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Statistical solutions and invariant measures in Hydrodynamics
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This thesis concerns a probabilistic approach to the problem of existence of solutions for two-dimensional models in hydrodynamics and the study of their properties. The solutions we refer to are almost everywhere defined with respect to infinite-dimensional probability measures. Roughly speaking, these measures give the probability of finding the dynamics in a certain configuration at a given time. In light of this, probabilistic solutions correspond to configurations of the dynamical system selected with probability one. Initial data belong to the support of the measures (consisting typically of irregular functions). We recall previous results about existence and uniqueness of infinite-dimensional random dynamical systems and present the two-dimensional models from hydrodynamics considered in this thesis: periodic averaged-Euler equations; non-periodic Euler equations; a modification of the Euler equation and stochastic Navier-Stokes equations. For the two dimensional averaged-Euler equation we define a Gaussian invariant measure and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the energy is also constructed, as well as the corresponding flow. Poincaré recurrence theorem is used to show that the flow returns infinitely many times in a neighborhood of the initial state. For the 2D Euler equation on the plane we construct Gaussian invariant measures. We obtain them as the weak limit of those previously considered in [3] for the torus. We show the existence of solution with initial conditions on the support of the measures. Continuity of the velocity flow is proved. Also, we consider a modified Euler equation on R2. We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are quasi-invariant for the modified Euler flow. Almost everywhere with respect to such measures (and, in particular, for less regular initial conditions), the flow is shown to be globally defined. Finally we study the limit of a perturbed Navier-Stokes flow when the viscosity coefficients converges to zero. We show the existence of a weak limit.
Descrição
Tese de doutoramento, Matemática (Física Matemática e Mecânica dos Meios Contínuos), Universidade de Lisboa, Faculdade de Ciências, 2019
Palavras-chave
Teses de doutoramento - 2019