Section: New Results

Various topics

A. Bakhta (École des Ponts) and V. Ehrlacher [28] have studied a system of PDEs modeling the cross-diffusion of different atomic species in a crystalline solid thin film during a Physical Vapor Deposition process, coupled with the evolution of the domain as external chemical species fluxes are absorbed at the surface of the solid layer. This model leads to a system of degenerate elliptic cross-diffusion equations. They proved the existence of a global weak solution to this system in arbitrary dimension in the case of a constant domain using analysis tools from gradient flow theory. The existence of a global weak solution in a one-dimensional case with external fluxes was also proved. Under the assumption that this solution is unique, the existence of optimal external fluxes in order to achieve desired concentration profiles of the different species in the thickness of the solid layer at the end of the process was also obtained.

Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. V. Ehrlacher, in a joint work with C. Ortner (U. of Warwick) and A. Shapeev (Skolkovo Institute of Science and Technology) [39] have studied a mathematical framework within which the accuracy of different types of boundary conditions can be precisely assessed.

T. Lelièvre together with F. Casenave (Safran) and A. Ern (École des Ponts) have proposed in the short note [36] an analysis of the Empirical Interpolation Method which highlights the symmetry played by the two variables (parameter and space variable). A variant of the Empirical Interpolation Method is introduced in order to deal with situations where some observations have to be discarded, and the number of observed values is thus different for the two variables.

In collaboration with P.-L. Lions (Collège de France), C. Le Bris has written an extensive set of lecture notes on parabolic equations with irregular data (initial conditions and parameter coefficients). These lecture notes correspond to joint works between the two authors and to an expanded version of the works by P.-L. Lions specifically exposed in his lectures delivered at Collège de France in 2012–2013. The application of the theory to the specific context of stochastic differential equations with irregular coefficients is also examined.