂ 2 u 1 + ∂ 1 u 2 the vorticity and j = ∇ × b = -∂ 2 b 1 + ∂ 1 b 2 the current density. Applying ∇× to 50
the equations of (2), we obtain the governing equations for ω and j

∂ t ω + (u • ∇)ω = (b • ∇)j ∂ t j + (u • ∇)j -∆j = (b • ∇)ω + T (∇u, ∇b). ( 15
)
where,

T (∇u, ∇b) = 2∂ 1 b 1 (∂ 2 u 1 + ∂ 1 u 2 ) + 2∂ 2 u 2 (∂ 2 b 1 + ∂ 1 b 2 ).
In this section we assume that the initial data (u 0 , b 0 ) ∈ H r σ (R 2 ) with r > 2. Then, we introduce ω 0 = ∇ × u 0 the vorticity of u 0 and j 0 = ∇ × b the current density of b 0 .

We assume that (u 0 , b 0 ) ∈ H r σ (R 2 ) with r > 2, thanks to Theorem 5.1 in [START_REF] Caflisch | Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD[END_REF] valid for all integer r ≥ 3 and by using the same arguments as in Proposition 4.3 of [START_REF] Agélas | Global regularity for logarithmically critical 2D MHD equations with zero viscosity[END_REF] valid for all real r > 2, we get that there exists a time of existence T > 0 such that there exists an unique strong solution (u, b) ∈ C([0, T [, H r σ (R 2 )) to the 2D MHD equations ( 2)-(3). Thanks to the Beale-Kato-Madja (BKM) criterion obtained in [START_REF] Caflisch | Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD[END_REF] for any integer r ≥ 3 and extended in Proposition 4.2 of [START_REF] Agélas | Global regularity for logarithmically critical 2D MHD equations with zero viscosity[END_REF] for any real r > 2, we get that if

(u, b) ∈ C([0, T ], H r σ (R 2 )), then we have T 0 (ω, j)(t) L ∞ dt = +∞. ( 16
)
From the first equation of (2), we can retrieve the pressure p from (u, b) with the formula,

p = -∆ -1 div((u • ∇)u -(b • ∇)b). (17)
Since ∇ • u = 0 and ∇ • b = 0, then we get (u

• ∇)u = ∇ • (u ⊗ u) and (b • ∇)b = ∇ • (b ⊗ b).
Then from [START_REF] Giga | Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains[END_REF], we get,

p = -∆ -1 div ∇ • (u ⊗ u -b ⊗ b). (18)
By introducing R

def = ∆ -1 div ∇• (19)
the Riesz transform on 2 × 2 matrix-valued functions on R 2 , we get

p = -R(u ⊗ u -b ⊗ b). (20)
Owing to (u, b) ∈ C([0, T [, H r (R 2 )), thanks to the L 2 -boundedness of the Riesz transforms and Lemma X4 in [START_REF] Kato | Commutator Estimates and the Euler and Navier-Stokes Equations[END_REF], from [START_REF] Jiu | Global regularity of 2D generalized MHD equations with magnetic diffusion[END_REF] we infer that p ∈ C([0, T [, H r (R 2 )).

Similarly as Proposition 4.1 in [START_REF] Agélas | Global regularity for logarithmically critical 2D MHD equations with zero viscosity[END_REF], we get the following local estimates in the higher Sobolev norm H r : there exists a real κ r > 0 depending only on r such that for all t ∈ [0, T [

(u, b)(t) H r ≤ (u 0 , b 0 ) H r e κr R t 0 (∇u,∇b)(τ ) ∞ dτ . (21)

Some estimates

In this section, we give some estimates related to the solutions of the 2D MHD equations 2.

Energy estimates

In this subsection, we recall some energies estimates. We give here the two following energy estimates given in [START_REF] Tran | On global regularity of 2D generalized magnetodydrodynamics equations[END_REF][START_REF] Lei | BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity[END_REF][START_REF] Agélas | Global regularity for logarithmically critical 2D MHD equations with zero viscosity[END_REF]:

for all t ∈ [0, T * [ u(t) 2 2 + b(t) 2 2 + 2 t 0 ∇b(τ ) 2 2 dτ = u 0 2 2 + b 0 2 2 (22)
and we get also that for all t ∈ [0, T * [

ω(t) 2 2 + j(t) 2 2 + t 0 ∇j(τ ) 2 2 dτ ≤ ( ω 0 2 2 + j 0 2
2 )e C( u0

2 2 + b0 2 2 ) , (23)
where C > 0 is an absolute constant.

Some estimates deriving from heat equation

In this subsection, in the Lemma just below, we give the details (often omitted) of the proof of some estimates deriving from the properties of the heat kernel.

Lemma 3.1. Let (u 0 , b 0 ) ∈ H r (R 2 ) with r > 2 and T > 0 be such that there

exists (u, b) ∈ C([0, T [, H r σ (R 2 )) solution of the 2D MHD equations (2)-(3). Then there exists a real C 1 > 0 depending only on (u 0 , b 0 ) H r , r and T such that b L ∞ (R 2 ×[0,T ]) ≤ C 1 .
For any real p > 1 and q > 2, we have also three real C 2 > 0, C 3 > 0 and C 4 > 0 depending only on (u 0 , b 0 ) H r , p, q, r and T such that,

∇b L ∞ ([0,T ]×L q (R 2 )) ≤ C 2 , ω L ∞ ([0,T ]×L q (R 2 )) ≤ C 3 , ∇ 2 b L p ([0,T ]×L q (R 2 )) ≤ C 4 .
Proof. For this, we write the second equation of 2 under its integral form, then we have for all t ∈ [0, T [

b(t) = e t∆ b 0 + t 0 e (t-s)∆ ((b • ∇)u(s) -(u • ∇)b(s)) ds. (24)
Then by using inequality (2.3) in [START_REF] Kato | Strong L q solutions of the Navier-Stokes equations in R m , with application to weak solutions[END_REF], we get

e (t-s)∆ ((b • ∇)u(s) -(u • ∇)b(s)) ∞ (t -s) -2 3 (b • ∇)u(s) -(u • ∇)b(s) 3 2 (t -s) -2 3 ( b(s) 6 ∇u(s) 2 + u(s) 6 ∇b(s) 2 ).
As a consequence, from (24) we get,

b(t) ∞ b 0 ∞ + t 0 (t -s) -2 3 ( b(s) 6 ∇u(s) 2 + u(s) 6 ∇b(s) 2 ) ds. ( 25
) Since b(s) 6 b(s) H 1 , u(s) 6 u(s) H 1 and ∇b(s) 2 j(s) 2 , ∇u(s) 2 ω(s) 2 (see Theorem 3.1.1 in [8]
), then thanks to ( 22) and ( 23), from [START_REF] Lei | BKM's Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity[END_REF] we deduce that there exists a real C 0 > 0 depending only on (u 0 , b 0 ) 2 , (ω 0 , j 0 ) 2 such that for all

t ∈ [0, T [, b(t) ∞ b 0 ∞ + C 0 (T ) 1 3 . (26)
Owing to [START_REF] Marsch | Non-Gaussian probability distributions of solar wind fluctuations[END_REF] and thanks to the Sobolev embedding H r (R 2 ) → L ∞ (R 2 ) since r > 2, we deduce that there exists a real C 1 > 0 depending only on (u 0 , b 0 ) H r , r and T such that for all t ∈ [0,

T [ b(t) ∞ ≤ C 1 , (27)
which concludes the first part of the proof. By virtue of (24), we get that for all t ∈ [0, T [

∇b(t) = e t∆ ∇b 0 + t 0 ∇e (t-s)∆ ((b • ∇)u(s) -(u • ∇)b(s)) ds. (28)
Then by using inequality (2.3') in [START_REF] Kato | Strong L q solutions of the Navier-Stokes equations in R m , with application to weak solutions[END_REF], from (28) we deduce,

∇b(t) q q ∇b 0 q + t 0 (t -s) -(1-1 q ) (b • ∇)u(s) -(u • ∇)b(s) 2 ds. ( 29
)
Thanks again to [START_REF] Kato | Strong L q solutions of the Navier-Stokes equations in R m , with application to weak solutions[END_REF] and [START_REF] Kerr | Evidence for a Singularity in Ideal Magnetohydrodynamics: Implications for Fast Reconnection[END_REF], from [START_REF] Priest | Solar Magnetohydrodynamics[END_REF] we get that for all t ∈ [0, T [, ∇b(t) q q ∇b 0 q + (u 0 , b 0 ) 2 (ω 0 , j 0 ) 2 e c (u0,b0) 2 2 qT

1 q . ( 30
)
Thanks to Gagliardo-Nirenberg inequality, for any q > 2 we have the following Sobolev embedding H r (R 2 ) → Ẇ 1,q (R 2 ) since r > 2, then owing to [START_REF] Priest | Magnetic reconnection, MHD theory and Applications[END_REF] we deduce that there exists a real C 2 > 0 depending only on (u 0 , b 0 ) H r , T , r and q such that for all t ∈ [0, T [,

∇b(t) q ≤ C 2 , (31)
which concludes the second part of the proof.

