Some new results on Lipschitz regularization for parabolic equations

Dirr, Nicolas ORCID: https://orcid.org/0000-0003-3634-7367 and Nguyen, Vinh Duc 2019. Some new results on Lipschitz regularization for parabolic equations. Journal of Evolution Equations 19 (4) , pp. 1149-1166. 10.1007/s00028-019-00512-w

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Abstract

It is well known that the bounded solution u(t, x) of the heat equation posed in RN×(0,T) for any continuous initial condition becomes Lipschitz continuous as soon as t>0, even if the initial datum is not Lipschitz continuous. We investigate this Lipschitz regularization for both strictly and degenerate parabolic equations of Hamilton–Jacobi type. We give proofs avoiding Bernstein’s method which leads to new, less restrictive conditions on the Hamiltonian, i.e., the first-order term. We discuss also whether the Lipschitz constant depends on the oscillation for the initial datum or not. Finally, some important applications of this Lipschitz regularization are presented.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Publisher:	Springer Verlag
ISSN:	1424-3199
Funders:	EPSRC
Date of First Compliant Deposit:	1 May 2019
Date of Acceptance:	13 March 2019
Last Modified:	06 Nov 2023 22:40
URI:	https://orca.cardiff.ac.uk/id/eprint/122089

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