On the Constantin–Lax–Majda Model with Convection

The well-known Constantin–Lax–Majda (CLM) equation, an important toy model of the 3D Euler equations without convection, can develop finite time singularities (Constantin et al. in Commun Pure Appl Math 38:715–724, 1985). De Gregorio modified the CLM model by adding a convective term (De Gregorio in Math Methods Appl Sci 19(15):1233–1255, 1996), which is known important for fluid dynamics (Hou and Lei in Commun Pure Appl Math 62(4):501–564, 2009; Okamoto in J Math Soc Jpn 65(4):1079–1099, 2013). Presented are two results on the De Gregorio model. The first one is the global well-posedness of such a model for general initial data with non-negative (or non-positive) vorticity which is based on a newly discovered conserved quantity. This verifies the numerical observations for such class of initial data. The second one is an exponential stability result of ground states, which is similar to the recent significant work of Jia et al. (Ration Mech Anal, 231:1269–1304, 2019), with the zero mean constraint on the initial data being removable. The novelty of the method is the introduction of the new solution space \({\mathcal {H}}_{DW}\) together with a new basis and an effective inner product of \({\mathcal {H}}_{DW}\).

The author was also in part supported by NSFC (Grant No. 11725102) and National Support Program for Young Top-Notch Talents.

Authors and Affiliations

1. School of Mathematical Sciences, LMNS and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

   Zhen Lei, Jie Liu & Xiao Ren

Corresponding author

Correspondence to Zhen Lei.

Communicated by C. Mouhot

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Appendix

Let us present a proof for the local wellposedness theory for (2.3), for completeness. We claim the following:

Lemma 5.1

Let \(k\ge 1\) be an integer, and \(\Omega = {\mathcal {S}}^1\) or \({\mathbb {R}}\). For any initial data \(f_{\scriptscriptstyle in} \in H^k(\Omega )\) with a compact support, there exists a time \(T=T(k,\Vert f_{\scriptscriptstyle in}\Vert _{H^1(\Omega )}) > 0\) and a unique solution \(f \in C([0,T];H^k(\Omega ))\) to (2.3), with \(\partial _t f \in C([0,T];H^{k-1}(\Omega ))\). Moreover, f satisfies the following identities for all \(0\le t \le T\),

and

Besides, if \(f_{\scriptscriptstyle in} \ge 0\), then the solution \(f \ge 0\) for \(0\le t \le T\).

Remark 5.1

In fact, one can even obtain similar results for fractional k by using Leibnitz rules for fractional derivatives, say, the Li’s law or the classical Kato-Ponce inequalities. See [12] for a complete presentation.

Proof

• Uniqueness. Assume that \(f,g \in C([0,T];H^1)\) solves (2.3) with the same initial data \(f_{\scriptscriptstyle in}\). By \(L^2\) energy estimate, we have

  $$\begin{aligned} \frac{d}{dt} \Vert f-g\Vert _{L^2}^2 \lesssim \Vert f-g\Vert _{L^2}^2 (\Vert f\Vert _{C([0,T];H^1)}^2+\Vert g\Vert _{C([0,T];H^1)}^2). \end{aligned}$$

  By Gronwall’s inequality, for \(0\le t \le T\),

  $$\begin{aligned} \Vert f-g\Vert _{L^2} \equiv 0. \end{aligned}$$

• Existence.Step 1. First, we work with smooth initial data \(f_{\scriptscriptstyle in} \in H^\infty = \cap _{N\ge 1} H^N\). We use the following iterative scheme to approximate the solution of (2.3):

  $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t f^{(n+1)} = - u^{(n)} \partial _\theta f^{(n+1)} + \frac{1}{2} f^{(n+1)} H((f^{(n)})^2), \\ f^{(n+1)}(t=0,\cdot ) = f_{\scriptscriptstyle in}. \end{array}\right. } \end{aligned}$$

  At each stage n, \(f^{(n+1)}\) can be solved using the method of characteristics, and is clearly smooth for all times. By energy estimates, we have

  $$\begin{aligned} \frac{d}{dt} \Vert f^{(n+1)}\Vert _{H^1}^2&\le C_\star \Vert H ((f^{(n)})^2)\Vert _{H^1} \Vert f^{(n+1)}\Vert _{H^1}^2 \\&\le C_\star \Vert f^{(n)}\Vert _{H^1}^2 \Vert f^{(n+1)}\Vert _{H^1}^2, \end{aligned}$$

  where \(C_\star > 1\) is an absolute positive constant whose meaning may change from line to line (in the later \(C_\star \) may depend on k). Gronwall’s inequality gives

  $$\begin{aligned} \Vert f^{(n+1)}\Vert _{H^1}^2 \le \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2 e^{C_\star \int _0^t \Vert f^{(n)}\Vert _{H^1}^2 ds}. \end{aligned}$$

