Hydrodynamic Limit of Nonlinear Diffusions with Fractional Laplacian Operators

2012/05/29 by Sylvia Serfaty, Serfaty, Sylvia, Juan Luis Vazquez +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #35K55 #35K65 #76S05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stochastic processes and financial applications #math.AP #msc:35K55 #msc:35K65 #msc:76S05

paper · pdf · doi:10.48550/arxiv.1205.6322

35 pages, second version

openalex publication_date 2012/05/29 · arxiv created 2012/06/28 · arxiv updated 2012/06/29 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a "mean field" equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the "elementary vortex patch".

Citations

• Some theorems on stable processes
• On explosions of solutions to a system of partial differential equations modelling chemotaxis
• Hyperbolic systems of conservation laws II
• Model for chemotaxis
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Related

• From non-local to local Navier-Stokes equations
• A microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions
• Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators
• Growing Solutions of the fractional p-Laplacian equation in the Fast\n Diffusion Range
• On the fractional diffusion for the linear Boltzmann equation with drift and general cross-section