Continuity equations and ODE flows with non-smooth velocity

Abstract

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable 'weak differentiability' assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

Renormalized Solutions of the 2D Euler Equations

“…We want to point out that the result will not be stated in the full generality but will be adapted to our setting. For a complete overview of the theory of renormalized solutions see for instance [2,3] and reference therein.…”

The Lagrangian Structure of the Vlasov–Poisson System in Domains with Specular Reflection

“…It is not a priori true that the trace of renormalized solutions can be taken in the strong sense, i.e., that the commutativity property holds for renormalized solutions. A counterexample can be found, for example, in [7,Remark 25] for traces taken in the temporal domain. Nevertheless, this will become true in this case once we introduce the theory of Lagrangian solutions.…”

On the Existence of a Scalar Pressure Field in the Brödinger Problem

“…With this definition, (ρ ε , c ε ) is clearly a solution to the continuity equation and the following inequality is classical (see formula (3.5) in [1] with Θ = | • | 2 /2):…”