/lp/unpaywall/infinite-superlinear-growth-of-the-gradient-for-the-two-dimensional-9nxIFEBizo
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- Publisher
- Unpaywall
- ISSN
- 1078-0947
- DOI
- 10.3934/dcds.2009.23.755
- Publisher site
- See Article on Publisher Site
Abstract
DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 23, Number 3, March 2009 pp. 1–XX INFINITE SUPERLINEAR GROWTH OF THE GRADIENT FOR THE TWO-DIMENSIONAL EULER EQUATION Sergey A. Denisov University of Wisconsin-Madison, Mathematics Department 480 Lincoln Dr. Madison, WI 53706-1388, USA (Communicated by Roger Temam) Abstract. For two-dimensional Euler equation on the torus, we prove that the L norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for finite time. 1. Introduction. In this note, we are dealing with two-dimensional Euler equation. We will write the equation for vorticity in the following form ⊥ −1 θ = ∇θ · u, u = ∇ ζ = (ζ ,−ζ ), ζ = Δ θ, θ(x, y, 0) = θ (x, y) y x 0 and θ is 2π–periodic in both x and y (e.g., the equation is considered on the torus −1 T). We assume that θ has zero average over T and then Δ is well-defined since the Euler flow is area-preserving and the average of θ(·, t) is zero as well. The global existence of the smooth solution for smooth initial data is well-known [2]. It
Journal
Discrete and Continuous Dynamical Systems – Unpaywall
Published: Nov 18, 2008