A scaling limit from the wave map to the heat flow into $\mathbb{S}^2$

Arghir Zarnescu (Basque Foundation for Science, Bilbao, Bizkaia, Spain; Basque Center for Applied Mathematics, Mazarredo, Bilbao, Spain; and Institute of the Romanian Academy, Bucharest, Romania)

Abstract

In this paper we study a limit connecting a scaled wave map with the heat flow into the unit sphere $\mathbb{S}^2$. We show quantitatively how the two equations are connected by means of an initial layer correction. This limit is motivated as a first step into understanding the limit of zero inertia for the hyperbolic-parabolic Ericksen–Leslie’s liquid crystal model.

Keywords

Ericksen–Leslie, wave map, heat flow, initial layer

2010 Mathematics Subject Classification

35L05, 35L30, 35L81, 37D50, 80A20

Full Text (PDF format)

The project of this paper was initialized when Ning Jiang visited Arghir Zarnescu at Basque Center for Applied Mathematics (BCAM) in March 2017. They appreciate the hospitality of BCAM. The activity of Ning Jiang on this work was supported by Chinese NSF grants 11471181 and 11731008. The research of S.J. Tang was supported by Chinese NSF grants 11601398 and 11671309.

The activity of A. Zarnescu on this work is partially supported by the Basque Government through the BERC 2018-2021 program; by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718 and through project MTM2017-82184-R funded by (AEI/FEDER, UE) and acronym “DESFLU”.

Received 12 January 2018

Received revised 6 December 2018

Accepted 6 December 2018

Published 8 July 2019