1 Introduction↩︎

1.1 Motivation↩︎

Let \(\Lambda\) be a two-dimensional lattice; that is, there exist two linearly independent vectors \(\boldsymbol{\xi}, \boldsymbol{\eta}\in\mathbb{R}^2\) such that \[\Lambda = \left\{ m \boldsymbol{\xi} + n \boldsymbol{\eta} \mid m, n \in \mathbb{Z} \right\}.\] The pair \((\boldsymbol{\xi}, \boldsymbol{\eta})\) is called a basis of \(\Lambda\). Consider the flat 2-torus \(\mathbb{T}_\Lambda:=\mathbb{R}^2/\Lambda\) and the incompressible Euler equation on \(\mathbb{T}_\Lambda\) in vorticity form: \[\label{euler} \left\{ \begin{align} &\partial_t \omega + \nabla^\perp\psi \cdot \nabla \omega = 0, \qquad t \in \mathbb{R},\;\mathbf{x} = (x_1,x_2) \in \mathbb{T}_\Lambda,\\ &\psi= (-\Delta)^{-1}\omega, \end{align} \right.\tag{1}\] where \(\omega\) is the scalar vorticity, \(\psi\) is the stream function, and \(\nabla^\perp:= (\partial_{x_2}, -\partial_{x_1})\). Note that \(\omega\) is assumed to have mean zero to ensure that \(\psi\) is well defined. The global well-posedness theory for 1 has been well developed in various function spaces [1]–[4]. In particular, global well-posedness holds in the class of mean-zero \(C^1\) functions. However, a comprehensive theory describing the evolution of solutions remains far from complete [5]. Among the many problems concerning evolution, a fundamental one is to study the long-time behavior of solutions near various steady states.

In this paper, we study the Lyapunov stability of a class of steady states, referred to as Laplacian eigenstates, whose stream functions are given by eigenfunctions of the following Laplacian eigenvalue problem: \[\require{physics} \label{lep} \begin{cases} -\Delta u = \lambda u, & \mathbf{x} \in \mathbb{T}_\Lambda,\\ \displaystyle \int_{\mathbb{T}_\Lambda} u \, \dd{\mathbf{x}} = 0. \end{cases}\tag{2}\] Note that all eigenfunctions of 2 are sinusoidal and can be obtained by computing the dual lattice of \(\Lambda\). For the first eigenspace, denoted by \(\mathbf{E}_1\), the following properties hold (see [6]):

• The dimension of \(\mathbf{E}_1\) is either \(2\), \(4\), or \(6\).

• \(\dim(\mathbf{E}_1) = 6\) if and only if \(\mathbb{T}_\Lambda\) is a hexagonal torus; that is, \(\Lambda\) admits a basis \((\boldsymbol{\xi}, \boldsymbol{\eta})\) such that \[|\boldsymbol{\xi}| = |\boldsymbol{\eta}|, \quad \angle(\boldsymbol{\xi}, \boldsymbol{\eta}) = \pi/3.\]

• If \(\mathbb{T}_\Lambda\) is a hexagonal torus, then there exist three nonzero vectors \(\mathbf{k}_1, \mathbf{k}_2, \mathbf{k}_3\) satisfying \[|\mathbf{k}_1| = |\mathbf{k}_2| = |\mathbf{k}_3|, \quad \mathbf{k}_3 = \mathbf{k}_1 + \mathbf{k}_2,\] such that \[\label{e1df} \mathbf{E}_1 = \operatorname{span}\left\{ \cos(\mathbf{k}_1 \cdot \mathbf{x}),\, \sin(\mathbf{k}_1 \cdot \mathbf{x}),\, \cos(\mathbf{k}_2 \cdot \mathbf{x}),\, \sin(\mathbf{k}_2 \cdot \mathbf{x}),\, \cos(\mathbf{k}_3 \cdot \mathbf{x}),\, \sin(\mathbf{k}_3 \cdot \mathbf{x}) \right\}.\tag{3}\]

For the square torus, orbital stability (up to translations) of the first Laplacian eigenstates has been established in [7]–[9], and has recently been extended to flat \(2\)-tori of arbitrary shape in [6].

However, the dependence of \(\delta\) on \(\varepsilon\) remains unknown, except in the special case of the square torus [7]. Our goal in this paper is to provide explicit estimates for this dependence on tori of arbitrary shape.

1.2 Main theorem↩︎

We present only the quantitative estimates for the most delicate hexagonal case as the main theorem below, and postpone the simpler non-hexagonal cases to the final section.

We give some remarks on Theorem 1 below.

1.3 Comments, proof strategy, and novelties↩︎

The Euler equation possesses a rich family of conservation laws, which provides a natural framework for stability analysis. In the case of a flat 2-torus, the following quantities are conserved:

• The kinetic energy \(\mathsf E\), which in terms of vorticity can be expressed as \[\require{physics} \label{defofe}\mathsf E(\omega)=\frac{1}{2}\int_{\mathbb{T}_\Lambda}\omega\,(-\Delta)^{-1}\omega\dd{\mathbf{x}}\footnote{Although (-\Delta)^{-1}\omega may differ by an arbitrary constant, this integral is well-defined since \omega has mean zero.};\tag{4}\]

• The Casimir functionals of the form \[\require{physics} \label{defofc} \mathsf C_F(\omega)=\int_{\mathbb{T}_\Lambda}F(\omega)\dd{\mathbf{x}},\tag{5}\] where \(F\in C(\mathbb{R})\). In particular, the \(k\)-th order Casimir \[\require{physics} \label{kcm} \mathsf C_k(\omega):=\int_{\mathbb{T}_\Lambda} \omega^k\dd{\mathbf{x}}\tag{6}\] is conserved, where \(k\) is a positive integer.

