Browsing by Subject "Integro-differential equations"
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Item Estimates on higher derivatives for the Navier-Stokes equations and Hölder continuity for integro-differential equations(2012-08) Choi, Kyudong; Vasseur, Alexis F.; Caffarelli, Luis; Figalli, Alessio; Gamba, Irene; Morrison, Philip; Pavlovic, NatasaThis thesis is divided into two independent parts. The first part concerns the 3D Navier-Stokes equations. The second part deals with regularity issues for a family of integro-differential equations. In the first part of this thesis, we consider weak solutions of the 3D Navier-Stokes equations with L² initial data. We prove that ([Nabla superscript alpha])u is locally integrable in space-time for any real [alpha] such that 1 < [alpha] < 3. Up to now, only the second derivative ([Nabla]²)u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in local weak-L[superscript (4/([alpha]+1))]. These estimates depend only on the L² norm of the initial data and on the domain of integration. Moreover, they are valid even for [alpha] ≥ 3 as long as u is smooth. The proof uses a standard approximation of Navier-Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced. In the second part of this thesis, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in C[superscript beta] for all time if its initial data lies in C[superscript beta]. Also, we prove a C[superscript beta]-regularization effect from [mathematical equation] initial data. It provides an alternative proof to the result of Caffarelli, Chan and Vasseur [10], which was based on De Giorgi techniques. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption [mathematical equation]on the divergent-free drift velocity. The proof is in the spirit of Kiselev and Nazarov [48] where they established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation by observing the evolution of a dual class of test functions.Item Integrodifference equations applied to plant competition and control(Texas Tech University, 1998-05) Kesinger, Jacob C.In this paper, the Allen Allen Atkinson two-dimensional model with control [3], a discrete-time continuous-space model, is derived and discussed. Some basic results are given, homogeneous steady states derived, and computer simulations performed.Item Numerical modeling and analysis of heat transfer in semitransparent media with combined radiation and conduction(Texas Tech University, 1999-12) Hu, ZirongThe governing equation of heat transfer in semitransparent media by coupled conduction and radiation is a very complicated integro-differential equation. This equation is a function of geometry (three variables), wavelength, and direction (two variables). It is a highly nonlinear equation. It becomes more complex if the thermal and radiative properties are temperature dependent. Most difficulties encountered while solving these problems are how to handle the complex integro-differential equation and complex geometries. Except for very special situations, numerical methods have to be used to solve problems involving radiative heat transfer, especially for semitransparent media with coupled radiation and conduction. Because of these complexities, some simplified methods cannot simulate realistic problems, and a great deal of computational time is required for the transient, inhomogeneous, anisotropic, scattering participating media problems. In such circumstances, it is intractable without high-performance supercomputers, even with many simplifications and hypotheses. This is especially true when dealing with multi-dimensional combined heat transfer mode problems. On the other hand, critical attributes of a successfully numerical analysis include efficiency of calculation, accuracy of results, compatibility with other energy transport or momentum transport mechanisms, and the ability to handle complex geometries. Many numerical methods have been used to analyze combined-mode heat transfer in participating media. After reviewing state-of-the-art methods and analyzing the attributes of the combined-mode heat transfer governing equations, a new methodology is presented in this dissertation. This methodology is applied to expedite the efficiency of finite element calculations. This methodology seeks to effectively handle the topics of efficiency, accuracy, and compatibility. By using the finite element method and the proposed Effective Optical Depth (EOD), it is easy to simulate complex geometries and other complexities with reasonable accuracy and high efficiency. The incorporation of the EOD makes the computing time decrease greatly with little sacrifice of accuracy under certain conditions. Furthermore, this methodology can be extended to multidimensional problems. A computational code is developed based on this methodology. and an one-dimensional plain parallel plates benchmark problem is used to demonstrate the validity of the code. The results of the code show great agreement with that obtained by other numerical methods published by other investigators. After the verification of the code, several parametric studies were conducted. The results of these studies verify that the EOD approach offers solutions with reasonable accuracy and high efficiency. The main contribution of this research is the development of a methodology for the numerical analysis of heat transfer in semitransparent media by employing an effective optical depth approximation and incorporating the EOD into a FEM formulation capable of treating integro-differential equations efficiently and accurately.Item Numerical solution of stochastic delay integrodifferential equations in population dynamics(Texas Tech University, 2004-08) Hopkins, TimA numerical procedure to approximate certain kinds of stochastic differential equations with time-delay is described. First, it is shown how a deterministic population model with continuously distributed time-delay can be transformed into a stochastic population model. A numerical method to solve the stochastic equations is described. An error analysis of the method is given. The numerical method is then applied to obtain approximations for four different systems of stochastic differential equations with time-delay, including three biological systems.Item Regularity for solutions of nonlocal fully nonlinear parabolic equations and free boundaries on two dimensional cones(2013-05) Chang Lara, Hector Andres; Caffarelli, Luis A.On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary.