# Control in fluid mechanics and boundary layers

Abstract : This thesis is devoted to the study of the controllability of non linear partial differential equations in fluid mechanics. We are mostly interested in Burgers equation and Navier-Stokes equation. Our main goal is to prove small-time global results, even in the presence of boundary layers. We prove that it is possible to obtain such results by introducing a new method named: well prepared dissipation''. This method proceeds in two phases: first, a quick phase using the inviscid behavior of the system, then a longer phase during which the boundary layer dissipates all by itself. Both for Burgers and for Navier-Stokes with Navier slip-with-friction boundary conditions, we prove that this dissipation is sufficient if it has been well prepared. Moreover, we study a question of local null controllability for the Burgers equation with a single scalar control. We prove by enhancing a second order kernel approach that the system is not small time locally null controllable.
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• HAL Id : tel-01434170, version 2

### Citation

Frédéric Marbach. Control in fluid mechanics and boundary layers. Analysis of PDEs [math.AP]. Université Pierre et Marie Curie - Paris VI, 2016. English. ⟨NNT : 2016PA066442⟩. ⟨tel-01434170v2⟩

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