Relevant bibliographies by topics / Weakly hyperbolic systems

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Weakly hyperbolic systems'

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Published: 25 May 2024

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Journal articles on the topic "Weakly hyperbolic systems"

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Arbieto, Alexander, André Junqueira, and Bruno Santiago. "On Weakly Hyperbolic Iterated Function Systems." Bulletin of the Brazilian Mathematical Society, New Series 48, no. 1 (2016): 111–40. http://dx.doi.org/10.1007/s00574-016-0018-4.

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YONEDA, GEN, and HISA-AKI SHINKAI. "CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY." International Journal of Modern Physics D 09, no. 01 (2000): 13–34. http://dx.doi.org/10.1142/s0218271800000037.

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Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.
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Krylovas, A., and R. Čiegis. "Asymptotic Approximation of Hyperbolic Weakly Nonlinear Systems." Journal of Nonlinear Mathematical Physics 8, no. 4 (2001): 458–70. http://dx.doi.org/10.2991/jnmp.2001.8.4.2.

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Spagnolo, Sergio, and Giovanni Taglialatela. "Analytic Propagation for Nonlinear Weakly Hyperbolic Systems." Communications in Partial Differential Equations 35, no. 12 (2010): 2123–63. http://dx.doi.org/10.1080/03605300903440490.

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Colombini, F., and Guy Métivier. "The Cauchy problem for weakly hyperbolic systems." Communications in Partial Differential Equations 43, no. 1 (2017): 25–46. http://dx.doi.org/10.1080/03605302.2017.1399906.

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Arbieto, Alexander, Carlos Matheus, and Maria José Pacifico. "The Bernoulli Property for Weakly Hyperbolic Systems." Journal of Statistical Physics 117, no. 1/2 (2004): 243–60. http://dx.doi.org/10.1023/b:joss.0000044058.99450.c9.

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D'Ancona, Piero, Tamotu Kinoshita, and Sergio Spagnolo. "Weakly hyperbolic systems with Hölder continuous coefficients." Journal of Differential Equations 203, no. 1 (2004): 64–81. http://dx.doi.org/10.1016/j.jde.2004.03.016.

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Souza, Rafael R. "Sub-actions for weakly hyperbolic one-dimensional systems." Dynamical Systems 18, no. 2 (2003): 165–79. http://dx.doi.org/10.1080/1468936031000136126.

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Alabau-Boussouira, Fatiha. "Indirect Boundary Stabilization of Weakly Coupled Hyperbolic Systems." SIAM Journal on Control and Optimization 41, no. 2 (2002): 511–41. http://dx.doi.org/10.1137/s0363012901385368.

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DREHER, MICHAEL, and INGO WITT. "ENERGY ESTIMATES FOR WEAKLY HYPERBOLIC SYSTEMS OF THE FIRST ORDER." Communications in Contemporary Mathematics 07, no. 06 (2005): 809–37. http://dx.doi.org/10.1142/s0219199705001969.

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For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp.
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Dissertations / Theses on the topic "Weakly hyperbolic systems"

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Gaito, Stephen Thomas. "Shadowing of weakly pseudo-hyperbolic pseudo-orbits in discrete dynamical systems." Thesis, University of Warwick, 1992. http://wrap.warwick.ac.uk/109461/.

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We consider Cr (r ≥ 1 +γ) diffeomorphisms of compact Riemannian manifolds. Our aim is to develop the analytic machinery required to describe the topological symbolic dynamics of sets of weakly hyperbolic orbits. The Pesin set is an example of such a set. For Axiom-A dynamical systems, that is, for diffeomorphisms which have a uniformly hyperbolic nonwandering set which is the closure of the periodic orbits, this analytic machinery is provided by the Shadowing Lemma. This lemma is a consequence of the Stable Manifold Theorem, and the local product structure of the nonwandering set of an Axiom-A diffeomorphism. Weakly hyperbolic invariant sets, such as the Pesin set, do not, in general, have local product structure. We can however, prove a generalization of the Shadowing Lemma by combining Anosov’s Stability Lemma with the Stable Manifold Theorem. In essence we prove a perturbed Stable Manifold Theorem. In order to deal with weakly hyperbolic orbits we use Pugh and Shub’s graph transform version of Pesin’s Stable Manifold Theorem. Normally, the contraction required to prove either Anosov’s Stability Lemma or the Stable Manifold Theorem, is derived from the hyperbolicity of a “supporting” invariant set. In fact neither of these proofs require this invariance; hyperbolic, or even pseudo-hyperbolic, families of pseudo-orbits are all that they require. This allows us to conclude the existence of shadowing orbits in the neighbourhood of “hyperbolic invariant sets” of numerical simulations of lowdimensional dynamical systems. In particular corresponding to any such numerical “hyperbolic invariant set”, there is a uniformly hyperbolic invariant set of the dynamical system itself.
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GRIFO', Gabriele. "Pattern formation in hyperbolic reaction-transport systems and applications to dryland ecology." Doctoral thesis, Università degli Studi di Palermo, 2023. https://hdl.handle.net/10447/580054.

