## Section: New Results

### Various topics

Participants : Sébastien Boyaval, Virginie Ehrlacher.

A new mathematical framework has been identified for the modelling of complex fluids in [42].
It allows one to incorporate rheological features of a real (non-ideal, visous and compressible) fluid and,
at the same time,
to compute flows as solution to a *hyperbolic system of conservation laws* complemented by an initial value.
In [42], the framework is specified for 2D hydrostatic flows of *Maxwell fluids*,
with a numerical finite-volume scheme preserving the positivity of mass and a Clausius-Duem inequality.
Formally, the (macroscopic) model has a (microscopic) molecular justification using a generalized Langevin equation for the distortion of the fluid texture.

On the other hand, recall that stochastic models are also used for the numerical simulation of hydrodynamical turbulence. In particular, a generalized Langevin equation can be used to model the "thermostated" velocity fluctuations in a "stationary" turbulent flow modelled as an invariant measure. But the interest for the effective numerical simulation of turbulent flows is not fully understood yet. In [43], S. Boyaval with S. Martel and J. Reygner (Ecole des Ponts) have studied the convergence of a discretization of a 1D stochastic scalar viscous conservation laws (a toy-model), for the numerical simulation of its invariant measure.

A. Benaceur (EDF), V. Ehrlacher and A. Ern (École des Ponts and Inria SERENA) developped a new EIM/reduced-basis method [10] for the reduction of parametrized variational inequalities with nonlinear constraints, and applied this method to the reduction of contact mechanics problems with non-coincident meshes.

V. Ehrlacher, D. Lombardi (Inria COMMEDIA), O. Mula (Université Paris-Dauphine) and F-X. Vialard (Université Paris-Est) developped new model-order reduction techniques based on the use of Wasserstein spaces for transport-dominated problems [51], which gives very encouraging results on several classes of conservative transport problems like the Burger's equation. Theoretical convergence rates are proved on some particular test cases.

J. Berendsen (Chemnitz, Germany), Martin Burger (Erlangen, Germany), V. Ehrlacher, J-F. Pietschmann (Chemnitz, Germany) proved the existence and uniqueness of strong solutions and weak-strong stability in a particular system of cross-diffusion equations [41]. It is in general very difficult to obtain such kind of results for general cross-diffusion systems. The proof for the particular system studied here relies on the fact that, when all the cross-diffusion coefficients of the system are equal to the same constant, the system boils down to a set of independent heat equations, for which uniqueness of strong solutions is trivial. The uniqueness of strong solutions was proved under the assumption that the cross-diffusion coefficients should not be close enough to one another.