In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial differential equations. The main purpose of this article is to prove an existence result for such type of stochastic differential equations with non-standard growth conditions. The main difficulty in the present problem is that the existence cannot be easily retrieved from the well known results under Lipschitz type of growth conditions [42]. / Dissertation (MSc)--University of Pretoria, 2010. / Mathematics and Applied Mathematics / unrestricted

Orientador: Simone Mazzini Bruschi / Banca: Cláudia Buttarello Gentile / Banca: Marta Cilene Gadoti / Resumo: Neste trabalho caracterizamos o conjunto dos pontos de equilíbrio de um problema parabólico quasilinear governado pelo p-Laplaciano, p > 2, e do problema parabólico governado pelo Laplaciano / Abstract: In this work we give a characterization set of the equilibrium points of a parabolic problem quasi-linear governed by the p-Laplacian, p > 2, and the a parabolic problem governed by the Laplacian / Mestre

Dans cette thèse, on s'intéresse au problème de Cauchy pour des équations quasi-linéaires dispersives. Pour une telle équation, l'enjeu est de montrer l'existence et l'unicité d'une solution de l'équation avec une donnée initiale prescrite dans un espace fonctionnel le plus large possible. Nous étudierons deux modèles décrivant l'évolution de la surface d'un fluide satisfaisant certaines conditions physiques.La première partie est consacrée à l'étude de l'équation de Kadomtsev-Petviashvili avec forte tension de surface (KP-I). Cette équation possède une structure Hamiltonienne et admet donc une fonctionnelle d'énergie préservée par le flot. Afin d'obtenir des solutions définies globalement en temps, on cherche donc à construire un flot dans l'espace de Banach naturellement associé à cette énergie. De plus, on se restreint à des espaces contenant des solutions particulières (les solitons linéaires de KdV), on impose donc une condition de périodicité dans la direction transverse à la propagation du fluide.On commence par illustrer le caractère quasi-linéaire de l'équation en montrant a priori que le flot dans cet espace ne peut pas être très régulier. Ceci restreint l'éventail des méthodes connues pour résoudre ce type de problème. On a donc recours à la méthode dite de restriction de la transformée de Fourier en temps petits développée récemment par Ionescu, Kenig et Tataru pour traiter ce même modèle sans condition de périodicité. On obtient ainsi l'existence globale et l'unicité de la solution du problème de Cauchy dans l'espace d'énergie. Enfin, on montre que le flot ainsi construit est continu mais pas uniformément continu sur les ensembles bornés de l'espace d'énergie.Une application intéressante de la construction d'un flot global sur l'espace d'énergie contenant les solitons linéaires est de lever une restriction sur les perturbations admissibles dans un résultat de Rousset-Tzvetkov sur la stabilité orbitale des solitons linéaires de faible vitesse.Dans la deuxième partie de la thèse, on s'intéresse à l'équation KP-I d'ordre cinq, qui est une alternative au modèle précédent dans le cas d'une tension de surface avoisinant une valeur critique pour laquelle l'effet dispersif devient plus faible. Pour cette équation, le comportement quasi-linéaire ne se manifeste que pour des données périodiques dans la direction transverse, et les autres cas avaient été étudiés précédemment dans les travaux de Saut et Tzvetkov. On considère ici des données également périodiques dans la direction de propagation. On montre que pour certains choix de périodes, le flot ne peut pas être régulier. Afin de traiter le problème indifféremment des périodes spatiales, on utilise donc une nouvelle fois la méthode précédente pour construire un flot global dans l'espace associé au Hamiltonien de ce modèle. / This thesis investigates the Cauchy problem for some quasilinear dispersive equations. Being given such an equation, the goal is then to construct a unique solution to this equation with a prescribed initial data belonging in a function space as large as possible. We will study two models describing the time evolution of the surface of a fluid in a particular regime.The first part of this thesis is devoted to the study of the Kadomtsev-Petviashvili equation in the case of strong surface tension (KP-I). This equation has a Hamiltonian structure, so it admits an energy functional which is preserved under the flow. In order to recover solutions which are globally defined in time, we thus seek to construct a flow map in the Banach sace naturally associated with the energy. In addition, we restrict ourself to spaces including some special solutions (the KdV line soliton), so we require the functions to be periodic in the transverse direction.We start by illustrating the quasilinear behaviour of the equation : we show that a flow map defined on this space cannot be too regular. This limits the range of applicable methods known to solve this kind of problem. We thus use the so-called small times Fourier restriction norm method recently developped by Ionescu, Kenig and Tataru to deal with the same model without the periodicity assumption. We thereby obtain the global existence and uniqueness of a solution to the Cauchy problem in the energy space. At last, we prove that the flow map constructed this way is continuous yet not uniformly continuous on the bounded sets of the energy space.An interesting application of the construction of a global flow on the energy space containing the line solitons is to get rid of an extra condition on admissible perturbations in a result of Rousset-Tzvetkov on the orbital stability of the small speed line solitons.In the second part of the thesis, we turn to the fifth-order KP-I equation, which is an alternative to the previous model should the tension surface come close to a critical value in which the dispersive effect becomes weaker. Regarding this equation, the quasilinear behaviour only manifests when solutions are periodic in the transverse direction, and the other cases were treated in the work of Saut and Tzvetkov. We study the case of functions which are also periodic in the direction of propagation, and we show that at least for some choice of periods the flow map fails to be smooth. In order to treat the problem regardless of the periods, we make another use of the method above to construct a global flow in the space associated to the Hamiltonian of the equation.

