The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems

Ambrosio, Luigi

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 25-41 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

n this note we describe some recent developments of the theory of flows associated to vector fields with a low regularity with respect to the spatial variables, for instance with a Sobolev or BV regularity. After the illustration of some applica- tions of this theory to conservation laws and PDE's in fluid dynamics, we give an axiomatic presentation of the problem, based on a probabilistic approach inspired by the work of L.C. Young. In the final part we discuss very recent results on the regularity of the flow itself with respect to the spatial variables.

Résumé

In questa nota descriviamo alcuni recenti sviluppi della teoria dei flussi as- sociati a campi vettoriali poco regolari rispetto alle variabili spaziali, ad esempio con regolarità di tipo Sobolev o BV. Dopo aver illustrato alcune applicazioni a leggi di conservazione e equazioni della fluidodinamica, diamo una presentazione di tipo assiomatico del problema, usando un linguaggio di tipo probabilistico ispirato dalla teoria di L.C. Young. Nella parte finale discutiamo dei risultati ancora più recenti sulla regolarità del flusso stesso rispetto alle variabili spaziali.

Détail

Publié le : 2007-02-01

MR 2310956 | Zbl 1178.34069

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