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Continuity equations with non smooth velocity: quantitative estimates and applications to nonlinear problems

English title Continuity equations with non smooth velocity: quantitative estimates and applications to nonlinear problems
Applicant Crippa Gianluca
Number 156112
Funding scheme Project funding (Div. I-III)
Research institution Fachbereich Mathematik Departement Mathematik und Informatik Universität Basel
Institution of higher education University of Basel - BS
Main discipline Mathematics
Start/End 01.10.2014 - 30.09.2018
Approved amount 285'585.00
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Keywords (6)

ordinary differential equation; continuity equation; nonlocal conservation laws; two-dimensional Euler equation; geometric measure theory; regular Lagrangian flow

Lay Summary (Italian)

Lead
L'equazione differenziale ordinaria (ODE) e l'equazione di continuità (PDE) appaiono in modo molto naturale in molti contesti fisici e applicati. L'analisi di questi problemi in un contesto non regolare ha avuto molto successo negli ultimi anni, anche a causa delle numerose applicazioni a problemi collegati.
Lay summary

In questo progetto investigheremo la buona positura (esistenza, unicità e stabilità) ed ulteriori proprietà per le soluzioni della ODE e della PDE. Molto spesso la regolarità a disposizione non è sufficiente per una trattazione "classica", e quindi strumenti specifici devono essere introdotti.

 Il tema guida è quello della ricerca di stime quantitative per la ODE: esse daranno risultati di stabilità, compattezza e regolarità con dei controlli espliciti delle quantità che appaiono nelle stime.

 Inoltre, applicheremo questi risultati a delle equazioni nonlineari: le equazioni di Eulero, di Vlasov-Poisson e di Burgers. Ci interesseremo soprattutto a risultati di esistenza di soluzioni, di continuità rispetto ai dati iniziali, di convergenza di approssimazioni singolari e di convergenza di approssimazioni nonlocali. 

Direct link to Lay Summary Last update: 30.09.2014

Responsible applicant and co-applicants

Employees

Publications

Publication
Lagrangian solutions to the Vlasov-Poisson system with a point charge
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.
Eulerian and Lagrangian solutions to the continuity and Euler equations with {$L^1$} vorticity
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.
Failure of the chain rule for the divergence of bounded vector fields
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.
Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system
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.
Optimality of integrability estimates for advection-diffusion equations
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.
Lagrangian flows for vector fields with anisotropic regularity
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.
Lagrangian solutions to the 2D Euler system with {$L^1$} vorticity and infinite energy
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.
Lagrangian solutions to the Vlasov-Poisson system with {$L^1$} density
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.
Logarithmic estimates for continuity equations
Colombo Maria, Crippa Gianluca, Spirito Stefano (2016), Logarithmic estimates for continuity equations, in Netw. Heterog. Media, 11(2), 301-311.
Non-uniqueness and prescribed energy for the continuity equation
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.
Renormalized solutions of the 2D Euler equations
Crippa Gianluca, Spirito Stefano (2015), Renormalized solutions of the 2D Euler equations, in Comm. Math. Phys., 339(1), 191-198.
Renormalized solutions to the continuity equation with an integrable damping term
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.
Strong continuity for the 2D Euler equations
Crippa Gianluca, Semenova Elizaveta, Spirito Stefano (2015), Strong continuity for the 2D Euler equations, in Kinet. Relat. Models, 8(4), 685-689.
Exponential self-similar mixing by incompressible flows
AlbertiGiovanni, CrippaGianluca, MazzucatoAnna, Exponential self-similar mixing by incompressible flows, in J. Amer. Math. Soc. , 913.

Collaboration

Group / person Country
Types of collaboration
Laura V. Spinolo, IMATI-CNR, Pavia Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Maria Colombo, EPFL Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Camillo De Lellis, Universitaet Zurich Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Stefano Spirito Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Luigi Ambrosio, Scuola Normale Superiore di Pisa Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Francois Bouchut, Université Paris-Est France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Stefano Bianchini, SISSA, Trieste Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication

Scientific events



Self-organised

Title Date Place
Irregular transport: analysis and applications 26.06.2017 Basel, Switzerland

Communication with the public

Communication Title Media Place Year
Talks/events/exhibitions Length, Area, Volume, and... Non-integer dimensions in measure theory Western Switzerland Italian-speaking Switzerland Rhaeto-Romanic Switzerland German-speaking Switzerland 2016

Awards

Title Year
Premio Bartolozzi 2013 (Unione Matematica Italiana) 2015

Associated projects

Number Title Start Funding scheme
140232 Continuity equations with non smooth velocity: fluid dynamics and further applications 01.04.2012 Project funding (Div. I-III)

Abstract

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.
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