arXiv subjects
Publications and source records attributed to Dylan Samuelian.
We consider $D$-equivariant solutions to the harmonic map heat flow from $B^2$ to $S^2$ under general time-dependent smooth boundary data and prove that there is no finite-time blow-up when $D \geq 2$.
We consider finite-time and $k$-equivariant solutions to the harmonic map heat flow from $B^2$ to $S^2$ under general time-dependent boundary data and prove that the bubble tree decomposition contains only one bubble. The method relies on the Maximum and Comparison Principle. We also exhibit solutions blowing up in infinite time for any $k \geq 1$.
We construct solutions $u(x,t)$ to the focusing, energy-critical, nonlinear wave equation \begin{equation} \partial_{tt}u - \Delta u - |u|^{p-1}u = 0, \quad t \geq 0, \ x \in \mathbb{R}^d, \ d \geq 3, \ p = (d+2)/(d-2) \end{equation} in dimension $d \in \{4,5\}$, exhibiting finite-time Type II blow-up precisely at $x = t = 0$ with a prescribed polynomial blow-up rate of $t^{-1-\nu}$, where $\nu > 1$ for $d = 4$ and $\nu > 3$ for $d = 5$. Such solutions have been constructed by Krieger-Schlag-Tataru for $d = 3$ and by Jendrej for $d = 5$. The work of Jendrej includes the extremal case $\nu = 3$, which our method does not address, and the regime $\nu > 8$. The major difference between dimensions $4$ and $5$ consists in the renormalization procedure. In $d = 4$, we essentially follow the Krieger-Schlag-Tataru scheme developed for the 3-dimensional equation. This scheme has been applied with success for other equations such as the 3D-critical NLS, Schr\"odinger maps or wave maps. In all of these cases, the polynomial structure of the nonlinearity permits the use of simple algebraic manipulations to control error terms. By contrast, the case $d = 5$ requires a modified setup due to the lower regularity of the nonlinearity, which complicates the treatment of nonlinear error terms.