Ingeniero Civil Matemático / Este trabajo consiste principalmente en dos resultados matemáticos, basados en el estudio de ecuaciones dispersivas no lineales, la estabilidad de ciertas soluciones de las mismas, como así también la posible explosión en tiempo finito. En una primera parte, Capítulo 1, presentamos una breve introducción a los tópicos tratados en esta memoria. Se hace especial énfasis en la descripción de los conceptos de ecuación dispersiva, buen colocamiento, 2-solitones, estabilidad y explosión. En el Capítulo 2 probaremos que las soluciones de tipo 2-soliton de la ecuación de sine-Gordon (SG) son orbitalmente estables en el espacio de energía, el espacio natural para resolver este problema. Las soluciones que estudiamos son los 2-kink, kink-antikink y breather de SG. Con el objetivo de probar este resultado, utilizaremos las transformaciones de Bäcklund implementadas gracias al Teorema de la Función Implícita. Estas transformaciones nos permitirán reducir el problema de estabilidad para cada una de la soluciones, al caso de la solución cero. Probaremos estos resultados siguiendo el espíritu de un paper de M. A. Alejo y C. Muñoz, que trata el caso de la ecuación de Korteweg-de Vries modificada. Sin embargo, más adelante veremos que el caso de la ecuación de SG presenta varias nuevas dificultades dado el carácter vectorial de sus soluciones. Este resultado mejora los anteriores probados por M. A. Alejo et al., y entrega una primera demostración rigurosa de la estabilidad de los 2-solitones de la ecuación de SG en el espacio de energía. En el Capítulo 3 nuestro principal objetivo será estudiar nuevas propiedades de blow-up dispersivo para el sistema de Schrödinger-Korteweg-de Vries. Más precisamente, probaremos explosión para datos iniciales en H^2-(R)xH^{3/2-}(R), como consecuencia de mostrar previamente una nueva propiedad de persistencia del flujo asociado al sistema, establecida sobre ciertos espacios de Sobolev con pesos fraccionarios cuidadosamente escogidos. / Este trabajo ha sido parcialmente financiado por los proyectos Fondecyt Regular 1150202 y CMM Conicyt PIA AFB170001

In this work we deal with the nonlinear o.d.e u"+ku = εu³ which represents a spring-mass system with no damping but perturbed by external force εu³. We want to know how the spring constant k and the perturbed term act on the equation. So we study this equation by the way: (I) u" + ku = 0 (II)u" = u³ (III) u" + ku = εu³ During the period of calculating, we find that k, ε and energy constant E(0) play important roles in the properties of the solutions of the equation. Finally we give the relation about them.

Let Md be the moduli space of algebraic maps (morphisms) of degree d from P^1 to a fixed Grassmannian. The main purpose of this thesis is to provide an explicit construction of a compactification of Md satisfying the following property: the compactification is a smooth projective variety and the boundary is a simple normal crossing divisor. The main tool of the construction is blowing-up. We start with a smooth compactification given by Quot scheme, which we denote by Qd. The boundary Qd\Md is singular and of high codimension. Next, we give a filtration of the boundary Qd\Md by closed subschemes: Zd,0 subset Zd,1 subset ... Zd,d-1=Qd\Md. Then we blow up the Quot scheme Qd along these subschemes succesively, and prove that the final outcome is a compactification satisfying the desired properties. The proof is based on the key observation that each Zd,r has a smooth projective variety which maps birationally onto it. This smooth projective variety, denoted by Qd,r, is a relative Quot scheme over the Quot-scheme compactification Qr for Mr. The map from Qd,r to Zd,r is an isomorphism when restricted to the preimage of Zd,r\ Zd,r-1. With the help of the Qd,r's, one can show that the final outcome of the successive blowing-up is a smooth compactification whose boundary is a simple normal crossing divisor.

L'éclatement est une transformation jouant un rôle important en géométrie, car il permet de résoudre des singularités, de relier des variétés birationnellement équivalentes, et de construire des variétés possédant des propriétés inédites. Ce mémoire présente d'abord l'éclatement tel que développé en géométrie algébrique classique. Nous l'étudierons pour le cas des variétés affines et (quasi-)projectives, en un point, et le long d'un idéal et d'une sous-variété. Nous poursuivrons en étudiant l'extension de cette construction à la catégorie différentiable, sur les corps réels et complexes, en un point et le long d'une sous-variété. Nous conclurons cette section en explorant un exemple de résolution de singularité. Ensuite nous passerons à la catégorie symplectique, où nous ferons la même chose que pour le cas différentiable complexe, en portant une attention particulière à la forme symplectique définie sur la variété. Nous terminerons en étudiant un théorème dû à François Lalonde, où l'éclatement joue un rôle clé dans la démonstration. Ce théorème affirme que toute 4-variété fibrée par des 2-sphères sur une surface de Riemann, et différente du produit cartésien de deux 2-sphères, peut être équipée d'une 2-forme qui lui confère une structure symplectique réglée par des courbes holomorphes par rapport à sa structure presque complexe, et telle que l'aire symplectique de la base est inférieure à la capacité de la variété. La preuve repose sur l'utilisation de l'éclatement symplectique. En effet, en éclatant symplectiquement une boule contenue dans la 4-variété, il est possible d'obtenir une fibration contenant deux sphères d'auto-intersection -1 distinctes: la pré-image du point où est fait l'éclatement complexe usuel, et la transformation propre de la fibre. Ces dernières sont dites exceptionnelles, et donc il est possible de procéder à l'inverse de l'éclatement - la contraction - sur chacune d'elles. En l'accomplissant sur la deuxième, nous obtenons une variété minimale, et en combinant les informations sur les aires symplectiques de ses classes d'homologies et de celles de la variété originale nous obtenons le résultat. / The blow-up is a transformation which plays an important role in geometry, because it can be used to resolve singularities, relate birationally equivalent varieties, and construct varieties with new properties. This thesis first presents blowing-up as developped in classical algebraic geometry. We will study it in the case of affine and (quasi-)projective varieties, on a point and along an ideal and a subvariety. Then a discussion about its extension to the differential category will be carried out, over the real and complex fields, on a point and along a submanifold. An example of a resolution of singularity will then follow. Subsequently we will discuss blowing-up in the symplectic category, where we will do the same as for complex manifolds, paying careful attention to the symplectic form. To conclude, we will study a theorem by François Lalonde, where the symplectic blow-up plays a major part in proof. This theorem states that any 4-variety fibered by 2-spheres over a Riemann surface, and different than the Cartesian product of two 2-spheres, can be equiped with a 2-form giving it a symplectic structure ruled by curves that are holomorphic with respect to its almost-complex structure, and such that the symplectic area of the base is smaller that the capacity of the variety. In the proof, we blow up a ball in the 4-variety, and obtain a fibration containing two distinct spheres with a self-intersection equal to -1: the pre-image of the point where the usual complex blow-up is done, and the proper transform of the fiber. These two are exceptional, so it is possible to do the inverse operation - the blow down - on each of them. By blowing down the latter, we get a minimal variety, and by combining information about the symplectic area of its homology classes and of those of the original variety, we obtain the result.

Cette thèse porte sur l'étude des phénomènes d'explosion pour certaines équations aux dérivées partielles dispersives et plus particulièrement pour l'équation de Schrodinger non linéaire. Ces phénomènes ont été beaucoup étudiés et notamment dans le cas Euclidien. On s'intéresse ici à des cas où l'espace n'est plus l'espace Euclidien. Cela comprend en particulier l'étude des trois prototypes : domaine de l'espace Euclidien, tore (courbure nulle), sphère (courbure positive) et espace hyperbolique (courbure négative). Concernant l'équation de Schrodinger, plusieurs résultats ont montré que la métrique pouvait influencer le comportement qualitatif des solutions, en particulier les propriétés dispersives des solutions et le seuil critique d'existence locale pour le problème de Cauchy. Plusieurs résultats concernant l'explosion sont ensuite venus confirmer ces phénomèmes. Dans cette thèse, on se propose de poursuivre cette étude. / In this thesis, we study blow-up behavior of solutions for dispersive equations, more precisely for the nonlinear Schr"odinger equation. This has been studied essentially in the Euclidean case. In this work, we are interested in the case where the equation is posed on a general manifold; this includes the case of a domain of the Euclidean space, torus (zero curvature); the sphere (non negative curvature) and the hyperbolic space (negative curvature). For the Schr"odinger equation, several results proved that the metric could change the qualitative behavior of the solutions, in particular dispersive properties and the critical threshold of existence for the Cauchy problem. Then, some results showed that blow-up theory is also concerned. In this work, we continue this study.

Made available in DSpace on 2014-06-12T18:29:03Z (GMT). No. of bitstreams: 2 arquivo6736_1.pdf: 1174894 bytes, checksum: 1469e3c3f5a8adcf9a070f5fa5cc9e27 (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2010 / Universidade Federal Rural de Pernambuco / Estudamos o sistema parabólico não-local acoplado ut &#8722; &#916;u = &#8747; t 0 (t &#8722; s)&#8722;1 |v|p&#8722;1v(s)ds, vt &#8722; &#916;v = &#8747; t 0 (t &#8722; s)&#8722;2 |u|q&#8722;1u(s)ds onde 0 &#8804; &#947;1, &#947;2 < 1 e p, q &#8805; 1. Consideramos o problema em (0, T)×RN e um problema de Dirichlet em (0, T)×&#937;, com &#937; &#8834; RN domínio limitado e fronteira regular. Admitimos que os dados iniciais u(0), v(0) &#8712; C0(RN) e u(0), v(0) &#8712; C0(&#937;), respectivamente. Encontramos condições que garantem a existência de solução global e a explosão num tempo finito de qualquer solução do sistema em questão