To get an estimate of ω L ∞ ([0,T ];L q ) , we borrow some arguments used in [START_REF] Jiu | Global regularity of 2D generalized MHD equations with magnetic diffusion[END_REF]. Thanks to the L p -L q maximal regularity of the Laplacian operator (see e.g [START_REF] Giga | Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains[END_REF]), from the second equation of 2, we get that for all t ∈ [0, T [, p > 1 and

q > 2 t 0 ∇ 2 b(s) p q p,q t 0 (b • ∇)u(s) -(u • ∇)b(s) p q ds p,q t 0 ( b(s) p ∞ ω(s) p q + u(s) p ∞ ∇b(s) p q ) ds, (32)
where we have used the fact that ∇u(s) q q ω(s) q (see Theorem 3.1.1 in [START_REF] Chemin | Perfect Incompressible Fluids[END_REF]). Then, we multiply the first equation of ( 15) by ω|ω| q-2 , integrate it over R 2 and use the fact that ∇ • u = 0 to obtain

75 1 q d dt ω(t) q q = R 2 b(x, t) • ∇j(x, t)ω(x, t)|ω(x, t)| q-2 dx ≤ b(t) ∞ ∇j(t) q ω(t) q-1 q , which yields to: for all t ∈ [0, T [, 1 2
d dt ω(t) 2 q ≤ b(t) ∞ ∇j(t) q ω(t) q .
After an integration over [0, t] of the inequality just above, we obtain

ω(t) 2 q ≤ ω 0 2 q + 2 t 0 b(s) ∞ ∇j(s) q ω(s) q ds ≤ ω 0 2 q + t 0 ( ∇j(s) 2 q + b(s) 2 ∞ ω(s) 2 q ) ds (33)
Then thanks to [START_REF] Tran | On global regularity of 2D generalized magnetodydrodynamics equations[END_REF], from [START_REF] Tran | Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number[END_REF] we infer that for all t ∈ [0, T [

ω(t) 2 q ω 0 2 q + t 0 ( u(s) 2 ∞ ∇b(s) 2 q + b(s) 2 ∞ ω(s) 2 q ) ds. ( 34
)
By using Gagliardo-Nirenberg inequalities, Young inequalities and the fact that ∇u(s) q q ω(s) q , we get

u(s) ∞ q u(s) 2 + ω(s) q . ( 35
)
By virtue of ( 34) and ( 35), we get that for all t ∈ [0, T [,

ω(t) 2 q q ω 0 2 q + t 0 ( u(s) 2 2 ∇b(s) 2 q + ( ∇b(s) 2 q + b(s) 2 ∞ ) ω(s) 2 q ) ds. ( 36
)
Thanks to ( 22), ( 27) and ( 31), we deduce that there exists a real C > 0 depending only on (u 0 , b 0 ) H r , T , r and q such that for all t ∈ [0, T [

ω(t) 2 q ≤ C + C t 0 ω(s) 2 q ds. ( 37
)
Thanks to Gronwall inequality, we infer that for all t ∈ [0, T [,

ω(t) 2 q ≤ Ce C T .
Therefore with

C 3 = √ Ce 1 2
C T , we obtain a real C 3 > 0 depending only on (u 0 , b 0 ) H r , T and q such that for all t ∈ [0, T [,

ω(t) q ≤ C 3 , (38)
which concludes the third part of the proof. By using ( 35) into ( 32) and thanks to ( 38), ( 22), ( 27) and ( 31), we completes the proof.

Some new estimates

In this section, we give a sketch of the proof of the Lemma 4.1 obtained in [START_REF] Yuan | Global Regularity of 2D almost resistive MHD Equations[END_REF] by exploiting the special structure of the 2D MHD equations [START_REF] Alfvén | Existence of electromagnetic-hydrodynamic waves[END_REF].

Lemma 4.1. Let (u 0 , b 0 ) ∈ H r (R 2 ) with r > 2 and T > 0 be such that there exists (u, b) ∈ C([0, T [, H r σ (R 2 )) solution of the 2D MHD equations (2)- (3)
. Then there exists a real C > 0 depending only on (u 0 , b 0 ) H r , r and

T such that ∆b + (b • ∇)u L ∞ ([0,T ];L ∞ (R 2 )) ≤ C, (39)
and we have also that for any real p ≥ 2 and q ≥ 2,

∇(∆b + (b • ∇)u) L p ([0,T ];L q (R 2 )) ≤ C. (40)
The proof of Lemma 4.1 given in [START_REF] Yuan | Global Regularity of 2D almost resistive MHD Equations[END_REF] is obtained by writing the equation satisfied by

F := ∆b + (b • ∇)u that is ∂ t F -∆F = -(b • ∇)P((u • ∇)u) + (b • ∇)P((b • ∇)b) -∆((u • ∇)b) -∇u (u • ∇)b + ∇u (b • ∇)u + ∇u ∆b. ( 41
)
This equation is obtained by applying (b • ∇) and ∆ respectively to the first equation of ( 14) and second equation of (2), and multiplying the second equation of ( 2) by ∇u and after adding the resulting equations together. Then, by writing equation ( 41) under its integral form and using the fact that ∇ • u = 0 and ∇ • b, we get for all t ∈ [0, T [

F(t) = e t∆ F(0) + t 0 ∇e (t-s)∆ (b(s) ⊗ P((u • ∇)u)(s) -b(s) ⊗ P((b • ∇)b)(s)) ds + t 0 ∇e (t-s)∆ ∇((u • ∇)b)(s) ds + t 0 e (t-s)∆ (-∇u(s) (u(s) • ∇)b(s) + ∇u(s) (b(s) • ∇)u(s) + ∇u(s) ∆b(s))ds. (42)
Then using inequalities (2.3) and (2.3') of [START_REF] Kato | Strong L q solutions of the Navier-Stokes equations in R m , with application to weak solutions[END_REF] stated for 1 < p ≤ q < +∞ but remaining true for q = ∞, we obtain for all t ∈ [0, T [

F(t) ∞ ≤ F(0) ∞ + t 0 (t -s) -5 6 (b(s) ⊗ P((u • ∇)u)(s) -b(s) ⊗ P((b • ∇)b)(s)) 3 ds + t 0 (t -s) -5 6 ∇((u • ∇)b)(s) 3 ds + t 0 (t -s) -1 2 -∇u(s) (u(s) • ∇)b(s) + ∇u(s) (b(s) • ∇)u(s) + ∇u(s) ∆b(s) 2 ds.
(43) By using the fact that P is a bounded operator from (vector-valued) L q to itself for all 1 < q < ∞ and Hölder inequality, we get

(b(s) ⊗ P((u • ∇)u)(s) -(b(s) ⊗ P((b • ∇)b)(s) 3 b(s) ∞ u(s) 6 ∇u(s) 6 + b(s) ∞ b(s) 6 ∇b(s) 6 , ∇((u • ∇)b)(s) 3 ∇u(s) 6 ∇b(s) 6 + u(s) 6 ∇ 2 b(s) 6 , -∇u(s) (u(s) • ∇)b(s) + ∇u(s) (b(s) • ∇)u(s) + ∇u(s) ∆b(s) 2 ∇u(s) 6 u(s) 6 ∇b(s) 6 + ∇u(s) 2 6 b(s) 6 + ∇u(s) 6 ∆b(s) 3 .
(44) Furthermore, thanks to Gagliardo-Nirenberg interpolation inequality and Theorem 3.1.1 in [START_REF] Chemin | Perfect Incompressible Fluids[END_REF], we get u(s) 6 u(s)

1 3 2 ω(s) 2 3
2 , b(s) 6 b(s)

1 3 2 j(s) 2 3
2 , ∇u(s) 6 ω(s) 6 .

