Maria Colombo:
"Flows of vector fields: existence and (non)uniqueness results"
Given a vector field in $ R^d$, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow (namely, the solution $X(t)$ of the ODE $X'(t) = b(t, X(t))$ from any initial datum $x \in R^d$) provided the vector field is sufficiently smooth. The theorem loses its validity as soon as v is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the ODE, and they show existence, uniqueness and stability for this notion of flow for much less regular vector fields. In this talk I will give an overview of the topic and provide a negative answer to the following long-standing open question: are the trajectories of the ODE unique for a.e. initial datum in R^d for vector fields as in Di Perna and Lions theorem? The result exploits the connection between the notion of flow and an associated PDE, the transport equation, and combines ingredients from probability theory, harmonic analysis, and the "convex integration" method for the construction of nonunique solutions to certain PDEs.
Tommaso Taddei:
"Registration-based model reduction of parameterized PDEs with sharp gradients"
We propose a model reduction procedure for rapid and reliable solution of parameterized advection-dominated problems. Accurate approximation of travelling parameter-dependent waves is extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov-Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We show numerical results for a one-dimensional shallow water model problem, to empirically demonstrate the potential of the method. We further discuss the extension to two-dimensional problems.
There will be a small break of 10 minutes between the two talks.