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Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In particular, starting from a quite general framework of reaction-transport model, three particular cases are derived. In detail, in the first case, the occurrence of stationary patterns is investigated in one-dimensional domains by looking for the inertial dependence of the main features that characterize the formation and stability process of the emerging patterns. In particular, the phenomenon of Eckhaus instability, in both supercritical and subcritical regimes, is studied by adopting linear and multiple-scale weakly-nonlinear analysis and the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips, is elucidated. Then, in the second topic, the focus is moved to oscillatory periodic patterns generated by wave (or oscillatory Turing) instability. This phenomenon is studied by considering 1D two-compartment hyperbolic reaction-transport systems where different transport mechanisms of the species here involved are taken into account. In these cases, by using linear and weakly nonlinear stability analysis techniques, the dependence of the non-stationary patterns on hyperbolicity is underlined at and close to the criticality. In particular, it is proven that inertial effects play a role, not only during transient regimes from the spatially-homogeneous steady state toward the patterned state but also in altering the amplitude, the wavelength, the migration speed, and even the stability of the travelling waves. Finally, in the last case, the formation and stability of stationary patterns are investigated in bi-dimensional domains. To this aim, a general class of two-species hyperbolic reaction-transport systems is deduced following the guidelines of Extended Thermodynamics theory. To characterize the emerging Turing patterns, linear and weakly nonlinear stability analysis on the uniform steady states are addressed for rhombic and hexagonal planform solutions. In order to gain some insight into the above-mentioned dynamics, the previous theoretical predictions are corroborated by numerical simulations carried out in the context of dryland ecology. In this context, patterns become a relevant tool to identify early warning signals toward desertification and to provide a measure of resilience of ecosystems under climate change. Such ecological implications are discussed in the context of the Klausmeier model, one of the easiest two-compartment (vegetation biomass and water) models able to describe the formation of patterns in semi-arid environments. Therefore, it will be also here discussed how the experimentally-observed inertia of vegetation affects the formation and stability of stationary and oscillatory periodic vegetation patterns.<br>Pattern formation and modulation is an active branch of mathematics, not only from the perspective of fundamental theory but also for its huge applications in many fields of physics, ecology, chemistry, biology, and other sciences. In this thesis, the occurrence of Turing and wave instabilities, giving rise to stationary and oscillatory patterns, respectively, is theoretically investigated by means of two-compartment reaction-transport hyperbolic systems. The goal is to elucidate the role of inertial times, which are introduced in hyperbolic models to account for the finite-time propagation of disturbances, in stationary and transient dynamics, in supercritical and subcritical regimes. In particular, starting from a quite general framework of reaction-transport model, three particular cases are derived. In detail, in the first case, the occurrence of stationary patterns is investigated in one-dimensional domains by looking for the inertial dependence of the main features that characterize the formation and stability process of the emerging patterns. In particular, the phenomenon of Eckhaus instability, in both supercritical and subcritical regimes, is studied by adopting linear and multiple-scale weakly-nonlinear analysis and the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips, is elucidated. Then, in the second topic, the focus is moved to oscillatory periodic patterns generated by wave (or oscillatory Turing) instability. This phenomenon is studied by considering 1D two-compartment hyperbolic reaction-transport systems where different transport mechanisms of the species here involved are taken into account. In these cases, by using linear and weakly nonlinear stability analysis techniques, the dependence of the non-stationary patterns on hyperbolicity is underlined at and close to the criticality. In particular, it is proven that inertial effects play a role, not only during transient regimes from the spatially-homogeneous steady state toward the patterned state but also in altering the amplitude, the wavelength, the migration speed, and even the stability of the travelling waves. Finally, in the last case, the formation and stability of stationary patterns are investigated in bi-dimensional domains. To this aim, a general class of two-species hyperbolic reaction-transport systems is deduced following the guidelines of Extended Thermodynamics theory. To characterize the emerging Turing patterns, linear and weakly nonlinear stability analysis on the uniform steady states are addressed for rhombic and hexagonal planform solutions. In order to gain some insight into the above-mentioned dynamics, the previous theoretical predictions are corroborated by numerical simulations carried out in the context of dryland ecology. In this context, patterns become a relevant tool to identify early warning signals toward desertification and to provide a measure of resilience of ecosystems under climate change. Such ecological implications are discussed in the context of the Klausmeier model, one of the easiest two-compartment (vegetation biomass and water) models able to describe the formation of patterns in semi-arid environments. Therefore, it will be also here discussed how the experimentally-observed inertia of vegetation affects the formation and stability of stationary and oscillatory periodic vegetation patterns.
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Chaisemartin, Stéphane de. "Modèles eulériens et simulation numérique de la dispersion turbulente de brouillards qui s'évaporent." Thesis, Châtenay-Malabry, Ecole centrale de Paris, 2009. http://www.theses.fr/2009ECAP0011/document.