(45) After plugging (45) into (44) and using Lemma 3.1 with the energy inequalities [START_REF] Kato | Strong L q solutions of the Navier-Stokes equations in R m , with application to weak solutions[END_REF], [START_REF] Kerr | Evidence for a Singularity in Ideal Magnetohydrodynamics: Implications for Fast Reconnection[END_REF], from (43) we infer that there exists a real C 0 > 0 depending only on (u 0 , b 0 ) H r , r and T such that for all t ∈ [0, T [

F(t) ∞ C 0 1 + t 0 (t -s) -5 6 (1 + ∇ 2 b(s) 6 ) + (t -s) -1 2 (1 + ∆b(s) 3 ) ds .
(46) Thanks to Hölder inequality used with the couples of exponents ( 76 , 7) and ( 3 2 , 3), from (46) we deduce that for all t ∈ [0, T [ , which yields to

F(t) ∞ C 0 + C 0 t 0 (t -s) -
F(t) ∞ C 0 1 + t 1 42 t 0
(1 + ∇ 2 b(s) 7 6 ) ds

1 7
+ t .

(47) Then, thanks again to Lemma 3.1, from (47) one obtains that there exists a real where β r > 0 is a real depending only on r. Let us give an estimate of the term 1 + log + u(t) H r . Thanks to [START_REF] Kato | Commutator Estimates and the Euler and Navier-Stokes Equations[END_REF], we get that there exists a real κ r > 0 depending only on r such that for all t ∈ [0, T [

(u, b)(t) H r ≤ (u 0 , b 0 ) H r e κr R t 0 (∇u,∇b)(τ ) ∞ dτ . ( 52
)
After taking the logarithm in the inequality (52), we observe that for all t ∈ [0, T [,

log + (u, b)(t) H r ≤ log + (u 0 , b 0 ) H r + κ r t 0 (∇u, ∇b)(τ ) ∞ dτ. (53)
Thanks to Lemma 3.1 and the Sobolev embedding

W 2,q (R 2 ) → W 1,∞ (R 2 )
with q > 2, we infer that there exists a real 0 > 0 depending only on r, T , (u 0 , b 0 ) H r such that

T 0 ∇b(σ) ∞ ≤ 0 . (54)
Then owing to (54), from (53) we infer that there exists a real 1 ≥ 1 depending only on r, T , (u 0 , b 0 ) H r such that for all t ∈ [0, T [,

1 + log + (u, b)(t) H r ≤ 1 + κ r t 0 ∇u(s) ∞ ds. (55)
Thus, by plugging (55) into (51), we deduce that there exists a real 2 > 0 depending only on (u 0 , b 0 ) H r , T and r such that for all t ∈ [0, T [,

∇u(t) ∞ ≤ β r + 2 ω(t) ∞ 1 + t 0 ∇u(s) ∞ ds . (56)
Now, let us estimate ω(t) ∞ . We observe that the first equation of ( 15) can be changed into

∂ t ω + u • ∇ω = F -b 1 b • ∇u 2 + b 2 b • ∇u 1 , (57)
where

F = b 1 (∆b 2 +b•∇u 2 )-b 2 (∆b 1 +b•∇u 1 )
. By using estimates for transport equations (see for instance Lemma 4.1 in [START_REF] Kato | Commutator Estimates and the Euler and Navier-Stokes Equations[END_REF]), we obtain that for all t ∈ [0, T [

ω(t) ∞ ≤ ω 0 ∞ + c t 0 F (s) ∞ ds + c t 0 b(s) 2 ∞ ∇u(s) ∞ ds, (58)
where c > 0 is a constant. Thanks to Lemmata 3.1 and 4.1, we deduce that there exist two real 3 > 0 and 4 > 0 depending only on (u 0 , b 0 ) H r , T and r such that for all t ∈ [0, T [,

c t 0 F (s) ∞ ds ≤ 3 b(t) 2 ∞ ≤ 4 . (59)
As a consequence of Gronwall Lemma, from (69) we get for all t ∈ [0, T [, log(e + J(t)) ≤ log(e + J(0)) exp

1 2 6 t 0 ∇u(s) 1 2
∞ log(e + ∇u(s)

1 2 ∞ ) ds . (70)
From (65), we get J(0) = 6 and thanks to (70), we thus obtain for all t ∈ [0, T [,

J(t) ≤ exp log(e + 1 2
6 ) exp

1 2 6 t 0 ∇u(s) 1 2
∞ log(e + ∇u(s)

1 2 ∞ ) ds . (71)
Owing to (66) and (71), we obtain that for all t ∈ [0, T [,

∇u(t) ∞ ≤ exp 2 log(e + 1 2
6 ) exp

1 2 6 t 0 ∇u(s) 1 2
∞ log(e + ∇u(s)

1 2 ∞ ) ds . (72)
Since e + ∇u(s)

1 2 ∞ ≥ (e + ∇u(s) ∞ ) 1
2 , then we get log(e + ∇u(s)

∞

log(e + ∇u(s) ∞ ) ds , which concludes the proof.

Remark 5.1. We observe that the expression of the estimate obtained in Lemma 5.1 for ∇u(t) ∞ makes appear a double exponential growth. This double exponential growth derives from the taking into account in the estimate of the term log(e + ∇u(t) ∞ ). We thus point out that we have also an upper bound of ∇u(t) ∞ for which we get only one single exponential growth. Indeed, from (64), thanks to Gronwall Lemma, we obtain that for all t ∈ [0, T [ Proof. Let us assume that we have

∇u(t) 1 2 ∞ ≤ 1 2 6 exp 1 2 6 t 0 ∇u(s) 1 2 ∞ ds , which yields to ∇u(t) ∞ ≤ 6 exp 2 1 2 6 t 0 ∇u(s)
T 0 ∇u(t) 1 2 ∞ log(e + ∇u(t) ∞ ) dt < +∞. ( 73
)
For a contradiction, we suppose that u ∈ C([0, T ], H r σ (R 2 ). Then we get [START_REF] Gala | An improved blow-up criterion for smooth solutions of the two-dimensional MHD equations[END_REF]. Thanks to Lemma 3.1 and the Sobolev embedding W 2,q (R 2 ) → W 1,∞ (R 2 ) with q > 2, we infer that T 0 j(t) ∞ dt < +∞. Then from ( 16), we get only

T 0 ω(t) ∞ dt = +∞. ( 74
)
Thanks to Lemma 5.1, there exists a real 1 > 0 depending only on (u 0 , b 0 ) H r , T and r such that for all t ∈ [0, T [,

∇u(t) ∞ ≤ exp 1 exp 1 t 0 ∇u(s) 1 2 ∞ log(e + ∇u(s) ∞ ) ds . (75)
Then by virtue of ( 75) and (73), we infer that

T 0 ∇u(t) ∞ dt < +∞ which 110 implies that T 0 ω(t) ∞ dt < +∞.
Then we obtain a contradiction with (74) and hence u ∈ C([0, T ], H r σ (R 2 ) which concludes the proof.

