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Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this thesis, we study long linear and weakly-nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition, which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1-dimensional cylindrical Korteweg-de Vries (cKdV)-type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant shear flow, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shape of the wavefronts: the wavefront of the surface wave is elongated in the shear flow direction while the wavefront of the interfacial wave is squeezed in this direction. We solve the derived 2+1-dimensional cKdV-type equation numerically using a finite-difference scheme. The effects of nonlinearity and dispersion are studied by considering numerical results for surface and interfacial ring waves generated from a localised source with and without shear flow and the 2D dam break problem. In these examples, the linear and nonlinear surface waves are faster than interfacial waves, the wave height decreases faster at the surface, the shear flow leads to the wave height decreasing slower downstream and faster upstream, and the effect becomes more prominent as the shear flow strengthens.

Le sujet principal de cette thèse est l'étude de la contrôlabilité exacte de deux équations dispersives, l'équation de Korteweg-de Vries et la "bonne" équation de Boussinesq. En ce qui concerne l'équation de Korteweg-de Vrie, on étend un résultat de Rosier en montrant la contrôlabilité exacte en tout temps de l'équation non linéaire autour d'une solution stationnaire proche de zéro mais non nulle, ce pour des longueurs de domaine spatial critiques. Cette démonstration utilise en particulier la méthode d'unicité hilbertienne couplée avec la méthode des multiplicateurs et un théorème de point fixe. Ensuite, nous étudions le problème de la contrôlabilité exacte de l'équation de Boussinesq pour deux contrôles différents. On utilise également la méthode d'unicité hilbertienne pour ces problèmes en appliquant une inégalité de Ingham. On obtient ainsi un résultat de contrôlabilité exacte pour des temps arbitrairement petits. Nous implémentons ensuite cette méthode de facon numérique pour l'équation de Boussinesq avec un contrôle portant sur la dérivée seconde a droite, tant sur le problème linéaire que non linéaire.

In this study, the method of deciding the water level of the initial tsunami is proposed by using spatial-temporal focusing (Coalescence) theory and waveform inversion reciprocal with Green function. Tsunami and earthquake are so closely bonded that the current tsunami numerical model is dependent on the parameters of the fault and the initial tsunami water level by calculating the theory of half flexibility. But in fact, it is not easy to have the parameters of seabed fault so that the initial tsunami water level is very hard to get a accurate value. On the other hand, although the parameters of fault can be speculated by seismic waves, because ground is uneven medium, therefore, it is still a lot of improvement to get the parameters of fault by using seismic waves. For the tsunami simulation, if you have the value of the initial tsunami water level, the fault parameters can be estimated.Since the propagation of tsunami in the ocean is a linear behavior, the propagating process is affected by the topography of the ocean and the nonlinear effect so minimal that it is to satisfy the linear shallow water equations and the requirement of reversibility;However, in fact, the values of the water level measured by the tide stations on the coast are influenced by the shoaling effect so that the reversibility of linear system can not be directly applied to Coastal areas.Therefore, the overall Inversion procedure on this study consists of two parts; the first one is that the usage of variable coefficient Korteweg-de Vries (vKdV) equation and the Coalescence theory inverses the data gathered by the Coastal tide stations to the water level data where the depth is more than 50m on the linear region, and compares the above results with the stimulation and confirms the accuracy of the inversed waveform;The second one is that according to the reversibility of the linear system the use of least squares and least squares QR- decomposition (LSQR) method reproduce the initial tsunami wave source that compares with the initial tsunami wave source by stimulating and has a very good conformity. The seismic parameters can be easily decided by the above results.

Cette Thèse se divise en deux parties. Dans la partie mathématique, nous étudions différentes edp d'ordre supérieur à 3 issues des neurosciences avec un point de vue probabiliste. Nous démontrons une formule de FK pour une grande classe de solutions de KdV (pas seulement les n-solitons), à l'aide des déterminants de Fredholm et des transformées de Laplace d'intégrales de Skorohod itérées. Concernant les edp d'ordre supérieur à 3, les processus itérés qui consistent en la composition de deux processus indépendants, l'un correspondant à la position et l'autre au temps, sont liés à leurs solutions. En effet, nous montrons une formule de FK pour des solutions d'edp d'ordre supérieur à 3 basée sur des fonctionnelles de processus itérés, même dans le cas non Markovien, étendant ainsi les résultats existants. Nous proposons aussi un schéma numérique pour la simulation de trajectoires de diffusions itérées basé sur le schéma d'Euler, qui converge p.s., uniformément en temps, avec un taux de convergence d'ordre $1/4$. Une estimation de l'erreur est proposée. Dans la partie biologique, nous avons collecté plusieurs articles en neuroscience et d'autres domaines de biologie, où les edp précédentes sont utilisées. En particulier, on s'intéresse à la simulation et à la propagation du potentiel d'action lorsque la capacité de la membrane cellulaire n'est pas supposée constante. Ces articles ont en commun le fait qu'ils remettent en question le fameux modèle d'Hodgkin-Huxley datant des années cinquante. En effet, même si ce modèle a été très efficace pour la compréhension du signal neuronal, il ne prend pas en compte tous les phénomènes résultants de la propagation du potentiel d'action. / This Thesis consists of two parts. In the mathematical part we study Korteweg--de Vries (KdV) equation and high-order pdes with a probabilistic point of view in order to obtain Feynman-Kac (FK) type formulas. This study was motivated by recent biological models. We prove a FK representation for a larger class of solutions of KdV equation (not only n-solitons), using Fredholm determinants and Laplace transforms of iterated Skorohod integrals. Regarding higher order pdes, iterated processes that consist in the composition of two independent processes, one corresponding to position and the other one to time, are naturally related to their solutions. Indeed, we prove FK formulas for solutions of high order pdes based on functionals of iterated processes even in the non Markovian case, thus extending the existing results. We also propose a scheme for the simulation of iterated diffusions trajectories based on Euler scheme, that converges a.s., uniformly in time, with a rate of convergence of order $1/4$. An estimation of the error is proposed. In the biological part, we have collected several papers in neuroscience and other fields of biology where the previous types of pdes are involved. In particular, we are interested in the simulation of the propagation of the action potential when the capacitance of the cell membrane is not assumed to be constant. These papers have in common the fact that they question the famous Hodgkin Huxley model dating back to the fifties. Indeed this model even if it has been very efficient for the understanding of neuronal signaling does not take into account all the phenomena that occur during the propagation of the action potential.

