An overview on some results concerning the transport equation and its applications to conservation laws
Gianluca Crippa Laura V. Spinolo
Communications on Pure & Applied Analysis 2010, 9(5): 1283-1293 doi: 10.3934/cpaa.2010.9.1283
We provide an informal overview on the theory of transport equa- tions with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.
keywords: BV functions. Continuity equation conservation laws
Leaf superposition property for integer rectifiable currents
Luigi Ambrosio Gianluca Crippa Philippe G. Lefloch
Networks & Heterogeneous Media 2008, 3(1): 85-95 doi: 10.3934/nhm.2008.3.85
We consider the class of integer rectifiable currents without boundary in $\R^n\times\R$ satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
keywords: Metric spaces valued $BV$ functions Multi-valued functions. Integer rectifiable currents Currents in metric spaces Cartesian currents
Lagrangian solutions to the Vlasov-Poisson system with a point charge
Gianluca Crippa Silvia Ligabue Chiara Saffirio
Kinetic & Related Models 2018, 11(6): 1277-1299 doi: 10.3934/krm.2018050

We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of [17], proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in [8] in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure.

keywords: Vlasov-Poisson system plasma physics Lagrangian solutions transport equations

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