2018/12/17 by Laura Caravenna, Caravenna, Laura, Gianluca Crippa +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1812.06817
openalex publication_date 2018/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a Lipschitz extension lemma in which the extension procedure\nsimultaneously preserves the Lipschitz continuity for two non-equivalent\ndistances. The two distances under consideration are the Euclidean distance\nand, roughly speaking, the geodesic distance along integral curves of a\n(possibly multi-valued) flow of a continuous vector field. The Lipschitz\nconstant for the geodesic distance of the extension can be estimated in terms\nof the Lipschitz constant for the geodesic distance of the original function.\nThis Lipschitz extension lemma allows us to remove the high integrability\nassumption on the solution needed for the uniqueness within the DiPerna-Lions\ntheory of continuity equations in the case of vector fields in the Sobolev\nspace W1,p, where p is larger than the space dimension, under the\nassumption that the so-called "forward-backward integral curves" associated to\nthe vector field are trivial for almost every starting point. More precisely,\nfor such vector fields we prove uniqueness and Lagrangianity for weak solutions\nof the continuity equation that are just locally integrable.\n