In the 1960s, inspired by the Lyapunov function method in ODE theory, Arnold [12], [13] introduced the energy-Casimir (EC) functional and established two types of stability theorems, now known as Arnold’s first and second stability theorems. For Laplacian eigenstates, the associated Casimir is the enstrophy, i.e., the \(L^2\) norm of the vorticity. Using the EC functional method, it can be proved that for any \(\bar\omega \in \mathbf{E}_1\), the first energy shell \[\mathbf{S}_{\bar\omega} := \left\{ v \in \mathbf{E}_1 \;\middle|\; \|v\|_{L^2(\mathbb{T}_{\Lambda})} = \|\bar\omega\|_{L^2(\mathbb{T}_{\Lambda})} \right\}\] is stable with respect to the enstrophy norm. However, this result cannot be improved using only the kinetic energy and the enstrophy, since these quantities do not distinguish between different states in \(\mathbf{S}_{\bar\omega}\). The first refinement was obtained by Wirosoetisno and Shepherd [9] for the square torus. They introduced higher-order Casimirs \(\mathsf C_k,\) \(k=2,4,6\), to distinguish elements of \(\mathbf{S}_{\bar\omega}\), and studied the stability of the translational orbit \(\mathbf{O}_{\bar\omega}\). Nevertheless, their formulation of stability relies on higher-order Casimirs, and complete orbital stability remains unresolved. A complete proof of the stability of \(\mathbf{O}_{\bar\omega}\) for both rectangular and square tori was given in [8], where they used a compactness argument originating from Burton’s work [14], [15], reducing the problem to the equimeasurable partition of the first eigenspace. The stability result in [8] is qualitative, and is measured by the \(L^p\) norm of the vorticity for any \(p\in(1,\infty)\). Later, Elgindi [7] established a quantitative estimate on the stability of \(\mathbf{O}_{\bar\omega}\) for the square torus in the \(L^2\) setting, building on and refining the approach of Wirosoetisno and Shepherd. Similar ideas have since been applied to the orbital stability of Laplacian eigenstates on the sphere [10] or in a disk [11]. Very recently, the result of [8] was extended to tori of arbitrary shape in [6], with the main contribution being the treatment of the hexagonal case.

The proof of Theorem 1 follows the framework introduced in [7], [9] (see also [10], [11] for further developments) and is divided into two main steps:

• We first establish the quantitative stability of the first eigenspace \(\mathbf{E}_1\). This step mainly relies on the conservation of the kinetic energy and the enstrophy, in combination with proper use of a Poincaré-type inequality.

• We then estimate the distance between the projection of the perturbed solution onto \(\mathbf{E}_1\) and the translational orbit \(\mathbf{O}_{\bar\omega}\). More specifically, we introduce modified integer-order Casimirs that are globally Lipschitz continuous in \(L^2\) and use them to control the variation of the amplitude and phase parameters. The problem is then reduced to the analysis of a possibly degenerate nonlinear map in finite dimensions.

The first step is standard and follows from an argument similar to that in [11]. The main challenges and differences lie in the second step. First, the structure of the first eigenspace and the characterization of translational orbits within the first eigenspace are more complicated, which makes the analysis of energy and vorticity transfer within it more difficult. Second, the nonlinear map arising in the second step may be highly degenerate at certain special configurations, leading to weaker stability. For example, the stability exponent \(\gamma\) may be as small as \(1/6\) when \(A_1=A_2=A_3\) and \(\sin\alpha=0\), whereas the smallest value of stability exponent appearing in [7], [10], [11] was \(1/2\). This strong degeneracy prevents us from directly applying the previous methods, such as the degenerate inverse function lemma established in [10] and the direct computations carried out in [11]. To overcome these difficulties, we make a key observation: the complicated nonlinear map in the second step can be suitably combined into a polynomial map given by the three elementary symmetric polynomials in three variables. This reduces the estimates for the amplitude parameters to a root-stability problem for a cubic polynomial under perturbations of its coefficients, which can be handled in a straightforward manner; see Lemma 2. Once the estimates for the amplitude parameters have been established, the estimates for the phase parameters can be obtained through careful and intricate computations.

1.4 Organization of the paper↩︎

The remainder of the paper is organized as follows. In Section 2, we establish a quantitative stability estimate for the first eigenspace \(\mathbf{E}_1\). In Section 3, we prove a technical lemma on the quantitative stability of the roots of a polynomial under perturbations of its coefficients. This lemma plays a crucial role in the proof of the main theorem. Section 4 is devoted to the proof of Theorem 1. Finally, in Section 5, we treat the remaining non-hexagonal cases and provide the corresponding quantitative stability results with brief proofs.