Conclusion

In this article, we obtained a new regularity criterion for the two-dimensional resistive magneto-hydrodynamic (MHD) equations less restrictive than the BKM's regularity criterion (see Theorem 5.1), by using the logarithmic Sobolev inequality. It is important to find some criteria less restrictive than the BKM's regularity criterion. Indeed, due to the quadratic nonlinearity of the MHD equations, we expect that the blow-up rate of ∇u(t) ∞ at a time T be at least faster than

O 1 T -t
. Thus, if one investigates numerically on the finite time singularities of the solutions of such system of equations and believe that its numerical solution computed leads to a finite-time blowup at some time T , then one may observe a blow-up rate at the time T for ∇u(t) of the form O 1 (T -t) γ , γ ≥ 1. Further, in all the recent numerical investigations performed to find finite time singularities of the 2D inviscid MHD equations, the results suggest blow-up rates at a time T for ∇u(t) ∞ of the form O 1 (T -t) α with 1 ≤ α < 2 (see [START_REF] Brachet | Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries[END_REF][START_REF] Kerr | Evidence for a Singularity in Ideal Magnetohydrodynamics: Implications for Fast Reconnection[END_REF]). Then, for these numerical cases, with the BKM's regularity criterion, one would conclude to an evidence for finite time singularity at some time T of the solutions of the 2D resistive MHD equations. However, with the use of our regularity criterion (see Theorem 5.1), we can confirm that in fact there is no blowup of the solution at this time T . Then, it is dangerous to interpret the blow-up of an under-resolved computation as evidence of finite-time singularities for the 2D resistive MHD equations. Indeed, computing 2D MHD singularities numerically is an extremely challenging task. First of all, it requires huge computational resources (see [START_REF] Brachet | Ideal evolution of magnetohydrodynamic turbulence when imposing Taylor-Green symmetries[END_REF]). Tremendous resolutions are required to capture the nearly singular behaviour of the 2D MHD equations. Secondly, one has to perform a careful convergence study. Furthermore, we notice also that our problem fits in the class of equations considered in [START_REF] Elgindi | L ∞ Ill-posedness for a class of equations arising in hydrodynamics[END_REF] in the study of L ∞ ill-posedness problem. We thus point out that by borrowing the same arguments used in this paper, we can establish the same regularity criterion to another interesting open problem in mathematical fluid dynamics mentioned in [START_REF] Elgindi | L ∞ Ill-posedness for a class of equations arising in hydrodynamics[END_REF], it is about the following type of equation in two-dimension:

∂ t u + (u • ∇)u + ∇p = Au, ∇ • u = 0, (76)
with initial conditions u 0 a free-divergence field vector and A is some constant matrix. Namely, as Theorem 5.1, we get the following Theorem for the system of equations (76):

  where the vorticity ω = ∇×u and the density j = ∇ × b. And so, the Beale-Kato-Majda's criterion ensures that the solution (u, b) of the inviscid MHD equations is smooth up to time T . Meanwhile the 2D Euler equation is globally well-posed for smooth initial data, however for the 2D inviscid MHD equations, the global wellposedness of classical solution is still a big open problem. Despite recent developments on regularity criteria (see

where 6 > 0 ∇u(t) 1 2 ∞

 602 0 is a real depending only on T , r and (u 0 , b 0 ) r . Let us establish now, a new regularity criterion in the Theorem just below. Theorem 5.1. Let (u 0 , b 0 ) ∈ H r σ (R 2 ) with r > 2 and T > 0 be such that there exists (u, b) ∈ C([0, T [, H r σ (R 2 )) solution of the 2D MHD equations (2)-(3). If T log(e + ∇u(t) ∞ ) dt < +∞ then there cannot be blowup of the solution u in H r (R 2 ) at the time T , in other words u ∈ C([0, T ], H r σ (R 2 )).

Theorem 5 . 2 . 0 ∇u(t) 1 2 ∞

 5202 Let u 0 ∈ H r σ (R 2 ) with r > 2 and T > 0 be such that there existsu ∈ C([0, T [, H r σ (R2)) solution of the equations (76).If T log(e + ∇u(t) ∞ ) dt <+∞ then there cannot be blowup of the solution u in H r (R 2 ) at the time T , in other words u ∈ C([0, T ], H r σ (R 2 )).

  we obtain a new estimate of ∇u(t) ∞ in Lemma 5.1 which leads to a new regularity criterion in Theorem 5.1. Our new regularity criterion states that

	if	T	∇u(t)	1 2
	0

∞

log(e + ∇u(t) ∞ ) dt < +∞ then the solution (u, b) of the 2D MHD equations (2) remains smooth up to time T . This new regularity criterion may appear less restrictive than the BKM regularity criterion which states that if

C 1 > 0 depending only on (u 0 , b 0 ) H r , r and T such that for all t ∈ [0, T [,

which gives us [START_REF] Ye | Global regularity of the two-dimensional incompressible generalized magnetohydrodynamics system[END_REF] the first inequality of Lemma 4.1. For the second inequality of Lemma 4.1, we use the L p -L q maximal regularity of the Laplacian operator [START_REF] Giga | Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains[END_REF], one has for any 1 < p < ∞, 1 < q < ∞ and

Then, with the expression of ∇F(t) obtained from [START_REF] Zweibel | Magnetic Fields in Galaxies and Beyond[END_REF] and by using (48), inequality (2.3') of [START_REF] Kato | Strong L q solutions of the Navier-Stokes equations in R m , with application to weak solutions[END_REF], Lemma 3.1 and the energy inequalities ( 22), [START_REF] Kerr | Evidence for a Singularity in Ideal Magnetohydrodynamics: Implications for Fast Reconnection[END_REF], we obtain in a similar way [START_REF] Yuan | Global Regularity of 2D almost resistive MHD Equations[END_REF] the second inequality of Lemma 4.1.

A new blowup criterion 90

In this section, we give a new estimate for ∇u(t) ∞ in Lemma 5.1 and from this estimate, we obtain a new regularity criterion in Theorem 5.1 less restrictive than the BKM regularity criterion.

)) solution of the 2D MHD equations (2)-(3). Then there exists a real γ 0 > 0 depending only on (u 0 , b 0 ) H r , T and r such that for all t ∈ [0, T [,

Proof. We begin the proof with the following logarithmic Sobolev inequality proved in [START_REF] Kozono | Bilinear estimates and critical Sobolev inequality in BMO, with applications to the Navier-Stokes and the Euler equations[END_REF] (see inequality (4.20)) and stands as an improved version of that in [START_REF] Beale | Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations[END_REF]:

the vorticity of f and log + x = max(0, log x) for any x > 0. Thus, by virtue of (50), we get that for all t ∈ [0, T [,

Thus by virtue of (59), from (58) we infer that for all t ∈ [0, T [

Furthermore, thanks to the Sobolev embedding

Hence, owing to (61), from (60) we deduce that there exists a real 5 > 0 depending only on (u 0 , b 0 ) H r , T and r such that for all t ∈ [0, T [,

By plugging (62) into (56), we infer that there exists a real 6 ≥ 1 depending only on r, T , (u 0 , b 0 ) H r such that for all t ∈ [0, T [,

which yields to

We thus introduce the real function J defined for all t ∈ [0, T [ by

On one hand, by virtue of (64), thanks to (65) we get that for all t ∈ [0, T [,

On the other hand, from (65), we infer that for any t ∈ [0, T [,

∞ log(e + ∇u(t)

∞ log(e + ∇u(t)

Then, owing to (66), from (67), we infer that for all t ∈ [0, T [,

6 ∇u(t)

∞

log(e + ∇u(t)

log(e + J(t)).

(69)


Jeanne Frere
email: jeanne.frere@univ-rennes1.fr

Alain Alcaras

Maxim Zhadobov

Christophe Lemoine

Gwenaël Le Cadre

Ronan Sauleau

Evaluation of Currents Induced in Human Body by Plane Wave Exposure At 1-90 MHz

Keywords: Human body equivalent impedance circuit, induced currents, HF, VHF

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Because of the increasing number of wireless systems, accurate dosimetric assessment of interactions between radiofrequency (RF) electromagnetic fields and human body is of importance. In the 1 -90 MHz range the limits established by ICNIRP [START_REF] Icnirp | Guidelines for limiting exposure to time-varying electric, magnetic and electromagnetic fields (up to 300 GHz)[END_REF] and IEEE [START_REF] Ieee | IEEE Standard for safety levels with respect to human exposure to radio frequency electromagnetic fields, 3 kHz to 300 GHz[END_REF] are defined in terms of external electric and magnetic fields, induced currents, current density and specific absorption rate (SAR). Tofani et al. reported that some limits are not fully consistent with each other, in particular in 90 -104 MHz range [START_REF] Tofani | Induced foot-currents in humans exposed to VHF radiofrequency EM fields[END_REF]. Although the limits on external electric field were not overpassed, the induced current exceeded the limits, which were the following: 45 mA for uncontrolled environment and 100 mA for controlled environment through one foot between 0.1 MHz and 100 MHz [START_REF] Ieee | IEEE Standard for safety levels with respect to human exposure to radio frequency electromagnetic fields, 3 kHz to 300 GHz[END_REF]. The same observation was made in a study by Gandhi et al. where the limits on currents induced by plane wave exposure up to 50 MHz were not overpassed although limits on SAR 1g were exceeded [START_REF] Gandhi | Currents induced in a human being for plane-wave exposure conditions 0-50 MHz and for RF sealers[END_REF].

