UCLA

Flow of an ideal gas through a homogeneous porous medium can be described by the well-known Porous Medium Equation $(PME)$. The key feature is that the pressure is proportional to some powers of the density, which corresponds to the anti-congestion effect given by the degenerate diffusion. This effect is widely seen in fluids, biological aggregation and population dynamics. If adding an advection, the equation can be naturally contextualized as a population moving with preferences or fluids in a porous medium moving with wind. Furthermore we may consider drifts that depend on the solution itself by a non-local convolution, which describe the interaction between particles in a swarm model or a model for chemotaxis. In this dissertation, we study those PDEs.

In the first two chapters, we consider local advection transportation driven by a known vector field. Chapter 1 is devoted to investigate the H\"{o}lder regularity of solutions in terms of bounds of the vector field in the space $L_x^{p}$. By a scaling argument, we find that $p=d$ is critical (where $d$ is the space dimension). Along with a De Giorgi-Nash-Moser type arguments, we prove H\"{o}lder regularity of solutions after time $0$ in the subcritical regime $p>d$. And we give examples showing the loss of uniform H\"{o}lder continuity of solutions in the critical regime even for divergence-free drifts.

In Chapter 2, we are interested in the geometric properties of the free boundary for the solution ($u$): $\partial\{u>0\}$. First it is shown that, if the initial data has super-quadratic growth at the free boundary, then the support strictly expands relative to the streamline. We then proceed to show the nondegeneracy and $C^{1,\alpha}$ regularity of the free boundary, when the solution is directionally monotone in space variable in a local neighborhood. The main challenge lies in establishing a local non-degeneracy estimate, which appears new even for the zero drift case.

In Chapter 3 and 4, we consider more general drifts which depends on the solution itself by a non-local convolution. If considering a swarm model or a model for chemotaxis, the non-local drift describes the interaction effect between particles as swarms of locusts or cells. Chapter 3 discusses the vanishing viscosity limit of the equation in a bounded and convex domain. The limit agrees with the first-order system with a projection operator on the boundary proposed by Carrillo, Slepcev and Wu. Thus our result gives another justification of their equation. We apply the gradient flow method and we explore bounded approximations of singular measures in the generalized Wasserstein distance, which I believe, is independently interesting and might be useful in other contexts.

Chapter 4 considers singular kernels of the form $(-\Delta)^{-s} u$ with $s\in (0,\frac{d}{2})$. With $s=1$ we recover the well-known Patlak-Keller-Seger equation which is an macroscopic description of the chemotaxis phenomenon. The competition between non-local attractive interactions and the diffusion is one of the core of subject of diffusion-aggregation equations. We study well-posedness, boundedness and H\"{o}lder regularity of solutions in most of the subcritical regime. Several open questions will be discussed.