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Wavelets and the well-posedness of incompressible magneto-hydrodynamic equations in Besov type \(Q\)-space. (English) Zbl 1309.35086

Summary: In this paper, we introduce a class of Besov type \(Q\)-spaces \(\dot B_{p,p}^{\gamma_1,\gamma_2}(\mathbb R^n)\) to study the well-posedness of the fractional magneto-hydrodynamic (FMHD) equations. Applying wavelets and multi-resolution analysis, we obtain the boundedness of a semigroup operator from \(\dot B_{p,p}^{\gamma_1,\gamma_2}(\mathbb R^n)\) to some tent spaces \(\mathbb B_{p,m,m'}^{\gamma_1,\gamma_2}\). As an application, we prove the global well-posedness of equations (FMHD) with data in \(\dot B_{p,p}^{\gamma_1,\gamma_2}(\mathbb R^n)\). Compared with the method of Fourier transform, the advantage of our method can be applied to the well-posedness with initial data in \(\dot B_{p,p}^{\gamma_1,\gamma_2}(\mathbb R^n)\), where \(p\neq 2\).

MSC:

35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
35R11 Fractional partial differential equations
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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