STUDY OF THE BLOW UP OF THE MAXIMAL SOLUTION TO THE THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC SYSTEM IN LEI-LIN-GEVREY SPACES

In this paper, we investigate existence, uniqueness and blowup in finite time of the local solution to the three dimensional magnetohydrodynamic system, in Gevrey-Lei-Lin spaces. To prove the blowup results and give the blow profile as a function of time, two key points are used. The first is a frequency decomposition of the spectrum of the initial data. This allows to use Leray theory. The second is a technical lemma we proved to state that the Lei-Lin space is an interpolation space between the Gevrey-Lei-Lin and the Lebesgue square integrable functions spaces. To prove uniqueness, we use a penalization procedure and energy methods. About existence, we split the initial condition into low frequencies part and high frequencies part. The former are considered as initial data to the linear part of the system. The latter will be taken as small as needed, so that smallness theory applies and allows to run a fixed point argument. Received February 8th, 2020; accepted February 28th, 2020; published May 1st, 2020. 2010 Mathematics Subject Classification. 35A01, 35A02, 35B40, 35B44, 35B45.

Let us consider the following three dimensional incompressible magnetohydrodynamic system, where u, b and p denote respectively the unknown velocity, the unknown magnetic field and the unknown pressure at the point (t, x). If the initial data u 0 and b 0 are quite regular, the divergence free conditions determine the pressure p. We aim to study the existence, uniqueness and blowup in finite time of local solution to the M HD system, in the framework of Gevrey-Lei-Lin spaces. These spaces are defined for the real numbers a > 0, σ > 1 and ρ, by |ξ| ρ e a|ξ| 1/σ |f (ξ)|dξ < ∞} and endowed with its naturel norm |ξ| ρ e a|ξ| 1/σ |f (ξ)|dξ, wheref denotes the Fourier transform of f .
In [15], the authors defined the Lei-Lin space by and endowed with its natural norm Here, L 1 loc (R 3 ) states for the set of locally R 3 -Lebesgue integrable distributions. In this critical space, the distinguishable fact was that to obtain global well-posedness to the Navier-Stokes equations, the norm of the initial data have to be exactly less than the viscosity of the fluid. However, in the wide fluid mechanic literature, it was always assumed that the initial data must be very small, especially smaller than the viscosity multiplied by a tiny positive constant. Such assumption is mandatory to run the smallness argument and to obtain global well-posedness; see for example [11][12][13][14] and a complete survey in [10]. For many fluid mechanics equations, well-posedness and asymptotic behavior, as time goes to infinity or as small parameter goes to zero, were investigated by the authors, in various spaces; see for example [6][7][8][9][18][19][20]. About blowup, it is worthwhile to emphasize that several authors studied this phenomena to the Navier-Stokes equations; see for example [1][2][3]17] and references therein. The author observed, in [4], that in the case of the Navier-Stokes equations, the blowup phenomena depends on the chosen space not on the nonlinear part. To do so, he used Fourier analysis in Sobolev-Gevrey spaces, for Sobolev index s > 3/2. His blowup result was improved later on, in [5], where the authors gave precisely an exponential type of the blowup profile, in Sobolev-Gevrey spaces but with less regularity on the initial data since they dealt with s = 1. In [16], authors studied the Cauchy problem for a two-components high-order Camassa-Holm system. First, they proved the local well-posedness of the system in Besov spaces. Then, using Littlewood-Paley theory, they derived a blowup criterion for the strong solution. Finally, they studied Gevrey regularity and analyticity of the solutions to the Camassa-Holm system in the Gevrey-Sobolev spaces.
In this paper, we begin by addressing the problem of local well-posedness. Our result is summarized in the following existence and uniqueness theorem.
Then, there exist a time T > 0 and a unique solution To prove uniqueness of solution, we use a penalisation method. This allows to put the problem in a form where Gronwall inequality can be applied. To establish existence of solution, we split the initial condition into low and high frequencies. The former will be considered as initial data to the linear part of the M HD system. The latter, taken as small as needed, will be the initial data to the remaining nonlinear part, for which smallness theory applies and allows to run a fixed point argument, in (X −1 a,σ (R 3 )) 2 . Then, we turn to the blowup result that we state in the following theorem.
be the maximal solution of MHD system, where T * < ∞. Then, there exists c 0 > 0, such that The proof is somewhat technical. Our idea is to use a suitable frequency decomposition and to impose "the problematic" large frequencies part to be a square integrable function, so that we fully profit from Leray theory. Some technical lemmas are specially derived to handle technical difficulties, mainly Lemma 1.1, where we proved that X 0 is an interpolation space between the Lei-Lin spaces X s and the space of Lebesgue square integrable functions. The structure of our proof is as follows. Starting with the energy estimates, we prove that the X −1 a,σ norm of the solution blows up, in finite time. Gronwall type inequality allows to infer that blowup holds also in X 0 a ,σ , for any a ∈ (0, a). Using a particular choice of parameter a , we deduce that our solution blows up in the X 0 norm, as a limit of X 0 a ,σ spaces. We split the initial data into two parts, a large frequencies one that belongs to X −1 a,σ ∩ L 2 and a small frequencies remainder that leis in X −1 a,σ . The smallness theory applies an leads to a global and continuous in time solution that belongs to X −1 a,σ . This continuity plays an important role. About the large frequencies part, the L 2 theory applies and allows to derive a Leray type energy estimates. We use the above two estimates to dominate the X 0 norm of the solution. Finally, Lemma 1.1 with a judicious choice of the index s, finish the proof of the blowup result, in X −1 a,σ and determine its profile as a function of time.
This paper is organized as follows. In section 2, we give some notations and useful preliminary results.
Section 3 is devoted to prove existence and uniqueness of local in time solution. In section 4, we establish the blowup result.

