Quelques problèmes mathématiques en hydrodynamique et physique quantique

Abstract : This thesis is organised in two relatively independent chapters. Chapter 1 is devoted to the study of some mathematical problems arising in the theory of hydrodynamic turbulence. Our results focus on questions related to the large deviations principle (LDP), Gallavotti–Cohen type symmetry, and ergodicity (existence and uniqueness of a stationary measure and its mixing properties) for a family of randomly forced PDE’s. We establish the LDP for parabolic PDE’s, such as the Navier–Stokes system or the complex Ginzburg–Landau equation, perturbed by a random kick force. Then we extend this result to the case of the damped nonlinear wave equation driven by a spatially regular white noise, by proving a local LDP. We establish a Gallavotti–Cohen type symmetry for the rate function of an entropy production functional for PDE’s with strong nonlinear dissipation, such as the Burgers equation. Finally, we prove a mixing property for the complex Ginzburg–Landau equation with a white-noise perturbation in any space dimension. In Chapter 2, we first consider the problem of controllability of a quantum particle by the amplitude of an electric field. The position of the particle is described by a wave function which obeys the bilinear Schrödinger equation. We are mainly interested in the global controllability problems of this equation. Using some variational methods, we establish approximate controllability, feedback stabilisation, and simultaneous controllability results. The second part of this chapter is concerned with the problem of controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force. We provide some examples of saturating spaces which ensure the approximate controllability of the system.
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Habilitation à diriger des recherches
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Vahagn Nersesyan. Quelques problèmes mathématiques en hydrodynamique et physique quantique. Analysis of PDEs [math.AP]. UVSQ, 2015. ⟨tel-01239580v1⟩

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