Several studies have used an equivalent antenna to model human body. Gandhi and Aslan have developed a humanequivalent antenna to measure the external current induced in a human body by RF exposure [START_REF] Gandhi | Human equivalent antenna for electromagnetic fields[END_REF]. The antenna impedance was approximated by the human body impedance in 3 kHz -110 MHz range. Using this antenna allows induced current measurements without human subject. In 2003, Poljak et al. developed a human equivalent antenna model for transient electromagnetic exposure. They used thick cylindrical antennas to represent the human body and thin-wire approximation to evaluate the current induced in human body for a body exposed to a pulsed radiation between 50 Hz and 110 MHz [START_REF] Poljak | Human equivalent antenna model for transient electromagnetic radiation exposure[END_REF]. Recently Kibret et al. characterized the human body as a monopole antenna from 10 MHz to 110 MHz [START_REF] Kibret | Characterizing the human body as a monopole antenna[END_REF]. They gave physical dimensions of the equivalent cylindrical antenna and characterized it experimentally (in terms of radiation efficiency, reflection coefficient and impedance).

The main objective of this work is to analyze theoretically the relationship between induced currents and incident electric fields in the 1 -90 MHz frequency range. The actual limits of induced current for a controlled environment are used, i.e. 100 mA in one foot in 1 -90 MHz range [START_REF] Ieee | IEEE Standard for safety levels with respect to human exposure to radio frequency electromagnetic fields, 3 kHz to 300 GHz[END_REF]. The human body is modeled as a thick dipole antenna to calculate induced currents from the external applied electric field for plane wave exposure. The coupling between the thick dipole and the incident plane wave is modeled using an equivalent Thevenin circuit consisted of a voltage source V th and impedance Z th (section III). This circuit allows calculating currents induced in the dipole. We validate the equivalent circuit by comparing induced currents computed in a realistic human body model with those calculated using the Thevenin circuit. Finally we use this equivalent circuit to suggest in the last section more consistent limits on induced currents in respect to the limits on external electric field.

II. MATERIALS AND METHODS

A. Anatomical human body model

We used the anatomical voxel body model Duke from Virtual Family [START_REF] Christ | The Virtual Family -Development of surface-based anatomical models of two adults and two children for dosimetric simulations[END_REF]. It is a 1.74 m high and 70 kg male model with a voxel size of 5 mm. In total 77 tissues are modeled in simulations. We used a heterogeneous and homogenized model with the same body geometry. For the heterogeneous phantom, the dielectric properties were taken from Gabriel database [START_REF] Gabriel | Compilation of the dielectric properties of body tisses at RF and microwave frequencies[END_REF]. For the homogeneous one, the equivalent tissue mass density was set at 1025 kg/m 3 ; its dispersive electromagnetic properties (i.e. relative permittivity and conductivity) are represented in Fig. 1. These are weighted average of the nine most present tissues from Hugo model of the Visible Human [START_REF]The Visible Human Project[END_REF].

B. Numerical modeling

The phantom is exposed to a vertically polarized plane wave in the frequency range 1-90 MHz with E rms field of 61 V/m. The numerical problems are solved using finite integration technique implemented in CST Microwave Studio. Matlab is used to calculate the induced current solving the Thevenin equivalent circuit.

III. HUMAN-EQUIVALENT IMPEDANCE AND VOLTAGE

CIRCUIT

In this section, based on the standard Thevenin equivalent circuit (Fig. 2), we develop a circuit equivalent to human body for a vertically polarized plane wave exposure between 1 and 90 MHz in free space. The circuit is composed of two elements. The first element is the Thevenin voltage V th that evaluates the voltage source at mid-height of the phantom equivalent to the coupling with the external applied electric field (E ext ). The second element is the Thevenin impedance Z th that evaluates the human equivalent antenna impedance. We assume that human body is conductive enough (i.e. σ > ωε 0 ε r in the considered frequency range) to consider it as equivalent to a homogeneous thick dipole antenna to design our circuits. The load in Fig. 2 is set as a short circuit. Then with this circuit, we calculate induced current at mid-height of the phantom.

A. Electric model of Thevenin voltage V th

The voltage at mid-height of the dipole antenna is the equivalent Thevenin voltage V th and it depends on the antenna effective height. Therefore to design the voltage V th in circuit, we have determined the effective height of Duke phantom depending on its physical parameters and frequency.

For frequencies below the Duke phantom resonance (i.e. f < 65 MHz), the effective height of the antenna is H/2 where H is the height of the antenna equal to the physical height of the Duke phantom. At resonance, the effective height is assumed to be λ/2 where λ is the wavelength in vacuum. Between 65 MHz and 200 MHz we assume that the effective height decreases with a -40 dB/decade slope. Therefore we can represent the voltage behavior in the thick dipole (i.e. in the phantom Duke) by a 2 nd order low-pass filter. V th is described by

2 2 1 1 1 1 0 0 1 1 2 ext ext th E E V jR C L C jm ω ω ω ω ω ω = = + -   + +     (1)
where E ext is the rms external electric field, ω = 2πf the angular frequency, ω 0 the angular frequency at resonance and m the damping factor, R 1 , C 1 and L 1 are the resistance, capacitor and inductance representing the lumped elements in the electric model of V th .

The voltage below and at resonance is defined by ( 2) and (3), respectively.

2 th ext H V E = (2) 2 th ext V E λ = (3)
After simulating the Thevenin voltage of Duke from 1 MHz to 300 MHz (Fig. 4a), we set the damping factor m = 0.35 and resistance R 1 = 100 Ω. Because the resonance frequency appears for H ~ λ/2, we can deduce R and C from equations ( 4) and [START_REF] Gandhi | Currents induced in a human being for plane-wave exposure conditions 0-50 MHz and for RF sealers[END_REF]. The electric model of V th is described by Fig. 3a where R 1 = 100 Ω, L 1 = 265 nH, C 1 = 13 pF and the source generator V 2 is the external rms electric field E ext of the plane wave.

0 1 1 1 L C ω = (4) 1 1 1 2 R C m L = (5)
As the dipole is thick, we have introduced a corrective factor of 0.74 to take into account the antenna thickness (shortening factor [START_REF] Schelkunoff | Theory of antennas of arbitrary size and shape[END_REF]). Therefore the voltage at mid-height of the phantom is given by

2 1 1 1 1 0.74 1 ext th E V jR C L C ω ω = + - (6)

B. Electric model of human antenna impedance

To design the electric model of the Thevenin impedance (equivalent to human body impedance), we simulated the Thevenin equivalent impedance Z th of Duke from 1 MHz to 300 MHz (solid black curve in Fig. 4b). This simulation allows us designing an electric model of Z th with the same impedance as Duke in the considered frequency range. At low frequencies (i.e. from 1 MHz to 65 MHz), the body impedance behaves as a capacitor impedance because the imaginary part of Z th is negative. Close to resonance, it becomes a LC series circuit with 75 Ω impedance at resonance. At higher frequencies (i.e. from 65 MHz to 300 MHz), the thick dipole behaves as a radiation resistance which mean impedance of 200 Ω.

The electric model of the antenna impedance Z th is shown in Fig. 3b. C hf , R hf and L hf are the components used for f < f resonance ; C vhf , R vhf and L vhf are those used around the resonance while R uhf and C uhf are used for f > f resonance . These values are determined by the following equations, where H is the height of Duke phantom, r its mean radius and c the speed of light in vacuum:

For

f < f resonance 17.63 120 ln( / ) hf H C p F c H r = = ( 7
)
75 hf R = Ω (8) 2 2 1
(2 ) 196 ( 2)

hf H L n H C c π = = (9)
For

f ~ f resonance vhf hf C C = ( 10
)
20 vhf R = Ω ( 1 1 ) 2 2
(2 ) 86.9 (3 )

vhf vhf H L n H C c π = = (12)
For f > f resonance

4 hf uhf C C = (13) 200 uhf R = Ω ( 1 4 )
To improve the agreement between the impedance values obtained with the heterogeneous Duke phantom in simulation and the impedance Z th obtained with the electric model in Fig. 3b, we tuned values of some components as follows: L hf = 288 nH, R vhf = 50 Ω, L vhf = 25 nH and C uhf = 8.8 pF.

