Self-similar finite-time blowups with singular profiles of the generalized Constantin-Lax-Majda model: theoretical and numerical investigations

Abstract

We investigate novel scenarios of self-similar finite-time blowups of the generalized Constantin-Lax-Majda model with a parameter $a$, which are induced by a new setting where the smooth initial data satisfy certain derivative degeneracy condition. In this setting, our numerical study reveals distinct self-similar blowup behaviors depending on the sign of $a$. For $a>0$, we observe one-scale self-similar blowups with regular profiles that have not been found in previous studies. In contrast, for $a\le 0$, we discover a novel two-scale self-similar blowup scenario where the outer profile converges to a singular function at the blowup time while the inner profile remains regular on a much smaller scale. Correspondingly, an $a$-parameterized family of singular self-similar profiles with explicit expressions are constructed for $a<0$ and shown to match nicely with the limiting profiles obtained in numerical simulation. In particular, for the specific case of $a=0$, we rigorously prove the convergence of the outer profile to an explicit singular function in self-similar coordinates. Furthermore, we demonstrate the two-scale nature of the blowup in this scenario by showing that the local inner profile behavior around the singularity point of the outer profile is governed by a traveling wave on a smaller scale. To support this observation, we rigorously establish the existence of such traveling wave solutions via a fixed-point method.

Figures (19)

• Figure 1.1: First row: Evolution of $\omega$ (in original spatial coordinate) for $a=-1$ (a) with non-degenerate odd-symmetric initial data and (b) with degenerate odd-symmetric initial data, respectively. Second row: Corresponding evolution of the profile (in dynamically rescaled coordinate) for $a=-1$ (c) with non-degenerate odd-symmetric initial data and (d) with degenerate odd-symmetric initial data, respectively. A regular profile arises from non-degenerate initial data, while a singular profile with singularities at $X=\pm 1$ arises from degenerate initial data. Here $X=x/(T-t)^\gamma$ is the dynamically rescaled spatial variable. Only the parts for $x\geq 0$ (or $X\geq 0$) are plotted due to the odd symmetry.
• Figure 4.1: Convergence of the profile for $a=0.5$ and $k=3$. (a) Evolution of the rescaled profile $\Omega(X,\tau)$ at different times. The dashed line plots the final steady state $\bar{\Omega}$, of which the residual has dropped below $1.5\times 10^{-8}$. (b) Evolution of $f(X,\tau)$, related to $\Omega(X,\tau)$ by $f(X,\tau)=\Omega(X,\tau)/X^k$. The preservation of $f(0)=1$ throughout the evolution demonstrates that the degeneracy of the solution at the origin is maintained numerically.
• Figure 4.2: Convergence of the scaling factors for $a=0.5$ and $k=3$. The stabilization of $c_l$, $c_\omega$, and $\gamma$ as $\tau$ increases supports convergence towards a one-scale self-similar blowup.
• Figure 4.3: Steady states of the profile $\bar{\Omega}(X)$ for $0.1\le a\le 1.5$ with fixed vanishing order $k=3$. The dashed curve corresponds to the critical value $a=a_{c,1}$ at which the profile transitions from being smooth with full-line support to being compactly supported.
• Figure 4.4: One-scale self-similar blowup profiles for fixed vanishing order $k=3$: (a) $a<a_{c,2}$ and (b) $a>a_{c,2}$. In (b), we plot the flipped profiles $\bar{\Omega}^{\mathrm{flip}}(X):=-\bar{\Omega}(X)$ since $\bar{c}_\omega>0$ in these cases; these $\bar{\Omega}^{\mathrm{flip}}$ correspond to unstable self-similar blowups.

Theorems & Definitions (35)

• Proposition 2.1
• proof
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6
• Theorem 2.7
• proof : Proof of Theorem \ref{['thm:convergence_a_0']}
• proof : Proof of Corollary \ref{['cor:convergence_a_0_halfline']}
• Proposition 7.1
• proof