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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
Article
  • 39 Downloads

Abstract

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 \(\alpha ,\beta \in (\frac{1}{2},1]\).

Keywords

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

Mathematics Subject Classification

35B44 35Q30 76D03 76D05 76W05 

Notes

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© Springer Nature Switzerland AG 2019

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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