On well-posedness of the Cauchy problem for 3D MHD system in critical Sobolev–Gevrey space

Abdelkerim Chaabani ()
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Abdelkerim Chaabani: University of Tunis El Manar

Partial Differential Equations and Applications, 2021, vol. 2, issue 2, 1-20

Abstract: Abstract This paper is devoted to the study of the three-dimensional periodic MHD system. We prove the local in time well-posedness for arbitrary large in $$H_{a,\sigma }^{1/2}(\mathbb {T}^3)$$ H a , σ 1 / 2 ( T 3 ) initial data as well as global in time well-posedness when initial data satisfies a smallness condition. We also provide an unusual global in time existence criteria. It is based on breaking up the Fourier sums $$\sum _{k\in \mathbb {Z}^3}|\hat{u}(t,k)|$$ ∑ k ∈ Z 3 | u ^ ( t , k ) | , $$\sum _{k\in \mathbb {Z}^3}|\hat{b}(t,k)|$$ ∑ k ∈ Z 3 | b ^ ( t , k ) | of the solution (u, b) into low frequency modes up to m and high frequency modes down to m. We prove that there exists a wave-number m such that if the map $$t\mapsto F_m(t)\,=\,\sum _{|k|\le m}|\hat{u}(t,k)|+\sum _{|k|\le m}|\hat{b}(t,k)|-\sum _{|k|> m}|\hat{u}(t,k)|-\sum _{|k|> m}|\hat{b}(t,k)|$$ t ↦ F m ( t ) = ∑ | k | ≤ m | u ^ ( t , k ) | + ∑ | k | ≤ m | b ^ ( t , k ) | - ∑ | k | > m | u ^ ( t , k ) | - ∑ | k | > m | b ^ ( t , k ) | keeps its sign constant on the time interval of existence [0, T), then the global in time existence of the solution follows.

Keywords: MHD system; Critical Sobolev–Gevrey space; Existence and uniqueness; Primary 35A01, 35A02; Secondary 35B05, 35B10, 35B35 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00081-z

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