## Inverse source problems and controllability for the stokes and navier-stokes equations

Tesis

Date

2016Metadata

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Osses Alvarado, Axel

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**Inverse source problems and controllability for the stokes and navier-stokes equations**

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Abstract/Review

This thesis is focused on the Navier{Stokes system for incompressible
uids with either
Dirichlet or nonlinear Navier{slip boundary conditions. For these systems, we exploit some
ideas in the context of the control theory and inverse source problems. The thesis is divided
in three parts.
In the rst part, we deal with the local null controllability for the Navier{Stokes system
with nonlinear Navier{slip conditions, where the internal controls have one vanishing component.
The novelty of the boundary conditions and the new estimates with respect to the
pressure term, has allowed us to extend previous results on controllability for the Navier{
Stokes system. The main ingredients to build our result are the following: a new regularity
result for the linearized system around the origin, and a suitable Carleman inequality for the
adjoint system associated to the linearized system. Finally, xed point arguments are used
in order to conclude the proof.
In the second part, we deal with an inverse source problem for the N- dimensional Stokes
system from local and missing velocity measurements. More precisely, our main result establishes
a reconstruction formula for the source F(x; t) = (t)f(x) from local observations of
N ����� 1 components of the velocity. We consider that f(x) is an unknown vectorial function,
meanwhile (t) is known. As a consequence, the uniqueness is achieved for f(x) in a suitable
Sobolev space. The main tools are the following: connection between null controllability and
inverse problems throughout a result on null controllability for the N- dimensional Stokes
system with N ����� 1 scalar controls, spectral analysis of the Stokes operator and Volterra integral
equations. We also implement this result and present several numerical experiments
that show the feasibility of the proposed recovering formula.
Finally, the last chapter of the thesis presents a partial result of stability for the Stokes
system when we consider a source F(x; t) = R(x; t)g(x), where R(x; t) is a known vectorial
function and g(x) is unknown. This result involves the Bukhgeim-Klibanov method for
solving inverse problems and some topics in degenerate Sobolev spaces.

General note

Doctor en Ciencias de la Ingeniería, Mención Modelación Matemática

Identifier

URI: http://repositorio.uchile.cl/handle/2250/141346
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