The first chapter of this thesis is the contents of the paper On Mix-norms and the Rate of Decay of Correlations, which was published June 2021 in Nonlinearity. There we explore two quantitative notions of mixing: the decay of correlations and the decay of the mix-norm - a negative Sobolev norm. The intensity of mixing can be measured by the rates of decay of these quantities. We show that the mix-norm and correlations decay at the same rate in the sense that they are Big Oh but not Little Oh of each other. This result bridges the connection between two fields: dynamical systems, where correlations are commonly used to study mixing in the context of ergodic theory, and fluid dynamics, where mix-norms are well suited for the partial differential equations context. The second chapter can be seen as a study of mixing rates in the absence of a fluid flow wherein we present a preprint of the paper Optimal Spatially Dependent Diffusion Coefficients under an Lp Constraint. We ask what spatially dependent diffusion coefficients will maximize the rate of convergence to equilibrium of solutions to the heat equation in inhomogeneous media. We formulate a variational problem for the optimal spectral gap. We solve a relaxed version of this variational problem which provides an upper bound for the optimal spectral gap. This solution is characterized in terms of the extremals of Sobolev and Poincaré type inequalities.