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1
Daghighi, Abtin. "Regularity and uniqueness-related properties of solutions with respect to locally integrable structures." Doctoral thesis, Mittuniversitetet, Avdelningen för ämnesdidaktik och matematik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-21641.
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We prove that a smooth generic embedded CR submanifold of C^n obeys the maximum principle for continuous CR functions if and only if it is weakly 1-concave. The proof of the maximum principle in the original manuscript has later been generalized to embedded weakly q-concave CR submanifolds of certain complex manifolds. We give a generalization of a known result regarding automatic smoothness of solutions to the homogeneous problem for the tangential CR vector fields given local holomorphic extension. This generalization ensures that a given locally integrable structure is hypocomplex at the origin if and only if it does not allow solutions near the origin which cannot be represented by a smooth function near the origin. We give a sufficient condition under which it holds true that if a smooth CR function f on a smooth generic embedded CR submanifold, M, of C^n, vanishes to infinite order along a C^infty-smooth curve \gamma in M, then f vanishes on an M-neighborhood of \gamma. We prove a local maximum principle for certain locally integrable structures.<br><p>Funding by FMB, based at Uppsala University.</p>
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Acevedo, Tapia Paul Andres. "Theorie L^p pour le système de boussinesq." Thesis, Pau, 2015. http://www.theses.fr/2015PAUU3027/document.
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Cette thèse est consacrée à l’étude du système de Boussinesq stationnaire:-νΔu+(u⋅∇)u+∇π=θg, div u=0,dans Ω(1a)-κΔθ+u⋅∇θ=h,dans Ω (1b)où Ω⊂R^3 est un ouvert, borné et connexe; les inconnues du système sont u,π et θ: la vitesse, la pression et la température du fluide, respectivement; ν>0 est la viscosité cinématique du fluide, κ>0 est la diffusivité thermique du fluide, g est l’accélération de la pesanteur et h est une source de chaleur appliquée au fluide.L’objectif de cette thèse est l’étude de la théorie L^p pour le système de Boussinesq en considérant deux différents types de conditions aux limites du champ de vitesse. En effet, dans une première partie, nous considérons une condition de Dirichlet non homogèneu=u_b, sur Γ (2)où Γ désigne la frontière du domaine. Dans une deuxième partie, nous considérons une condition de Navier non homogèneu⋅n=0,2[D(u)n]_τ+αu_τ=a,sur Γ(3)où D(u)=1/2 (∇u+(∇u)^T ) est le tenseur de déformation associé au champ de vitesse u, n est le vecteur normal unitaire extérieur, τ est le correspondant vecteur tangent unitaire, α et a sont une fonction scalaire de friction et un champ de vecteur tangentiel donnés sur la frontière, respectivement. De plus, la condition aux limites pour la température sera, dans les deux premières parties, une condition aux limites de Dirichlet non homogèneθ=θ_b, sur Γ. (4)Alors, premièrement, nous étudions l’existence et l’unicité d’une solution faible pour le problème (1), (2) et (4) dans le cas hilbertien. Également, l’existence de solutions généralisées pour p≥3/2 et des solutions fortes pour 12 et des solutions fortes pour p≥6/5 pour le problème (1), (3) et (4). Notez que l’hypothèse d’une frontière non-connexe, mentionnée précédemment, ne figurait pas dans cette partie du travail en raison de la restriction d’imperméabilité de la frontière.Enfin, dans la dernière partie de cette thèse, nous étudions la théorie L^p pour les équations de Stokes avec la condition de Navier (3). Plus précisément, nous examinons la régularité W^(1,p) pour p≥2 et la régularité W^(2,p) pour p≥6/5.Mots clés: système de Boussinesq; régularité L^p; solutions faibles; solutions fortes; solutions très faibles<br>This thesis is dedicated to the study of the stationary Boussinesq system:-νΔu+(u⋅∇)u+∇π=θg, div u=0,in Ω(1a)-κΔθ+u⋅∇θ=h,in Ω (1b)where Ω⊂R^3 is an open bounded connected set; u,π and θ are the velocity field, pressure and temperature of the fluid, respectively, and stand for the unknowns of the system; ν>0 is the kinematic viscosity of the fluid, κ>0 is the thermal diffusivity of the fluid, g is the gravitational acceleration and h is a heat source applied to the fluid.The aim of this thesis is the study of the L^p-theory for the stationary Boussinesq system in the context of two different types of boundary conditions for the velocity field. Indeed, in the first part of the thesis, we will consider a non-homogeneous Dirichlet boundary conditionu=u_b, on Γ (2)where Γ denotes the boundary of the domain; meanwhile in the second part, the velocity field will be prescribed through a non-homogeneous Navier boundary conditionu⋅n=0,2[D(u)n]_τ+αu_τ=a,on Γ(3)where D(u)=1/2 (∇u+(∇u)^T ) is the strain tensor associated with the velocity field u, n is the unit outward normal vector, τ is the corresponding unit tangent vector, α and a are a friction scalar function and a tangential vector field defined both on the boundary, respectively. Further, the boundary condition for the temperature will be, in the first two parts of the thesis, a non-homogeneous Dirichlet boundary conditionθ=θ_b, on Γ. (4)Then, firstly, we study the existence and uniqueness of the weak solution for the problem (1), (2) and (4) in the hilbertian case. Also, the existence of generalized solutions for p≥3/2 and strong solutions for 12 and strong solutions for p≥6/5 for the problem (1), (3) and (4). Note that the assumption of a non-connected boundary, which was mentioned before, will not appear here due to the impermeability restriction on the boundary.Finally, in the last part of this thesis, we study the L^p-theory for the Stokes equations with Navier boundary condition (3). Specifically, we deal with the W^(1,p)-regularity for p≥2 and the W^(2,p)-regularity for p≥6/5.Keywords: Boussinesq system; L^p-regularity; weak solutions; strong solutions; very weak solutions
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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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Gibbeson, Dominic. "Low regularity solutions of nonlinear wave equations." Thesis, University of Edinburgh, 2004. http://hdl.handle.net/1842/14900.
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We investigate solutions of the coupled Diral Klein-Gordon Equations in one and three space dimensions. Through analysis of the Fourier representations of the solutions to these equations, we introduce the ‘Null Structure’ as developed by Klainerman and Machedon. This structure allows us to prove the necessary estimates, both fixed time and bilinear space-time, that allow us to show existence of solutions of these equations with initial data of lower regularity than previously required. We also study global existence for a two dimensional wave equation with a critical non-linearity.
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Wolf, Jörg. "Regularität schwacher Lösungen nichtlinearer elliptischer und parabolischer Systeme partieller Differentialgleichungen mit Entartung." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2002. http://dx.doi.org/10.18452/14792.
