The unique solvability of second order elliptic and parabolic equations (in either divergence form or non-divergence form) in Sobolev spaces is well known if the leading coefficients are, for example, uniformly continuous.
However, in general, it is not possible to solve equations in Sobolev spaces unless the coefficients have some regularity assumptions.
In this talk we will discuss some possible classes of discontinuous coefficients with which elliptic and parabolic equations are uniquely solvable in Sobolev spaces.
Especially, as our main results, we will focus on the unique solvability of equations with coefficients only measurable in one spatial variable and having small mean oscillations in the other variables (called partially BMO coefficients).
We will also discuss some applications of our results as well as a new approach to a priori L_p estimates.
Most of the talk is based on joint work with Nicolai Krylov and with Hongjie Dong.

I will discuss the metric behavior of the Kahler-Ricci flow on Hirzebruch surfaces assuming that the initial metric is invariant under a maximal compact subgroup of the automorphism group. I will describe how, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to P^1 or contracts an exceptional divisor. This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work with Jian Song.

In processes involving multi-phase flow in highly heterogeneous media, such as oil recovery by gas injection, mobile phases will seek high-permeability flow paths. Therefore, it is essential that models for such processes effectively account for these paths. For this purpose, we have developed a computational framework for flow solvers based on adapted Cartesian grids that are equipped with multi-point flux approximations obtained with specialized transmissibility upscaling methods.

For gridding, we propose using Cartesian Cell-based Anisotropically Refined (CCAR) grids, which inherit the ease of Cartesian grids while providing rapid transition between coarse and fine scales to resolve fine-scale features accurately and efficiently. We present an iterative algorithm for automatically generating such grids based on geological data and information from global coarse-scale flow simulations.

For upscaling, we discuss a local transmissibility upscaling method, called Variable Compact Multi-Point (VCMP), that uses spatially varying and compact multi-point flux stencils. The stencil weights are chosen so as to reproduce generic local flow problems accurately, while remaining as close as possible to a two-point flux for the sake of robustness. The inherent flexibility of VCMP can also be exploited to ensure that the solution of the resulting system satisfies a discrete maximum principle.

We conclude with application of these gridding, upscaling and discretization methods, originally designed for single-phase flow, to two-phase flow, which requires enhancing our adaptive mesh refinement scheme in order to accurately resolve rapidly advanacing saturation fronts. We show that adaptivity allows such accurate resolution by upscaling only single-phase parameters, thus avoiding the significant computational expense of multi-phase upscaling.