2 Quantitative stability of \(\mathbf{E}_1\)↩︎

Denote by \(\mathring L^2(\mathbb{T}_\Lambda)\) the closed subspace of mean-zero functions in \(L^2(\mathbb{T}_\Lambda)\), i.e., \[\require{physics} \label{defoflo} \mathring L^2(\mathbb{T}_\Lambda):=\left\{v\in L^2(\mathbb{T}_\Lambda) \;\middle|\; \int_{\mathbb{T}_\Lambda} v \dd{\mathbf{x}}=0\right\},\tag{7}\] and denote by \(\mathbf{E}_1^\perp\) the orthogonal complement of \(\mathbf{E}_1\) in \(\mathring L^2(\mathbb{T}_\Lambda)\).

To prove Proposition 6, we need the following Poincaré-type inequality associated with \(\mathbf{E}_1\).

The proof of the above lemma is standard and follows by repeating the argument in [11].

3 A technical lemma↩︎

Let \(n \ge 2\) be an integer. Denote by \(f_1, \dots, f_n\) the \(n\) elementary symmetric polynomials in \(n\) variables, i.e., \[f_1(\mathbf{a})=\sum_{i=1}^n a_i,\quad f_2(\mathbf{a})=\sum_{1\le i<j\le n} a_i a_j,\quad \dots,\quad f_n(\mathbf{a})=\prod_{i=1}^n a_i,\] where \(\mathbf{a} = (a_1,a_2, \dots, a_n) \in \mathbb{R}^n.\)

The following lemma gives quantitative bounds on \(|b_k-a_k|\) in terms of \(\sum_{i=1}^n |f_i(\mathbf{a}) - f_i(\mathbf{b})|\), and plays a key role in proving the main theorem.

In the rest of this paper, we only need Lemma 2 in the cases \(n=2\) and \(n=3\), which we state as the following two corollaries.

4 Proof of main theorem↩︎

For simplicity, write \(\mathbf{O} = \mathbf{O}_{\bar\omega}\), where \(\bar\omega\) is as in Theorem 1. Let \(\omega_t := \omega(t,\cdot)\) be a mean-zero \(C^1\) solution of the Euler equation 1 . Our goal is to estimate \({\rm dist}_2(\omega_t,\mathbf{O})\). Denote by \(v_t\) the orthogonal projection of \(\omega_t\) onto \(\mathbf{E}_1\). Then \[{\rm dist}_2(\omega_t,\mathbf{O})^2 = {\rm dist}_2(\omega_t,\mathbf{E}_1)^2 + {\rm dist}_2(v_t,\mathbf{O})^2.\] In view of Proposition 6, it suffices to estimate \({\rm dist}_2(v_t,\mathbf{O})\). To this end, we proceed in several steps.

4.1 Reduction to estimates of amplitudes and phases↩︎

Assume that \(v_t\) has the form \[v_t(\mathbf{x})=\sum_{i=1}^{3} B_i(t) \cos\left(\mathbf{k}_i \cdot \mathbf{x} + \beta_i(t)\right),\qquad B_i(t)\geq 0, \quad \beta_i(t)\in\mathbb{R}.\] Recall that \(\mathbf{O}\) is determined by \(A_1, A_2, A_3\) and \(\alpha_1, \alpha_2, \alpha_3\), as stated in Theorem 1. The following lemma shows that \({\rm dist}_2(v_t,\mathbf{O})\) can be expressed explicitly in terms of the amplitudes \(A_i, B_i(t)\) and the phases \(\alpha_i, \beta_i(t)\), where \(i=1,2,3\).

Throughout this section, for notational simplicity, we use \(C\) to denote various positive constants depending only on \(\Lambda\) and \(\|\bar\omega\|_{L^2(\mathbb{T}_\Lambda)}\), whose values may vary from line to line. We also sometimes write \(B_i = B_i(t)\) and \(\beta_i = \beta_i(t)\) when convenient.

4.2 Casimirs and estimates involving elementary symmetric polynomials↩︎

Given that \({\rm dist}_2(\omega_0,\mathbf{O})<\varepsilon\), by replacing \(\bar\omega\) with another element of \(\mathbf{O}\) if necessary, we may assume that \[{\rm dist}_2(\omega_0,\mathbf{O})= \|\omega_0-\bar\omega\|_{L^2(\mathbb{T}_\Lambda)}<\varepsilon.\] Without loss of generality, we also assume that \(0<\varepsilon<1.\)

Let \(f_1,f_2,f_3\) be the elementary symmetric polynomials in three variables. Denote \[\mathbf{a}=(A_1^2,A_2^2,A_3^2),\qquad \mathbf{b}=(B_1^2,B_2^2,B_3^2).\] From Lemma 4, we can prove the following lemma.

4.3 Estimate of \(|b_k(t)-a_k|\)↩︎

Let \(k\in\{1,2,3\}\) be fixed. Combining Lemma 5 with Corollary 2, we obtain the following estimate of \(|b_k(t)- a_k|\) for all \(t\in\mathbb{R}\).

