Exact periodic solutions of the generalized Constantin-Lax-Majda equation with dissipation

TL;DR

This work constructs exact periodic pole-dynamics solutions for the generalized Constantin-Lax-Majda equation with dissipation on the circle, focusing on advection parameters $a=0$ and $a=\tfrac{1}{2}$ with dissipation exponents $\sigma=0$ or $1$. For $a=0$ the authors derive closed-form single-pair solutions and establish global well-posedness for small data in Wiener-type spaces, while revealing finite-time blow-up for $\sigma=0$ even at small $L^2$ norms; they also extend the Wiener algebra theory to $\sigma>0$ in the periodic setting. For $a=\tfrac{1}{2}$, they construct a mixed-pole ansatz (double and simple poles) to cancel logarithmic terms, obtaining an implicit solution for $\sigma=0$ and a phase-plane analysis for $\sigma=1$ that delineates global existence versus finite-time singularity via invariant regions and blow-up curves. Overall, the results illuminate how dissipation and advection shape self-similar blow-up in periodic gCLM models, connect periodic and real-line theories, and refine the criteria for global existence in the Wiener and Wiener-type norms.

Abstract

We present exact pole dynamics solutions to the generalized Constantin-Lax-Majda (gCLM) equation in a periodic geometry with dissipation $-Λ^σ$, where its spatial Fourier transform is $\widehat{Λ^σ}=|k|^σ$. The gCLM equation is a simplified model for singularity formation in the 3D incompressible Euler equations. It includes an advection term with parameter $a$, which allows different relative weights for advection and vortex stretching. There has been intense interest in the gCLM equation, and it has served as a proving ground for the development of methods to study singularity formation in the 3D Euler equations. Several exact solutions for the problem on the real line have been previously found by the method of pole dynamics, but only one such solution has been reported for the periodic geometry. We derive new periodic solutions for $a=0$ and $1/2$ and $σ=0$ and $1$, for which a closed collection of (periodically repeated) poles evolve in the complex plane. Self-similar finite-time blow-up of the solutions is analyzed and compared for the different values of $σ$, and to a global-in-time well-posedness theory for solutions with small data presented in a previous paper of the authors. Motivated by the exact solutions, the well-posedness theory is extended to include the case $a=0$, $σ\geq 0$. Several interesting features of the solutions are discussed.

Figures (14)

• Figure 1: Pole trajectories for $N=1,a=0,\sigma=0$ in $\tan(\frac{x}{2})$-space. Initial data is $v_c(0)=0.5 - \mathrm{i}$ and $\omega_{-1}(0) = 0.5 - 0.25 \mathrm{i}$, with different $\nu$ and $\omega_{av}(0)$. (a) $\nu = 0.2, \omega_{av}(0)=0$: The pole crosses the real-$\tan(\frac{x}{2})$ line with $\frac{d Re[v_c(t_c)]}{dt} \neq 0$. (b) $\nu \approx 0.2865, \omega_{av}(0)=0$: The pole tends to the real line as $t \rightarrow \infty$. (c) $\nu =0.1, \omega_{av}(0) \approx 0.543$: The pole touches the real line with $\frac{d Re[v_c(t)]}{dt} = 0$. (d) $\nu =0.01, \omega_{av}(0)\approx 0.543$: The pole traverses the circle touching the real line multiple times before it settles at $\tan(\frac{x}{2}) \approx 0.0806 + 0.0086\mathrm{i}$. Grid points shown on the trajectories are equispaced in $t$ and cluster near the final location of the trajectories.
• Figure 1: Summary of our results on global existence versus finite-time singularity formation in $L^2$ and $B_0$. FTS = finite-time singularity formation for arbitrarily small data; GEPD = global existence of pole dynamics solutions for small data; GEIVP = global existence of solutions to initial value problem for small data; NT = neither pole dynamics solutions nor our general theory applies. The pole dynamics solutions blow up for sufficiently large data.
• Figure 2: Degenerate pole trajectory for $N=1,a=0,\sigma=0$. Initial data is $v_c(0)=0.5 - \mathrm{i}$, $\omega_{-1}(0) = 1.5 - \mathrm{i}$, $\omega_{av}(0)=0, \nu = 0$. (a) The trajectory in $\tan(\frac{x}{2})$-space. The pole moves to $Re[v_c(t)] \rightarrow \infty$ as $t \rightarrow t_c$ (blue line) and continues from $Re[v_c(t)] = -\infty$ to its final point $v_c(\infty)=-1$ as $t \rightarrow \infty$ (red line). (b) The same trajectory in $x$-space. The pole trajectory $x_c(t)$ crosses the real line at $x=\pm \pi$ with $\frac{d Im[x_c(t_c)]}{dt} \neq 0$. Grid points shown are as in (a).
• Figure 3: $N=2$ solution for $\omega_{-1,1}(0)=-2.2$, $\omega_{-1,2}(0)=2$, $v_{c,1}(0)=0.0732644,~v_{c,2}(0)=0.17$. (a) Time evolution of the pole amplitudes. (b) Time evolution of the pole positions. Inset: $|\omega(0,t)|$ versus $dv_{c,1}/dt \sim (t_c-t)$ (dashed blue curve), which approaches slope $-2$ (solid red curve) as $t \rightarrow t_c$. This shows the similarity exponent $\beta=2$ in this example. (c) Plot of $\omega$ at $t=0.029,~0.051,~0.079$.
• Figure 4: Plot of $f_0(v_c(0))$ from \ref{['f-firstTime']} with $x=v_{c}(0)$.

Theorems & Definitions (24)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof