On local existence, uniqueness and blow-up of solutions for the generalized MHD equations in Lei–Lin spaces

  • Wilberclay G. MeloEmail author
  • Cilon Perusato
  • Natã Firmino Rocha


This paper establishes the existence and uniqueness, and also presents a specific blow-up criterion, for solutions of the generalized magnetohydrodynamics (GMHD) equations in Lei–Lin spaces \(\mathcal {X}^s(\mathbb {R}^3)\), by considering appropriate values for s. More precisely, if it is assumed that the initial data \((u_0,b_0)\) belong to \(\mathcal {X}^{s}(\mathbb {R}^3)\), we demonstrate that there exists an instant of time \(T>0\) such that \((u,b)\in [C_{T}(\mathcal {X}^s(\mathbb {R}^3))\cap L^1_{T}({\mathcal {X}}^{s+2\alpha }(\mathbb {R}^3))]\times [C_{T}(\mathcal {X}^s(\mathbb {R}^3))\cap L^1_{T}({\mathcal {X}}^{s+2\beta }(\mathbb {R}^3))]\), provided that \(\alpha ,\beta \in (\frac{1}{2},1]\) and \(\max \big \{\frac{\alpha (1-2\beta )}{\beta },\frac{\beta (1-2\alpha )}{\alpha }\big \} \le s<0\) (here \(\alpha \) and \(\beta \) are related to the fractional Laplacian that appears in the GMHD system). Furthermore, we prove that if \(T^*\) (finite) is the first blow-up instant of the solution (ub)(xt), then \( \lim _{t\nearrow T^*}\Vert (u,b)(t)\Vert _{\mathcal {X}^s(\mathbb {R}^3)}=\infty \), whether \(\max \big \{1-2\alpha ,1-2\beta ,\frac{\alpha (1-2\beta )}{\beta },\frac{\beta (1-2\alpha )}{\alpha }\big \}< s<0\) and \(\alpha ,\beta \in (\frac{1}{2},1]\).


GMHD equations Local existence and uniqueness of solutions Limit superior Lei–Lin spaces 

Mathematics Subject Classification

35B44 35Q30 76D03 76D05 76W05 



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Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de SergipeSão CristóvãoBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de PernambucoRecifeBrazil
  3. 3.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

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