4.4 Concluding the proof: \(A_1A_2A_3=0\)↩︎

Without loss of generality, we may assume that \(A_3=0\). Then 14 becomes \[\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr)^2 =C\left(\left|B_1-A_1\right|^2 + \left|B_2-A_2\right|^2 + \left| B_3\right|^2\right).\] So \[\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) <C\left(\left|B_1-A_1\right| + \left|B_2-A_2\right| + B_3 \right).\] Now we estimate \(\left|B_1-A_1\right|\), \(\left|B_2-A_2\right|\) and \(B_3\) based on Lemma 6 (recall that \(\mathbf{a}=\left(A_1^2,A_2^2,A_3^2\right), \mathbf{b}=\left(B_1^2,B_2^2,B_3^2\right)\)). We distinguish three cases.

In this case, \[\left|B_1^2 - A_1^2\right| < \frac{C\varepsilon}{\left|\left(A_2^2 - A_1^2\right)\left(A_3^2 - A_1^2\right)\right|},\quad \left|B_2^2 - A_2^2\right| < \frac{C\varepsilon}{\left|\left(A_1^2 - A_2^2\right)\left(A_3^2 - A_2^2\right)\right|},\quad B_3^2 < \frac{C\varepsilon}{A_1^2 A_2^2},\] which yields \[\left|B_1 - A_1\right| < \frac{C\varepsilon}{\left|\left(A_2^2 - A_1^2\right)\left(A_3^2 - A_1^2\right)\left(B_1 + A_1\right)\right|}<\frac{C\varepsilon}{\left|\left(A_2^2 - A_1^2\right)\left(A_3^2 - A_1^2\right)\right| A_1 },\] \[\left|B_2 - A_2\right| < \frac{C\varepsilon}{\left|\left(A_1^2 - A_2^2\right)\left(A_3^2 - A_2^2\right)\left(B_2 + A_2\right)\right|}<\frac{C\varepsilon}{\left|\left(A_1^2 - A_2^2\right)\left(A_3^2 - A_2^2\right)\right| A_2 },\] and \[B_3 < \frac{C\varepsilon^{1/2}}{A_1 A_2 }.\] Thus \[\begin{align} \mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) &<\frac{C \varepsilon }{\left|\left(A_2^2-A_1^2\right) \left(A_3^2-A_1^2\right)\right| A_1 }+\frac{C \varepsilon }{\left|\left(A_1^2-A_2^2\right) \left(A_3^2-A_2^2\right)\right| A_2 }+\frac{C \varepsilon^{1/2}}{ A_1A_2}\\ &\leq \frac{C \varepsilon^{1/2}}{ A_1A_2}. \end{align}\] Note that we may need to choose a smaller \(\varepsilon_0\) so that the last inequality holds.

In this case, \[\left|B_1^2-A_1^2\right|+\left|B_2^2-A_2^2\right|< \frac{C\varepsilon^{1/2}}{A_1}, \qquad B_3^2< \frac{C \varepsilon}{A_1^4},\] which yields \[\left|B_1 -A_1\right| < \frac{C\varepsilon^{1/2}}{A_1^2},\quad \left|B_2 -A_2\right| < \frac{C\varepsilon^{1/2}}{A_1^2},\quad B_3<\frac{C\varepsilon^{1/2}}{A_1^2}.\] Thus \[\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) <\frac{C\varepsilon^{1/2}}{A_1^2}.\]

Without loss of generality, we may assume that \(A_2=0\). Then \[\left|B_1^2-A_1^2\right|<\frac{C \varepsilon}{A_1^4}, \quad B_2^2+B_3^2< \frac{C \varepsilon^{1/2}}{A_1},\] which gives \[\left|B_1 -A_1 \right|<\frac{C \varepsilon}{A_1^5}, \quad B_2 < \frac{C \varepsilon^{1/4}}{A^{1/2}_1},\quad B_3 < \frac{C \varepsilon^{1/4}}{A^{1/2}_1}.\] Thus, by choosing a smaller \(\varepsilon_0\) if necessary, we get \[\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) <\frac{C\varepsilon^{1/4}}{A_1^{1/2}}.\] This completes the proof of Theorem 1 in the case \(A_1A_2A_3=0\).

4.5 Concluding the proof: \(A_1A_2A_3\neq 0\)↩︎

In this case, we further estimate \(\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr)\) according to 14 as follows: \[\label{min95control} \begin{align} &\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr)^2\\ =&C\min_{\theta_1, \theta_2 \in \mathbb{R}} \left( \left|B_1-A_1 e^{i\theta_1}\right|^2 + \left|B_2-A_2 e^{i\theta_2}\right|^2 + \left|B_3-A_3 e^{i(\theta_1 + \theta_2 +\beta-\alpha)}\right|^2\right)\\ =&C\min_{\theta_1, \theta_2 \in \mathbb{R}} \left( \left|B_1-A_1 e^{i\theta_1}\right|^2 + \left|B_2-A_2 e^{i\theta_2}\right|^2 + \left|B_3e^{-i(\beta-\alpha)}-A_3 e^{i(\theta_1 + \theta_2)}\right|^2\right)\\ \leq&C\min_{\theta_1, \theta_2 \in \mathbb{R}} \left( \left|B_1-A_1 e^{i\theta_1}\right|^2 + \left|B_2-A_2 e^{i\theta_2}\right|^2 + \left( \left|B_3e^{-i(\beta-\alpha)}-A_3\right|+ \left|A_3-A_3 e^{i(\theta_1 + \theta_2)}\right|\right)^2\right)\\ =&C \left( \left|B_1 -A_1\right|^2 + \left|B_2 -A_2\right|^2 + \left|B_3e^{-i\beta}-A_3e^{-i\alpha}\right|^2\right). \end{align}\tag{33}\] So \[\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) \leq C \left( \left|B_1 -A_1\right| + \left|B_2 -A_2\right| + \left|B_3e^{-i\beta}-A_3e^{-i\alpha}\right|\right).\] To estimate \(\left|B_1-A_1\right|\), \(\left|B_2 -A_2\right|\) and \(\left|B_3e^{-i\beta}-A_3e^{-i\alpha}\right|\), we distinguish three cases.

