Convex integration solutions to the transport equation with full dimensional concentration

2019/02/22 by Stefano Modena, Gabriel Sattig · 59 citations
Mathematics · #Geometry #Mathematical analysis #Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Regular polygon #math.AP

paper · pdf · doi:10.1016/j.anihpc.2020.03.002

published in Annales de l Institut Henri Poincaré C Analyse Non Linéaire 37(5), 1075-1108 (Elsevier BV) · 33 pages

arxiv created 2019/02/22 · openalex publication_date 2020/03/18 · arxiv updated 2020/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We construct infinitely many incompressible Sobolev vector fields u ∈ C_tW_x^{1, p} on the periodic domain \mathbbT^d for which uniqueness of solutions to the transport equation fails in the class of densities ρ ∈ C_tL_x^p , provided 1/ p + 1/ p> 1 + 1/ d . The same result applies to the transport-diffusion equation, if, in addition, p^′ < d .

Citations

• Elliptic Partial Differential Equations of Second Order
• A proof of Onsager's conjecture
• Stationary and discontinuous weak solutions of the Navier-Stokes equations