Made available in DSpace on 2014-06-11T19:24:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-05-07Bitstream added on 2014-06-13T20:07:58Z : No. of bitstreams: 1 martins_j_me_rcla.pdf: 650990 bytes, checksum: 197553843285b3dcdd899cddf89d0ab2 (MD5) / Universidade Estadual Paulista (UNESP) / Neste trabalho caracterizamos o conjunto dos pontos de equilíbrio de um problema parabólico quasilinear governado pelo p-Laplaciano, p > 2, e do problema parabólico governado pelo Laplaciano / In this work we give a characterization set of the equilibrium points of a parabolic problem quasi-linear governed by the p-Laplacian, p > 2, and the a parabolic problem governed by the Laplacian

No description available.

No description available.

Experimental results show that collagen fibers exhibit stress relaxation under tension and a highly anisotropic distribution. To further develop the earlier model of Stender [1], the collagen constituent was updated to reflect its intrinsic viscoelasticity and anisotropic distribution, and integrated with an existing mixture model with glycosaminoglycans and ground substance matrix. A two-term Prony series expansion of the quasi-linear viscoelastic model was chosen to model the viscoelastic properties of the collagen fibers. Material parameters were determined by using the simplex method to minimize the sum of squared errors between model results and experimental stress relaxation data of tissue in tension. Collagen elastic fiber modulus was calculated by fitting to the equilibrium data and viscoelastic parameters were determined by fitting to the relaxation curve. Results of newborn (~1-3 week old) untreated bovine articular cartilage explants from the patellar femoral groove as well as explants cultured in transforming growth factor-β1 (TGF-β1), from both the superficial (~0-0.5 mm from the articular surface) and middle (~0.5-1.0 mm from the articular surface) layers were compared to examine the effects of TGF- β1. TGF-β1 has been shown to maintain or even enhance mechanical properties of articular cartilage in compression and tension [2, 3] and this study continues with the hope that it may be used to improve tissue engineering of mature cartilage to better survive implantation in vivo for the successful repair of articular cartilage defects. Results show that TGF-β1 has a maturational effect on collagen, causing the tissue to become stiffer through an increase in elastic collagen fiber modulus and less viscous through shorter relaxation time and less stress relaxation (tissue retained a higher percentage of residual stress). The results of this study further advance the understanding of the effects of location and treatment with TGF-β1.

We consider the approach to regularity structures introduced by Otto, Sauer, Smith and Weber to obtain a priori bounds for quasi-linear SPDEs. This approach replaces the index set of trees, used in the original constructions of Hairer et. al., by multi-indices describing products of derivatives of the corresponding nonlinearity. The two tasks of this thesis are: - Construction and estimates of the model. We first provide the construction of a model in the regular, deterministic setting, where negative renormalization can be avoided. We later extend these ideas to the singular case, incorporating BPHZ-renormalization under spectral gap assumptions as a convenient input for an automated proof of the stochastic estimates of the singular model in the full subcritical regime. - Characterization of the algebraic structures generated by the multi-index setting. We consider natural actions on functionals of the nonlinearity and build a (pre-)Lie algebra from them. We use this as the starting point of an algebraic path towards the structure group, which as in the regularity structures literature is based on a Hopf algebra. This approach further allows us to explore the relation between multi-indices and trees, which we express through pre-Lie and Hopf algebra morphisms, in certain semi-linear equations. All the results are based on a series of joint works with Otto, Tempelmayr and Tsatsoulis.

