Orientador: Waldemar Donizete Bastos / Coorientador: Olimpio Hiroshi Miyagaki / Banca: Everaldo Souto Medeiros / Banca: German Jesus Lozada Cruz / Banca: Juliana Conceição Precioso Pereira / Banca: Sérgio Henrique Monari Soares / Resumo: Neste trabalho estudamos problemas elípticos do seguinte tipo: (P) Lu+V(x)|x|−ap* |u|p−2u = K(x)|x|−ap* f(u), em RN, em que V,K :RN → R são potenciais não negativos que podem tender a zero no infinito, f :RN →R tem crescimento subcrítico e Lu é um operador elíptico. Quando Lu é o operador p-Laplaciano com peso, isto é, Lu = Lapu = −div(|x|−ap|∇u|p−2∇u), provamos resultados de existência de solução positiva para K(x) ≡ 1 em RN e de solução positiva de energia mínima para K podendo tender a zero no infinito. No primeiro caso a técnica é baseada num argumento de truncamento, introduzido por del Pino e Felmer em [34] e usado por Alves e Souto em [10], que nos permite uma abordagem variacional. No segundo caso, usamos novamente a abordagem variacional e o principal argumento, usado por Alves e Souto em [11], é considerar convenientes condições de crescimento sobre os potenciais para obter imersões compactas no espaço todo. Esta última técnica foi adaptada para obter resultados de existência de solução de energia mínima não trivial para o operador Lu = ∆2u = ∆(∆u) / Abstract: In this work we studied elliptic problems of the following type: (P) Lu+V(x)|x|−ap* |u|p−2u = K(x)|x|−ap* f(u), em RN, em que V,K :RN → R are nonnegative potentials that can vanish at infinity, f :RN →R has a subcritical growth and Lu is an elliptic operator. When Lu is the weighted p-laplacian operator, namely, Lu = Lapu = −div(|x|−ap|∇u|p−2∇u), we prove existence results of positive solution for K(x) ≡ 1 in RN and positive ground state solution for the case when K may tend to zero in infinity. In the first case the technique is a truncation argument, introduced by del Pino and Felmer, in [34], and used by Alves and Souto, in [10], that allows us to use a variational approach. In the second case, we also use the variational approach and the main argument, used by Alves and Souto, in [11], is to consider suitable growth conditions on the potentials to obtain compact embedded in the whole space. This last technique was adapted to obtain existence of nontrivial ground state solution for operator Lu = ∆2u = ∆(∆u) / Doutor

Orientador: German Jesus Lozada Cruz / Banca: Waldemar Donizete Bastos / Banca: Marcia Cristina Anderson Braz Federson / Resumo: Sejam X um espaço de Banach,\alpha um número complexo tal que Re\alpha > 0 e A um operador linear fechado, não negativo, com domínio e imagem em X. O objetivo deste trabalho é definir o objeto A^\alpha de modo que as propriedades de potência de números complexos sejam preservadas, ou seja, (i) A ^\alpha A^\beta = A^(\alpha+\beta) ; (aditividade) (ii) A^1 = A; (iii) (A^\alpha )^\beta = A (quando o primeiro membro faz sentido). Como aplicação da teoria, caracterizamos o dom ínio da potência fracionária de um operador de nido matricialmente a partir da seguinte Equação Diferencial Parcial abstrata em espaço de Hilbert, prototipo utilizado para modelar sistemas elásticos com forte (ou estrutural) amortecimento: x ' + A^\alpha x' + Ax = 0; 0 < \alpha <= 1; com A sendo um operador positivo e autoadjunto / Abstract: Let X be a Banach space, \alpha a complex number such that Re \alpha > 0 and A a non-negative closed linear operator with domain and range in X. The purpose of this work is to de fine the object A^\alpha in a way that the properties of powers of complex numbers be preserved, i.e, (i) A ^\alpha A^\beta = A^(\alpha+\beta) ; (additivity) (ii) A^1 = A; (iii) (A^\alpha )^\beta = A (when the fi rst member makes sense). As an application of theory, we characterized the domain of fractional power of a matrix-valued operator from the abstract Partial Di erential Equation in Hilbert space, prototype used to model elastic systems with strong/structural damping: x' + A^\alpha x' + Ax = 0; 0<\alpha <= 1; with A being a positive self-adjoint operator / Mestre