Case 1: \(A_1, A_2, A_3\) are pairwise distinct. In this case, based on Lemma 6, we have the following estimates on \(|B_k-A_k|\), \(k=1,2,3\): \[\label{bkak1} \begin{align} &\left|B_1^2-A_1^2\right| <\frac{C\varepsilon}{\left|A_2^2-A_1^2\right| \,\left|A_3^2-A_1^2\right|} \quad\Longrightarrow\quad \left|B_1 -A_1 \right| <\frac{C\varepsilon}{\left|A_2^2-A_1^2\right| \,\left|A_3^2-A_1^2\right|A_1},\\ &\left|B_2^2-A_2^2\right|<\frac{C\varepsilon}{\left|A_1^2-A_2^2\right| \,\left|A_3^2-A_2^2\right|}\quad\Longrightarrow\quad \left|B_2 -A_2 \right|<\frac{C\varepsilon}{\left|A_1^2-A_2^2\right| \,\left|A_3^2-A_2^2\right|A_2},\\ &\left|B_3^2-A_3^2\right|<\frac{C\varepsilon}{\left|A_1^2-A_3^2\right|\,\left|A_2^2-A_3^2\right|}\quad\Longrightarrow\quad \left|B_3 -A_3 \right|<\frac{C\varepsilon}{\left|A_1^2-A_3^2\right|\,\left|A_2^2-A_3^2\right|A_3}. \end{align}\tag{34}\] To estimate \(|B_3 e^{-i\beta}-A_3e^{-i\alpha}|\), notice that \[|B_3 e^{-i\beta}-A_3e^{-i\alpha}| \leq |B_3\cos\beta-A_3\cos\alpha|+|B_3\sin\beta-A_3\sin\alpha|.\] So it suffices to estimate \(|B_3\cos\beta-A_3\cos\alpha|\) and \(|B_3\sin\beta-A_3\sin\alpha|\). To estimate \(|B_3\cos\beta-A_3\cos\alpha|\), notice that \[\begin{align} \left|B_3\cos\beta-A_3\cos\alpha\right| &=\frac{ \left|B_1B_2B_3\cos\beta-A_1A_2A_3\cos\alpha +A_1A_2A_3\cos\alpha -B_1B_2A_3\cos\alpha\right| }{ B_1B_2 }\\ &\leq \frac{ \left|B_1B_2B_3\cos\beta-A_1A_2A_3\cos\alpha\right| + A_3\left|B_1B_2-A_1A_2\right| }{ B_1B_2 }\\ &\leq \frac{ \left|B_1B_2B_3\cos\beta-A_1A_2A_3\cos\alpha\right| + A_3\left(\left|B_1B_2-A_1B_2\right|+\left|A_1B_2-A_1A_2\right|\right) }{ B_1B_2 }. \end{align}\] By choosing a smaller \(\varepsilon_0\) if necessary, we may assume that \(A_i/2<B_i< 3A_i/2\), \(i=1,2,3\). Then, in view of 21 and 34 , \[\label{BACOS1} \left|B_3\cos\beta-A_3\cos\alpha\right| \leq \frac{C\varepsilon}{A_1A_2}+\frac{CA_3\varepsilon}{A_1^2\left|A_2^2-A_1^2\right| \,\left|A_3^2-A_1^2\right|} +\frac{CA_3\varepsilon}{A_2^2\left|A_1^2-A_2^2\right| \,\left|A_3^2-A_2^2\right|}.\tag{35}\] We now turn to the estimate of \(|B_3\sin\beta-A_3\sin\alpha|\). Notice that \[\begin{align} \left| \left(B_3\sin\beta\right)^2-\left(A_3\sin\alpha\right)^2\right| =&\left| B_3^2-A_3^2+ \left(A_3\cos\alpha\right)^2-\left(B_3\cos\beta\right)^2 \right|\\ \leq & \left(B_3+A_3\right) \left(\left|B_3-A_3 \right| +\left|B_3\cos\beta-A_3\cos\alpha \right|\right). \end{align}\] Taking into account 34 \(_3\) and 35 , \[\label{b2a2} \begin{align} &\left| B_3\sin\beta-A_3\sin\alpha\right|\,\left| B_3\sin\beta+A_3\sin\alpha\right|\\ \leq& C\left( \frac{1}{\left|A_1^2-A_3^2\right| \,\left|A_2^2-A_3^2\right|} +\frac{A_3^2}{\left|A_2^2-A_1^2\right| \,\left|A_3^2-A_1^2\right|A_1^2} +\frac{A_3^2}{\left|A_1^2-A_2^2\right| \,\left|A_3^2-A_2^2\right|A_2^2}+ \frac{A_3}{A_1A_2}\right)\varepsilon. \end{align}\tag{36}\] We further distinguish two cases.