Technical lemmas
In this section, we prove some technical lemmas that will be used later on.
By Cauchy-Schwarz inequality, we prove the following lemma.
Using the Dominated Convergence Theorem, we get As it will be seen below, for instance, the stability condition of the fixed point argument requires a choice of ε > 0 such that ε 1/2 u 0 1/2 X −1 a,σ < 1 18 . Moreover, for the operator Ψ to be a contraction mapping, ε has to fulfill the supplementary condition ( (u 0 , b 0 ) X −1 a,σ + r) 1/2 ε 1/2 < 1 32 . For this choice of ε, by (2.8), there exists a time T = T (ε) > 0 such that v L 1 T (X 1 a,σ ) < ε. (2.9) Put w = u − v and d = b − c, the 3+3 components vector (w, d) satisfies, for all (t, x), the following nonlinear system denoted (MHDNL), To run a fixed point argument, we introduce the following operator Ψ defined for all (w, d) T by the right hand side of the following integral equation and we consider the space In a first step, let us prove that X T is stable under the operator Ψ. To do so, we denote by B r the subset of X T defined by For (w, d) ∈ B r , we have ψ((w, d)) ∈ B r . In fact, it holds that Ψ(w, d)(t) X −1 a,σ ≤ I ww + I wv + I vw + I vv + I dd + I dc + I cd + I cc + I wd + I wc + I vd + I vc + I dw + I dv + I cw + I cv , where we denoted, for any divergence free vector field υ and ω, To estimate Ψ(w, d)(t) X −1 a,σ , we recall that according to the choice of N , we have Using divergence free condition and lemma 1.5, we obtain that The same holds for I cc , I vc and I cv . Moreover, and the same holds for I dd , I wd and I dw . Furthermore, and the same holds for the seven remaining integrals. Finally, we obtain Let us estimate Ψ(w)(t) L 1 (X 1 a,σ ) . As above, we have where we denoted, for any divergence free vector field υ and ω, By the facts that v L 1 Using lemma 1.6 and the fact that w ∈ B r , we get and so on for J cc , J vc and J cv . Also, we get and so on for J dd , J wd and J dw . Moreover, and so on for the seven remaining integrals. Thus, Combining (2.11) and (2.13), we deduce that Ψ(B r ) ⊂ B r .
In a second step, to prove that Ψ is a contraction mapping on B r . One has Or equivalently, in an adequate form to be estimated, one has It follows that and so on for the other integrals. Using lemma 1.5, triangle inequality, the fact that w i belongs to B r , inequalities (2.7) and (2.9), we infer that Thus, (2.14) To estimate the L 1 T (X 1 a,σ ) norm, we proceed as above; where and so on for the other integrals. Using lemma 1.6, triangle inequality, the fact that w i belongs to B r , inequalities (2.7) and (2.9), we infer that The same holds for L (2.15) By (2.14) and (2.15), we infer that This implies that and Ψ is a contraction mapping. The fixed point theorem implies that there is a unique (w, d) ∈ B r , such

Blowup results
In this section we prove theorem 0.2. First of all, the following energy estimates holds in X −1 a,σ , where L υω = div (υω) X −1 a,σ . By lemma 1.2, we have It follows that Using Lemma 1.3 and product Young inequality, we obtain However, a direct computation implies that (u, b) X 0 By continuity of (u, b), we have T ∈ ]0, T * ] and We infer that If we consider the magnetohydrodynamic system starting at initial time t, with the data (u, b)(t) , we get Therefore, we infers that Applying Gronwall inequality to inequality (3.1), we get By equation (3.2), we obtain This implies that At this point, we proved that X 0 a σ ,σ norm of the solution blows up, in finite time. Let a = a σ ∈ (0, a), using the same method, we obtain where R xy (t) = x.∇y X 0 a ,σ . Using Lemma 1.4, we get It follows that By lemma 1.3, we obtain Using product Young inequality, we get Gronwall lemma gives Or equivalently, Integrating over [0, T * ) and using (3.4), we infer that Since a σ n < ... < a σ < a, by the same method we used for a ∈ (0, a), we prove that By Dominate Convergence Theorem, we obtain Consider the (M HD) system starting at t ∈ [0, T * ), by time translation, we have At this point, we proved that the X 0 norm of the solution blows up, in finite time.
Let k ∈ N * , we consider the subset A k defined by A k = {ξ ∈ R 3 ; |ξ| ≤ k and | u 0 (ξ)| ≤ k} and v 0 and c 0 in , one has lim k→∞ (w 0 k , d 0 k ) X −1 a,σ = 0. So, there exists k ∈ N, such that (w 0 k , d 0 k ) X −1 a,σ < 1 16 . Using smallness theory, we prove that a unique and global in time solution (w k , d k ) to the system ) and satisfies for t ≥ 0, ) and satisfies, for all (t, x) ∈ R × R 3 , the following (M HD 2 ) system, Having that (v 0 k , c 0 k ) ∈ L 2 (R 3 ), we take the scalar product and use L 2 theory. Under divergence free condition, we infers that The Gronwall lemma implies that .