The aim of this diploma thesis is to assess the applicability of pulse wave propagation monitoring in the cardiovascular system in the field of prediction and early diagnosis of abdominal aortic aneurysm (AAA). The very first part is focused on description of heart and blood vessels with its pathological changes in presence of aneurysm. For this reason, current methods of monitoring and surgical treating of AAA were mentioned. Due to their difficult clinical use widely in the population, new methods based on pulse wave monitoring were presented. Using an analytical approach we estimated the difference in the arrival of the pulse wave at measurable locations between healthy and pathological aorta in the order of miliseconds. By experimental monitoring using photoplethysmographic sensors, we observed significant changes of pulse wave velocity with respect to the mechanical properties of the artery wall (mainly associated with age), which we tried to implement by hyperelastic material models used in computational simulations of pulse wave proagation on simplified geometries by fluid structure interaction method. These analyzes should verify applicability of FSI simulations in further development of diagnostic methods of AAA.

No description available.

Dans cette thèse, nous considérons le problème d'Euler surface libre sur un domaine à fond non plat, dans le cadre du régime d'ondes longues de faible amplitude. L'objectif est de construire, justifier et comparer de nouveaux modèles asymptotiques pour ce problème, permettant de prendre en compte les effets liés aux variations bathymétriques. En premier lieu, nous construisons rigoureusement deux classes de modèles de Boussinesq symétriques dans le cadre de deux régimes topographiques distincts, celui de faible variations bathymétriques et celui de fortes variations. Dans un second temps, nous retrouvons et discutons dans le cas de faibles variations topographiques l'approximation classique de Korteweg-de Vries, et proposons une nouvelle approximation via l'ajout de termes bathymétriques. Dans une troisième partie, ces deux modèles, ainsi que les modèles de Boussinesq construits dans la première partie, sont simulés numériquement et comparés sur des cas tests de topographie. Enfin, il est présenté une étude numérique des équations de Green-Naghdi, dont le domaine de validité physique est plus étendu, ainsi qu'une comparaison numérique de ce modèle avec les modèles précédents sur des bathymétries spécifiques.

Flow of granular materials is usually initiated when the shear stress is large enough and exceeds certain critical value. This can result in the presence of the dead-zones in which the flow itself does not take place. Motions of such materials are frequently described by Bingham model. Flows of granular fluids are frequently connected with the presence of free surface. In the thesis Bingham model is incorporated into a more general framework of Bingham-Korteweg fluids, which is a suitable way how to transfer free- boundary problems into the problems on fixed domains. A part of the thesis concerns mathematical analysis of interesting relevant problems for incompressible fluids. 1

We present two possible thermodynamical approaches towards a derivation of a model, proposed by Korteweg at the beginning of the 20th century, that is suitable to describe phase transitions liquid-vapor with non-sharp interfaces. The first approach (Dunn, Serrin (1985)) is based on classical rational continuum thermodynamics. The second approach (Heida, Málek (2010)) stems from the principles of classical nonequilibrium continuum thermodynamics. We compare both approaches in favor of the second one. The considered constitutive equation for the Cauchy stress is nonlinear. Nonlinearity and higher order derivatives of the density makes the analysis of relevant problems for the Navier-Stokes- Korteweg (NSK) fluid more difficult in comparison to problems concerning Navier-Stokes equations. Special attention is devoted to the appropriate choice of the boundary conditions. We also investigate the influence of compressibility on the stability of bubbles by comparing numerical simulations for compressible NSK fluid and its incompressible variant. Instabilities observed for a compressible NSK fluid are due to the pressure that has a different meaning for incompressible fluid. Powered by TCPDF (www.tcpdf.org)