Abstract
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.
Highlights
(u, b)(0) = (u0, b0), x ∈ R3, where u, b and p denote respectively the unknown velocity, the unknown magnetic field and the unknown pressure at the point (t, x)
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
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
Summary
To prove uniqueness of solution to the (M HD), we consider two solutions (u1, b1) and (u2, b2) that belong to (C([0, T ], Xa−,σ1(R3)))2 ∩ (L1([0, T ], Xa1,σ(R3))) and have the same initial data. Taking the Fourier transform and using divergence free condition, one infers that. Substituting (2.5) in (2.4), dividing by |δ|2 + |η|2 + ε2, integrating with respect to time, letting ε → 0 and using that |δ|√+|η| ≤ |δ|2 + |η|2, we infer that t. For the operator Ψ to be a contraction mapping, ε has to fulfill the supplementary condition. For this choice of ε, by (2.8), there exists a time T = T (ε) > 0 such that v L1T (Xa1,σ) < ε. 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 w. T e(t−τ)∆(w2 − w1)∇(w1 + v)dτ , Xa−,σ1 and so on for the other integrals.
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