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In der vorliegenden Arbeit untersuchen wir schwache Lösungen, die zu einem geeigneten Sobolevraum gehören, q-elliptischer und parabolischer Systeme partieller Differentialgleichungen auf deren Regularität für den Fall 1<br>In the present work we study the regularity of weak solution to q-elliptic and parabolic systems partial differential equations in appropriate Sobolev spaces in case 1
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Raynor, Sarah Groff 1977. "Regularity of Neumann solutions to an elliptic free boundary problem." Thesis, Massachusetts Institute of Technology, 2003. http://hdl.handle.net/1721.1/29353.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003.<br>Includes bibliographical references (p. 57-58).<br>We examine the regularity properties of solutions to an elliptic free boundary problem, near a Neumann fixed boundary. Consider a nonnegative function u which minimizes the functional ... on a bounded, convex domain ... This function u is harmonic in its positive phase and satisfies ... along the free boundary ... , in a weak sense. We prove various basic properties of such a minimizer near the portion of the boundary ... on which ... weakly. These results include up-to-the boundary gradient estimates on harmonic functions with Neumann boundary conditions on convex domains. The main result is that the minimizer u is Lipschitz continuous. The proof in dimension 2 is by means of conformal mapping as well as a simplified monotonicity formula. In higher dimensions, the proof is via a maximum principle estimate for ...<br>by Sarah Groff Raynor.<br>Ph.D.
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Ivarsson, Björn. "Regularity and boundary behavior of solutions to complex Monge–Ampère equations." Doctoral thesis, Uppsala University, Department of Mathematics, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-1603.
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<p>In the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α < 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.</p>
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Flad, Heinz-Jürgen, Reinhold Schneider, and Bert-Wolfgang Schulze. "Asymptotic regularity of solutions of Hartree-Fock equations with coulomb potential." Universität Potsdam, 2007. http://opus.kobv.de/ubp/volltexte/2009/3026/.
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We study the asymptotic regularity of solutions of Hartree-Fock equations for Coulomb systems. In order to deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo-differential operators on conical manifolds. First, the non-self-consistent-field case is considered which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator which guarantee ellipticity for the local part of the Fock operator on the open stretched cone R+ × S². This proves existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed-point approach based on Cances and Le Bris analysis of the level-shifting algorithm, we show via another bootstrap argument, that the corresponding self-consistent-field solutions of the Hartree-Fock equation have the same type of asymptotic regularity.
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Ivarsson, Björn. "Regularity and boundary behaviour of solutions to complex Monge-Ampère equations /." Uppsala : Matematiska institutionen, Univ. [distributör], 2002. http://publications.uu.se/theses/91-506-1533-5/.
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De, Silva Daniela. "Existence and regularity of monotone solutions to a free boundary problem." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/31160.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.<br>Includes bibliographical references (p. 71-72).<br>In the first part of this dissertation, we provide the first example of a singular energy minimizing free boundary. This singular solution occurs in dimension 7 and higher, and in fact it is conjectured that there are no singular minimizers in dimension lower than 7. Our example is the analogue of the 8-dimensional Simons cone in the theory of minimal surfaces. The minimality of the Simons cone is closely related to the existence of a complete minimal graph in dimension 9, which is not a hyperplane. The first step toward solving the analogous problem in the free boundary context, consists in developing a local existence and regularity theory for monotone solutions to a free boundary problem. This is the objective of the second part of our thesis. We also provide a partial result in the global context..<br>by Daniela De Silva.<br>Ph.D.
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Ricciotti, Diego. "Regularity of solutions of the p-Laplace equation in the Heisenberg group." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2013. http://amslaurea.unibo.it/5708/.
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MONTEIRO, GABRIEL DE LIMA. "WEAK SOLUTIONS FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2018. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=36023@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO<br>PROGRAMA DE EXCELENCIA ACADEMICA<br>Esse trabalho tem como objetivo ser uma introdução ao estudo da existência e unicidade de soluções fracas para equações diferenciais parciais elípticas. Começamos definindo o espaço de Sobolev para, a partir da definição, provarmos algumas propriedades básicas que nos ajudarão no estudo das equações diferenciais parciais elípticas. Finalizamos com o desenvolvimento do Teorema de Lax-Milgram e de Stampacchia que permitirão o uso de técnicas de Análise Funcional para estudarmos alguns exemplos de equações elípticas.<br>This dissertation aims to be an introduction to the study of the existence and uniqueness of weak solutions for elliptic partial differential equations. We begin by defining the Sobolev spaces and proving some basics properties that will assist in the study of the elliptical equations. Lastly, we develop the Theorems of Lax-Milgram and Stampacchia that allow the use of Functional Analysis for the studying of some examples of elliptic equations.
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Kawagoe, Daisuke. "Regularity of solutions to the stationary transport equation with the incoming boundary data." Kyoto University, 2018. http://hdl.handle.net/2433/232413.
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Scarinci, Teresa. "Sensitivity Relations and Regularity of Solutions of HJB Equations arising in Optimal Control." Thesis, Paris 6, 2015. http://www.theses.fr/2015PA066573.
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Dans cette thèse nous étudions une classe d’équations de Hamilton-Jacobi-Bellman provenant de la théorie du contrôle optimal des équations différentielles ordinaires. Nous nous intéressons principalement à l’analyse de la sensibilité de la fonction valeur des problèmes de contrôle optimal associés à de telles équations de H-J-B. Dans la littérature, les relations de sensibilité fournissent une “mesure” de la robustesse des stratégies optimales par rapport aux variations de la variable d’état. Ces résultats sont des outils très importants pour le contrôle appliqué, parce qu’ils permettent d’étudier les effets que des approximations des données du système peuvent avoir sur les politiques optimales. Cette thèse est dédiée également à l’étude des problèmes de Mayer et de temps minimal. Nous supposons que la dynamique du problème soit une inclusion différentielle, afin de permettre aux données d’être non régulières et d’embrasser un ensemble plus grand d’applications. Néanmoins, cette tâche rend notre analyse plus difficile. La première contribution de cette étude est une extension de quelques résultats classiques de la théorie de la sensibilité au domaine des problèmes non paramétrées. Ces relations prennent la forme d’inclusions d’état adjoint, figurant dans le principe du maximum de Pontryagin, dans certains gradients généralisés de la fonction valeur évalués le long des trajectoires optimales. En deuxième lieu, nous développons des nouvelles relations de sensibilité impliquant des approximations du deuxième ordre de la fonction valeur. Cette analyse mène à de nouvelles applications concernant la propagation, tant ponctuel que local, de la régularité de la fonction valeur le long des trajectoires optimales. Nous proposons également des applications aux conditions d’optimalité<br>This dissertation investigates a class of Hamilton-Jacobi-Bellman equations arising in optimal control of O.D.E.. We mainly focus on the sensitivity analysis of the optimal value function associated with the underlying control problems. In the literature, sensitivity relations provide a measure of the robustness of optimal control strategies with respect to variations of the state variable. This is a central tool in applied control, since it allows to study the effects that approximations of the inputs of the system may produce on the optimal policies. In this thesis, we deal whit problems in the Mayer or in the minimum time form. We assume that the dynamic is described by a differential inclusion, in order to allow data to be nonsmooth and to embrace a large area of concrete applications. Nevertheless, this task makes our analysis more challenging. Our main contribution is twofold. We first extend some classical results on sensitivity analysis to the field of nonparameterized problems. These relations take the form of inclusions of the co-state, featuring in the Pontryagin maximum principle, into suitable gradients of the value function evaluated along optimal trajectories. Furthermore, we develop new second-order sensitivity relations involving suitable second order approximations of the optimal value function. Besides being of intrinsic interest, this analysis leads to new consequences regarding the propagation of both pointwise and local regularity of the optimal value functions along optimal trajectories. As applications, we also provide refined necessary optimality conditions for some class of differential inclusions
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Akramov, Ibrokhimbek [Verfasser]. "Conservation vs. Dissipation for Weak Solutions in Fluid Dynamics / Ibrokhimbek Akramov." Ulm : Universität Ulm, 2020. http://d-nb.info/1236842065/34.