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Le modèle multi-fluide permet de décrire par une approche Eulérienne les sprays polydispersés et apparaît donc comme une méthode indiquée pour les applications de combustion diphasique. Sa pertinence pour la simulation à l’échelle d’applications industrielles est évaluée dans ce travail, par sa mise en oeuvre dans des configurations bi-dimensionnelle et tri-dimensionnelle plus représentatives de ce type de simulations. Cette évaluation couple une étude de faisabilité en terme de coût de calcul avec une analyse de la précision obtenue, par des comparaisons avec les résultats de méthodes de références pour la description des sprays. Afin de définir une telle référence, une hiérarchisation des modèles de spray est proposée dans ce travail, soulignant les niveaux de modélisation associée aux diverses méthodes. Une première configuration d’écoulements tourbillonnaires est utilisée pour caractériser la méthode multi-fluide. L’étude de la structure mathématique du système de lois de conservation permet d’analyser la formation de singularités et de fournir les outils permettant d’évaluer leur impact sur la modélisation. Cette étude permet également de dériver un schéma numérique robuste et efficace pour des configurations bi- et tri-dimensionnelle. La description des dynamiques de gouttes conditionnées par la taille est évaluée dans ces configurations tourbillonnaires au moyen de comparaisons quantitatives, sur des champs instantanés, où le multi-fluide est confronté à une méthode Lagrangienne, ainsi qu’à des résultats expérimentaux. Afin d’évaluer le comportement de la méthode multi-fluide dans des configurations plus représentatives des problématiques industrielles, le solveur MUSES3D est développé, permettant, entre autres, une évaluation fine des méthodes de résolution des sprays. Une implémentation originale de la méthode multi-fluide, conciliant généricité et efficacité pour le calcul parallèle, est réalisée. Le couplage de ce solveur avec le code ASPHODELE, développé au CORIA, permet d’effectuer une évaluation opérationnelle des approches Euler/Lagrange et Euler/Euler pour la description des écoulements diphasiques à inclusions dispersées. Finalement, le comportement de la méthode multi-fluide dans des jets bi-dimensionnels et dans une turbulence homogène isotrope tri-dimensionnelle permet de montrer sa précision pour la description de la dynamique de sprays évaporant dans des configurations plus complexes. La résolution de la polydispersion du spray permet de décrire précisément la fraction massique de combustible en phase vapeur, un élément clé pour les applications de combustion. De plus, l’efficacité du calcul parallèle par décomposition de domaine avec la méthode multi-fluide permet d’envisager son utilisation à l’échelle d’applications industrielles<br>The multi-fluid model, providing a Eulerian description of polydisperse sprays, appears as an interesting method for two-phase combustion applications. Its relevance as a numerical tool for industrial device simulations is evaluated in this work. This evaluation assesses the feasibility of multi-fluid simulations in terms of computational cost and analyzes their precision through comparisons with reference methods for spray resolution. In order to define such a reference, the link between the available methods for spray resolution is provided, highlighting their corresponding level of modeling. A first framework of 2-D vortical flows is used to assess the mathematical structure of the multi-fluid model governing system of equations. The link between the mathematical peculiarities and the physical modeling is provided, and a robust numerical scheme efficient for 2-D/3-D configurations is designed. This framework is also used to evaluate the multi-fluid description of evaporating spray sizeconditioned dynamics through quantitative, time-resolved, comparisons with a Lagrangian reference and with experimental data. In order to assess the multi-fluid efficiency in configurations more representative of industrial devices, a numerical solver is designed, providing a framework devoted to spray method evaluation. An original implementation of the multifluid method, combining genericity and efficiency in a parallel framework, is achieved. The coupling with a Eulerian/Lagrangian solver for dispersed two-phase flows, developed at CORIA, is conducted. It allows a precise evaluation of Euler/Lagrange versus Euler/Euler approaches, in terms of precision and computational cost. Finally, the behavior of the multi-fluid model is assessed in 2D-jets and 3-D Homogeneous Isotropic Turbulence. It illustrates the ability of the method to capture evaporating spray dynamics in more complex configurations. The method is shown to describe accurately the fuel vapor mass fraction, a key issue for combustion applications. Furthermore, the method is shown to be efficient in domain decomposition parallel computing framework, a key issue for simulations at the scale of industrial devices
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Dongmo, Nguepi Guissel Lagnol. "Modèles mathématiques et numériques avancés pour la simulation du polymère dans les réservoirs pétroliers." Electronic Thesis or Diss., université Paris-Saclay, 2021. http://www.theses.fr/2021UPASG077.

