Séminaire avec Sébastien Boyaval (MATHERIALS) et Karine Beauchard (ENS Rennes)
14h - Sébastien Boyaval (MATHERIALS) - Symmetric-hyperbolic conservation laws to model viscoelastic flows as Maxwell fluids
14h - Karine Beauchard (ENS Rennes) - Lie brackets and interpolation for controllability
Résumé de S. Boyaval : Among the many Partial Differential Equations (PDEs) proposed to govern the motions of various materials,
Maxwell's equations for viscoelastic flows are particularly interesting insofar as asymptotically, they can model both elastic solids as well as Newtonian fluids.
However, computations of multi-dimensional viscoelastic flows using Maxwell's PDEs have remained limited, at least without additional diffusion that blurs the hyperbolic character of Maxwell's PDEs. (Maxwell proposed only seminal hyperbolic PDEs for 1D shear waves).
We propose a new system of PDEs to model 3D viscoelastic flows of Maxwell fluids.
Our system, quasilinear and symmetric-hyperbolic, unequivocally models smooth flows on small times, while ensuring propagation of waves at finite-speed.
A key feature is the introduction of a structural tensor variable.
Our system rigorously unifies fluid models with elasto-dynamics for compressible solids.
Variations are possible for applications to environmental hydraulics (shallow-water flows) or materials engineering (non-isothermal flows).
Résumé de K. Beauchard : This talk will survey old and recent results on the local controllability of control systems modeled by ODEs, focussing on results stated using Lie brackets of the vector fields defining the dynamics. We will propose a unified approach to determine and prove obstructions to local controllability. This approach relies on a recent Magnus-type representation formula of the state, a new Hall basis of the free Lie algebra over two generators and Gagliardo-Nirenberg interpolation inequalities. This approach allows to recover the known necessary conditions, but also to prove a conjecture of 1986 due to Kawski and many other new necessary conditions. Finally, we will see how these results translate for PDEs, in particular the Schrödinger equation. This is a joint work with Frederic Marbach and Jeremy Le Borgne.