Fig. 4a represents the Thevenin equivalent voltage in the heterogeneous Duke phantom and the one calculated with the electric model of V th presented in Fig. 3a. Fig. 4b shows the Thevenin equivalent impedance in the heterogeneous Duke phantom and the Thevenin equivalent impedance calculated with the electric model given in Fig. 3b. The results calculated with the proposed Thevenin equivalent circuit are consistent with those calculated using Duke, validating the two electric models proposed for V th and Z th . Then the induced current in the thick dipole at mid-height is calculated as V th /Z th ratio (Fig. 5).

In [START_REF] Chen | RF Currents induced in an anatomicallybased midel of a human for plane-wave exposures (20 -100 MHz)[END_REF], the authors studied induced currents in human body for plane-wave exposure in 20 -100 MHz range in two cases: isolated from ground (same as in our study) and with feet in contact with the ground. In the isolated case, the induced current had a sinusoidal variation. King [START_REF] King | Electric current and electric field induced in the human body when exposed to an incident electric field near the resonant frequency[END_REF] developed analytical formulas to calculate electric current and electric field in the body using a cylindrical model. With this formulation, induced current calculated near the resonance had also a sinusoidal variation. Therefore we assume that the induced current along the human body in free space has a sinusoidal variation given by [START_REF] Frere | Influence de l'impédance d'onde sur les valeurs de débit d'absorption spécifique pour un fantôme de l'humain en bande HF[END_REF] where I max is the induced current at mid-height of the body calculated by our circuits and z the coordinate along the body.

( ) max ( ) sin 0.26 2 I z I z π = + (15)
Because the body is not a perfect antenna, the current is not zero at the extremities, so we introduce a corrective coefficient 0.26 in [START_REF] Frere | Influence de l'impédance d'onde sur les valeurs de débit d'absorption spécifique pour un fantôme de l'humain en bande HF[END_REF]. This coefficient is determined based on simulated induced current in Duke and induced currents calculated with the above described circuits.

C. Validation of the circuits

To validate the designed Thevenin circuit, we expose Duke phantom in free space to a vertically polarized plane-wave from 1 to 90 MHz (E rms field is 61 V/m). Then we compare the computed induced current at mid-height (i.e. at z = H/2) of the phantom to the currents calculated as V th /Z th of the Thevenin equivalent circuit.

The mid-height induced current in the homogeneous and heterogeneous Duke have the similar behavior, being slightly higher in the heterogeneous phantom (maximum deviation is of 3 dB) (Fig. 5). The current induced along the body in homogeneous Duke is nearly identical to the current induced in heterogeneous phantom (difference < 1 dB, Fig. 6). This suggests that a homogeneous antenna can be used to represent induced current in human body.

The current induced at mid-height and calculated using the Thevenin equivalent circuit is close to those in Duke (maximum deviation of 1 dB) (Fig. 5). Because the electric proposed models of V th and Z th give consistent values of induced current, voltage and impedance with Duke phantom, we consider that the Thevenin equivalent circuit is validated.

As the mid-height (H/2) induced current calculated by the Thevenin equivalent circuit is validated, we study in the following current distribution described by [START_REF] Frere | Influence de l'impédance d'onde sur les valeurs de débit d'absorption spécifique pour un fantôme de l'humain en bande HF[END_REF] and along the simulated Duke phantom. We compare induced currents along the body computed using heterogeneous and homogeneous Duke to those calculated by the Thevenin equivalent circuit and equation [START_REF] Frere | Influence de l'impédance d'onde sur les valeurs de débit d'absorption spécifique pour un fantôme de l'humain en bande HF[END_REF].

For the four frequencies represented in Fig. 6 (2 MHz, 30 MHz, 62 MHz and 90 MHz), the distribution of induced current along the body is sinusoidal for heterogeneous Duke as well as for the equivalent circuit. The induced current 15) is in a good agreement with that computed in the heterogeneous phantom (maximum deviation is 1 dB). The induced currents values at the feet (z between 0 and 20 cm) from the equivalent circuit and [START_REF] Frere | Influence de l'impédance d'onde sur les valeurs de débit d'absorption spécifique pour un fantôme de l'humain en bande HF[END_REF] are slightly lower than those of Duke (maximum deviation is 1.5 dB). For the head (z between 160 and 180 cm), the difference is higher (up to 8 dB at 2 MHz), but the limits on induced current given by ICNIRP [START_REF] Icnirp | Guidelines for limiting exposure to time-varying electric, magnetic and electromagnetic fields (up to 300 GHz)[END_REF] and IEEE [START_REF] Ieee | IEEE Standard for safety levels with respect to human exposure to radio frequency electromagnetic fields, 3 kHz to 300 GHz[END_REF] concern feet, so we consider our model still acceptable.

These results suggest that Thevenin equivalent circuit is an appropriate model for determining the induced current along the body for a plane wave exposure in 1 -90 MHz range, particularly from feet to torso. From the Thevenin equivalent circuit (via V th ), we can describe induced current along the body as a function of the external electric field. In the next section, the equivalent circuit and equation ( 15) are used to propose new limits on induced current more consistent with external electric field limits reported in [START_REF] Ieee | IEEE Standard for safety levels with respect to human exposure to radio frequency electromagnetic fields, 3 kHz to 300 GHz[END_REF].

IV. LIMITS ON INDUCED CURRENT DEPENDING ON EXISTING EXTERNAL ELECTRIC FIELD LIMITS

The limits on induced currents are defined for currents induced trough foot, so we consider that the induced current in foot can be measured at the ankle (z = 10 cm). For a controlled environment the standard limit through a foot is I limit = 100 mA [START_REF] Ieee | IEEE Standard for safety levels with respect to human exposure to radio frequency electromagnetic fields, 3 kHz to 300 GHz[END_REF].

Using [START_REF] Frere | Influence de l'impédance d'onde sur les valeurs de débit d'absorption spécifique pour un fantôme de l'humain en bande HF[END_REF] and the proposed models, Fig. 7 demonstrates that to respect the actual limit on external electric field at 10 MHz (i.e.184 V/m), the induced current has to be lower than 80 mA. Between 30 and 90 MHz, the limit on external electric field is 61 V/m thus to respect those limits, the induced current in ankle has to be lower than 95 mA at 30 MHz and lower than 180 mA at 50 MHz, 70 MHz and 90 MHz.

Therefore the actual limits on induced currents are more restrictive than external electric field for frequencies above 40 MHz. For frequencies below 40 MHz, the limits on external electric field are more restrictive. However we did not take into account the coupling with the magnetic field because the phantom is isolated thus only sensitive to electric field [START_REF] Frere | Influence de l'impédance d'onde sur les valeurs de débit d'absorption spécifique pour un fantôme de l'humain en bande HF[END_REF]. Therefore the same study with the magnetic field coupling could be done because for frequencies below 10 MHz, the ratio E/H from ICNIRP limits is low impedance (lower than 377 Ω) [START_REF] Icnirp | Guidelines for limiting exposure to time-varying electric, magnetic and electromagnetic fields (up to 300 GHz)[END_REF]. An equivalent Thevenin circuit has been proposed to evaluate human body impedance and voltage at mid-height for a vertically polarized plane wave exposure in 1 -90 MHz range. This circuit allows evaluating induced current along the human body depending on the external electric field for plane wave exposure. It provides an accurate estimate of the induced current compared to numerical results for a realistic human body model, particularly from feet to torso.

Finally, based on the proposed circuit and current distribution along the body, updated limits on induced current have been proposed depending on actual external electric field IEEE limits. This allows harmonizing induced current and external electric field limits, because currently induced currents limits are more restrictive than those of external electric field at frequencies above 40 MHz.

Fig. 1 .

 1 Fig. 1. Electromagnetic properties of the equivalent tissue used to homogenize Duke phantom

Fig. 2 .

 2 Fig. 2. Thevenin equivalent circuit modeling the voltage and impedance at mid-height of the phantom Duke

Fig. 3 .Fig. 4 .

 34 Fig. 3. Electric models of (a) voltage at mid-heigth of the dipole (Thevenin voltage V th ) and (b) human equivalent impedance (Thevenin impedance Z th )

Fig. 5 .Fig. 6 .

 56 Fig. 5. Mid-length induced current in the heterogeneous, homogeneous Duke and in the equivalent dipole from 1 to 100 MHz

Fig. 7 .

 7 Fig. 7. Induced current in ankle as a fonction of external electric field at 10 MHz, 30 MHz, 50 MHz, 70 MHz and 90 MHz

ACKNOWLEDGMENT This work was supported by Thales Communications & Security, French National Association of Research and Technology (ANRT) and French National Center for Scientific Research (CNRS).


