# The Foias Lectures

## Spring 2024

Date: | March 4, 2024 |

Time: | 4:00pm |

Location: | BLOC 117 |

Speaker: | Professor Camillo De Lellis, Institute of Advanced Studies |

Title: | Flows of nonsmooth vector fields |

Abstract: | Consider a vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field is Lipschitz in space, for every initial datum $x$ there is a unique trajectory $\gamma$ starting at $x$ at time $0$ and solving the ODE $\dot \gamma (t) = v (t, \gamma (t))$. The theorem looses its validity as soon as $v$ is slightly less regular. However, if we bundle all trajectories into a global map allowing $x$ to vary, a celebrated theory started by DiPerna and Lions in the 1980es shows that there is a unique such flow under very reasonable conditions and for much less regular vector fields. This has a lot of repercussions to several important partial differential equations where the idea of ``following the trajectories of particles'' plays a fundamental role. In this lecture I will review the fundamental ideas of the original theory and an alternative approach due to Gianluca Crippa and myself. |

Date: | March 5, 2024 |

Time: | 4:00pm |

Location: | BLOC 117 |

Speaker: | Professor Camillo De Lellis, Institute of Advanced Studies |

Title: | DiPerna-Lions theory and convex integration |

Abstract: | After reviewing the fundamental theorems of DiPerna-Lions and Ambrosio on flows of Sobolev vector fields, we will explore a number of sharpness questions related to them. Many of these questions have been answered in the last few years using "convex integration" methods, exported by Modena and Sz\'ekelyhidi to the context of transport equations from that of incompressible fluid dynamics (where they were first introduced 17 years ago by Sz\'ekelyhidi and myself). I will in particular touch upon a striking application: there are divergence-free Sobolev vector fields for which uniqueness of the trajectories fails for a positive measure set of initial conditions, while there is a unique sensible choice of one ``good'' trajectory for almost all initial condition, given by the DiPerna-Lions flow. This theorem was first proved in a joint work of Bru\'e, Colombo, and myself with convex integration techniques and later improved by Kumar using different techniques. |

Date: | March 7, 2024 |

Time: | 4:00pm |

Location: | BLOC 117 |

Speaker: | Professor Camillo De Lellis, Institute of Advanced Studies |

Title: | Anomalous dissipation and flows of rough vector fields |

Abstract: | Consider smooth solutions to the 3d Navier-Stokes for divergence-free vector fields $u^\varepsilon$: \[ \partial_t u^\varepsilon + {\rm div}\, (u^\varepsilon \otimes u^\varepsilon ) + \nabla p^\varepsilon = \varepsilon \Delta u^\varepsilon \] While the balance of the energy is \[ \frac{d}{dt} \int |u^\varepsilon|^2 (x,t)\, dx = - 2 \varepsilon \int |Du^\varepsilon|^2 (x,t)\, dx\, , \] it is a tenet of the theory of fully developed turbulence that in a variety of situations the size of the left hand side should typically be independent of $\varepsilon$. However this ``anomalous'' dissipation should not be triggered by an hypothetical initial datum $u^\varepsilon (\cdot, 0)$ in which we ignite lots of oscillations: a rule of thumb would be that, if the {\em linear} Stokes equations exhibit a similar behavior, then the roughness of the initial data is certainly excessive. Producing rigorous mathematical examples of this prediction is hard: a tight control on the smoothness of the initial data forces the solutions to converge to classical solutions of the Euler equation and therefore rules out any possible dissipation, unless one shows a quite severe blow-up for smooth solutions of Euler; for sufficiently rough initial data this obstruction is absent, but proving something about the solutions of Navier-Stokes becomes very challenging. In a recent joint work with Elia Bru\'e we study what happens if we introduce a forcing term $f^\varepsilon$. The problem is compatively easier and it can be proved that there is an exact regularity threshold below which anomalous dissipation happens in some examples and above which it would only be possible through a blow-up scenario. Surprisingly one side of the problem is linked with some fundamental questions about solving ODEs with rough coefficients. |