Made available in DSpace on 2015-05-15T11:46:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 454059 bytes, checksum: d0cd2983cb53098030085d8bc6449f12 (MD5) Previous issue date: 2010-03-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work is dedicated to the study of existence, uniqueness and stability for the nonlinear equation for Kawahara ut + ux + uxxx + upux - uxxxxx = 0 (p = 1; 2) , on a bounded domain. To prove the existence and uniqueness, we use techniques of nite di¤erences for the case p = 1 and semigroup theory for the case p = 2. Under e¤ect of a localized damping mechanism, we obtain an exponential decay (as t ! 1) for the energy associated to solutions of Kawahara equation. Combining energy estimatives, multipliers and compacteness argument, the stabilization result was reduced to prove a unique continuation property for the Kawahara equation. This property was proved using a result due to J. C. Saut and B. Sheurer (see [38]). / Este trabalho é dedicado ao estudo da existência, unicidade e estabilidade para a equação não linear de Kawahara ut + ux + uxxx + upux - uxxxxx = 0 (p = 1; 2) , em um domínio limitado. Para provar a existência e unicidade, usamos técnicas de semi-discretização para o caso p = 1 e, para o caso p = 2, utilizamos a teoria de semigrupos. Ao adicionarmos uma dissipação localizada, obtemos um decaimento exponencial (quando t ! 1) da energia associada às soluções da equação de Kawahara. Isto foi feito combinando estimativas de energia, técnicas de multiplicadores e argumentos de compacidade, fazendo com que o resultado de estabilização ficasse reduzido a provar uma propriedade de continuação única para a equação de Kawahara. Tal propriedade foi provada usando um resultado devido a J. C. Saut e B. Sheurer (ver [38]).

The thesis consists of two parts. The first part discusses the initial value problem of a fifth-order Korteweg-de Vries type of equation w_t + w_xxx - w_xxxxx - ⁿ&#8721;_j=1 a_jw^jw_x = 0, w(x, 0) = w₀(x) posed on a periodic domain x &#8712; [0, 2&#960;] with boundary conditions w_ix(0, t) = w_ix(2&#960;, t), i = 0, 2, 3, 4 and an L²-stabilizing feedback control law w_x(2&#960;, t) = &#945;w_x(0, t) + (1 - &#945;)w_xxx(0; t) where n is a fixed positive integer, a_j, j = 1, 2, ... n, &#945; are real constants, and |&#945;| < 1. It is shown that for w₀(x) &#8712; H¹_&#945;(0, 2&#960;) with the boundary conditions described above, the problem is locally well-posed for w &#8712; C([0, T]; H¹_&#945;(0, 2&#960;)) with a conserved volume of w, [w] = &#8747;^2&#960;₀ w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w₀(x)]/(2&#960;) as t &#8594; + &#8734;. The second part concerns wave motions on water in a rectangular basin with a wave generator mounted on a side wall. The linear governing equations are used and it is assumed that the surface tension on the free surface is not zero. Two types of generators are considered, flexible and rigid. For the flexible case, it is shown that the system is exactly controllable. For the rigid case, the system is not exactly controllable in a finite-time interval. However, it is approximately controllable. The stability problem of the system with the rigid generator controlled by a static feedback is also studied and it is proved that the system is strongly stable for this case. / Ph. D.

Dans cette thèse on étudie la stabilité orbitale des ondes solitaires de deux types d’équations d’évolution non linéaires: l’équation de Degasperis-Procesi (DP), qui est une équation du type Camassa-Holm, et l’équation de Kawahara généralisée (gKW), qui correspond à une équation de Korteweg-de Vries généralisée (gKdV) supplémentée d’un terme d’ordre 5. Sur le modèle DP on apporte une amélioration significative de la preuve de la stabilité d’un peakon donnée par Lin et Liu. Puis, en utilisant la méthode de Martel-Merle-Tsai adaptée par El Dika-Molinet dans le cas de l’équation de Camassa-Holm, on montre que la somme de N peakons, de vitesses croissantes et suffisamment distants les uns des autres à l’instant initial, est orbitalement stable. Sur le modèle de Kawahara généralisé, on prouve l’existence de deux branches d’ondes solitaires : l’une construite en appliquant le théorème des fonctions implicites au voisinage d’une onde solitaire explicite de gKW découverte par Dey. al., l’autre construite en résolvant un problème de minimisation sur R, avec une contrainte qui force la famille à converger vers le soliton explicite de l’équation de Korteweg-de Vries généralisée (gKdV) lorsque le coefficient devant l’opérateur d’ordre 5 tend vers 0. Par remise à l’échelle, on obtient ainsi une branche constituée d’ondes solitaires voyageant à faibles vitesses. On prouve ensuite que les ondes solitaires constituant ces deux branches sont orbitalement stables en appliquant la méthode spectrale introduite par Benjamin et des arguments de continuité. / No summary available