  An induction argument on n gives, for \(0\le t\le T_1=1/(2eC_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2)\),

  $$\begin{aligned} \Vert f^{(n+1)}\Vert _{H^1}^2 \le 2e\Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2. \end{aligned}$$

  More generally, using the classical Gagliardo–Nirenberg inequality, we have

  $$\begin{aligned} \frac{d}{dt} \Vert f^{(n+1)}\Vert _{H^k}^2&\le C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2 \Vert f^{(n+1)}\Vert _{H^k}^2 + C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2 \Vert f^{(n)}\Vert _{H^k} \Vert f^{(n+1)}\Vert _{H^k}\\&\le C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2 \Vert f^{(n+1)}\Vert _{H^k}^2 + C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2 \Vert f^{(n)}\Vert _{H^k}^2. \end{aligned}$$

  Gronwall’s inequality gives

  $$\begin{aligned} \Vert f^{(n+1)}\Vert _{H^k}^2 \le (\Vert f_{\scriptscriptstyle in}\Vert _{H^k}^2 + C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2 \int _0^t\Vert f^{(n)}\Vert _{H^k}^2ds ) e^{C_\star \int _0^t \Vert f^{(n)}\Vert _{H^1}^2 ds}. \end{aligned}$$

  Induction on n gives, for \(0\le t \le T_k = 1/(2eC_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2)\),

  $$\begin{aligned} \Vert f^{(n+1)}\Vert _{H^k}^2 \le 2e \Vert f_{\scriptscriptstyle in}\Vert _{H^k}^2. \end{aligned}$$

  Hence \(f^{(n)}\) is uniformly bounded in \(C([0,T_k];H^k)\) for \(n = 0,1,2,\cdots \). Next we claim that \(f^{(n)}\) is a Cauchy sequence in \(C([0,T_0],L^2)\) for some \(T_0>0\). Note that

  $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t (f^{(n+2)} - f^{(n+1)}) = - (u^{(n+1)}- u^{(n)}) \partial _\theta f^{(n+2)} - u^{(n)} \partial _\theta (f^{(n+2)} - f^{(n+1)}) \\ \qquad \qquad \qquad \qquad \qquad + \frac{1}{2} (f^{(n+2)} - f^{(n+1)}) H((f^{(n+1)})^2) \\ \qquad \qquad \qquad \qquad \qquad + \frac{1}{2} (f^{(n+1)}) H((f^{(n+1)})^2-(f^{(n)})^2) \\ (f^{(n+2)} - f^{(n+1)})(t=0,\cdot ) = 0. \end{array}\right. } \end{aligned}$$

  Hence \(L^2\) estimate gives

  $$\begin{aligned} \frac{d}{dt} \Vert f^{(n+2)} - f^{(n+1)}\Vert _{L^2} \le C_\star (\Vert f^{(n+2)} - f^{(n+1)}\Vert _{L^2}+ \Vert f^{(n+1)} - f^{(n)}\Vert _{L^2}) \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2. \end{aligned}$$

  Gronwall’s inequality gives

  $$\begin{aligned} \Vert f^{(n+2)} - f^{(n+1)}\Vert _{L^2} \le C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2 \int _0^t \Vert f^{(n+1)} - f^{(n)}\Vert _{L^2} ds \,\,e^{C_\star t \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2}. \end{aligned}$$

  Thus, for \(0\le t\le T_0=1/(2eC_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2)\), we have

  $$\begin{aligned} \sup _{0\le t\le T_0}\Vert f^{(n+2)} - f^{(n+1)}\Vert _{L^2} \le \frac{1}{2} \sup _{0\le t\le T_0} \Vert f^{(n+1)} - f^{(n)}\Vert _{ L^2}. \end{aligned}$$

  This proves the claim. Let us denote the limit function by f, then \(f\in C([0,\tilde{T}_k];L^2)\cap L^\infty ([0,\tilde{T}_k]; H^k)\), with \(\tilde{T}_k = \min \{T_0,T_1,T_k\}\). Moreover, we have, for \(0\le t \le \tilde{T}_k\),

  $$\begin{aligned} \Vert f\Vert _{H^k}^2 \le 2e \Vert f_{\scriptscriptstyle in}\Vert _{H^k}^2. \end{aligned}$$

  It is easy to check that f satisfies the equation (2.3) in the sense of distributions. Using the equation (2.3), we have \(\partial _t f \in L^\infty ([0,\tilde{T}_k];H^{k-1})\) and \(f \in C([0,\tilde{T}_k];H^{k-1})\).

