ordinary differential equation; continuity equation; nonlocal conservation laws; two-dimensional Euler equation; geometric measure theory; regular Lagrangian flow
Crippa Gianluca, Ligabue Silvia, Saffirio Chiara (2018), Lagrangian solutions to the Vlasov-Poisson system with a point charge, in
Kinet. Relat. Models, 11(6), 1277-1299.
Crippa Gianluca, Nobili Camilla, Seis Christian, Spirito Stefano (2017), Eulerian and Lagrangian solutions to the continuity and Euler equations with {$L^1$} vorticity, in
SIAM J. Math. Anal., 49(5), 3973-3998.
Crippa Gianluca, Gusev Nikolay, Spirito Stefano, Wiedemann Emil (2017), Failure of the chain rule for the divergence of bounded vector fields, in
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 17(1), 1-18.
Choudhury Anupam Pal, Crippa Gianluca, Spinolo Laura V. (2017), Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system, in
Z. Angew. Math. Phys., 68(6), 138-21.
Bianchini Stefano, Colombo Maria, Crippa Gianluca, Spinolo Laura V. (2017), Optimality of integrability estimates for advection-diffusion equations, in
NoDEA Nonlinear Differential Equations Appl., 24(4), 33-19.
Bohun Anna, Bouchut Françcois, Crippa Gianluca (2016), Lagrangian flows for vector fields with anisotropic regularity, in
Ann. Inst. H. Poincaré Anal. Non Linéaire, 33(6), 1409-1429.
Bohun Anna, Bouchut Françcois, Crippa Gianluca (2016), Lagrangian solutions to the 2D Euler system with {$L^1$} vorticity and infinite energy, in
Nonlinear Anal., 132, 160-172.
Bohun Anna, Bouchut Françcois, 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
Netw. Heterog. Media, 11(2), 301-311.
Crippa Gianluca, Gusev Nikolay, Spirito Stefano, Wiedemann Emil (2015), Non-uniqueness and prescribed energy for the continuity equation, in
Commun. Math. Sci., 13(7), 1937-1947.
Crippa Gianluca, Spirito Stefano (2015), Renormalized solutions of the 2D Euler equations, in
Comm. Math. Phys., 339(1), 191-198.
Colombo Maria, Crippa Gianluca, Spirito Stefano (2015), Renormalized solutions to the continuity equation with an integrable damping term, in
Calc. Var. Partial Differential Equations, 54(2), 1831-1845.
Crippa Gianluca, Semenova Elizaveta, Spirito Stefano (2015), Strong continuity for the 2D Euler equations, in
Kinet. Relat. Models, 8(4), 685-689.
AlbertiGiovanni, CrippaGianluca, MazzucatoAnna, Exponential self-similar mixing by incompressible flows, in
J. Amer. Math. Soc. , 913.
We investigate well-posedness and further properties for the continuity equation and for the ordinary differential equation out of the classical smooth (Lipschitz regular) framework. The leading theme is the search for quantitative estimates: stability, compactness and regularity statements in which we give an explicit control of the quantities under analysis in terms of natural bounds on the data.This quantitative analysis allows us to study several applications to nonlinear partial differential equations: we address various questions about existence, continuity with respect to initial data, convergence of singular approximations and convergence of nonlocal approximations for the Euler equation, the Vlasov-Poisson equation, and the Burgers' equation.