On the fractional approach to quadratic nonlinear parabolic systems

TL;DR

This paper develops a method to analyze quadratic nonlinear parabolic systems with fractional diffusion, demonstrating convergence to classical diffusion models and revealing new phenomena, with applications to fluid dynamics and biological systems.

Contribution

Introduces a rigorous framework for fractional diffusion in nonlinear parabolic systems and establishes convergence rates to classical models, applicable to several important physical and biological equations.

Findings

Established convergence of fractional to classical diffusion models with explicit rates

Revealed unexpected phenomena in the convergence process

Applied results to Navier-Stokes, Magneto-hydrodynamics, Boussinesq, and Keller-Segel systems

Abstract

We introduce a general coupled system of parabolic equations with quadratic nonlinear terms and diffusion terms defined by fractional powers of the Laplacian operator. We develop a method to establish the rigorous convergence of the fractional diffusion case to the classical diffusion case in the strong topology of Sobolev spaces, with explicit convergence rates that reveal some unexpected phenomena. These results apply to several relevant real-world models included in the general system, such as the Navier-Stokes equations, the Magneto-hydrodynamics equations, the Boussinesq system, and the Keller-Segel system. For these specific models, this fractional approach is further motivated by previous numerical and experimental studies.