geometric measure theory; continuity equation; two-dimensional Euler equation; transport equation; ordinary differential equation; hyperbolic conservation laws; vortex dynamics
Crippa Gianluca, Lopes Filho Milton, Miot Evelyne, Nussenzveig Lopes Helena (2016), Flows of vector fields with point singularities and the vortex-wave system, in Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
, 36(5), 2405-2417.
Bohun Anna, Bouchut Francois, Crippa Gianluca (2016), Lagrangian flows for vector fields with anisotropic regularity, in Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
, 33(6), 1409-1429.
Bohun Anna, Bouchut Francois, Crippa Gianluca (2016), Lagrangian solutions to the 2D Euler system with $L^1$ vorticity and infinite energy, in Nonlinear Analysis: Theory, Methods & Applications
, 132, 160-172.
Bohun Anna, Bouchut Francois, Crippa Gianluca (2016), Lagrangian solutions to the Vlasov-Poisson system with $L^1$ density, in J. Differential Equations
, 260(4), 3576-3597.
Colombo Maria, Crippa Gianluca, Spirito Stefano (2016), Logarithmic estimates for continuity equations, in Networks and Heterogeneous Media
, 11(2), 301-311.
Crippa Gianluca, Gusset Nikolay, Spirito Stefano, Wiedemann Emil (2015), Non-uniqueness and prescribed energy for the continuity equation, in Communications in Mathematical Sciences
, 13(7), 1937-1947.
Crippa Gianluca, Spirito Stefano (2015), Renormalized Solutions of the 2D Euler Equations, in Communications in Mathematical Physics
, 339(1), 191-198.
Colombo Maria, Crippa Gianluca, Spirito Stefano (2015), Renormalized solutions to the continuity equation with an integrable damping term, in Calculus of Variations and Partial Differential Equations
, 54(2), 1831-1845.
Crippa Gianluca, Semenova Elizaveta, Spirito Stefano (2015), Strong continuity for the 2D Euler equations, in Kinetic and Related Models
, 8(4), 685-689.
Crippa Gianluca, Donadello Carlotta, Spinolo Laura Valentina (2014), A note on the initial-boundary value problem for continuity equations with rough coefficients, in Hyperbolic Problems: Theory, Numerics, Applications. HYP2012
, AIMS, United States.
Alberti Giovanni, Bianchini Stefano, Crippa Gianluca (2014), A uniqueness result for the continuity equation in two dimensions, in Journal of the European Mathematical Society (JEMS)
, 16(2), 201-234.
Ambrosio Luigi, Crippa Gianluca (2014), Continuity equations and ODE flows with non-smooth velocity, in Proceedings of the Royal Society of Edinburgh, Section A: Mathematics
Alberti Giovanni, Crippa Gianluca, Mazzucato Anna (2014), Exponential self-similar mixing and loss of regularity for continuity equations, in Comptes Rendus Mathematique
, 352(11), 901-906.
Crippa Gianluca, Donadello Carlotta, Spinolo Laura Valentina (2014), Initial-boundary Value Problems for Continuity Equations with BV Coefficients, in Journal de Mathematiques Pures et Appliquees
, 102, 79-98.
Alberti Giovanni, Bianchini Stefano, Crippa Gianluca (2014), On the L^p differentiability of certain classes of functions, in Revista Matematica Iberoamericana
, 30(1), 349-367.
Crippa Gianluca (2014), Ordinary Differential Equations and Singular Integrals, in Hyperbolic Problems: Theory, Numerics, Applications. HYP2012
, AIMS, United States.
Crippa Gianluca, Lecureux-Mercier Magali (2013), Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, in Nonlinear Differential Equations and Applications NoDEA
, 20(3), 523-537.
Bouchut Francois, Crippa Gianluca (2013), Lagrangian flows for vector fields with gradient given by a singular integral, in Journal of Hyperbolic Differential Equations
, 10(2), 235-282.
Alberti Giovanni, Bianchini Stefano, Crippa Gianluca (2013), Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples, in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze
, XII, 863-902.
Crippa Gianluca, Gusev Nikolay, Spirito Stefano, Wiedemann Emil, Failure of the chain rule for the divergence of bounded vector fields, in Annali della Scuola Normale Superiore di Pisa
We will address various open problems related to the behaviour of the continuity equation and of the associated ordinary differential equation when the vector field governing the transport process lacks the usual (Lipschitz) regularity properties. The motivations come from the applications of such results to nonlinear problems, originating in fluid dynamics or in the theory of conservation laws. Besides exploiting hyperbolic PDEs techniques, the analysis requires new tools from geometric measure theory, properly adapted in order to describe and control the irregular behaviours under consideration. One first line of work regards a precise understanding of further suitable weak settings in which the continuity equation and the ordinary differential equation are well-posed and enjoy additional properties (compactness or regularity of solutions, for instance). A second line will address some questions on two-dimensional incompressible nonviscous fluids, mainly in the framework of measure-valued vorticity.