Collagen is the most abundant protein in mammals and has special mechanical behavior that enables it to play an important role in the structural integrity of many tissues, e.g., skin, tendon, bone, cartilage and blood vessels. The mechanical properties of collagen are governed by hierarchical mechanisms in different length-scales from molecule to tissue level. Currently, there is no multi-scale model that can predict the mechanical properties of collagen at macroscopic length scales from the behavior of microstructural elements at smaller length scales. This dissertation aimed at developing a multi-scale model using a bottom-up approach to predict the elastic and viscoelastic behaviors of collagen at length scales spanning from nano to microscale. Creep simulations were performed using steered molecular dynamics (SMD) method on collagen molecules, cross-link, and micro-fibrils with various lengths. A micro-fibril is considered as a combination of two collagen molecules connected by a cross-link. The strain time histories for force levels in the range of 10 to 4000 pN were characterized using quasilinear viscoelastic models. These models were utilized to make a reduced model of a micro-fibril and the reduced models, in turn, were combined to make a model of a fibril up to 300 micrometers in length. The micro-fibril and fibril models were validated with available experimental measurements. Hydrogen bonds rupture and formation of collagen molecule played a central role in its viscoelastic behavior and were used to estimate the creep growth rate. The propagation of force wave in the molecule was shown to be an important factor in providing the time-dependent properties of the fibrils. This propagation was modeled with delay elements and this allowed reducing the micro-fibril model to only three degrees of freedom. In conclusion, the results confirmed that the combination of molecular dynamics simulations and viscoelastic theory could be successfully utilized to investigate the viscoelastic behavior of collagen at small scales. The model reported in this dissertation, lays the groundwork for future studies on collagen, particularly in elucidating how each particular level of hierarchy affects the overall tissue behavior. / Mechanical Engineering

L'application d'Abel-Jacobi fait le lien entre la forme de Weierstrass d'une courbe elliptique définie sur C et le tore complexe qui lui est associé. Il est possible de la calculer en un nombre d'opérations quasi-linéaire en la précision voulue, c'est à dire en temps O(M(P) log P). Son inverse est donné par la fonction p de Weierstrass, qui s'exprime en fonction de thêta, une fonction importante en théorie des nombres. L'algorithme naturel d'évaluation de thêta nécessite O(M(P) sqrt(P)) opérations, mais certaines valeurs (les thêta-constantes) peuvent être calculées en O(M(P) log P) opérations en exploitant les liens avec la moyenne arithmético-géométrique (AGM). Dans ce manuscrit, nous généralisons cet algorithme afin de calculer thêta en O(M(P) log P). Nous exhibons une fonction F qui a des propriétés similaires à l'AGM. D'une façon similaire à l'algorithme pour les thêta-constantes, nous pouvons alors utiliser la méthode de Newton pour calculer la valeur de thêta. Nous avons implanté cet algorithme, qui est plus rapide que la méthode naïve pour des précisions supérieures à 300 000 chiffres décimaux. Nous montrons comment généraliser cet algorithme en genre supérieur, et en particulier comment généraliser la fonction F. En genre 2, nous sommes parvenus à prouver que la même méthode mène à un algorithme qui évalue thêta en O(M(P) log P) opérations ; la même complexité s'applique aussi à l'application d'Abel-Jacobi. Cet algorithme est plus rapide que la méthode naïve pour des précisions plus faibles qu'en genre 1, de l'ordre de 3 000 chiffres décimaux. Nous esquissons également des pistes pour obtenir la même complexité en genre quelconque. Enfin, nous exhibons un nouvel algorithme permettant de calculer une isogénie de courbes elliptiques de noyau donné. Cet algorithme utilise l'application d'Abel-Jacobi, car il est facile d'évaluer l'isogénie sur le tore ; il est sans doute possible de le généraliser au genre supérieur / The Abel-Jacobi map links the short Weierstrass form of a complex elliptic curve to the complex torus associated to it. One can compute it with a number of operations which is quasi-linear in the target precision, i.e. in time O(M(P) log P). Its inverse is given by Weierstrass's p-function, which can be written as a function of theta, an important function in number theory. The natural algorithm for evaluating theta requires O(M(P) sqrt(P)) operations, but some values (the theta-constants) can be computed in O(M(P) log P) operations by exploiting the links with the arithmetico-geometric mean (AGM). In this manuscript, we generalize this algorithm in order to compute theta in O(M(P) log P). We give a function F which has similar properties to the AGM. As with the algorithm for theta-constants, we can then use Newton's method to compute the value of theta. We implemented this algorithm, which is faster than the naive method for precisions larger than 300,000 decimal digits. We then study the generalization of this algorithm in higher genus, and in particular how to generalize the F function. In genus 2, we managed to prove that the same method leads to a O(M(P) log P) algorithm for theta; the same complexity applies to the Abel-Jacobi map. This algorithm is faster than the naive method for precisions smaller than in genus 1, of about 3,000 decimal digits. We also outline a way one could reach the same complexity in any genus. Finally, we study a new algorithm which computes an isogeny of elliptic curves with given kernel. This algorithm uses the Abel-Jacobi map because it is easy to evaluate the isogeny on the complex torus; this algorithm may be generalizable to higher genera