  According to the proof of (2.1) and (2.2), we have the identities

  $$\begin{aligned} \Vert f\Vert _{L^2(\Omega )} = \Vert f_{\scriptscriptstyle in}\Vert _{L^2(\Omega )}, \end{aligned}$$

  (5.1)

  and

  $$\begin{aligned} \Vert f\Vert _{H^1(\Omega )} = \Vert f_{\scriptscriptstyle in}\Vert _{H^1(\Omega )} \end{aligned}$$

  (5.2)

  for \(0\le t \le \tilde{T}_k\). We emphasize here that \(\tilde{T}_k\) depends only on k and \(\Vert f_{\scriptscriptstyle in}\Vert _{H^1}\).

  Step 2. Next, given a general initial data \(f_{\scriptscriptstyle in} \in H^k\), we approximate \(f_{\scriptscriptstyle in}\) with \(P_{\le N} f_{\scriptscriptstyle in} \) and prove a \(H^k\) stability result for (2.3). Here N is a dyadic number (i.e., \(\log _2 N\) is an integer) and \(P_{\le N}\) is the standard Littlewood-Paley projection operator. Note that

  $$\begin{aligned} \Vert P_{\le N} f_{\scriptscriptstyle in}\Vert _{H^k} \le \Vert f_{\scriptscriptstyle in}\Vert _{H^k}. \end{aligned}$$

  Since \(P_{\le N} f_{\scriptscriptstyle in} \in H^\infty \), by Step 1, there exists \(T=\tilde{T}_{k+3}>0\) depending on \(\Vert f_{\scriptscriptstyle in}\Vert _{H^1}\) and k, and a sequence of solutions \(f_N\in C([0,T];H^{k+2})\) with the initial data \(P_{\le N} f_{\scriptscriptstyle in}\), which satisfy (5.1) and (5.2) with \(f_{\scriptscriptstyle in}\) being replaced by \(P_{\le N}f_{\scriptscriptstyle in}\). We are going to take the limit \(N \rightarrow \infty \).

  Let \(N'>N\). By energy estimate, we have

  $$\begin{aligned} \frac{d}{dt} \Vert f_N -f_{N'}\Vert _{L^2}^2 \le C_\star \Vert f_N -f_{N'}\Vert _{L^2}^2 \Vert f_{\scriptscriptstyle in}\Vert _{H^1}^2. \end{aligned}$$

  This implies that for \(0\le t \le T\),

  $$\begin{aligned} \Vert f_N -f_{N'}\Vert _{L^2}^2 \le C_\star \Vert P_{\le N}f_{\scriptscriptstyle in} - P_{\le {N'}} f_{\scriptscriptstyle in}\Vert _{L^2}^2. \end{aligned}$$

  More generally, we have

  $$\begin{aligned}&\frac{d}{dt} \Vert f_N-f_{N'}\Vert _{H^k}^2\\&\quad \le C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^k}^2 \Vert f_N-f_{N'}\Vert _{H^k}^2\\&\qquad + C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^k} \Vert P_{\le {N}} f_{\scriptscriptstyle in}\Vert _{H^{k+1}} \Vert f_N-f_{N'}\Vert _{L^2} \Vert f_N-f_{N'}\Vert _{H^k}\\&\quad \le C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^k}^2 \Vert f_N-f_{N'}\Vert _{H^k}^2\\&\qquad + C_\star \Vert f_{\scriptscriptstyle in}\Vert _{H^k} N\Vert P_{\le N} f_{\scriptscriptstyle in}\Vert _{H^{k}} \Vert P_{\le N}f_{\scriptscriptstyle in} - P_{\le {N'}} f_{\scriptscriptstyle in}\Vert _{L^2} \Vert f_N-f_{N'}\Vert _{H^k}. \end{aligned}$$

  Note that as \(N,N'\rightarrow \infty \),

  $$\begin{aligned} N\Vert P_{\le N}f_{\scriptscriptstyle in} - P_{\le N'} f_{\scriptscriptstyle in}\Vert _{L^2} \le \Vert P_{N \le \cdot \le N'}f_{\scriptscriptstyle in}\Vert _{H^1} \rightarrow 0. \end{aligned}$$

  Using Gronwall’s inequality we obtain

  $$\begin{aligned} \sup _{0\le t \le T}\Vert f_N-f_{N'}\Vert _{H^k}^2\rightarrow 0. \end{aligned}$$

  Hence \(f_N\) converges strongly in \(C([0,T];H^k)\) to a limit function \(f\in C([0,T];H^k)\) which satisfies the Eq. (2.3) with initial data \({f_{\scriptscriptstyle in}}\). The identities (5.1) and (5.2) for \(f_N\) and \(P_{\le N}f_{\scriptscriptstyle in}\) can be passed to the limit.

  Finally, we remark that if \(f_{\scriptscriptstyle in}\) is non-negative, then using the method of characteristics, f is also non-negative. \(\quad \square \)

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