M.Sc. (Applied Mathematics) / Explicit schemes for integrating ODEs and time–dependent partial differential equations (in the method of lines–MoL–approach) are very well–known to be stable as long as the maximum sizes of their timesteps remain below a certain minimum value of the spatial grid spacing. This is the Courant– Friedrich’s–Lewy (CFL) condition. These schemes are the ones traditionally being used for performing simulations in Numerical Relativity (NR). However, due to the above restriction on the timestep, these schemes tend to be so much inadequate for simulating some of the highly probable and astrophysically interesting phenomenae. So, it is of interest this currernt moment to seek or find integrating schemes that may help numerical relativists to somehow circumvent the CFL restriction inherent in the use of explicit schemes. In this quest, a more natural starting point appears to be implicit schemes. These schemes possess a highly desireable stability property – they are unconditionally stable. There also exists a combination of implicit and explicit (IMEX) schemes. Some researchers have already started exploring (since 2009, 2011) these for NR purposes. We report on the implementation of two implicit schemes (implicit Euler, and implicit midpoint rule) for Einstein’s evolution equations. For low computational costs, we concentrated on spherical symmetry. The integration schemes were successfully implemented and showed satisfactory second order convergence patterns on the systems considered. In particular, the Implicit Midpoint Rule proved to be a little superior to the implicit Euler scheme.

Ce manuscrit présente des résultats d’analyse mathématique autour de deux exemples de problèmes singuliers d’équations aux dérivées partielles issus de la modélisation. I. Diffusion croisée en dynamique des populations. En dynamique des populations, les systèmes de réaction –diffusion croisée modélisent l’évolution de populations d’espèces en compétition avec un effet répulsif entre les individus. Pour ces systèmes fortement couplés, souvent non linéaires, une question aussi fondamentale que l’existence de solutions se révèle extrêmement complexe. Dans ce manuscrit, on introduit une approche basée sur des extensions récentes de lemmes de dualité et sur des méthodes d’entropie. On démontre l’existence de solutions faibles dans un cadre général de systèmes de réaction-diffusion croisée, ainsi que certaines propriétés qualitatives des solutions. II. Équation de Boltzmann en domaine borné. L’équation de Boltzmann, introduite en 1872, modélise la dynamique des gaz raréfiés hors équilibre. Malgré les nombreux résultats autour de la question de l’existence de solutions fortes proches de l’équilibre, très peu concernent le cas d’un domaine borné, situation pourtant fréquente dans les applications. Une raison de la difficulté du problème est l’irruption des singularités le long des trajectoires rasant le bord du domaine. Dans ce manuscrit, on présente une théorie de la régulation de l’équation de Boltzmann en domaine borné. Grâce à l’introduction d’une distance cinétique qui compense les singularités au bord, on montre des résultats de propagation de normes de Sobolev et de propagation C^1 en domaine convexe. En domaine non convexe, on montre un résultat de propagation de régularité BV. / This manuscript presents results of mathematical analysis concerning two singular problems of partial differential equations coming from the modeling. I. Cross-diffusion in Population dynamics. In Population dynamics, reaction-cross diffusion systems model the evolution of the populations of competing species with a repulsive effect between individuals. For these strongly coupled, often non linear systems, a question as basic as the existence of solutions appears to be extremely complex. In this manuscript, we introduce an approach based on the most recent extensions of duality lemmas and on entropy methods. We prove the existence of weak solutions in a general setting of reaction-cross diffusion systems, as well as some qualitative properties of the solutions. II. Boltzmann equation in bounded domains The Boltzmann equation, introduced in 1872, model the evolution of a rarefied gas out of equilibrium. Despite the numerous results concerning the existence of strong solutions close to equilibrium, very few concern the case of bounded domain, though this situation is very useful in applications. A crucial reason of the difficulty of this problem is the formation of a singularity on the trajectories grazing the boundary. In this manuscript, we present a theory of the regularity of the Boltzmann equation in bounded domains. Thanks to the introduction of a kinetic distance which balances the singularity, we obtain results of propagation of Sobolev norms and C^1 propagation in convex domains. In non convex domains, we obtain the propagation of BV regularity.

No description available.

Enclosure theorems for the eigenvalues and representational formulae for the eigenfunctions of a linear, elliptic, second order partial differential operator will be established for specific domain perturbations to which the classical theory cannot be applied. In particular, the perturbation of n-dimensional Euclidean space E[superscript]n to an n-disk D[subscript]a of radius a is considered in Chapter I and the perturbation of the upper half-space H[superscript]n of E[superscript]n to the upper half of D[subscript]a, S[subscript]a, is discussed in Chapter II. In each case a general self-adjoint boundary condition is adjoined on the bounding surface of the perturbed domain. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis presents a multi-grid scheme applied to the solution of transport equations in turbulent flow associated with heat transfer. The multi-grid scheme is then applied to flow which occurs in the film cooling of turbine blades. The governing equations are discretized on a staggered grid with the hybrid differencing scheme. The momentum and continuity equations are solved by a nonlinear full multi-grid scheme with the SIMPLE algorithm as a relaxation smoother. The turbulence k — Є equations and the thermal energy equation are solved on each grid without multi-grid correction. Observation shows that the multi-grid scheme has a faster convergence rate in solving the Navier-Stokes equations and that the rate is not sensitive to the number of mesh points or the Reynolds number. A significant acceleration of convergence is also produced for the k — Є and the thermal energy equations, even though the multi-grid correction is not applied to these equations. The multi-grid method provides a stable and efficient means for local mesh refinement with only little additional computational and.memory costs. Driven cavity flows at high Reynolds numbers are computed on a number of fine meshes for both the multi-grid scheme and the local mesh-refinement scheme. Two-dimensional film cooling flow is studied using multi-grid processing and significant improvements in the results are obtained. The non-uniformity of the flow at the slot exit and its influence on the film cooling are investigated with the fine grid resolution. A near-wall turbulence model is used. Film cooling results are presented for slot injection with different mass flow ratios. / Science, Faculty of / Mathematics, Department of / Graduate

The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries. This dissertation develops methods for the accurate computation of high-order polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05]. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

The use of the non-linear SPDEs are inevitable in both physics and applied mathematics since many of the physical phenomena in nature can be effectively modeled in random and non-linear way. The Kardar-Parisi-Zhang (KPZ) equation is well-known for its applications in describing various statistical mechanical models including randomly growing surfaces, directed polymers and interacting particle systems. We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation. We provide the first tight bounds on the lower tail probability of the one point distribution of the KPZ equation with narrow wedge initial data. Our bounds hold for all sufficiently large times $T$ and demonstrates a crossover between super-exponential decay with exponent $\tfrac{5}{2}$ (and leading pre-factor $\frac{4}{15\pi} T^{1/3}$) for tail depth greater than $T^{2/3}$ (deep tail), and exponent $3$ (with leading pre-factor at least $\frac{1}{12}$) for tail depth less than $T^{2/3}$ (shallow tail). We also consider the case when the initial data is drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $\frac{5}{2}$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $\frac{3}{2}$ at all depths in the tail. We study the correlation of fluctuations of the narrow wedge solution to the KPZ equation at two different times. We show that when the times are close to each other, the correlation approaches one at a power-law rate with exponent $\frac{2}{3}$, while when the two times are remote from each other, the correlation tends to zero at a power-law rate with exponent $-\frac{1}{3}$.

Degree awarded with distinction on 6 December 1995. A research report submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Masters. Johannesburg, 1995. / In this work we give a brief overview of the existing group classification methods of partial differential equations by means of examples. On top of these methods we introduce another new method which classify according to low-dimensional Lie elgebras, One can ask: What is the aim of introducing a new method whilst there are existing methods? This question is answered in the following paragraph. Firstly we classify our system of non-linear partial differential equations using the preliminary group classification method (one of the existing methods). The results are not different from what; Euler, Steeb and Mulsor have obtained in 1991 and 1992. That is, this method does not yield new information. This new method which classifies according to low-dimensional Lie algebras is used to classify a general system of equations from plasma physics. Finally, using this method we completely classify our system for four-dimensionnl algebras. For a partial differential equation to be completely classified using this method, it must admit a low-dimensional Lie algebra.