Christian Commault

Jean-Michel Dion

Sameh Yacoub Agha

Michel Dion

Sameh Yacoub

SENSOR CLASSIFICATION FOR THE FAULT DETECTION AND ISOLATION PROBLEM

Keywords: Linear systems, Structured systems, Fault detection and isolation, Sensor classification

In this paper we deal with the observer based Fault Detection and Isolation problem. When this FDI problem is solvable with the existing sensors, we will classify these sensors according to their importance for the solvability of the considered FDI problem. The sensors will be classified into essential sensors, useless sensors and useful sensors. We tackle this problem when the system under consideration is structured, that is, the entries of the system matrices are either fixed zeros or free parameters. With such structured systems one can associate a graph. Simple algorithms allow to get from the graph the above sensor classification.

INTRODUCTION

The Fault Detection and Isolation (FDI) problem has received considerable attention in the past ten years [START_REF] Chen | Robust modelbased fault diagnosis for dynamic systems[END_REF][START_REF] Frank | Analytical and qualitative model-based fault diagnosis -a survey and some new results[END_REF]. It consists of building residuals from the available data and isolating, whenever possible, the faults using the residuals. When this FDI problem is solvable with the existing sensors, we will classify these sensors according to their importance in the solvability of the considered FDI problem. The sensors will be classified into essential sensors, useless sensors and useful sensors. In simple words the failure of an essential sensor will lead to the non solvability of the FDI problem while the failure of a useless sensor will not affect the solvability of the FDI problem. The useful sensors are simply those which are not useless. We consider the FDI problem in the framework of structured linear systems which represent a large class of parameter dependent systems [START_REF] Lin | Structural controllability[END_REF][START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF], that is the entries of the system matrices are either fixed zeros or free parameters. With such structured systems one can associate a graph and simple properties of these graphs will allow us to get interesting structural information on the system. The FDI problem has been tackled in this context, see [START_REF] Commault | Observer-based fault detection and isolation for structured systems[END_REF][START_REF] Verde | Faults isolation for nonlinear Hessenberg systems[END_REF]. When the FDI problem has no solution with the existing sensors a structural analysis has allowed to determine the number and the location of needed additional sensors [START_REF] Commault | Sensor location for diagnosis in linear systems: a structural analysis[END_REF][START_REF] Commault | Structural analysis for sensor location in fault detection and isolation[END_REF]. Using the graph associated with the system, we will prove that useless sensors correspond to output vertices which are not connected to faults by a path. One can determine in the system associated graph a set of bottleneck vertices called "essential vertices", the output vertices in this set indeed correspond to the essential sensors. The sensor classification can be performed using simple polynomial complexity algorithms. In (Commault et al., 2006a) a similar sensor classification has been proposed for the case of generic observability. The paper is structured as follows. The problem is formulated in Section 2. Structured systems are presented in Section 3. In Section 4, we provide with a characterization of useless and essential sensors for the observer-based FDI problem. Numerical aspects are considered in Section 5. Concluding remarks end the paper.

PROBLEM FORMULATION

Observer-based FDI problem

Let us consider the following linear time-invariant system :

Σ ẋ(t) = Ax(t) + Lf (t) + Bu(t) y(t) = Cx(t) + M f (t) . ( 1
)
where x(t) ∈ R n is the state vector, f (t) ∈ R r the fault vector, u(t) ∈ R m is the control input vector and y(t) ∈ R p the measured output vector. Each output variable is provided by a specific sensor.

For the sake of simplicity, in the following we will denote by y i either the ith sensor or the ith output variable. A, C, B, L and M are matrices of appropriate dimensions.

A dedicated residual set is designed using a bank of r observers for system (1), according to the dedicated observer scheme [START_REF] Chen | Robust modelbased fault diagnosis for dynamic systems[END_REF].

The ith observer of this bank of r observers is designed for a system of type (1) as follows:

ẋi (t) = Ax i (t) + K i (y(t) -C xi (t)) + Bu(t), (2)
where xi (t) ∈ R n is the state of the ith observer, K i is the observer gain to be designed such that xi (t) asymptotically converges to x(t), when f (t) = 0. The residuals are defined as :

r i (t) = Q i (y(t) -C xi (t)), for i = 1, . . . , r, (3)
where Q i is a 1 × p matrix.

Definition 1. Let Σ be a linear system as in Equation (1) with the set of sensors Y = {y 1 , . . . , y p }.

The bank of observer-based FDI problem associated with Y consists in finding, if possible, matrices K i and Q i , such that, for i = 1, 2, . . . , r, A -K i C is stable, and the fault to residual transfer matrix is non zero, proper and diagonal.

It is clear that the solvability of this problem relies on the available sensors, a sufficient amount of information is needed to detect and isolate the faults via the dedicated observer scheme.

Sensor classification for FDI

In this section we provide with a classification of the system sensors with respect to their importance concerning the bank of observer based FDI problem of Definition 1. In this study we suppose that the F DI problem has a solution with the given sensors. The sensors will be classified into essential sensors, useless sensors and useful sensors. In simple words the failure of an essential sensor will lead to the non solvability of the FDI problem while the failure of an useless sensor will not affect the solvability of the FDI problem.

Let us first define what we mean by a solution for the FDI problem with a given subset of sensors.

Definition 2. Let Σ be the linear system of Equation (1) and assume that the FDI problem of Definition 1 has a solution with the sensor set Y . Let V be a subset of Y . V is a solution of the bank of observer based FDI problem of Definition 1, if the considered FDI problem remains solvable with the subset of sensors V .

Define now the essential and useless sensors:

Definition 3. Consider the linear system Σ of Equation (1). Let y i be the ith sensor of the system.

The sensor y i is essential if y i belongs to any solution of the FDI problem in the sense of Definition 2.

Definition 4. Consider the linear system Σ of Equation ( 1). Let y i be the ith sensor of the system and V /{y i } the set V from which the sensor y i has been removed. The sensor y i is useless if for any solution V containing y i of the FDI problem in the sense of Definition 2, V /{y i } is still a solution.

LINEAR STRUCTURED SYSTEMS

Definitions and basic properties

We will consider models based on the available physical knowledge on the system. These models capture the relations between internal variables but without fixing the precise value of the parameters. Frequently, starting from a nonlinear dynamical model of a system one gets linearized models around different set points which share the same structure but with different parameter values. A distillation column example is worked out in (Commault and Dion, 2006). This is illustrated in Figure 1 where a nonlinear model is linearized around different set points. Such models which incorporate prior knowledge [START_REF] Lin | Structural controllability[END_REF][START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF] In this paper we will consider linear structured systems as in [START_REF] Lin | Structural controllability[END_REF]. We consider linear systems as described in (1), but with parameterized entries and denoted by Σ

Λ Σ Λ ẋ(t) = Ax(t) + Lf (t) + Bu(t) y(t) = Cx(t) + M f (t) . ( 4
)
This system is called a linear structured system if the entries of the composite matrix J = A L B C M 0 are either fixed zeros or independent parameters (not related by algebraic equations). Λ = {λ 1 , λ 2 , . . . , λ k } denotes the set of independent parameters of the composite matrix J. More details can be found in [START_REF] Dion | Generic properties and control of linear structured systems: a survey[END_REF].

For such systems one can study generic properties i.e. properties which are true for almost all values of the parameters collected in Λ [START_REF] Murota | Systems Analysis by Graphs and Matroids[END_REF].

A directed graph G(Σ Λ ) = (V, W ) can be easily associated with the structured system Σ Λ of type (4) where the matrix A L B C M 0 is structured:

• the vertex set is V = U ∪ F ∪ X ∪ Y where U , F , X
W = {(u i , x j )|B ji = 0} ∪ {(f i , x j )|L ji = 0}∪{(x i , x j )|A ji = 0}∪{(x i , y j )|C ji = 0} ∪ {(f i , y j )| M ji = 0}
, where A ji (resp. B ji C ji ,L ji ,M ji ) denotes the entry (j, i) of the matrix A (resp. B, C,L,M ).

Recall that a directed path in G(Σ Λ ) from a vertex i 0 to a vertex i q is a sequence of arcs (i 0 , i 1 ), (i 1 , i 2 ), . . . , (i q-1 , i q ) such that i t ∈ V for t = 0, 1, . . . , q and (i t-1 , i t ) ∈ W for t = 1, 2, . . . , q. If i 0 ∈ F and, i q ∈ Y , P is called a fault-output path.

