Moving Mesh for the Axisymmetric Harmonic Map Flow

We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method. Mathematics Subject Classification. 35A05, 35K55, 65N30, 65N50, 65N99. Received: June 10, 2004. Revised: March 2, 2005. Introduction Let D be the unit disc in R and let S be the two dimensional unit sphere in R. In this paper, we compute solutions u : D × [0,+∞) → S to the harmonic map flow: ut = ∆u + u|∇u|2 in D × (0,+∞), (1) u = u0 on ∂D × [0,+∞), (2) u = u0 on D × {0}, (3) where the initial condition u0 is a map from D to S. The harmonic map heat flow has been introduced by Eells and Sampson in [7] as the L gradient of the Dirichlet energy: E(u) = 1 2 ∫ D |∇u|2. A result of Struwe [15], completed by Chang [6] for domains with boundary, asserts that if u0 belongs to H(D,S) with u0|∂D in H (∂D,S2), then problem (1)–(3) has a weak solution u ∈ Hloc(D × [0,+∞)), which is smooth away from finitely many singular points (xi, ti)1≤i≤N ⊂ D × [0,+∞). Its Dirichlet energy