In this case, \(|B_3\sin\beta \pm A_3\sin\alpha| =|B_3\sin\beta|\). Thus, in view of 36 , \[\begin{align} &|B_3\sin\beta-A_3\sin\alpha|\\ \leq& C\left( \frac{1}{|A_1^2-A_3^2||A_2^2-A_3^2|}+ \frac{A_3^2}{|A_2^2-A_1^2||A_3^2-A_1^2|A_1^2} +\frac{A_3^2}{|A_1^2-A_2^2||A_3^2-A_2^2|A_2^2} +\frac{A_3}{A_1A_2} \right)^{1/2}\varepsilon^{1/2}. \end{align}\] Combining this estimate with 34 and 35 , and choosing \(\varepsilon_0>0\) smaller if necessary, we obtain \[\begin{align} &\mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr)\\ \leq&C\left( \frac{1}{|A_1^2-A_3^2||A_2^2-A_3^2|}+ \frac{A_3^2}{|A_2^2-A_1^2||A_3^2-A_1^2|A_1^2} +\frac{A_3^2}{|A_1^2-A_2^2||A_3^2-A_2^2|A_2^2} +\frac{A_3}{A_1A_2} \right)^{1/2}\varepsilon^{1/2}. \end{align}\]

In this case, we have \[\label{positive95lower95bound} 0<2A_3|\sin\alpha| \leq \left|B_3\sin\beta-A_3\sin\alpha\right| + \left|B_3\sin\beta+A_3\sin\alpha\right|.\tag{37}\] Combining 36 with 37 , we obtain the following pointwise-in-time dichotomy. For every \(t\in\mathbb{R}\), at least one of the following two alternatives holds: \[\label{alternative-new-1} \begin{align} &\left|B_3\sin\beta-A_3\sin\alpha\right|\\ \leq& \frac{C\varepsilon}{|\sin\alpha|} \Bigg( \frac{1}{A_3|A_1^2-A_3^2||A_2^2-A_3^2|} + \frac{A_3}{|A_2^2-A_1^2||A_3^2-A_1^2|A_1^2} + \frac{A_3}{|A_1^2-A_2^2||A_3^2-A_2^2|A_2^2} + \frac{1}{A_1A_2} \Bigg), \end{align}\tag{38}\] or \[\label{alternative-new-2} \begin{align} &\left|B_3\sin\beta+A_3\sin\alpha\right|\\ \leq& \frac{C\varepsilon}{|\sin\alpha|} \Bigg( \frac{1}{A_3|A_1^2-A_3^2||A_2^2-A_3^2|} + \frac{A_3}{|A_2^2-A_1^2||A_3^2-A_1^2|A_1^2} + \frac{A_3}{|A_1^2-A_2^2||A_3^2-A_2^2|A_2^2} + \frac{1}{A_1A_2} \Bigg). \end{align}\tag{39}\] Indeed, by 37 , at least one of \(\left|B_3\sin\beta-A_3\sin\alpha\right|\), \(\left|B_3\sin\beta+A_3\sin\alpha\right|\) is not smaller than \(A_3|\sin\alpha|\). Dividing the product estimate 36 by this lower bound gives either 38 or 39 . The constants in this dichotomy are independent of \(t\), but the dichotomy is, at this stage, only pointwise in time.

We next show that, for sufficiently small \(\varepsilon\), the second alternative cannot occur. If 38 holds at a time \(t\), then combining 34 , 35 , and 38 , we obtain \[\label{dist-O-case12} \begin{align} &\mathrm{dist}_2(v_t,\mathbf{O})\\ \leq\;& \frac{C\varepsilon}{|A_2^2-A_1^2||A_3^2-A_1^2|} \left( \frac{1}{A_1} + \frac{A_3}{A_1^2} + \frac{A_3}{A_1^2|\sin\alpha|} \right) + \frac{C\varepsilon}{|A_1^2-A_2^2||A_3^2-A_2^2|} \left( \frac{1}{A_2} + \frac{A_3}{A_2^2} + \frac{A_3}{A_2^2|\sin\alpha|} \right)\\ &+ \frac{C\varepsilon}{|A_1^2-A_3^2||A_2^2-A_3^2|A_3|\sin\alpha|} + \frac{C\varepsilon}{A_1A_2} + \frac{C\varepsilon}{A_1A_2|\sin\alpha|}. \end{align}\tag{40}\] On the other hand, if 39 holds at a time \(t\), then the same argument, applied to the conjugate phase configuration, gives \[\label{dist-conj-case12} \begin{align} &\mathrm{dist}_2(v_t,\widetilde{\mathbf{O}})\\ \leq\;& \frac{C\varepsilon}{|A_2^2-A_1^2||A_3^2-A_1^2|} \left( \frac{1}{A_1} + \frac{A_3}{A_1^2} + \frac{A_3}{A_1^2|\sin\alpha|} \right) + \frac{C\varepsilon}{|A_1^2-A_2^2||A_3^2-A_2^2|} \left( \frac{1}{A_2} + \frac{A_3}{A_2^2} + \frac{A_3}{A_2^2|\sin\alpha|} \right) \\ &+ \frac{C\varepsilon}{|A_1^2-A_3^2||A_2^2-A_3^2|A_3|\sin\alpha|} + \frac{C\varepsilon}{A_1A_2} + \frac{C\varepsilon}{A_1A_2|\sin\alpha|}, \end{align}\tag{41}\] where \[\widetilde{\mathbf{O}} := \left\{ \widetilde{\omega}(\cdot-\mathbf{p}) \mid \mathbf{p}\in\mathbb{T}_{\Lambda} \right\}, \qquad \widetilde{\omega}(\mathbf{x}) = \sum_{i=1}^3 A_i\cos(\mathbf{k}_i\cdot\mathbf{x}-\alpha_i).\] Since \(\sin\alpha\neq0\), \(\mathbf{O}\neq \widetilde{\mathbf{O}}.\) By Lemma B.1(3) in [6], the two compact orbits are separated in \(L^2\). Thus \[d_0:=\mathrm{dist}_2(\mathbf{O},\widetilde{\mathbf{O}})>0.\] Choose \(\varepsilon_0>0\) sufficiently small such that, whenever \(0<\varepsilon<\varepsilon_0\), the right-hand sides of both 40 and 41 are smaller than \(d_0/3\). We also choose \(\varepsilon_0\) so that \[\mathrm{dist}_2(v_0,\mathbf{O})<\frac{d_0}{3}.\]

