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Theses

Fluids, graphs and Fourier transform : three incarnations of the laplacian

Abstract : This thesis is devoted to the study of the laplacian properties in three fully distinct contexts.In a first part, it will be used to smooth solutions of equations coming from incompressible fluid mechanics.As an application, we will show a result in the spirit of J. Serrin and his continuators' theorem.In a second part, the laplacien is seen as the stationary counterpart of the wave operator on a graph, whose eigenmodes and eigenfrequencies determine the propagation of perturbations on the graph.We explore and disentangle the ties between the graph's topology, its shape and its first nonzero eigenfrequency.In the last part, the laplacian is thought of as a linear operator which we wish to diagonalize in an appropriate basis, a goal which is intimately tied to the Fourier transform.Two major difficulties appear in our context : the noncommutativity of the groups of interest on the one hand, the appearance of a singular limit in the Fourier transform on the other hand.
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Guillaume Lévy. Fluids, graphs and Fourier transform : three incarnations of the laplacian. Analysis of PDEs [math.AP]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066321⟩. ⟨tel-01724100⟩

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