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Une technique efficace pour accroître la production d’un champ pétrolier consiste à y injecter un mélange d’eau et de polymère. La viscosité du polymère réduit en effet la mobilité de l’eau, qui pousse alors mieux l’huile, d’où un taux d’extraction plus élevé. La simulation numérique d’un tel procédé de récupération d’hydrocarbures revêt donc d’une importance capitale. Or, malgré des décennies de recherche, la modélisation des écoulements avec polymère en milieu poreux et sa résolution numérique demeurent un sujet difficile. D’une part, les modèles habituellement employés par les ingénieurs de réservoir présentent, au mieux, des singularités de type résonance qui les rend faiblement hyperboliques. Ce défaut donne lieu à certaines complicationsD’une part, les modèles habituellement employés par les ingénieurs de réservoir présentent, au mieux, des singularités de type résonance qui les rend faiblement hyperboliques. Ce défaut donne lieu à certaines complications mais reste acceptable. Au pire, quand on veut incorporer l’effet du volume de pore inaccessible (IPV), les modèles deviennent non hyperboliques, ce qui aggrave les instabilités numériques susceptibles d’apparaître.D’autre part, les schémas numériques classiques ne conduisent pas à des résultats satisfaisants. Sans IPV, la diffusion excessive autour de l’onde de contact fait perdre les informations pertinentes. Avec IPV, l’existence des valeurs propres complexes crée des instabilités exponentielles au niveau continu qu’il faut traiter au niveau discret sous peine d’arrêt prématuré du code.L’objectif de cette thèse est de remédier à ces difficultés. Au niveau des modèles, nous analysons plusieurs lois d’IPV et établissons une équivalence entre deux d’entre elles. Nous proposons de surcroît des conditions suffisantes raisonnables sur la loi d’IPV en vue de l’hyperbolicité faible du système d’écoulement. Au niveau des schémas pour le problème sans IPV, nous préconisons une correction afin d’améliorer la précision des discontinuités de contact. Pour le problème avec IPV,nous élaborons une méthode de relaxation qui garantit la stabilité des calculs quelle que soit la loi IPV<br>An effective technique to increase production in an oil field is to inject a mixture of water and polymer. The viscosity of polymer reduces the mobility of water, which then pushes oil better, resulting in a higher extraction rate. The numerical simulation of such an enhanced oil recovery is therefore of paramount importance. However, despite decades of research, the modeling of polymer flows in porous media and its numerical resolution remains a difficult subject.On the one hand, the models traditionally used by reservoir engineers exhibit, in the best case, resonance-like singularities that make them weakly hyperbolic. Thisdefect gives rise to some complications but remains acceptable. In the worst case, when we wish to incorporate the effect of the inaccessible pore volume (IPV), themodels become non-hyperbolic, which exacerbates the numerical instabilities that are likely to appear.On the other hand, classical numerical schemes do not yield satisfactory results. Without IPV, the excessive diffusion around the contact wave causes the most relevant information to be lost. With IPV, the existence of complex eigenvalues generates exponential instabilities at the continuous level that must be addressed at the discrete level to avoid a premature stop of the code.The objective of this thesis is to remedy these difficulties. Regarding models, we analyze several IPV laws and show an equivalence between two of them. Furthermore, we propose reasonable sufficient conditions on the IPV law to enforce weak hyperbolicity of the flow system. Regarding schemes for the problem without IPV, we advocate a correction to improve the accuracy of contact discontinuities. For the problem with IPV, we design a relaxation method that guarantees the stability of the calculations for all IPV laws
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Fougeirol, Jérémie. "Structure de variété de Hilbert et masse sur l'ensemble des données initiales relativistes faiblement asymptotiquement hyperboliques." Thesis, Avignon, 2017. http://www.theses.fr/2017AVIG0417/document.

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La relativité générale est une théorie physique de la gravitation élaborée il y a un siècle, dans laquelle l'univers est modélisé par une variété Lorentzienne (N,gamma) de dimension 4 appelée espace-temps et vérifiant les équations d'Einstein. Lorsque l'on sépare la dimension temporelle des trois dimensions spatiales, les équations de contrainte découlent naturellement de la décomposition 3+1 des équations d'Einstein. Elles constituent une condition nécessaire et suffisante pour pouvoir considérer l'espace-temps N comme l'évolution temporelle d'une hypersurface Riemannienne (m,g) plongée dans N avec une seconde forme fondamentale K. Le triplet (m,g,K) constitue alors une donnée initiale solution des équations de contrainte dont on note C l'ensemble. Dans cette thèse, nous utilisons la méthode de Robert Bartnik pour établir la structure de sous-variété de Hilbert de C pour des données initiales faiblement asymptotiquement hyperboliques, dont la régularité peut être reliée à la conjecture de courbure L^{2} bornée. Les difficultés inhérentes au cas faiblement AH ont nécessité l'introduction de deux opérateurs différentiels d'ordre deux et l'obtention d'estimées de type Poincaré et Korn pour ces opérateurs. Une fois la structure de Hilbert obtenue, nous définissons une fonctionnelle masse lisse sur la sous-variété C et compatible avec nos conditions de faible régularité. L'invariance géométrique de la masse est étudiée et montrée, modulo une conjecture en faible régularité relative au changement de cartes au voisinage de l'infini. Enfin, nous faisons le lien entre les points critiques de la masse et les métriques statiques<br>General relativity is a gravitational theory born a century ago, in which the universe is a 4-dimensional Lorentzian manifold (N,gamma) called spacetime and satisfying Einstein's field equations. When we separate the time dimension from the three spatial ones, constraint equations naturally follow on from the 3+1 décomposition of Einstein's equations. Constraint equations constitute a necessary condition,as well as sufficient, to consider the spacetime N as the time evolution of a Riemannian hypersurface (m,g) embeded into N with the second fundamental form K. (m,g,K) is then an element of C, the set of initial data solutions to the constraint equations. In this work, we use Robert Bartnik's method to provide a Hilbert submanifold structure on C for weakly asymptotically hyperbolic initial data, whose regularity can be related to the bounded L^{2} curvature conjecture. Difficulties arising from the weakly AH case led us to introduce two second order differential operators and we obtain Poincaré and Korn-type estimates for them. Once the Hilbert structure is properly described, we define a mass functional smooth on the submanifold C and compatible with our weak regularity assumptions. The geometrical invariance of the mass is studied and proven, only up to a weak regularity conjecture about coordinate changes near infinity. Finally, we make a correspondance between critical points of the mass and static metrics
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Leguil, Martin. "Cocycle dynamics and problems of ergodicity." Thesis, Sorbonne Paris Cité, 2017. http://www.theses.fr/2017USPCC159/document.

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Le travail qui suit comporte quatre chapitres : le premier est centré autour de la propriété de mélange faible pour les échanges d'intervalles et flots de translation. On y présente des résultats obtenus avec Artur Avila qui renforcent des résultats précédents dus à Artur Avila et Giovanni Forni. Le deuxième chapitre est consacré à un travail en commun avec Zhiyuan Zhang et concerne les propriétés d'ergodicité et d'accessibilité stables pour des systèmes partiellement hyperboliques de dimension centrale au moins égale à deux. On montre que sous des hypothèses de cohérence dynamique, center bunching et pincement fort, la propriété d'accessibilité stable est dense en topologie C^r, r&gt;1, et même prévalente au sens de Kolmogorov. Dans le troisième chapitre, on expose les résultats d'un travail réalisé en collaboration avec Julie Déserti, consacré à l'étude d'une famille à un paramètre d'automorphismes polynomiaux de C^3 ; on montre que de nouveaux phénomènes apparaissent par rapport à ce qui était connu dans le cas de la dimension deux. En particulier, on étudie les vitesses d'échappement à l'infini, en montrant qu'une transition s'opère pour une certaine valeur du paramètre. Le dernier chapitre est issu d'un travail en collaboration avec Jiangong You, Zhiyan Zhao et Qi Zhou ; on s'intéresse à des estimées asymptotiques sur la taille des trous spectraux des opérateurs de Schrödinger quasi-périodiques dans le cadre analytique. On obtient des bornes supérieures exponentielles dans le régime sous-critique, ce qui renforce un résultat précédent de Sana Ben Hadj Amor. Dans le cas particulier des opérateurs presque Mathieu, on montre également des bornes inférieures exponentielles, qui donnent des estimées quantitatives en lien avec le problème dit "des dix Martinis". Comme conséquences de nos résultats, on présente des applications à l'homogénéité du spectre de tels opérateurs ainsi qu'à la conjecture de Deift<br>The following work contains four chapters: the first one is centered around the weak mixing property for interval exchange transformations and translation flows. It is based on the results obtained together with Artur Avila which strengthen previous results due to Artur Avila and Giovanni Forni. The second chapter is dedicated to a joint work with Zhiyuan Zhang, in which we study the properties of stable ergodicity and accessibility for partially hyperbolic systems with center dimension at least two. We show that for dynamically coherent partially hyperbolic diffeomorphisms and under certain assumptions of center bunching and strong pinching, the property of stable accessibility is dense in C^r topology, r&gt;1, and even prevalent in the sense of Kolmogorov. In the third chapter, we explain the results obtained together with Julie Déserti on the properties of a one-parameter family of polynomial automorphisms of C^3; we show that new behaviours can be observed in comparison with the two-dimensional case. In particular, we study the escape speed of points to infinity and show that a transition exists for a certain value of the parameter. The last chapter is based on a joint work with Jiangong You, Zhiyan Zhao and Qi Zhou; we get asymptotic estimates on the size of spectral gaps for quasi-periodic Schrödinger operators in the analytic case. We obtain exponential upper bounds in the subcritical regime, which strengthens a previous result due to Sana Ben Hadj Amor. In the particular case of almost Mathieu operators, we also show exponential lower bounds, which provides quantitative estimates in connection with the so-called "Dry ten Martinis problem". As consequences of our results, we show applications to the homogeneity of the spectrum of such operators, and to Deift's conjecture
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Fino, Ahmad. "Contributions aux problèmes d'évolution." Phd thesis, Université de La Rochelle, 2010. http://tel.archives-ouvertes.fr/tel-00437141.

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Dans cette thèse, nous nous intéressons à l'étude de trois équations aux dérivées partielles et d'évolution non-locales en espace et en temps. Les solutions de ces trois solutions peuvent exploser en temps fini. Dans une première partie de cette thèse, nous considérons l'équation de la chaleur nonlinéaire avec une puissance fractionnaire du laplacien, et obtenons notamment que, dans le cas d'exposant sur-critique, le comportement asymptotique de la solution lorsque $t\rightarrow+\infty$ est déterminé par le terme de diffusion anormale. D'autre part, dans le cas d'exposant sous-critique, l'effet du terme non-linéaire domine. Dans une deuxième partie, nous étudions une équation parabolique avec le laplacien fractionnaire et un terme non-linéaire et non-local en temps. On montre que la solution est globale dans le cas sur-critique pour toute donnée initiale ayant une mesure assez petite, tandis que dans le cas sous-critique, on montre que la solution explose en temps fini $T_{\max}>0$ pour toute condition initiale positive et non-triviale. Dans ce dernier cas, on cherche le comportement de la norme $L^1$ de la solution en précisant le taux d'explosion lorsque $t$ s'approche du temps d'explosion $T_{\max}.$ Nous cherchons encore les conditions nécessaires à l'existence locale et globale de la solution. Une toisième partie est consacré à une généralisation de la deuxième partie au cas de systèmes $2\times 2$ avec le laplacien ordinaire. On étudie l'existence locale de la solution ainsi qu'un résultat sur l'explosion de la solution avec les mêmes propriétés étudiées dans le troisième chapitre. Dans la dernière partie, nous étudions une équation hyperbolique dans $\mathbb{R}^N,$ pour tout $N\geq2,$ avec un terme non-linéaire non-local en temps. Nous obtenons un résultat d'existence locale de la solution sous des conditions restrictives sur les données initiales, la dimension de l'espace et les exposants du terme non-linéaire. De plus on obtient, sous certaines conditions sur les exposants, que la solution explose en temps fini, pour toute condition initiale ayant de moyenne strictement positive.
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Chen, Shih Tzung, and 陳世宗. "A WEAK AND NUMERICAL METHOD FOR SYSTEM OF HYPERBOLIC EQUATION." Thesis, 1996. http://ndltd.ncl.edu.tw/handle/41475657908484436065.

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9

Su, Ying-Chin, and 蘇萾欽. "Global Existence of Weak Solutions to the Initial-BoundaryValue Problem of Inhomogeneous Hyperbolic Systems of Conservation Laws." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/24884536149658156044.

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博士<br>國立中央大學<br>數學研究所<br>96<br>In this article we provide a generalized version of Glimm scheme to study the global existence of weak solutions to the initial-boundary value problem of 2 by 2 hyperbolic systems of conservation laws with source terms. Due to the structure of source terms, we extend the methods invented in [10,13] to construct the weak solutions of Riemann and boundary Riemann problems, which can be dopted as a building block of the approximate solution by Glimm scheme. By modifying the results in [7] and showing the weak convergence of residuals, we establish the stability and consistency of scheme. In addition we investigate the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax method and generalized Glimm scheme, we construct the approximate solutions of initial-boundary Riemann problem near the boundary and perturbed Riemann problem away the boundary. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.
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Bira, Bibekananda. "Lie group analysis and evolution of weak waves for certain hyperbolic system of partial differential equations." Thesis, 2014. http://ethesis.nitrkl.ac.in/6604/1/B._BIRA_Ph._D._THESIS.pdf.

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In the present thesis, we study the applications of Lie group theory to system of quasilinear hyperbolic partial differential equations (PDEs), which are governed by many physical phenomena and having various important physical significance in the real life. Our primary objective in this thesis is to identify the symmetries of system of PDEs in order to obtain certain classes of group invariant solutions. The investigations carried out in this thesis are confined to the applications of Lie group method to the system of quasilinear hyperbolic PDEs arising in magnetogasdynamics, two phase flows and other scientific fields. We organize the whole thesis into 7 chapters, described as follows. First chapter is introductory and deals with a short background history of Lie group of transformations and symmetries along with some of their important features which are of great importance in the work of proceeding chapters and the motivation behind our interest. In the second chapter, we obtain exact solutions to the quasilinear system of PDEs, describing the one dimensional unsteady simple flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. Lie group of point transformations are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities. The next chapter deals with system of PDEs, governing the one dimensional unsteady flow of inviscid and perfectly conducting compressible fluid in the presence of magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuity.
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Books on the topic "Weakly hyperbolic systems"

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Gaito, Stephen Thomas. Shadowing of weakly pseudo-hyperbolic pseudo-orbits in discrete dynamical systems. typescript, 1992.

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Zeitlin, Vladimir. Rotating Shallow-Water Models as Quasilinear Hyperbolic Systems, and Related Numerical Methods. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0007.

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The chapter contains the mathematical background necessary to understand the properties of RSW models and numerical methods for their simulations. Mathematics of RSW model is presented by using their one-dimensional reductions, which are necessarily’one-and-a-half’ dimensional, due to rotation and include velocity in the second direction. Basic notions of quasi-linear hyperbolic systems are recalled. The notions of weak solutions, wave breaking, and shock formation are introduced and explained on the example of simple-wave equation. Lagrangian description of RSW is used to demonstrate that rotation does not prevent wave-breaking. Hydraulic theory and Rankine–Hugoniot jump conditions are formulated for RSW models. In the two-layer case it is shown that the system loses hyperbolicity in the presence of shear instability. Ideas of construction of well-balanced (i.e. maintaining equilibria) shock-resolving finite-volume numerical methods are explained and these methods are briefly presented, with illustrations on nonlinear evolution of equatorial waves.
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Kaloshin, Vadim, and Ke Zhang. Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691202525.001.0001.

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Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. This book provides the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom). This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. The book follows Mather's strategy but emphasizes a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, the book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.
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Book chapters on the topic "Weakly hyperbolic systems"

1

Korsch, Andrea. "Weakly Coupled Systems of Conservation Laws on Moving Surfaces." In Theory, Numerics and Applications of Hyperbolic Problems II. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91548-7_18.

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Pesin, Ya B., and Ya G. Sinai. "Space-time chaos in the system of weakly interacting hyperbolic systems." In Selecta. Springer New York, 2010. http://dx.doi.org/10.1007/978-0-387-87870-6_15.

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Jakobsen, E. R., K. H. Karlsen, and N. H. Risebro. "On the Convergence Rate of Operator Splitting for Weakly Coupled Systems of Hamilton-Jacobi Equations." In Hyperbolic Problems: Theory, Numerics, Applications. Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8372-6_9.

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Hawerkamp, Maryse, Dietmar Kröner, and Hanna Moenius. "Optimal Controls in Flux, Source, and Initial Terms for Weakly Coupled Hyperbolic Systems." In Theory, Numerics and Applications of Hyperbolic Problems I. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-91545-6_52.

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Hsiao, Ling, and Hailiang Li. "Asymptotic Behavior of Entropy Weak Solution for Hyperbolic System with Damping." In Hyperbolic Problems: Theory, Numerics, Applications. Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8372-6_7.

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Liebscher, Stefan. "Stable, Oscillatory Viscous Profiles of Weak Shocks in Systems of Stiff Balance Laws." In Hyperbolic Problems: Theory, Numerics, Applications. Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8372-6_20.

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Brio, M. "Admissibility Conditions for Weak Solutions of Nonstrictly Hyperbolic Systems." In Nonlinear Hyperbolic Equations — Theory, Computation Methods, and Applications. Vieweg+Teubner Verlag, 1989. http://dx.doi.org/10.1007/978-3-322-87869-4_5.

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Feireisl, E. "Asymptotic Properties of a Class of Weak Solutions to the Navier–Stokes–Fourier System." In Hyperbolic Problems: Theory, Numerics, Applications. Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-75712-2_49.

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Floch, Philippe. "Entropy Weak Solutions to Nonlinear Hyperbolic Systems in Nonconservation Form." In Nonlinear Hyperbolic Equations — Theory, Computation Methods, and Applications. Vieweg+Teubner Verlag, 1989. http://dx.doi.org/10.1007/978-3-322-87869-4_37.

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Ndjinga, Michaël. "Weak Convergence of Nonlinear Finite Volume Schemes for Linear Hyperbolic Systems." In Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05684-5_40.

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Conference papers on the topic "Weakly hyperbolic systems"

1

ORIVE, R. "WEAKLY NONLINEAR LONG-TIME BEHAVIOR OF SOLUTIONS TO A HYPERBOLIC RELAXATION SYSTEMS." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0111.

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POPIVANOV, PETAR, and IORDAN IORDANOV. "ANOMALOUS SINGULARITIES OF THE SOLUTIONS TO SEVERAL CLASSES OF WEAKLY HYPERBOLIC SEMILINEAR SYSTEMS: EXAMPLES." In Proceedings of the 3rd ISAAC Congress. World Scientific Publishing Company, 2003. http://dx.doi.org/10.1142/9789812794253_0079.

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Popivanov, Petar, and Iordan Iordanov. "On the anomalous singularities of the solutions to some classes of weakly hyperbolic semilinear systems. Examples." In Evolution Equations Propagation Phenomena - Global Existence - Influence of Non-Linearities. Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc60-0-16.

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Davlatov, Jasur, Kholmatjon Imomnazarov, and Abdulkhamid Kholmurodov. "Weak approximation method for the Cauchy problem for one-dimensional hyperbolic system." In INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE ON ACTUAL PROBLEMS OF MATHEMATICAL MODELING AND INFORMATION TECHNOLOGY. AIP Publishing, 2024. http://dx.doi.org/10.1063/5.0210419.

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Provotorov, V. V. "Unique weak solvability of a hyperbolic systems with distributed parameters on the graph." In 2018 14th International Conference "Stability and Oscillations of Nonlinear Control Systems" (Pyatnitskiy's Conference) (STAB). IEEE, 2018. http://dx.doi.org/10.1109/stab.2018.8408390.

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Bock, Igor. "On the Dynamic Contact Problem for a Viscoelastic Plate." In ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2010. http://dx.doi.org/10.1115/esda2010-24130.

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We deal with an initial-boundary value problem describing the perpendicular vibrations of an anisotropic viscoelastic plate free on its boundary and with a rigid inner obstacle. A weak formulation of the problem is in the form of the hyperbolic variational inequality. We solve the problem using the discretizing the time variable. The elliptic variational inequalities for every time level are uniquely solved. We derive the a priori estimates and the convergence of the sequence of segment line functions to a variational solution of the considered problem.
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Haarer, D., and H. Maier. "Tunneling Dynamics and Spectral Diffusion in the Millikelvin Regime." In Spectral Hole-Burning and Related Spectroscopies: Science and Applications. Optica Publishing Group, 1994. http://dx.doi.org/10.1364/shbs.1994.thd3.

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The tunneling model [1] is the theoretical basis for several anomalous and time dependent phenomena in amorphous materials, which are caused by a broad distribution of relaxation rates of the so-called two-level systems (TLS). This model has also been applied to interpret spectral diffusion in glasses and to explain the observation of time dependent spectral linewidths [2]. In terms of spectral hole-burning, the TLS dynamics leads to a logarithmic hole broadening for times larger than the minimum TLS relaxation time, while the hole widths approach a constant value for times larger than the maximum TLS relaxation time [3]. This functional dependence is caused by a hyperbolic distribution of relaxation rates; all the above described phenomena follow from the “weak coupling” model: In polymer glasses a logarithmic broadening has been found on time scales between milliseconds and days [4, 5].
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