Exponential self-similar mixing and loss of regularity for continuity equations

2014/07/09 by Giovanni Alberti, Alberti, Giovanni, Gianluca Crippa +3 · 1 citation
Engineering · Mathematics · #35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #math.AP #msc:35

paper · pdf · doi:10.48550/arxiv.1407.2631

8 pages, 3 figures, statement of Theorem 11 slightly revised

openalex publication_date 2014/07/09 · arxiv created 2014/09/18 · arxiv updated 2014/09/22 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.

Cited by

• Introduction to the theory of mixing for incompressible flows

Related

• Loss of regularity for the continuity equation with non-Lipschitz\n velocity field
• Exponential self-similar mixing by incompressible flows
• Cellular mixing with bounded palenstrophy
• Critical non Sobolev regularity for continuity equations with rough force fields
• Propagation of regularity for transport equations. A Littlewood-Paley approach