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Gradient flows of energy functionals on the space of probability measures with Wasserstein metric has proved to be a strong tool in studying certain mass conserving evolution equations. Such gradient flows provide an alternate formulation for the solutions of the corresponding evolution equations. An important condition, which is known to guarantees existence, uniqueness, and continuous dependence on initial data is that the corresponding energy functional be displacement convex. We introduce a relaxed notion of displacement convexity and we show that it still guarantees short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals which are not displacement convex in the standard sense. This extends the applicability of the gradient flow approach to larger family of energies. As an application, local and global well-posedness of different higher order non-linear evolution equations are derived. Examples include the thin-film equation and the quantum drift diffusion equation in one spatial variable.

In a Hilbert space setting homogenization of evolutionary equations is discussed. In order to do so, a suitable topology on material laws is introduced and several properties of that topology are shown. With those properties homogenization theorems of a large class of linear evolutionary problems of classical mathematical physics can be obtained. The results are exemplified by the equations of piezo-electro-magnetism.

The results of this thesis are motivated by the investigation of abstract Cauchy problems. Our primary contribution is encapsulated in two new theorems. The first main theorem is a generalisation of a result of E. M. Stein. In particular, we show that every symmetric diffusion semigroup acting on a complex-valued Lebesgue space has a tensor product extension to a UMD-valued Lebesgue space that can be continued analytically to sectors of the complex plane. Moreover, this analytic continuation exhibits pointwise convergence almost everywhere. Both conclusions hold provided that the UMD space satisfies a geometric condition that is weak enough to include many classical spaces. The theorem is proved by showing that every symmetric diffusion semigroup is dominated by a positive symmetric diffusion semigoup. This allows us to obtain (a) the existence of the semigroup's tensor extension, (b) a vector-valued version of the Hopf--Dunford--Schwartz ergodic theorem and (c) an holomorphic functional calculus for the extension's generator. The ergodic theorem is used to prove a vector-valued version of a maximal theorem by Stein, which, when combined with the functional calculus, proves the pointwise convergence theorem. The second part of the thesis proves the existence of abstract Strichartz estimates for any evolution family of operators that satisfies an abstract energy and dispersive estimate. Some of these Strichartz estimates were already announced, without proof, by M. Keel and T. Tao. Those estimates which are not included in their result are new, and are an abstract extension of inhomogeneous estimates recently obtained by D. Foschi. When applied to physical problems, our abstract estimates give new inhomogeneous Strichartz estimates for the wave equation, extend the range of inhomogeneous estimates obtained by M. Nakamura and T. Ozawa for a class of Klein--Gordon equations, and recover the inhomogeneous estimates for the Schr??dinger equation obtained independently by Foschi and M. Vilela. These abstract estimates are applicable to a range of other problems, such as the Schr??dinger equation with a certain class of potentials.

Thesis (MSc.(Physics)--University of Pretoria, 2007. / Includes bibliographical references.

Thesis (Ph. D.)--West Virginia University, 1998. / Title from document title page. Document formatted into pages; contains xi, 116 p. : ill. (some col.) Includes abstract. Includes bibliographical references (p. 94-98).

Thesis (Ph.D.)--University of North Carolina at Charlotte, 2005. / Includes bibliographical references (leaves 71-79).

Thesis (Ph.D.)--Ohio University, June, 2005. / Includes bibliographical references (p. 96-102)

Thesis (Ph.D.)--Ohio University, June, 2005. / Title from PDF t.p. Includes bibliographical references (p. 96-102)

Thesis (PhD.(Mathematics))-University of Pretoria, 2005. / Includes bibliographical references. Available on the Internet via the World Wide Web.