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Self-similar solutions for active scalar equations in Fourier–Besov–Morrey spaces

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Abstract

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of positive-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier–Besov–Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large \(L^{2}\)-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and couplings. Also, we show that solutions are asymptotically self-similar at infinity. Our results can be applied in a unified way for a number of active scalar PDEs like 1D models on dislocation dynamics in crystals, Burgers’ equation, 2D vorticity equation, 2D generalized SQG, 3D magneto-geostrophic equations, among others.

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Authors and Affiliations

  1. Departamento de Matemática, IMECC, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, CEP 13083-859, Brazil

    Lucas C. F. Ferreira & Lidiane S. M. Lima

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  1. Lucas C. F. Ferreira
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  2. Lidiane S. M. Lima
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Correspondence to Lucas C. F. Ferreira.

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Communicated by D. Lannes.

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Ferreira, L.C.F., Lima, L.S.M. Self-similar solutions for active scalar equations in Fourier–Besov–Morrey spaces. Monatsh Math 175, 491–509 (2014). https://doi.org/10.1007/s00605-014-0659-6

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