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A recent study (Lee et al., 2021) has shown that contrails are the main contributor to aviation-related radiative forcing. However, the same study shows that this contribution is highly imprecise due to numerous uncertainties. Among the most important are the numerous contingencies regarding the vertical and horizontal extent of ice plumes, as well as their altitude, which may differ from the flight level of the emitting aircraft, rising to hundreds of meters. This uncertainty is largely due to its interaction with the aircraft’s dynamic wake, which, very soon after the aircraft’s passage, is reduced to two counter-rotating vortices known as wingtip vortices.

Our goal is to highlight some of the deep links between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes a non-reversible dynamic, so that one is interested in schemes only involving forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control~$u$ which is a finite sum of Dirac masses. The general goal is then to find a control such that the flow of $f_0 + u(t) f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results concerning numerical splitting methods, and we prove a handful of new ones, with an emphasis on splittings with additional positivity conditions on the coefficients. First, we show that there exist numerical schemes of any arbitrary order involving only forward flows of $f_0$ if one allows complex coefficients for the flows of $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to the small-time local controllability of a system. Second, for real-valued coefficients, we show that the well-known order restrictions are linked with so-called "bad" Lie brackets from control theory, which are known to yield obstructions to small-time local controllability. We use our recent basis of the free Lie algebra to precisely identify the conditions under which high-order methods exist.

Les travaux présentés dans ce mémoire concernent différentes équations de la théorie cinétique des gaz (Boltzmann, Landau, Fokker-Planck) et s'articulent autour de deux grands axes :- développer une théorie de Cauchy dans un cadre perturbatif de solutions fortes et étudier le comportement en temps grand des solutions,- étudier des problèmes de limites hydrodynamiques.Qu'elles s'inscrivent dans l'un ou l'autre de ces deux axes, les études menées reposent sur l'analyse de problèmes linéaires au moyen de différentes théories : théorie des semi- groupes, théorie spectrale, hypocoercivité, élargissement d'espace, argument de perturbation, hypoellipticité etc... Ainsi, nous appliquons et/ou étendons ces techniques à de nouvelles situations : EDP de nouvelle nature, équations discrètes, nouveau cadre fonctionnel notamment.