Regularity and uniqueness of a class of biharmonic map heat flows

Autor: Changyou Wang, Tao Huang, Jay Hineman
Rok vydání: 2013
Předmět:
Zdroj: Calculus of Variations and Partial Differential Equations. 50:491-524
ISSN: 1432-0835
0944-2669
DOI: 10.1007/s00526-013-0644-2
Popis: We consider a class of weak solutions of the heat flow of biharmonic maps from \(\Omega \subset \mathbb{R }^n\) to the unit sphere \(\mathbb{S }^L\subset \mathbb{R }^{L+1}\), that have small renormalized total energies locally at each interior point. For any such a weak solution, we prove the interior smoothness, and the properties of uniqueness, convexity of hessian energy, and unique limit at \(t=\infty \). We verify that any weak solution \(u\) to the heat flow of biharmonic maps from \(\Omega \) to a compact Riemannian manifold \(N\) without boundary, with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12), has small renormalized total energy locally and hence enjoys both the interior smoothness and uniqueness property. Finally, if an initial data \(u_0\in W^{2,r}(\mathbb{R }^n, N)\) for some \(r>\frac{n}{2}\), then we establish the local existence of heat flow of biharmonic maps \(u\), with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12).
Databáze: OpenAIRE
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