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Dylan Samuelian

Publications and source records attributed to Dylan Samuelian.

4 recordsLinked to original sources
arXiv2026

On blow-up dynamics for the nonlinear wave equation and the harmonic map heat flow

This Ph.D. thesis concerns the blow-up analysis, and more precisely, the asymptotic decomposition into profiles of solutions to nonlinear partial differential equations. In Chapter 1, we present a unifying framework for finding profile decompositions in Hilbert spaces using the notion of dislocation spaces. The next chapters are then devoted to my publications. In the second chapter, we construct solutions to the focusing, energy-critical, nonlinear wave equation in dimension 4 and 5, exhibiting finite-time Type II blow-up precisely at the origin with a single profile and a prescribed polynomial blow-up rate of order $-1-ν$, where $ν> 1$ in 4D and $ν> 3$ in 5D. The major difference between dimension 4 and 5 consists in the renormalization procedure. In 4D, we essentially follow the Krieger-Schlag-Tataru scheme developed for the 3-dimensional equation, where the polynomial structure of the nonlinearity permits the use of simple algebraic manipulations to control error terms. By contrast, the 5D case requires a modified setup due to the lower regularity of the nonlinearity, which complicates the treatment of nonlinear error terms. In Chapter 3, we provide a modern presentation of classical Lp and Schauder estimates for parabolic equations. Then, in Chapter 4, we consider finite-time and $k$-equivariant solutions to the harmonic map heat flow from a disk to a sphere. Under general time-dependent, $k$-equivariant and smooth boundary data, we prove that the profile decomposition at the singular time contains only one bubble. The method relies on the maximum and comparison principle. We also prove a global existence criterion and exhibit solutions blowing up in infinite time for any degree of equivariance. Finally, the appendix presents the profile decomposition for the harmonic map heat flow proved by Struwe and Qing using energy arguments and the Palais-Smale condition.

math.AP↗
arXiv2026

On blow-up trees for the harmonic map heat flow from $B^2$ to $S^2$

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 $k \geq 1$.

math.AP↗
arXiv2025

Construction of blow-up solutions for the focusing energy-critical nonlinear wave equation in $\mathbb{R}^4$ and $\mathbb{R}^5$

We construct solutions $u(x,t)$ to the focusing, energy-critical, nonlinear wave equation \begin{equation} \partial_{tt}u - Δu - |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 $t^{-1-ν}$, where $ν> 1$ for $d = 4$ and $ν> 3$ 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 $ν= 3$, which our method does not address, and the regime $ν> 8$. 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, Schrödinger 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.

math.AP↗