2013/09/20 by Lucas C. F. Ferreira, Ferreira, Lucas C. F., Lidiane S. M. Lima +1
Mathematics · #35A01 #35B40 #35C06 #35Q35 #35R11 #42B35 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A01 #msc:35B40 #msc:35C06 #msc:35Q35 #msc:35R11 #msc:42B35
paper · pdf · doi:10.48550/arxiv.1309.5369
arxiv created 2014/02/13 · arxiv updated 2014/02/14
We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-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 L2-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, Burguer's equations, 2D vorticity equation, 2D generalized SQG, 3D magneto geostrophic equations, among others.