## Contact

Swiss National Science Foundation (SNSF)

Wildhainweg 3P.O. Box

CH-3001 Bern

Phone +41 31 308 22 22

Applicant | Crippa Gianluca |
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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 |

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

Lead |
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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 |

Name | Institute |
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Name | Institute |
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Publication |
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On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws |

Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field |

Failure of the chain rule for the divergence of bounded vector fields |

Flows of vector fields with point singularities and the vortex-wave system |

Lagrangian flows for vector fields with anisotropic regularity |

Lagrangian solutions to the 2D Euler system with $L^1$ vorticity and infinite energy |

Lagrangian solutions to the Vlasov-Poisson system with $L^1$ density |

Logarithmic estimates for continuity equations |

Non-uniqueness and prescribed energy for the continuity equation |

Renormalized Solutions of the 2D Euler Equations |

Renormalized solutions to the continuity equation with an integrable damping term |

Strong continuity for the 2D Euler equations |

A note on the initial-boundary value problem for continuity equations with rough coefficients |

A uniqueness result for the continuity equation in two dimensions |

Continuity equations and ODE flows with non-smooth velocity |

Exponential self-similar mixing and loss of regularity for continuity equations |

Initial-boundary Value Problems for Continuity Equations with BV Coefficients |

On the L^p differentiability of certain classes of functions |

Ordinary Differential Equations and Singular Integrals |

Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow |

Lagrangian flows for vector fields with gradient given by a singular integral |

Structure of level sets and Sard-type properties of Lipschitz maps: results and counterexamples |

Group / person | Country |
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Types of collaboration |
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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 |

Title | Date | Place |
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Summer School on Geometric Measure Theory and Geometric Analysis | 23.06.2014 | Basel, Switzerland |

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 |

HYP2012: 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications | 25.06.2012 | Padova, Italy, Italy |

Title | Year |
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Premio Bartolozzi | 2013 |

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

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.

Swiss National Science Foundation (SNSF)

Wildhainweg 3P.O. Box

CH-3001 Bern

Phone +41 31 308 22 22

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