On the Slightly Perturbed De Gregorio Model on $S^1$

Abstract

It is conjectured that the generalization of the Constantin–Lax–Majda model (gCLM) $\omega _t + a u\omega _x = u_x \omega$, due to Okamoto, Sakajo and Wunsch, can develop a finite time singularity from smooth initial data for $a < 1$. For the endpoint case where a is close to and less than 1, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection a is slightly larger than 1, we prove that the solution exists globally with $|| \omega (t)||_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.

Appendix A

Lemma A.1

Suppose that $f \in L^2( (\sin \frac{x}{2})^{-2} )$. We have

$$\begin{aligned} \int _{S^1} \cot \frac{x}{2} \cdot f \cdot Hf \,{\mathrm{d}}x = - \pi H f(0)^2. \end{aligned}$$

(3.9)

Proof

Firstly, we consider $f \in C^{\infty }$. Using the Tricomi identity of the Hilbert transform (see for example [4, 12]), we obtain

$$\begin{aligned} \int _{S^1} \cot \frac{x}{2} f \cdot Hf \,{\mathrm{d}}x = - 2 \pi H( f \cdot Hf)(0) = - \pi ( (Hf(0))^2 - f(0)^2). \end{aligned}$$

Since $f \in L^2( ( \sin \frac{x}{2})^{-2} )$, we have $f(0) = 0$ and obtain (A.1) for $f \in C^{\infty }$. For general f, we can find a sequence $f_n \in C^{\infty }$ such that $f_n \rightarrow f$ in $L^2( ( \sin \frac{x}{2})^{-2} )$. Clearly, we have $H f_n \rightarrow Hf$ and $f_n \cot \frac{x}{2} \rightarrow f \cot \frac{x}{2}$ in $L^2$. Using the Cauchy–Schwarz inequality, we get $Hf_n(0) \rightarrow Hf(0)$. Applying (A.1) to $f_n$ and then taking $n\rightarrow \infty$ concludes the proof. $\square$

Next, we prove Lemmas 2.6 and 2.7 .

Proof of Lemma 2.6

Applying integration by parts yields

$$\begin{aligned} \begin{aligned} D_x Hf(x)&=\frac{1}{2\pi } \int \sin x \cdot f(y) \cdot \partial _x \cot \frac{x-y}{2} \,{\mathrm{d}}y\\&= -\frac{1}{2\pi } \int \sin x \cdot f(y) \partial _y \cot \frac{x-y}{2} \,{\mathrm{d}}y \\&= \frac{1}{2\pi } \int \sin x \cdot f_y(y) \cot \frac{x-y}{2} \,{\mathrm{d}}y . \\ \end{aligned} \end{aligned}$$

It follows

$$\begin{aligned} \begin{aligned} D_x H f(x) - H(D_x f )(x)&= \frac{1}{2\pi } \int (\sin x -\sin y) f_y(y) \cot \frac{x-y}{2} \,{\mathrm{d}}y \\ \end{aligned} \end{aligned}$$

Note that $(\sin x - \sin y) \cot \frac{x-y}{2} = 2 \sin \frac{x-y}{2} \cos \frac{x+y}{2} \cot \frac{x-y}{2} = 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2} = \cos x + \cos y.$ We conclude

$$\begin{aligned} D_x H f(x) - H(D_x f )(x) = \frac{1}{2\pi } \int (\cos x + \cos y) f_y(y) \,{\mathrm{d}}y = \frac{1}{2\pi } \int \sin y \cdot \omega (y) \,{\mathrm{d}}y. \end{aligned}$$

$\square$

Proof of Lemma 2.7

Using the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} \Big |\frac{ \omega }{\sin \frac{x}{2}} \Big |= \frac{1}{ |\sin \frac{x}{2}| } |\int _0^x \omega _x \,{\mathrm{d}}x | \lesssim \frac{1}{ |\sin \frac{x}{2}| } (\int _0^x \sin ^2 \frac{x}{2} \,{\mathrm{d}}x )^{1/2} || \omega ||_{{{\mathcal {H}}}} \lesssim || \omega ||_{{{\mathcal {H}}}} \end{aligned}$$

for any $x \in S^1$, which concludes the proof. $\square$