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Rao, Arvind Satya. "Weak solutions to a Monge-Ampère type equation on Kähler surfaces." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/582.
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In the context of moment maps and diffeomorphisms of Kähler manifolds, Donaldson introduced a fully nonlinear Monge-Ampère type equation. Among the conjectures he made about this equation is that the existence of solutions is equivalent to a positivity condition on the initial data. Weinkove later affirmed Donaldson's conjecture using a gradient flow for the equation in the space of Kähler potentials of the initial data. The topic of this thesis is the case when the initial data is merely semipositive and the domain is a closed Kähler surface. Regularity techniques for degenerate Monge-Ampère equations, specifically those coming from pluripotential theory, are used to prove the existence of a bounded, unique, weak solution. With the aid of a Nakai criterion, due to Lamari and Buchdahl, it is shown that this solution is smooth away from some curves of negative self-intersection.
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Forcillo, Nicolò. "Free Boundary Regularity of Some Non-Homogeneous Problems." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/16392/.
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In this work, we deal with the study of some free boundary problems governed by non-homogeneous equations. In particular, we are interested in the regularity of the free boundaries for solutions of one-phase problems associated with non-divergence elliptic operators with variable coefficients.
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Chen, Hua, and Ke Li. "The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations." Universität Potsdam, 2007. http://opus.kobv.de/ubp/volltexte/2009/3024/.
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Let X = (X1,.....,Xm) be an infinitely degenerate system of vector fields, we study the existence and regularity of multiple solutions of Dirichelt problem for a class of semi-linear infinitely degenerate elliptic operators associated
with the sum of square operator Δx = ∑m(j=1) Xj* Xj.
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Wiedemann, Emil [Verfasser]. "Weak and measure-valued solutions of the incompressible Euler equations / Emil Wiedemann." Bonn : Universitäts- und Landesbibliothek Bonn, 2012. http://d-nb.info/1043911308/34.
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Couchman, Benjamin Luke Streatfield. "On the convergence of higher-order finite element methods to weak solutions." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/115685.
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Thesis: S.M., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2018.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 77-79).<br>The ability to handle discontinuities appropriately is essential when solving nonlinear hyperbolic partial differential equations (PDEs). Discrete solutions to the PDE must converge to weak solutions in order for the discontinuity propagation speed to be correct. As shown by the Lax-Wendroff theorem, one method to guarantee that convergence, if it occurs, will be to a weak solution is to use a discretely conservative scheme. However, discrete conservation is not a strict requirement for convergence to a weak solution. This suggests a hierarchy of discretizations, where discretely conservative schemes are a subset of the larger class of methods that converge to the weak solution. We show here that a range of finite element methods converge to the weak solution without using discrete conservation arguments. The effect of using quadrature rules to approximate integrals is also considered. In addition, we show that solutions using non-conservation working variables also converge to weak solutions.<br>by Benjamin Luke Streatfield Couchman.<br>S.M.
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Quiroga, Gonzáles Cruz Sonia, Juan Limaco, Pedro Gamboa, and Rioco K. Barreto. "Weak and periodical solutions of the navier-stokes equation in noncylindrical domains." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/95081.
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Ali, Zakaria Idriss. "Stochastic quasilinear parabolic equations with non standard growth : weak and strong solutions." Thesis, University of Pretoria, 2015. http://hdl.handle.net/2263/53502.
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This thesis consists of two main parts.
The rst part concerns the existence of weak probabilistic solutions (called elsewhere
martingale solutions) for a stochastic quasilinear parabolic equation of generalized
polytropic ltration, characterized by the presence of a nonlinear elliptic part admitting
nonstandard growth.
The deterministic version of the equation was rst introduced and studied by Samokhin
in [178] as a generalized model for polytropic ltration. Our objective is to investigate
the corresponding stochastic counterpart in the functional setting of generalized
Lebesgue and Sobolev spaces. We establish an existence result of weak probabilistic
solutions when the forcing terms do not satisfy Lipschitz conditions and the noise
involves cylindrical Wiener processes.
The second part is devoted to the existence and uniqueness results for a class of
strongly nonlinear stochastic parabolic partial di erential equations.
This part aims to treat an important class of higher-order stochastic quasilinear
parabolic equations involving unbounded perturbation of zeroth order. The deterministic
case was studied by Brezis and Browder (Proc. Natl. Acad. Sci. USA,
76(1): 38-40, 1979). Our main goal is to provide a detailed study of the corresponding
stochastic problem. We establish the existence of a probabilistic weak solution
and a unique strong probabilistic solution. The main tools used in this part of the
thesis are a regularization through a truncation procedure which enables us to adapt
the work of Krylov and Rozosvkii (Journal of Soviet Mathematics, 14: 1233-1277,
1981), combined with analytic and probabilistic compactness results (Prokhorov and
Skorokhod Theorems), the theory of pseudomonotone operators, and a Banach space
version of Yamada-Watanabe's theorem due to R ockner, Schmuland and Zhang.
The study undertaken in this thesis is in some sense pioneering since both classes
of stochastic partial di erential equations have not been the object of previous investigation,
to the best of our knowledge.
The results obtained are therefore original and constitute in our view signi cant
contribution to the nonlinear theory of stochastic parabolic equations.<br>Thesis (PhD)--University of Pretoria, 2015.<br>Mathematics and Applied Mathematics<br>PhD<br>Unrestricted
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Dumas, Thomas. "Existence de solutions pour des équations apparentées au 1 Laplacien anisotrope." Thesis, Cergy-Pontoise, 2018. http://www.theses.fr/2018CERG0963/document.
