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

Abstract

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 $L^2$-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, Burguer's equations, 2D vorticity equation, 2D generalized SQG, 3D magneto geostrophic equations, among others.

References in corpus (3)

• On the regularity of a class of generalized quasi-geostrophic equations
• On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
• Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations