Local gradient estimate for porous medium and fast diffusion equations by Martingale method

Thesis

This thesis focuses on a certain type of nonlinear parabolic partial differential equations, i.e. PME and FDE, which can be written as ∂_tu = Δu^m with m > 0. Our aim is to derive better gradient bound in the form of

|∇u|2 (t,·) ≤ C

||u(0,·)||k∞ − uk (t,·) t

,  and

|∇u|2 uk1

,  (t,·) −

∂tu uk2

,  (t,·) ≤

C1 t

·

Chapter 1 consists of a survey on results related to PME and FDE, and a short review on some works about deriving gradient estimates in probabilistic ways.

In Chapter 2 we estimate gradient on space variables of solutions to the heat equation on Euclidean space. The main idea is to construct two semimartingales by letting the solution and its gradient running backward on the path space of a diffusion process. Estimates derived from decompositions of those two semimartingales are then combined to give rise to an upper bound on gradient that only involves the maximum of the initial data and time variable. In particular, it is independent of the dimension.

In Chapter 3 we carry the idea in Chapter 2 onto the study of positive solutions to PME or FDE, and obtained a similar type of bound on |∇u| for local solutions to PME or FDE on Euclidean space. In existing literature there have always been constraints on m. By considering a more general form of transformation on u and introducing a family of equivalent measures on path space, we add more flexibility to our method. Thus our result is valid for a larger range of m. For global solutions, when m violates our constraint, we need two-sided bound on u to control |∇u|.

In Chapter 4 we utilize maximum principle to derive Li-Yau type gradient estimate for PME on a compact Riemannian manifold with Ricci curvature bounded from below. Our result is able to yield a Harnack inequality possessing the right order in time variable when the lower bound of Ricci curvature is negative.

Authors: Zhang, Z

• Publication date: 2014
• DOI: 10.5287/ora-gavy6eoa4
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Aronson-Benilan estimate; Porous medium equation; Fast diffusion equation; Martingale; Curvature-dimension condition