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Nous étudions des équations relatives au p-Laplacien anisotrope lorsque certaines composantes du vecteur p sont égales à 1<br>We study anisotropic p-Laplacian equations when some components of p are equal to 1
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Prazeres, Disson Soares dos. "Improved regularity estimates in nonlinear elliptic equations." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=13536.
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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior<br>Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico<br>In this work we establish local regularity estimates for
at solutions to non-convex fully nonlinear elliptic equations and we study cavitation type equations modeled within coef-
icients bounded and measurable.<br>Neste trabalho estabelecemos estimativas de regularidade local para soluÃÃes "flat" de equaÃÃes elÃpticas totalmente nÃo-lineares nÃo-convexas e estudamos equations do tipo cavidade com coeficientes meramente mensurÃveis.
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Ran, Hongjun. "A Framework for the Determination of Weak Pareto Frontier Solutions under Probabilistic Constraints." Diss., Georgia Institute of Technology, 2007. http://hdl.handle.net/1853/14511.
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A framework is proposed that combines separately developed multidisciplinary optimization, multi-objective optimization, and joint probability assessment methods together but in a decoupled way, to solve joint probabilistic constraint, multi-objective, multidisciplinary optimization problems that are representative of realistic conceptual design problems of design alternative generation and selection. The intent here is to find the Weak Pareto Frontier (WPF) solutions that include additional compromised solutions besides the ones identified by a conventional Pareto frontier. This framework starts with constructing fast and accurate surrogate models of different disciplinary analyses. A new hybrid method is formed that consists of the second order Response Surface Methodology (RSM) and the Support Vector Regression (SVR) method. The three parameters needed by SVR to be pre-specified are automatically selected using a modified information criterion based on model fitting error, predicting error, and model complexity information. The model predicting error is estimated inexpensively with a new method called Random Cross Validation. This modified information criterion is also used to select the best surrogate model for a given problem out of the RSM, SVR, and the hybrid methods. A new neighborhood search method based on Monte Carlo simulation is proposed to find valid designs that satisfy the deterministic constraints and are consistent for the coupling variables featured in a multidisciplinary design problem, and at the same time decouple the three loops required by the multidisciplinary, multi-objective, and probabilistic features. Two schemes have been developed. One scheme finds the WPF by finding a large enough number of valid design solutions such that some WPF solutions are included in those valid solutions. Another scheme finds the WPF by directly finding the WPF of each consistent design zone. Then the probabilities of the PCs are estimated, and the WPF and corresponding design solutions are found. Various examples demonstrate the feasibility of this framework.
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Vink, Claire Bridget. "Time resolved infrared studies of weak magnetic field effects on radical pair solutions." Thesis, University of Leicester, 2007. http://hdl.handle.net/2381/30001.
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The thesis describes the development, implementation and application of the first magnetic field time resolved infrared (MFTRIR) spectrometer, capable of studying the kinetics of radical reactions and the effect of weak magnetic fields upon them. The thesis commences with an overview of the theoretical concepts associated with free radicals and their reactions. The idea of the radical pair is presented and the magnetic field susceptibility of this species is discussed in the context of the radical pair mechanism. The construction and development of the MFTRIR spectrometer is then examined, highlighting the important design features and operational principles. The remaining chapters comprise an in depth analysis of the results obtained from the first studies performed using the spectrometer. The radical recombination kinetics and magnetic field dependence are studied for a range of precursor molecules with insights drawn into the structural factures that influence both the magnitude and form of the observed magnetic field effects. Later chapters explore the effects of the radical pair environment by investigating reactions in both isotropic solution and reverse micelle environments. A number of key findings are generated by this work. First is the excellent capability of the new instrument in measuring very small perturbations on radical reactions with weak magnetic fields. Second is the correlation between the size of the observed field effects and the magnetic parameters of the radicals. Third is the ability to remove the effects at very weak magnetic fields (approximately 2mT) without perturbing the effects at higher magnetic fields, through the selective removal of longer lived radical pairs.
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Mazzoleni, Dario [Verfasser], Aldo [Akademischer Betreuer] Pratelli, Dorin [Akademischer Betreuer] Bucur, and Giuseppe [Akademischer Betreuer] Buttazzo. "Existence and regularity results for solutions of spectral problems / Dario Mazzoleni. Gutachter: Aldo Pratelli ; Dorin Bucur ; Giuseppe Buttazzo." Erlangen : Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), 2014. http://d-nb.info/1065270380/34.
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Bauer, Martin [Verfasser], and Thilo [Akademischer Betreuer] Meyer-Brandis. "Mean-field stochastic differential equations with irregular coefficients : solutions and regularity properties / Martin Bauer ; Betreuer: Thilo Meyer-Brandis." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2020. http://d-nb.info/1215499787/34.
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Riechwald, Paul Felix [Verfasser]. "Very Weak Solutions to the Navier-Stokes Equations in General Unbounded Domains / Paul Felix Riechwald." München : Verlag Dr. Hut, 2011. http://d-nb.info/1018982213/34.
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Lucic, Vladimir. "On uniqueness and weak convergence of solutions for the stochastic differential equations of nonlinear filtering." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ60554.pdf.
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Zhang, Xin. "Étude qualitative des solutions du système de Navier-Stokes incompressible à densité variable." Thesis, Paris Est, 2017. http://www.theses.fr/2017PESC1215/document.
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Dans cette thèse, on s'intéresse à deux problèmes provenant de l'étude mathématique des fluides incompressibles visqueux : la propagation de la régularité tangentielle et le mouvement d'une surface libre.La première question concerne plus particulièrement l'étude qualitative de l'évolution de quantités thermodynamiques telles que la température dans l'équation de Boussinesq sans diffusion et la densité dans le système de Navier-Stokes non homogène. Typiquement, on suppose que ces deux quantités sont, à l'instant initial, discontinues le long d'une interface à régularité h"oldérienne. Comme conséquence de résultats de propagation de régularité tangentielle pour le champ de vitesses, on établit que la régularité des interfaces persiste pour tout temps aussi bien en dimension deux d'espace, qu'en dimension supérieure (avec condition de petitesse). Notre approche suit celle du travail de J.-Y. Chemin dans les années 90 pour le problème des poches de tourbillon dans les fluides incompressiblesparfaits.Dans le cas présent, outre cette hypothèse de régularité tangentielle, nous n'avons besoin que d'une régularité critique sur le champ de vitesses.La démonstration repose sur le calcul para-différentiel et les espaces de multiplicateurs.Dans la dernière partie de la thèse, on considère le problème à frontière libre pour le système de Navier-Stokes incompressible à deux phases. Ce système permet de décrire l'évolution d'un mélange de deux fluides non miscibles tels que l'huile et l'eau par exemple. Différents cas de figure sont étudiés : le cas d'un réservoir borné, d'une goutte ou d'une rivière à profondeur finie.On établit l'existence et l'unicité à temps petit pour ce problème. Notre démonstration repose fortement sur des propriétés de régularité maximale parabolique de type $L_p$-$L_q<br>This thesis is dedicated to two different problems in the mathematical study of the viscous incompressible fluids: the persistence of tangential regularity and the motion of a free surface.The first problem concerns the study of the qualitative properties of some thermodynamical quantities in incompressible fluid models, such as the temperature for Boussinesq system with no diffusion and the density for the non-homogeneous Navier-Stokes system. Typically, we assume those two quantities to be initially piecewise constant along an interface with H"older regularity.As a consequence of stability of certain directional smoothness of the velocity field, we establish that the regularity of the interfaces persist globally with respect to time both in the two dimensional and higher dimensional cases (under some smallness condition). Our strategy is borrowed from the pioneering works by J.-Y.Chemin in 1990s on the vortex patch problem for ideal fluids.Let us emphasize that, apart from the directional regularity, we only impose rough (critical) regularity on the velocity field. The proof requires tools from para-differential calculus and multiplier space theory.In the last part of this thesis, we are concerned with the free boundary value problem for two-phase density-dependent Navier-Stokes system.This model is used to describe the motion of two immiscible liquids, like the oil and the water. Such mixture may occur in different situations, such as in a fixed bounded container, in a moving bounded droplet or in a river with finite depth. We establish the short time well-posedness for this problem. Our result strongly relies on the $L_p$-$L_q$ maximal regularity theoryfor parabolic equations
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Coutinho, Francisco Edson Gama. "Universal moduli of continuity for solutions to fully nonlinear elliptic equations." Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11427.
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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior<br>In this paper we provide a universal solution for continuity module in the direction of the viscosity of fully nonlinear elliptic equations considering properties of the function f integrable in different situations. Established inner estimate for the solutions of these equations based on some conditions the norm of the function f. To obtain regularity in solutions of these inhomogeneous equations and coefficients of variables we use a method of compactness, which consists essentially of approximating solutions of inhomogeneous equations for a solution of a homogeneous equation in order to "inherit" the regularity that those
equations possess.<br>Neste trabalho fornecemos mÃdulo de continuidade universal para soluÃÃes, no sentido da viscosidade,de equaÃÃes elÃpticas totalmente nÃo lineares, considerando propriedades de integrabilidade da funÃÃo f em diferentes situaÃÃes. Estabelecemos estimativa interior para as soluÃÃes dessas equaÃÃes baseadas em algumas condiÃÃes da norma da funÃÃo f. Para se obter regularidade nas soluÃÃes dessas equacÃes nÃo homogÃneas e de coeficientes variÃveis usamos um mÃtodo de compacidade, o qual consiste, essencialmente, em aproximar soluÃÃes de equaÃÃes nÃo homogÃneas por uma soluÃÃo de uma equaÃÃo homogÃnea com o objetivo de âherdarâ a regularidade que essas equaÃÃes possuem.
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Krämer, Jan Martin [Verfasser], Bernd [Gutachter] Kawohl, and Guido [Gutachter] Sweers. "Regularity and Symmetry Results for Ground State Solutions of Quasilinear Elliptic Equations / Jan Martin Krämer ; Gutachter: Bernd Kawohl, Guido Sweers." Köln : Universitäts- und Stadtbibliothek Köln, 2020. http://d-nb.info/1221718398/34.
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Schnieders, Inka [Verfasser], Guido [Gutachter] Sweers, and Dirk [Gutachter] Horstmann. "Positivity and regularity of solutions to higher order Dirichlet problems on smooth domains / Inka Schnieders ; Gutachter: Guido Sweers, Dirk Horstmann." Köln : Universitäts- und Stadtbibliothek Köln, 2021. http://d-nb.info/1226933378/34.
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Huicheng, Yin, and Ingo Witt. "Global singularity structure of weak solutions to 3-D semilinear dispersive wave equations with discontinuous initial data." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2639/.
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Bakhtavoryan, Rafael Gagik. "Endogenous variables and weak instruments in cross-sectional nutrient demand and health information analysis: a comparison of solutions." Thesis, Texas A&M University, 2003. http://hdl.handle.net/1969.1/191.
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In recent years, increasing attention has turned toward the effect of health information or health knowledge on nutrient intake. In determining the effect of health information on nutrient demand, researchers face the estimation problem of dealing with the endogeneity of health information knowledge. The standard approach for dealing with this problem is an instrumental variables (IV) procedure. Unfortunately, recent research has demonstrated that the IV procedure may not be reliable in the types of data sets that contain health information and nutrient intakes because the instruments are not sufficiently correlated with the endogenous variables (i.e., instruments are weak). This thesis compares the reliability of the IV procedure (and the Hausman test) with a relatively new procedure, directed graphs, given weak instruments. The goal is to determine if the method of directed graphs performs better in identifying an endogenous variable and also relevant instruments. The performance of the Hausman test and directed graphs are first assessed through conducting a Monte-Carlo sampling experiment containing weak instruments. Because the structure of the model is known in the Monte-Carlo experiment, these results are used as a guideline to determine which procedure would be more reliable in a real world setting. The procedures are then applied to a real-world cross-sectional dataset on nutrient intake. This thesis provides empirical evidence that neither the IV estimator (and Hausman test) or the directed graphs are reliable when instruments are weak, as in a cross-sectional dataset.
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Stachura, Eric Christopher. "On Generalized Solutions to Some Problems in Electromagnetism and Geometric Optics." Diss., Temple University Libraries, 2016. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/403050.
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Mathematics<br>Ph.D.<br>The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\`ere type arises here.<br>Temple University--Theses
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Lazarou, Georgia. "Development of the SAFT-γ Mie equation of state for predicting the thermodynamic behaviour of strong and weak electrolyte solutions". Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/60588.
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The thermodynamic modelling of fluid mixtures containing electrolytes using the SAFT-γ Mie equation of state is addressed in detail in this thesis. The SAFT-γ Mie approach allows the implementation of heteronuclear molecules using a group-contribution formalism, and offers a versatile framework for developing models to describe molecules of varying chemical functionality for a broad range of physical properties. In the present work, the SAFT-γ Mie equation of state is extended to electrolyte mixtures with the incorporation of the primitive unrestricted mean spherical approximation (MSA-PM) for describing the Coulombic ion–ion interactions, and the Born solvation free energy to implicitly treat ion– solvent polar interactions. Novel reformulations of the MSA-PM and Born theories within a group-contribution framework are proposed in order to enable ionic species of any size and chemical structure to be modelled, from small inorganic ions to large non-spherical charged molecules. Taking carboxylate anions in linear alkyl chain molecules as an illustrative case study, the proposed theory is shown to effectively account for localised charge effects arising from the structural topology of the charged species. A holistic description of electrolyte solutions is employed in this work; in addition to the short-range dispersion forces and the long-range Coulombic interactions which are pertinent to such mixtures, the models developed here also account for the formation of hydrogen bonds, ion-pairing phenomena, and electrolyte dissociation equilibria. The proposed SAFT-γ Mie equation of state is used to model aqueous solutions of strong electrolytes including alkali halide salts, hydrogen halide acids, and alkali hydroxide bases. Aqueous solutions of sulphuric acid and nitric acid are studied in detail by modelling these as speciating weak electrolytes. Finally, the treatment of ion-pairing phenomena is investigated through a consideration of aqueous alkali nitrate salt solutions. This work presents a new theoretical formulation and SAFT-γ Mie group models for twenty species in total.
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Rodney, Scott W. Sawyer E. T. "Existence of weak solutions to subelliptic partial differential equations in divergence form and the necessity of the Sobolev and Poincare inequalities." *McMaster only, 2007.
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Wang, Xince. "Quasilinear PDEs and forward-backward stochastic differential equations." Thesis, Loughborough University, 2015. https://dspace.lboro.ac.uk/2134/17383.
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In this thesis, first we study the unique classical solution of quasi-linear second order parabolic partial differential equations (PDEs). For this, we study the existence and uniqueness of the $L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{d}) \otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k})\otimes L^2_{\rho}( \mathbb{R}^{d}; \mathbb{R}^{k\times d})$ valued solution of forward backward stochastic differential equations (FBSDEs) with finite horizon, the regularity property of the solution of FBSDEs and the connection between the solution of FBSDEs and the solution of quasi-linear parabolic PDEs. Then we establish their connection in the Sobolev weak sense, in order to give the weak solution of the quasi-linear parabolic PDEs. Finally, we study the unique weak solution of quasi-linear second order elliptic PDEs through the stationary solution of the FBSDEs with infinite horizon.
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JÃnior, Elzon CÃzar Bezerra. "Um estudo sobre regularidade de soluÃÃes de equaÃÃes diferenciais parciais elÃpticas." Universidade Federal do CearÃ, 2016. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=17187.
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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico<br>O objetivo principal deste trabalho à o estudo da regularidade de soluÃÃes de equaÃÃes diferenciais parciais elÃpticas de segunda ordem, serÃo usadas tÃcnicas tais como o princÃpio do mÃximo, estimativas a priori e a desigualdade de Harnack. Por fim generalizamos o conceito de soluÃÃo buscando soluÃÃes no espaÃo de Sobolev W2,p(Ω).<br>The main objective of this work is to study the regularity of solutions of elliptic partial differential equations of second order, will be used techniques such as the principle of
maximum estimates a priori and the unequal Harnack. Finally generalize the solution concept seeking solutions in the Sobolev space W2,p((Ω).
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Gyurko, Lajos Gergely. "Numerical methods for approximating solutions to rough differential equations." Thesis, University of Oxford, 2008. http://ora.ox.ac.uk/objects/uuid:d977be17-76c6-46d6-8691-6d3b7bd51f7a.
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The main motivation behind writing this thesis was to construct numerical methods to approximate solutions to differential equations driven by rough paths, where the solution is considered in the rough path-sense. Rough paths of inhomogeneous degree of smoothness as driving noise are considered. We also aimed to find applications of these numerical methods to stochastic differential equations. After sketching the core ideas of the Rough Paths Theory in Chapter 1, the versions of the core theorems corresponding to the inhomogeneous degree of smoothness case are stated and proved in Chapter 2 along with some auxiliary claims on the continuity of the solution in a certain sense, including an RDE-version of Gronwall's lemma. In Chapter 3, numerical schemes for approximating solutions to differential equations driven by rough paths of inhomogeneous degree of smoothness are constructed. We start with setting up some principles of approximations. Then a general class of local approximations is introduced. This class is used to construct global approximations by pasting together the local ones. A general sufficient condition on the local approximations implying global convergence is given and proved. The next step is to construct particular local approximations in finite dimensions based on solutions to ordinary differential equations derived locally and satisfying the sufficient condition for global convergence. These local approximations require strong conditions on the one-form defining the rough differential equation. Finally, we show that when the local ODE-based schemes are applied in combination with rough polynomial approximations, the conditions on the one-form can be weakened. In Chapter 4, the results of Gyurko & Lyons (2010) on path-wise approximation of solutions to stochastic differential equations are recalled and extended to the truncated signature level of the solution. Furthermore, some practical considerations related to the implementation of high order schemes are described. The effectiveness of the derived schemes is demonstrated on numerical examples. In Chapter 5, the background theory of the Kusuoka-Lyons-Victoir (KLV) family of weak approximations is recalled and linked to the results of Chapter 4. We highlight how the different versions of the KLV family are related. Finally, a numerical evaluation of the autonomous ODE-based versions of the family is carried out, focusing on SDEs in dimensions up to 4, using cubature formulas of different degrees and several high order numerical ODE solvers. We demonstrate the effectiveness and the occasional non-effectiveness of the numerical approximations in cases when the KLV family is used in its original version and also when used in combination with partial sampling methods (Monte-Carlo, TBBA) and Romberg extrapolation.
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Brandman, Jeremy. "A level-set method for solving elliptic eigenvalue problems on hypersurfaces ; and, Finite-time blow-up of L[superscript infty] weak solutions of an aggregation equation." Diss., Restricted to subscribing institutions, 2008. http://proquest.umi.com/pqdweb?did=1619423481&sid=1&Fmt=2&clientId=1564&RQT=309&VName=PQD.
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Maltese, David. "Quelques résultats en analyse théorique et numérique pour les équations de Navier-Stokes compressibles." Thesis, Toulon, 2016. http://www.theses.fr/2016TOUL0005/document.
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Dans cette thèse, nous nous intéressons à l’analyse mathématique théorique et numérique des équations deNavier-Stokes compressibles en régime barotrope. La plupart des travaux présentés ici combinent desméthodes d’analyse des équations aux dérivées partielles et des méthodes d’analyse numérique afin de clarifierla notion de solution faible ainsi que les mécanismes de convergence de méthodes numériques approximant cessolutions faibles. En effet les équations de Navier-Stokes compressibles sont fortement non linéaires et leuranalyse mathématique repose nécessairement sur la structure de ces équations. Plus précisément, nousprouvons dans la partie théorique l’existence de solutions faibles pour un modèle d’écoulement compressibled’entropie variable où l’entropie du système est transportée. Nous utilisons les méthodes classiques permettantde prouver l’existence de solutions faibles aux équations de Navier-Stokes compressibles en regime barotrope.Nous étudions aussi dans cette partie la réduction de dimension 3D/2D dans les équations de Navier-Stokescompressibles en utilisant la méthode d’énergie relative. Dans la partie numérique nous nous intéressons auxestimations d’erreur inconditionnelles pour des schémas numériques approximant les solutions faibles deséquations de Navier-Stokes compressibles. Ces estimations d’erreur sont obtenues à l’aide d’une versiondiscrète de l’énergie relative satisfaite par les solutions discrètes de ces schémas. Ces estimations d’erreur sontobtenues pour un schéma numérique académique de type volumes finis/éléments finis ainsi que pour le schémanumérique Marker-and-Cell. Nous prouvons aussi que le schéma Marker-and-Cell est inconditionnellement etuniformément asymptotiquement stable en régime bas Mach. Ces résultats constituent les premiers résultatsd’estimations d’erreur inconditionnelles pour des schémas numériques pour les équations de Navier-Stokescompressibles en régime barorope<br>In this thesis, we deal with mathematical and numerical analysis of compressible Navier-Stokes equations inbarotropic regime. Most of these works presented here combine mathematical analysis of partial differentialequations and numerical methods with aim to shred more light on the construction of weak solutions on oneside and on the convergence mechanisms of numerical methods approximating these weak solutions on theother side. Indeed, the compressible Navier-Stokes equations are strongly nonlinear and their mathematicalanalysis necessarily relies on the structure of equations. More precisely, we prove in the theorical part theexistence of weak solutions for a model a flow of compressible viscous fluid with variable entropy where theentropy is transported. We use the classical techniques to prove the existence of weak solutions for thecompressible Navier-Stokes equations in barotropic regime. We also investigate the 3D/2D dimensionreduction in the compressible Navier-Stokes equations using the relative energy method. In the numerical wedeal with unconditionally error estimates for numerical schemes approximating weak solutions of thecompressible Navier-Stokes equations. These error estimates are obtained by using the discrete version of therelative energy method. These error estimates are obtained for a academic finite volume/finite element schemeand for the Marker-and-Cell scheme. We also prove that the Marker-and-cell scheme is unconditionally anduniformly asymptotically stable at the Low Mach number regime. These are the first results onunconditionally error estimates for numerical schemes approximating the compressible Navier-Stokesequations in barotropic regime
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Filho, NarcÃlio Silva de Oliveira. "Regularidade para equaÃÃes quase lineares em conjuntos singulares degenerados." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=14681.
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FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico<br>CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior<br>We will study a new universal gradient continuity estimate for solutions to quasi-linear equations with varying coefficients at singular set of degeneracy: S(u) := {X : Du(X) = 0}. Ourmain theorem reveals that along S(u), u is asymptotic as regular as solutions to constant coefficient equations. In particular, along the critical set S(u),u enjoys a modulus of continuity much superior than the possibly low, continuity feature of the coefficients. The results are new even in the context of linear elliptic equations, where it is herein shown that H^1- weak solutions to div (a(X,Du))= 0 with aij elliptic and dinicontinuous are actually C ^{1,1^{-}} along S(u). The results and insights of this work foster a new understanding os smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.<br>Neste trabalho estudaremos uma nova estimativa universal para a continuidade do gradiente de soluÃÃes para equaÃÃes quase lineares com coeficientes variÃveis em conjuntos singulares degenerados que serÃo denotados por S(u) := {X : Du(X) = 0} .
O resultado principal deste trabalho revela que ao longo de S(u), u à assintoticamente tÃo regular quanto as soluÃÃes das equaÃÃes com coeficientes constantes. Em particular, ao longo do conjunto S(u), Du tem um mÃdulo de continuidade superior a baixa caracterÃstica de continuidade de seus coeficientes. Os
resultados sÃo novos e mesmo no contexto de equaÃÃes diferenciais lineares onde se mostra que soluÃÃes H^1- fracas da equaÃÃo div(a(X, Du)) = 0 com os aij elÃpicos e Dini-ContÃnuos sÃo realmente C ^{1,1^{-}} ao longo de S(u). Os resultados e as perspectivas deste trabalho promovem um novo entendimento sobre as propriedades suavidade de soluÃÃes para equaÃÃes singulares, ou degeneradas, alÃm de estimativas tÃpicas
sobre regularidade elÃpticas, precisamente onde temos os atributos de difusÃo do equaÃÃo do colapso.
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Barker, Tobias. "Uniqueness results for viscous incompressible fluids." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:db1b3bb9-a764-406d-a186-5482827d64e8.
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First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calderón. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L<sub>∞</sub>(-1; 0; L<sup>3, β</sup>(B(1) ⋂ ℝ<sup>3</sup> <sub>+</sub>)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and Šverák [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new ϵ-regularity criterion. Third, we show that if a weak Leray-Hopf solution in ℝ<sup>3</sup> <sub>+</sub>×]0,∞[ has a finite blow-up time T, then necessarily lim<sub>t↑T</sub>||v(·, t)||<sub>L<sup>3,β</sup>(ℝ<sup>3</sup> <sub>+</sub>)</sub> = ∞ with 3 < β < ∞. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and Šverák [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in ℝ<sup>3</sup>, with solenoidal initial data in the critical Besov space ?<sup>-1/4</sup><sub>4,∞</sub>(ℝ<sup>3</sup>), which has certain continuity properties with respect to weak* convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in ℝ<sup>3</sup>×]0,∞[ has a finite blow-up time T, then necessarily lim<sub>t↑T</sub> ||v(·, t)||<sub>L<sub>3</sub>(ℝ<sup>3</sup>)</sub> = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.
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Redwane, Hicham. "Solutions normalisées de problèmes paraboliques et elliptiques non linéaires." Rouen, 1997. http://www.theses.fr/1997ROUES059.
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Cette thèse est consacrée à l'étude de problèmes elliptiques ou paraboliques non linéaires qui sont, d'une façon générale, mal posés dans le cadre des solutions faibles (c'est-à-dire des solutions au sens des distributions). Pour surmonter cette difficulté, on va s'intéresser à une autre classe de solutions : les solutions renormalisées. Cette notion a été introduite par R. -J. Di Perna et P. -L. Lions pour l'étude des équations de Boltzmann, et les équations du premier ordre.
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Jesslé, Didier. "Quelques résultats mathématiques en thermodynamique des fluides compressibles." Phd thesis, Toulon, 2013. http://tel.archives-ouvertes.fr/tel-00860854.
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Dans cette thèse, nous étudions les écoulements de fluides compressibles décrits par les équations de Navier-Stokes-Fourier dans les cas stationnaire et instationnaire et avec des conditions de bord assurant l'isolation thermique et mécanique du fluide. On commence par le cas stationnaire barotrope et des conditions de Navier à la frontière du domaine. La pression est donc de la forme p(%) = % où est appelé coefficient adiabatique et nous arrivons à montrer l'existence de solutions faibles pour > 1.On généralise ensuite ce résultat aux équations de Navier-Stokes-Fourier avec conduction de la chaleur et glissement (partiel ou total) au bord, toujours dans le cas stationnaire. On montre cette fois-ci l'existence de solutions faibles particulières appelées solutions entropiques variationnelles respectant l'inégalité d'entropie pour > 1 et l'existence de solutions faibles respectant le bilan de l'énergie totale au sens faible pour > 5/4. On travaille ensuite sur les écoulements instationnaires décrits par les équations de Navier-Stokes-Fourier sur une large variété de domaines non bornés, tout d'abord pour des conditions de bord d'adhérence puis pour des conditions de Navier à la frontière (ce qui restreintquelque peu la diversité des domaines non bornés admissibles). On arrive à montrer l'existence de solutions faibles particulières respectant l'inégalité d'entropie et une inégalité de dissipation remplaçant l'égalité de conservation d'énergie totale dans le volume qui n'a plus de sens dans les domaines non bornés. Par après, on met en place une inégalité dite d'entropie relative dont on montre qu'elle est respectée par certaines des solutions faibles exhibées auparavant. Ces solutions sont appelées solutions dissipatives. On parvient à prouver que pour chaque donnée initiale, il existe au moins une solution dissipative. Cette inégalité d'entropie relative nous permet de démontrer le principe d'unicité forte-faiblepour nos solutions dissipatives. Précisément, cela signifie qu'une solution dissipative et une solution forte issues des mêmes données initiales coïncident sur le temps maximal d'existence de la solution forte. La propriété d'unicité forte-faible donne un fondement à la notion de solution dissipative pour les domaines non bornés.
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Ciomaga, Adina. "Analytical properties of viscosity solutions for integro-differential equations : image visualization and restoration by curvature motions." Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2011. http://tel.archives-ouvertes.fr/tel-00624378.
Full textAbstract:
Le manuscrit est constitué de deux parties indépendantes.Propriétés des Solutions de Viscosité des Equations Integro-Différentielles.Nous considérons des équations intégro-différentielles elliptiques et paraboliques non-linéaires (EID), où les termes non-locaux sont associés à des processus de Lévy. Ce travail est motivé par l'étude du Comportement en temps long des solutions de viscosité des EID, dans le cas périodique. Le résultat classique nous dit que la solution u(¢, t ) du problème de Dirichlet pour EID se comporte comme ?t Åv(x)Åo(1) quand t !1, où v est la solution du problème ergodique stationaire qui correspond à une unique constante ergodique ?.En général, l'étude du comportement asymptotique est basé sur deux arguments: la régularité de solutions et le principe de maximumfort.Dans un premier temps, nous étudions le Principe de Maximum Fort pour les solutions de viscosité semicontinues des équations intégro-différentielles non-linéaires. Nous l'utilisons ensuite pour déduire un résultat de comparaison fort entre sous et sur-solutions des équations intégro-différentielles, qui va assurer l'unicité des solutions du problème ergodique à une constante additive près. De plus, pour des équationssuper-quadratiques le principe de maximum fort et en conséquence le comportement en temps grand exige la régularité Lipschitzienne.Dans une deuxième partie, nous établissons de nouvelles estimations Hölderiennes et Lipschitziennes pour les solutions de viscosité d'une large classe d'équations intégro-différentielles non-linéaires, par la méthode classique de Ishii-Lions. Les résultats de régularité aident de plus à la résolution du problème ergodique et sont utilisés pour fournir existence des solutions périodiques des EID.Nos résultats s'appliquent à une nouvelle classe d'équations non-locales que nous appelons équations intégro-différentielles mixtes. Ces équations sont particulièrement intéressantes, car elles sont dégénérées à la fois dans le terme local et non-local, mais leur comportement global est conduit par l'interaction locale - non-locale, par exemple la diffusion fractionnaire peut donner l'ellipticité dans une direction et la diffusion classique dans la direction orthogonale.Visualisation et Restauration d'Images par Mouvements de CourbureLe rôle de la courbure dans la perception visuelle remonte à 1954, et on le doit à Attneave. Des arguments neurologiques expliquent que le cerveau humain ne pourrait pas possiblement utiliser toutes les informations fournies par des états de simulation. Mais en réalité on enregistre des régions où la couleur change brusquement (des contours) et en outre les angles et les extremas de courbure. Pourtant, un calcul direct de courbures sur une image est impossible. Nous montrons comment les courbures peuvent être précisément évaluées, à résolution sous-pixelique par un calcul sur les lignes de niveau après leur lissage indépendant.Pour cela, nous construisons un algorithme que nous appelons Level Lines (Affine) Shortening, simulant une évolution sous-pixelique d'une image par mouvement de courbure moyenne ou affine. Aussi bien dans le cadre analytique que numérique, LLS (respectivement LLAS) extrait toutes les lignes de niveau d'une image, lisse indépendamment et simultanément toutes ces lignes de niveau par Curve Shortening(CS) (respectivement Affine Shortening (AS)) et reconstruit une nouvelle image. Nousmontrons que LL(A)S calcule explicitement une solution de viscosité pour le le Mouvement de Courbure Moyenne (respectivement Mouvement par Courbure Affine), ce qui donne une équivalence avec le mouvement géométrique.Basé sur le raccourcissement de lignes de niveau simultané, nous fournissons un outil de visualisation précis des courbures d'une image, que nous appelons un Microscope de Courbure d'Image. En tant que application, nous donnons quelques exemples explicatifs de visualisation et restauration d'image : du bruit, des artefacts JPEG, de l'aliasing seront atténués par un mouvement de courbure sous-pixelique
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Amorim, Charles Braga. "Existência e simetrias para uma equação elíptica não-linear com potencial monopolar e anisotrópico." Universidade Federal de Sergipe, 2015. https://ri.ufs.br/handle/riufs/5810.
Full textAbstract:
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>This master thesis is concerned to nonlinear elliptic problem with mono-polar anisotropic potential
u + u|u|p−1 + v (x)u + f(x) = 0 in Rn
u(x) - 0, as |x| - 00
provided n > 3 and p > n
n−2 . These results, between others things, deals with sub-critical, critical and
super-critical nonlinearity. We obtain well-posedness of solutions, regularity in c2(Rn), symmetries and
asymptotic behavior of solutions in singular spaces Hk. We employ Banach fixed technique and a theorem
of regularity elliptic to get those results, this technique does not need of the Hardy type inequalities and
variational methods.<br>Nesta dissertação estudamos o problema elíptico
u + u|u|p−1 + v (x)u + f(x) = 0 em Rn
u(x) - 0, quando |x| - 00 sujeito a restrições n > 3 e p > n
n−2 , cobrindo os casos sub-críticos, críticos e super-críticos. Obtemos
boa-colocação de soluções, regularidade, simetrias de soluções e comportamento assintótico em espaços
singulares Hk. Empregamos um argumento de ponto fixo em Hk e Ek ao invés de usar desigualdades do
tipo Hardy e métodos variacionais.