Analysis
Department of Mathematics and Statistics
Texas Tech University
abstract pdf
We generalize Einstein’s probabilistic method for the Brownian motion to study compressible fluids in porous media. The multi-dimensional case is considered with general probability distribution functions. By relating the expected displacement per unit time with the velocity of the fluid, we derive an anisotropic diffusion equation in non-divergence form that contains a transport term. Under the Darcy law assumption, a corresponding nonlinear partial differential equations for the density function is obtained. The classical solutions of this equation are studied, and the maximum and strong maximum principles are established. We also obtain exponential decay estimates for the solutions for all time, and particularly, their exponential convergence as time tends to infinity. Our analysis uses some transformations of the Bernstein-Cole–Hopf type which are explicitly constructed even for very general equations of state. Moreover, the Lemma of Growth in time is proved and utilized in order to achieve the above decaying estimates.
This is joint work with Akif Ibragimov (Texas Tech University, and Oil and Gas Institute of the Russian Academy of Science).
We generalize Einstein’s probabilistic method for the Brownian motion to study compressible fluids in porous media. The multi-dimensional case is considered with general probability distribution functions. By relating the expected displacement per unit time with the velocity of the fluid, we derive an anisotropic diffusion equation in non-divergence form that contains a transport term. Under the Darcy law assumption, a corresponding nonlinear partial differential equations for the density function is obtained. The classical solutions of this equation are studied, and the maximum and strong maximum principles are established. We also obtain exponential decay estimates for the solutions for all time, and particularly, their exponential convergence as time tends to infinity. Our analysis uses some transformations of the Bernstein-Cole–Hopf type which are explicitly constructed even for very general equations of state. Moreover, the Lemma of Growth in time is proved and utilized in order to achieve the above decaying estimates.
This is joint work with Akif Ibragimov (Texas Tech University, and Oil and Gas Institute of the Russian Academy of Science).
An under-appreciated notion from Mathematics is that of a mapping space. In many circumstances, a class of maps between two topological spaces might be a topological space of its own. This notion is generalized into the notion of an internal hom-object in several branches of Math. For example in Measure theory, this allows mapping spaces between measurable spaces to be treated as new measurable spaces, and in higher-order logic, it is a bedrock of the Curry-Howard-Lambek correspondence. I will present the structural interpretation of a mapping space, some examples and challenges in realizing them for ordinary topological or measurable spaces. As an application, I will show how this naturally leads to the notion of a path-space and shift-space for dynamical systems, and how they are defined uniquely by universal properties.