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New bounds for the Kuramoto‐Sivashinsky equation
Author(s) -
Giacomelli Lorenzo,
Otto Felix
Publication year - 2005
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20031
Subject(s) - mathematics , burgers' equation , bounded function , mathematical analysis , inviscid flow , perturbation (astronomy) , mathematical physics , pure mathematics , partial differential equation , classical mechanics , physics , quantum mechanics
Abstract We show that every L ‐periodic mean‐zero solution u of the Kuramoto‐Sivashinsky equation is on average o ( L ) for L ≫ 1, in the sense that for any T > 0 the space average of | u ( t ) | is bounded by $L \over T$ for any t > T and any L sufficiently large. For this we argue that on large spatial scales, the solution behaves like an entropy solution of the inviscid Burgers equation. The analysis of this non‐standard perturbation of the Burgers equation is based on a “div‐curl” argument. © 2004 Wiley Periodicals, Inc.