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Continuity equations with non smooth velocity: fluid dynamics and further applications

Applicant Crippa Gianluca
Number 140232
Funding scheme Project funding
Research institution Fachbereich Mathematik Departement Mathematik und Informatik Universität Basel
Institution of higher education University of Basel - BS
Main discipline Mathematics
Start/End 01.04.2012 - 30.09.2014
Approved amount 150'713.00
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Keywords (7)

geometric measure theory; continuity equation; two-dimensional Euler equation; transport equation; ordinary differential equation; hyperbolic conservation laws; vortex dynamics

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 abbiamo analizzato vari problemi relativi alla ODE e alla PDE in un contesto non regolare. In particolare, abbiamo esteso alcune stime rilevanti per la buona positura a contesti meno sommabili o meno regolari e applicato tali risultai ad alcune equazioni non lineari. Inoltre abbiamo studiato una caratterizzazione della buona positura in due dimensioni, la buona positura per soluzioni misura di equazioni di continuità non locali, il problema dei valori al bordo per l'equazione di continuità su un dominio, e l'esistenza di soluzioni lagrangiane per il "vortex-wave system".

 

Direct link to Lay Summary Last update: 11.11.2014

Responsible applicant and co-applicants

Employees

Name Institute

Publications

Publication
On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws
Colombo Maria, Crippa Gianluca, Graff Marie, Spinolo Laura V. (2021), On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws, in ESAIM: Mathematical Modelling and Numerical Analysis, 55(6), 2705-2723.
Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field
Alberti Giovanni, Crippa Gianluca, Mazzucato Anna L. (2019), Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field, in Annals of PDE, 5(1), 9-9.
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 ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 1-18.
Flows of vector fields with point singularities and the vortex-wave system
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.
Lagrangian flows for vector fields with anisotropic regularity
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.
Lagrangian solutions to the 2D Euler system with $L^1$ vorticity and infinite energy
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.
Lagrangian solutions to the Vlasov-Poisson system with $L^1$ density
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.
Logarithmic estimates for continuity equations
Colombo Maria, Crippa Gianluca, Spirito Stefano (2016), Logarithmic estimates for continuity equations, in Networks and Heterogeneous Media, 11(2), 301-311.
Non-uniqueness and prescribed energy for the continuity equation
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.
Renormalized Solutions of the 2D Euler Equations
Crippa Gianluca, Spirito Stefano (2015), Renormalized Solutions of the 2D Euler Equations, in Communications in Mathematical Physics, 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 Calculus of Variations and 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 Kinetic and Related Models, 8(4), 685-689.
A note on the initial-boundary value problem for continuity equations with rough coefficients
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.
A uniqueness result for the continuity equation in two dimensions
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.
Continuity equations and ODE flows with non-smooth velocity
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, 1191-1244.
Exponential self-similar mixing and loss of regularity for continuity equations
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.
Initial-boundary Value Problems for Continuity Equations with BV Coefficients
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.
On the L^p differentiability of certain classes of functions
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.
Ordinary Differential Equations and Singular Integrals
Crippa Gianluca (2014), Ordinary Differential Equations and Singular Integrals, in Hyperbolic Problems: Theory, Numerics, Applications. HYP2012.
Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow
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.
Lagrangian flows for vector fields with gradient given by a singular integral
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.
Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples
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.

Collaboration

Group / person Country
Types of collaboration
Emil Wiedemann Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Francois Bouchut, Universite 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
Giovenni Alberti, Universita' di Pisa Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Laura V. Spinolo Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Evelyne Miot, CNRS & Universite d'Orsay France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Maria Colombo 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
Anna L. Mazzucato United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Stefano Spirito Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Carlotta Donadello, Université de Franche-Comté France (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

Scientific events



Self-organised

Title Date Place
Two days on Hyperbolic PDEs, Geometric Measure Theory and Optimal Transport' 28.10.2013 SISSA Trieste, Italy
10th Meeting on Hyperbolic Conservation Laws and Fluid Dynamics 11.07.2013 L'Aquila, Italy
Basel Junior Symposium in Analysis 12.02.2013 Basel, Switzerland

Awards

Title Year
Premio Bartolozzi 2013

Associated projects

Number Title Start Funding scheme
156112 Continuity equations with non smooth velocity: quantitative estimates and applications to nonlinear problems 01.10.2014 Project funding

Abstract

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.
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