Stefano Spirito

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

We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures.

On the construction of suitable weak solutions to the 3D Navier–Stokes equations in a bounded domain by an artificial compressibility method

In this paper we will prove that suitable weak solutions of three dimensional Navier-Stokes equations in bounded domain can be constructed by a particular type of artificial compressibility approximation.

Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with

In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an

L^1

L^1

Vorticity

function, extending the Lagrangian theory in cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in cite{Seis16a} and some tools from harmonic analysis introduced in cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~

On the Vanishing Viscosity Limit of 3D Navier-Stokes Equations under Slip Boundary Conditions in General Domains

L^1

Renormalized solutions to the continuity equation with an integrable damping term

, extending the proof for the

Renormalized Solutions of the 2D Euler Equations

L^p

NON-UNIQUENESS AND PRESCRIBED ENERGY FOR THE CONTINUITY EQUATION

case in cite{CrippaSpirito15}.