One of the most prominent open problems in mathematical physics is determining whether solutions to the incompressible three-dimensional (3D) Navier-Stokes system, corresponding to arbitrarily large smooth initial data, remain regular for arbitrarily long times. A promising approach to this problem relies on the fact that both the smoothness of classical solutions and the uniqueness of weak solutions in 3D flows are ultimately controlled by the growth properties of the $H^1$ seminorm of the velocity field U, also known as the enstrophy. In this context, the sharpness of analytic estimates for the instantaneous rate of growth of the $H^2$ seminorm of U in two-dimensional (2D) flows, also known as palinstrophy, and for the instantaneous rate of growth of enstrophy in 3D flows, is assessed by numerically solving suitable constrained optimization problems. It is found that the instantaneous estimates for both 2D and 3D flows are saturated by highly localized vortex structures. Moreover, finite-time estimates for the total growth of palinstrophy in 2D and enstrophy in 3D are obtained from the corresponding instantaneous estimates and, by using the (instantaneously) optimal vortex structures as initial conditions in the Navier-Stokes system and numerically computing their time evolution, the finite-time estimates are found to be uniformly sharp for 2D flows, and sharp over increasingly short time intervals for 3D flows. Although computational in essence, these results indicate a possible route for finding an extreme initial condition for the Navier-Stokes system that could lead to the formation of a singularity in finite time. / Thesis / Doctor of Philosophy (PhD)

Neste trabalho abordamos dois exemplos de equações de difusão não locais do tipo Neumann: o problema linear homogêneo e um semilinear com termo de reação representado pela função f(u) = u|u|^(p-1). Em ambos os casos, apresentamos condições de existência e unicidade de soluções e analisamos seu comportamento em relação ao tempo. Estudamos uma discretização para o problema linear e a utilizamos para realizar simulações numéricas nas quais podemos verificar algumas das propriedades demonstradas. Também simulamos o problema semilinear observando o comportamento de suas soluções mesmo em casos em que as hipóteses dos teoremas apresentados não são todas satisfeitas. / In this work we approach two examples of nonlocal diffusion equations of Neumann type: the homogeneous linear problem and a semilinear with a reaction term represented by the function f(u) = u|u|^(p-1). In both cases, we present conditions of existence and uniqueness of solutions and we analyze their behavior with respect to time. We study a discretization to the linear problem and use it to perform numerical experiments in order to illustrate some of the demonstrated properties. We also simulate the semilinear problem observing the behavior of its solutions even in cases where the hypothesis of the presented theorems are not all satisfied.

Relaxation oscillations are highly non-linear oscillations, which appear to feature many important biological phenomena such as heartbeat, neuronal activity, and population cycles of predator-prey type. They are characterized by repeated switching of slow and fast motions and occur naturally in singularly perturbed ordinary differential equations, which exhibit dynamics on different time scales. Traditionally, slow-fast systems and the related oscillatory phenomena -- such as relaxation oscillations -- have been studied by the method of the matched asymptotic expansions, techniques from non-standard analysis, and recently a more qualitative approach known as geometric singular perturbation theory. It turns out that relaxation oscillations can be found in a more general setting; in particular, in slow-fast systems, which are not written in the standard form. Systems in which separation into slow and fast variables is not given a priori, arise frequently in applications. Many of these systems include additionally various parameters of different orders of magnitude and complicated (non-polynomial) non-linearities. This poses several mathematical challenges, since the application of singular perturbation arguments is not at all straightforward. For that reason most of such systems have been studied only numerically guided by phase-space analysis arguments or analyzed in a rather non-rigorous way. It turns out that the main idea of singular perturbation approach can also be applied in such non-standard cases. This thesis is concerned with the application of concepts from geometric singular perturbation theory and geometric desingularization based on the blow-up method to the study of relaxation oscillations in slow-fast systems beyond the standard form. A detailed geometric analysis of oscillatory mechanisms in three mathematical models describing biochemical processes is presented. In all the three cases the aim is to detect the presence of an isolated periodic movement represented by a limit cycle. By using geometric arguments from the perspective of dynamical systems theory and geometric desingularization based on the blow-up method analytic proofs of the existence of limit cycles in the models are provided. This work shows -- in the context of non-trivial applications -- that the geometric approach, in particular the blow-up method, is valuable for the understanding of the dynamics of systems with no explicit splitting into slow and fast variables, and for systems depending singularly on several parameters.

在這篇論文中，我們討論具有初始值條件的二階微分方程 □□□□□ 和系統微分方程 □□□□□等問題。 我們利用能量方法（即能量是常數的特性）來探討上述方程解之特性。例如：生成時間、爆破和爆破速率以及局部解的漸進行為。 / In this paper we shall consider the initial value problem for second order differential equation of the form □□□□□ and the system □□□□□ . We shall discuss the blow-up properties, such as the life-span, the blow-up rate and blow-up constants, and the asymptotic behavior of the global solution by using the energy method.