Define \[I_1:= \left\{ t\in\mathbb{R}: \mathrm{dist}_2(v_t,\mathbf{O})<\frac{d_0}{3} \right\}, \qquad I_2:= \left\{ t\in\mathbb{R}: \mathrm{dist}_2(v_t,\widetilde{\mathbf{O}})<\frac{d_0}{3} \right\}.\] The pointwise dichotomy above implies that \(I_1\cup I_2=\mathbb{R}.\) Moreover, \[I_1\cap I_2=\varnothing.\] Indeed, if \(t\in I_1\cap I_2\), then \[d_0 = \mathrm{dist}_2(\mathbf{O},\widetilde{\mathbf{O}}) \leq \mathrm{dist}_2(v_t,\mathbf{O}) + \mathrm{dist}_2(v_t,\widetilde{\mathbf{O}}) < \frac{2d_0}{3},\] which is impossible. Since \(t\mapsto v_t\) is continuous in \(L^2\), both \(I_1\) and \(I_2\) are open subsets of \(\mathbb{R}\). Furthermore, \(0\in I_1\). Since \(\mathbb{R}\) is connected, we must have \[I_2=\varnothing.\] Therefore the second alternative 39 never occurs, and 38 holds for all \(t\in\mathbb{R}\). Combining this with 34 and 35 , we conclude that 40 holds for all \(t\in\mathbb{R}\).

Case 2: Exactly two of \(A_1, A_2, A_3\) are equal. Without loss of generality, we may assume that \(A_1=A_2\), and then, based on Lemma 6, we have the following estimates on \(|B_k-A_k|\), \(k=1,2,3\). \[\label{bkak952} \begin{align} &\left|B_1^2-A_1^2\right| <\frac{C\varepsilon^{1/2}}{\left|A_3^2-A_1^2\right|^{1/2}} \quad\Longrightarrow\quad \left|B_1 -A_1 \right| <\frac{C\varepsilon^{1/2}}{\left|A_3^2-A_1^2\right|^{1/2}A_1},\\ &\left|B_2^2-A_2^2\right|<\frac{C\varepsilon^{1/2}}{\left|A_3^2-A_1^2\right|^{1/2}}\quad\Longrightarrow\quad \left|B_2 -A_2 \right|<\frac{C\varepsilon^{1/2}}{\left|A_3^2-A_1^2\right|^{1/2}A_1},\\ &\left|B_3^2-A_3^2\right|<\frac{C\varepsilon}{\left|A_1^2-A_3^2\right|^2}\quad\Longrightarrow\quad \left|B_3 -A_3 \right|<\frac{C\varepsilon}{\left|A_1^2-A_3^2\right|^2A_3}. \end{align}\tag{42}\] The same estimate as in the previous case, with the bounds in 21 and 42 substituted, gives

\[\label{BACOS952} \begin{align} |B_3\cos\beta-A_3\cos\alpha| \leq \frac{C\varepsilon}{A_1^2} +\frac{CA_3\varepsilon^{1/2}}{A_1^2\left|A_3^2-A_1^2\right|^{1/2}}. \end{align}\tag{43}\] The same argument as above gives \[\label{b2a2952} \left|B_3\sin\beta-A_3\sin\alpha\right|\,\left|B_3\sin\beta+A_3\sin\alpha\right| \leq \frac{C A_3^2\varepsilon^{1/2}}{A_1^2\left|A_3^2-A_1^2\right|^{1/2}} +\frac{C\varepsilon}{\left|A_1^2-A_3^2\right|^2}+ \frac{CA_3\varepsilon}{A_1^2}.\tag{44}\]

By 44 , \[\begin{align} |B_3\sin\beta-A_3\sin\alpha| \leq \frac{C A_3\varepsilon^{1/4}}{A_1\left|A_3^2-A_1^2\right|^{1/4}} +\frac{C\varepsilon^{1/2}}{\left|A_1^2-A_3^2\right|}+ \frac{CA_3^{1/2}\varepsilon^{1/2}}{A_1}. \end{align}\] Choosing \(\varepsilon_0>0\) smaller if necessary, we have \[\begin{align} \mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) \leq & \frac{CA_3\varepsilon^{1/4}}{\left|A_3^2-A_1^2\right|^{1/4} A_1}. \end{align}\]