If i 0 ∈ V 1 and, i l ∈ V 2 , where V 1 and V 2 are two subsets of V , P is called a V 1 -V 2
path. Moreover, if i 0 is the only vertex of P which belongs to V 1 and i l is the only vertex of P which belongs to V 2 , P is called a direct V 1 -V 2 path. A set of paths with no common vertex is said to be vertex

disjoint. A V 1 -V 2 linking of size k is a set of k vertex disjoint V 1 -V 2 paths. A linking is maximal when k is maximal.
In this paper we will consider that Σ Λ is structurally observable [START_REF] Lin | Structural controllability[END_REF][START_REF] Murota | Systems Analysis by Graphs and Matroids[END_REF]. This implies that any state vertex is linked to output (sensor) vertices by a path in G(Σ Λ ).

Give now the result concerning the diagonal FDI problem by using a bank of observers, which was stated first in [START_REF] Commault | Observer-based fault detection and isolation for structured systems[END_REF].

Theorem 5. Consider the structurally observable system with r faults Σ Λ as defined in (4) and the associated graph G(Σ Λ ). The bank of observerbased diagonal FDI problem of Definition 1, is generically solvable if and only if:

k = r, ( 5
)
where k is the size of a maximal fault-output linking in G(Σ Λ ).

The essential vertices and the minimal output separator

Most of the basic material of this subsection is based on (van der [START_REF] Van Der Woude | The generic number of invariant zeros of a structured linear system[END_REF]. A separator S of G(Σ Λ ) is a set of vertices such that any fault-output path has at least one vertex in S.

Separators with a minimal number of vertices are called minimal. A classical result is that the minimal size of a separator is the maximal size of a fault-output linking.

Definition 6. The set of essential vertices V ess of G(Σ Λ ) is the set of vertices which belong to any maximal size fault-output linking.

Definition 7. The minimal output separator of G(Σ Λ ), S * , is the set of begin vertices of all direct V ess -Y paths.

It can be shown that S * is a separator of minimal dimension. S * is indeed the last bottleneck between faults and outputs. S * may contain fault, state and output vertices.

Example 1

To illustrate the previous notions, let us deal with a first example with no control input, 6 faults and 8 sensors, which graph G(Σ Λ ) is given in Figure 2. One can show that the corresponding set of essential vertices is V ess = {f 1 , . . . , f 6 , x 3 , x 4 , x 5 , x 6 , y 1 , y 2 , y 3 , y 5 }, see Figure 3. The corresponding minimal output separator is S * = {x 4 , x 6 , y 1 , y 2 , y 3 , y 5 }, see Figure 4. Proposition 8. Consider the structurally observable system Σ Λ with r faults and sensor set Y as defined in (4) and the associated graph G(Σ Λ ). Let V be a subset of Y . V is a solution of the generic bank of observer based diagonal FDI problem if and only if there exists an F -V linking of size r in G(Σ Λ ).

The useless sensors of Definition 4 for the FDI can then be characterized as follows.

Theorem 9. Consider the linear structured system Σ Λ with sensor set Y and its associated graph G(Σ Λ ). A sensor y i ∈ Y is useless if and only if there is no F -{y i } path in G(Σ Λ ), where F is the fault set.

Proof

Sufficiency Assume that there is no F -{y i } path in G(Σ Λ ) and let V be a solution such that y i ∈ V .

V being a solution, from Proposition 8 there is an F -V linking of size r in G(Σ Λ ). This linking is not incident to y i since there is no F -{y i } path. Therefore this linking is also a size r linking from F to V /{y i } then V /{y i } is a solution which proves that y i is useless.

Necessity Assume now that there is an F -{y i } F V path P in G(Σ Λ ) and consider an F -Y linking L of size r in G(Σ Λ ), which is then maximal. Denote by V the subset of Y consisting of all the outputs incident to L. V is clearly a solution for the FDI problem. Assume first that y i ∈ V , then V /{y i } is not a solution because it has cardinality (r -1). Therefore y i is not useless. Suppose now that y i does not belong to V . A given F -{y i } path P must have common vertices with the linking L, otherwise L would not be maximal. Concentrate now on the F -Y path L j of L which contains the last common vertex z between L and P when travelling along P from F to y i , see Figure 5. By concatenating the part of L j from F to z and the part of P from z to y i we get a new path P ′ which is incident to y i and having no common vertex with L/L j . We get a new linking L ′ = P ′ ∪(L/L j ) of size r which is incident to y i , see Figure 6. Therefore y i belongs to a solution V ′ such that V ′ /{y i } is not a solution, then y i is not useless.

Theorem 10. Consider the linear structured system Σ Λ with its associated graph G(Σ Λ ). The set of essential sensors is given by Y e =Y ∩ V ess , where Y is the sensor set and V ess is the set of essential vertices of G(Σ Λ ).

Proof

Sufficiency Assume that y i ∈ Y e =Y ∩ V ess . Then any F -Y linking of size r is incident to y i and any linking F -{Y /y i } has size less than or equal to r -1. This is a fortiori true for for any F -{V /y i } linking with V ⊆ Y therefore y i must belong to any solution and y i is essential.

Necessity If y i / ∈ Y e =Y ∩ V ess , then y i does not belong to any maximal matching. Therefore there exists a size r linking which is not incident to y i . The set V of end vertices of the paths of this linking is a solution of the FDI problem which does not contain y i , then y i is not essential .

Example 2

Consider the linear structured system with 3 control inputs, 6 faults and 10 sensors illustrated in Figure 7. The corresponding set of essential vertices is V ess = {y 1 , y 2 , f 1 , . . . , f 6 , x 1 , . . . , x 6 , x 9 , . . . , x 14 }, see Figure 8.

-The sensors y 1 and y 2 are essential sensors because they belong to Y ∩ V ess .

-The sensors y 9 and y 10 are useless sensors because they belong to no F -Y path.

-The sensors y 1 , . . . , y 8 , are useful since they are not useless.

COMPUTATIONAL ASPECTS

In this section we give some indications on the way for computing the sets of useless and essential sensors. Theorem 10 which characterizes the essential sensors can be reformulated as follows using the concept of minimal output separator.

Proposition 11. Consider the linear structured system Σ Λ with its associated graph G(Σ Λ ). The set of essential sensors is given by Y e =Y ∩ S * , where Y is the sensor set and S * is the minimal output separator of G(Σ Λ ).

It turns out that the set S * can be computed using max flow techniques as the Ford and Fulkerson algorithm [START_REF] Murota | Systems Analysis by Graphs and Matroids[END_REF]. This kind of algorithm is of polynomial complexity allowing to deal with large scale systems. The set of useless sensors can be obtained by using a marking procedure starting from the fault vertices and walking along F -Y paths. The output vertices which will not be marked will correspond to useless sensors. This very simple algorithm is also polynomial.

CONCLUDING REMARKS

In this paper we have classified the sensors with respect to their importance for the solvability of the observer based FDI problem. The proposed analysis is mainly based on the system structure and is parameter independent. This analysis reveals the critical sensors which should be comforted in order to perform a robust FDI. The number of useful but non essential sensors give a measurement of their degree of redundancy concerning the FDI problem.

  Fig.1. Nonlinear system on the structure have been often used in the literature see e.g[START_REF] Lin | Structural controllability[END_REF][START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF] In this paper we will consider linear structured systems as in[START_REF] Lin | Structural controllability[END_REF]. We consider linear systems as described in (1), but with parameterized entries and denoted by Σ Λ

Fig. 3 .

 3 Fig. 2. Graph G(Σ Λ ) of Example 1.

Fig. 4 .

 4 Fig. 4. Output separator of Example 1.

Fig. 5 .Fig. 6 .

 56 Fig. 5. The linking L and the path P .

Fig. 7 .

 7 Fig. 7. The graph G(Σ Λ ) of Example 2.

Fig. 8 .

 8 Fig. 8. Essential vertices of Example 2.

  and Y are the control input, fault, state and output sets given by {u 1 , u 2 , . . . , u m }, {f 1 , f 2 , . . . , f r }, {x 1 , x 2 , . . . , x n } and {y 1 , y 2 , . . . , y p } respectively,• the arc set is