Local Smoothness of Weak Solutions to the Magnetohydrodynamics Equations via Blowup Methods (original) (raw)
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arXiv (Cornell University), 2014
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2016
In this paper, we prove a blow-up criterion in terms of the magnetic field H and the mass density ρ for the strong solutions to the 3D compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum. More precisely, we show that the upper bounds of (H, ρ) control the possible blow-up (see [26][32][36]) for strong solutions.
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In this paper, we consider the 3-D compressible isentropic MHD equations with infinity electric conductivity. The existence of unique local classical solutions is established when the initial data is arbitrarily large, contains vacuum and satisfies some initial layer compatibility condition. The initial mass density needs not be bounded away from zero and may vanish in some open set or decay at infinity. Moreover, we prove that the L ∞ norm of the deformation tensor of velocity gradients controls the possible blow-up (see [16][22]) for classical (or strong) solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the deformation tensor as the critical time approaches. Our criterion (see (1.12)) is the same as Ponce's criterion for 3-D incompressible Euler equations [15] and Huang-Li-Xin's criterion for the 3-D compressible Navier-stokes equations [9].
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Discrete & Continuous Dynamical Systems - B, 2017
This work establishes local existence and uniqueness as well as blow-up criteria for solutions (u, b)(x, t) of the Magneto-Hydrodynamic equations in Sobolev-Gevrey spacesḢ s a,σ (R 3). More precisely, we prove that there is a time T > 0 such that (u, b) ∈ C([0, T ];Ḣ s a,σ (R 3)) for a > 0, σ ≥ 1 and 1 2 < s < 3 2. If the maximal time interval of existence is finite, 0 ≤ t < T * , then the blow-up inequality C 1 exp{C 2 (T * − t) − 1 3σ } (T * − t) q ≤ (u, b)(t) Ḣs a,σ (R 3) with q = 2(sσ + σ 0) + 1 6σ holds for 0 ≤ t < T * , 1 2 < s < 3 2 , a > 0, σ > 1 (2σ 0 is the integer part of 2σ).
SIAM Journal on Mathematical Analysis, 2013
We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamics (MHD) equations in terms of the velocity field. The result means that the velocity field plays a more important role than the magnetic field in the local regularity theory of the MHD equations. This also gives a positive answer to the problem proposed by Kang and Lee in [J.
Journal of Mathematical Sciences, 2006
The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.
International Journal of Analysis and Applications, 2020
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.