As in Case 1.2, the pointwise dichotomy obtained from 44 and 37 has two branches. The branch corresponding to the conjugate orbit is excluded by the continuity of \(t\mapsto v_t\) in \(L^2\), the positive distance between \(\mathbf{O}\) and the conjugate orbit, and the initial closeness to \(\mathbf{O}\). From 44 , we have that \[\left|B_3\sin\beta-A_3\sin\alpha\right| \leq \frac{C A_3\varepsilon^{1/2}}{A_1^2\left|A_3^2-A_1^2\right|^{1/2}\left|\sin\alpha\right|} +\frac{C\varepsilon}{\left|A_1^2-A_3^2\right|^2A_3\left|\sin\alpha\right|}+ \frac{C\varepsilon}{A_1^2\left|\sin\alpha\right|}.\] Then, choosing \(\varepsilon_0>0\) smaller if necessary, we obtain \[\begin{align} \mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) \leq \frac{C\varepsilon^{1/2}}{\left|A_3^2-A_1^2\right|^{1/2}}\left(\frac{A_3}{A_1^2\left|\sin\alpha\right|}+\frac{A_3}{A_1^2}+\frac{1}{A_1}\right). \end{align}\]

Case 3: \(A_1=A_2=A_3\). Based on Lemma 6, we have \[\label{bkak953} \begin{align} &\left|B_k^2-A_k^2\right| <{C\varepsilon^{1/3}} \quad\Longrightarrow\quad \left|B_k -A_k \right| <\frac{C\varepsilon^{1/3}}{A_1},\quad k=1,2,3. \end{align}\tag{45}\] Repeating the argument leading to 35 , we obtain \[\label{BACOS953} \begin{align} |B_3\cos\beta-A_3\cos\alpha| \leq \frac{C\varepsilon}{A_1^2} +\frac{C\varepsilon^{1/3}}{A_1}. \end{align}\tag{46}\] Repeating the argument leading to 36 , we obtain \[\label{b2a2953} \begin{align} \left|B_3\sin\beta-A_3\sin\alpha\right|\,\left|B_3\sin\beta+A_3\sin\alpha\right| \leq C \varepsilon^{1/3}+\frac{C\varepsilon}{A_1}. \end{align}\tag{47}\]

By 47 , \[\begin{align} |B_3\sin\beta-A_3\sin\alpha|\leq C \varepsilon^{1/6}+\frac{C\varepsilon^{1/2}}{A_1^{1/2}}. \end{align}\] Therefore, \[\begin{align} \mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) \leq C\varepsilon^{1/6}, \end{align}\] after choosing \(\varepsilon_0\) smaller if necessary.

As in Case 1.2, the branch corresponding to the conjugate orbit is excluded by the continuity of \(t\mapsto v_t\) in \(L^2\), the positive distance between \(\mathbf{O}\) and the conjugate orbit, and the initial closeness to \(\mathbf{O}\). From 47 , we have that \[\begin{align} \left|B_3\sin\beta-A_3\sin\alpha\right| \leq \frac{C \varepsilon^{1/3}}{\left|\sin\alpha\right|A_1}+\frac{C \varepsilon}{\left|\sin\alpha\right|A_1^2}. \end{align}\] Therefore, \[\begin{align} \mathrm{dist}_2\bigl(v_t,\mathbf{O}\bigr) \leq &\frac{C \varepsilon^{1/3}}{\left|\sin\alpha\right|\,A_1}+\frac{C \varepsilon^{1/3}}{A_1}, \end{align}\] after choosing \(\varepsilon_0\) smaller if necessary.

5 Non-hexagonal tori↩︎

In this section, we consider the remaining cases of flat two-dimensional tori for which \(\dim(\mathbf{E}_1)=2\) or \(4\). For these two cases, by [6], we have the following:

• If \(\dim(\mathbf{E}_1) = 2\), then there exists some nonzero vector \(\mathbf{k}\) such that \[\mathbf{E}_1={\rm span}\left\{\cos(\mathbf{k} \cdot \mathbf{x}), \sin(\mathbf{k} \cdot\mathbf{x})\right\}.\]

• If \(\dim(\mathbf{E}_1) = 4\), then there exist two linearly independent vectors \(\mathbf{k}_1, \mathbf{k}_2\) satisfying \(|\mathbf{k}_1| = |\mathbf{k}_2|\) such that \[\mathbf{E}_1={\rm span}\left\{ \cos(\mathbf{k}_1 \cdot\mathbf{x}), \sin(\mathbf{k}_1 \cdot\mathbf{x}), \cos(\mathbf{k}_2 \cdot\mathbf{x}), \sin(\mathbf{k}_2 \cdot\mathbf{x})\right\}.\]

5.1 2D case↩︎

5.2 4D case↩︎

G. Wang was supported by National Natural Science Foundation of China (Grant No. 12471101) and Fundamental Research Funds for the Central Universities (Grant No. DUT23RC(3)077).

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

The authors declare that they have no conflict of interest to this work.

References↩︎