Strong solutions to a modified Michelson-Sivashinsky equation

We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson–Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space $W^{1,\infty} (\mathbb{R}^d)$ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity. The proof of the latter result utilizes ideas previously introduced by Kiselev, Nazarov, Volberg and Shterenberg to handle the critically dissipative surface quasi-geostrophic equation and the critically dissipative fractional Burgers equation. Namely, the global regularity result is achieved by constructing a time-dependent modulus of continuity that must be obeyed by the solution of the initial-value problem for all time, preventing blowup of the gradient of the solution. This work provides an example where regularity is shown to persist even when a priori bounds are not available.

Keywords

global regularity, Michelson–Sivashinsky, nonlinear-nonlocal parabolic equation, maximum principle

2010 Mathematics Subject Classification

35B50, 35B65, 35K55

Full Text (PDF format)

Received 23 April 2020

Accepted 8 December 2020

Published 18 June 2021