On the generalized Buckley-Leverett equation

TL;DR

This paper investigates the generalized Buckley-Leverett equation with nonlocal regularizing terms, establishing conditions for global solutions and exploring potential finite-time blow-up scenarios through analysis and numerical simulations.

Contribution

It provides new theoretical results on global existence of solutions depending on the order of regularizing terms and introduces numerical evidence for finite-time blow-up when the diffusion order is less than one.

Findings

Global strong solutions exist for regularizing terms of order greater than one.

Solutions exist under size restrictions when combined regularizing order is one.

Numerical simulations suggest finite-time blow-up for diffusion order between 0 and 1.

Abstract

In this paper we study the generalized Buckley-Leverett equation with nonlocal regularizing terms. One of these regularizing terms is diffusive, while the other one is conservative. We prove that if the regularizing terms have order higher than one (combined), there exists a global strong solution for arbitrarily large initial data. In the case where the regularizing terms have combined order one, we prove the global existence of solution under some size restriction for the initial data. Moreover, in the case where the conservative regularizing term vanishes, regardless of the order of the diffusion and under certain hypothesis on the initial data, we also prove the global existence of strong solution and we obtain some new entropy balances. Finally, we provide numerics suggesting that, if the order of the diffusion is $0< \alpha<1$, a finite time blow up of the solution is possible.