Gianluca Crippa

Estimates and regularity results for the DiPerna-Lions flow

Abstract In this paper we derive new simple estimates for ordinary differential equations with Sobolev coefficients. These estimates not only allow to recover some old and recent results in a simple direct way, but they also have some new interesting corollaries.

Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields

The aim of this book is to provide a self-contained introduction and an up-to-date survey on many aspects of the theory of transport equations and ordinary differential equations with non-smooth velocity fields. The interest in this topic is motivated by important issues in nonlinear PDEs, in particular conservation laws and fluid mechanics. A fascinating feature of this research area, which is currently of concern in mathematics, is the interplay between PDE techniques and geometric measure theory techniques. Several masterpieces appear in the related literature, balancing completely rigorous proofs with more heuristic arguments. A consistent part of the book is based on results obtained by the author in collaboration with other mathematicians. After a short introduction to the classical smooth theory, the book is divided into two parts. The first part focuses on the PDE aspect of the problem, presenting some general tools of this analysis, many well-posedness results, an abstract characterization of the well-posedness, and some examples showing the sharpness of the assumptions made. The second part, instead, deals with the ODE aspect of the problem, respectively by an abstract connection with the PDE, and by some direct and simple (but powerful) a priori estimates.

Uniqueness, renormalization, and smooth approximations for linear transport equations

Transport equations arise in various areas of fluid mechanics, but the precise conditions on the vector field for them to be well‐posed are still not fully understood. The renormalized theory of DiPerna and Lions for linear transport equations with an unsmooth coefficient uses the tools of approximation of an arbitrary weak solution by smooth functions, and also uses the renormalization property; that is, the possibility of writing an equation on a nonlinear function of the solution. Under some

A uniqueness result for the continuity equation in two dimensions

W^{1,1}

Some New Well-Posedness Results for Continuity and Transport Equations, and Applications to the Chromatography System

regularity assumption on the coefficient, well‐posedness holds. In this paper, we establish that these properties are indeed equivalent to the uniqueness of weak solutions to the Cauchy problem, without any regularity assumption on the coefficient. Coefficients with unbounded divergence but with bounded compression are also considered.

Continuity equations and ODE flows with non-smooth velocity

We characterize the autonomous, divergence-free vector elds b on the plane such that the Cauchy problem for the continuity equation @tu + div (bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain non-autonomous vector elds b with bounded divergence.

LAGRANGIAN FLOWS FOR VECTOR FIELDS WITH GRADIENT GIVEN BY A SINGULAR INTEGRAL

We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blow-up of the

Structure of level sets and Sard-type properties of Lipschitz maps

BV

Failure of the Chain Rule for the Divergence of Bounded Vector Fields

norm at the initial time. We apply these results (valid in any space dimension) to the

A Note on Two-Dimensional Transport with Bounded Divergence

k\times k