Type II Singularity Formation in Harmonic Map Flow

We consider the harmonic map flow from a bounded two-dimensional domain to $S^2$: $$u: \Omega \to S^2,~ u_t=\Delta u+|\nabla u|^2 u.$$ Here we don't assume any symmetry on the domain. We construct Type II blow up solutions and prove that the blow rate $ (T-t)/\log^2 (T-t)$ is universal and stable in general domains. We also construct multiple and reverse bubblings. As a consequence we can perform a new geometric surgery for continuation of solutions after bubbling. If time permits, other Type II blow up problems, such as Keller-Segel and critical exponent problems will be discussed as well. (Joint work with Manuel del Pino and Juan Davila.)

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HOST: Huiqiang Jiang