A Fourth-Order Dispersive Flow Equation for Closed Curves on Compact Riemann Surfaces

Abstract

A fourth-order dispersive flow equation for closed curves on the canonical two-dimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is considered, where the solutions are supposed to take values in compact Riemann surfaces. As a main result, time-local existence and uniqueness of a solution to the initial value problem are established under the assumption that the sectional curvature of the Riemann surface is constant. The analytic difficulty comes from the so-called loss of derivatives and the absence of the local smoothing effect. The proof is based on the geometric energy method combined with a kind of gauge transformation to eliminate the loss of derivatives. Specifically, to show the uniqueness of the solution, the detailed geometric analysis of the solvable structure for the equation is presented.

Introduction

Dispersive partial differential equations have been extensively studied in mathematical research. Many studies have paid attention to real- or complex-valued functions as solutions to these equations. However, some nonlinear dispersive partial differential equations in contexts in classical mechanics and fluid mechanics require their solutions to take values in a (curved) Riemannian manifold. In general, their nonlinear structures depend on the geometric setting of the manifold. Therefore, concerning how to solve their initial value problem, geometric analysis of the relationship between their solvable structure and the geometric setting of the manifold plays an essential role.

In this field, after the pioneering work of Koiso [11], the method of geometric analysis for the so-called one-dimensional Schrödinger flow equation, the higher-dimensional generalization, and a third-order analogue has been developed extensively. Many results on how to solve their initial value problem have been established mainly from the following three points of view: analysis of the solvable structure of dispersive partial differential equations (systems), application of Riemannian geometry, and analysis of nonlinear partial differential equations with physical backgrounds; see, e.g., [2,3,4,5, 10, 11, 13,14,15,16,17], and references therein. In this paper, we study a fourth-order analogue, the solutions of which are required to take values in a compact Riemann surface. This is a continuation of [6, 18] and presents the answer to the problem suggested in [4].

The setting of our problem is stated as follows: Given a compact Riemann surface N with the complex structure J and with a Hermitian metric g, consider the following initial value problem:

$$\begin{aligned}&\displaystyle u_t = a\,J_u\nabla _x^3u_x + \{\lambda + b\, g(u_x,u_x)\}J_u\nabla _xu_x + c\,g(\nabla _xu_x,u_x)J_uu_x \quad \text {in}\quad \mathbb {R}{\times } \mathbb {T}, \nonumber \\ \end{aligned}$$
(1.1)
$$\begin{aligned}&\displaystyle u(0,x) = u_0(x) \quad \text {in}\quad \mathbb {T}. \end{aligned}$$
(1.2)

Here \(\mathbb {T}=\mathbb {R}/2\pi \mathbb {Z}\) is the one-dimensional flat torus, \(u=u(t,x):\mathbb {R}\times \mathbb {T}\rightarrow N\) is the unknown map describing the deformation of closed curves lying on N parameterized by t, \(u_0=u_0(x):\mathbb {T}\rightarrow N\) is the given initial map, \(u_t=\mathrm{d}u(\frac{\partial }{\partial t})\), \(u_x=\mathrm{d}u(\frac{\partial }{\partial x})\), \(\mathrm{d}u\) is the differential of the map u, \(\nabla _x\) is the covariant derivative along u in x, \(J_u:T_uN\rightarrow T_uN\) is the complex structure at \(u\in N\), and a, b, c, and \(\lambda \) are real constants. If \(a,b,c=0\) and \(\lambda =1\), then (1.1) is reduced to the second-order dispersive equation of the form:

$$\begin{aligned} u_t=J_u\nabla _xu_x, \end{aligned}$$
(1.3)

which is called a one-dimensional Schrödinger flow equation. As a fourth-order analogue of (1.3), we call (1.1) with \(a\ne 0\) a fourth-order dispersive flow equation. Hereafter it is assumed that \(a\ne 0\).

An example of (1.1) with \(a\ne 0\) arises in two areas of physics, where N is supposed to be the canonical two-dimensional unit sphere \(\mathbb {S}^2\). Indeed, if \(N=\mathbb {S}^2\) equipped with the complex structure acting as \(\pi /2\)-degree rotation on each tangent plane and with the canonical metric induced from the Euclidean metric in \(\mathbb {R}^3\), then (1.1) is described by

$$\begin{aligned}&u_t = u\wedge \left[ a\,\partial _x^3u_{x} + \{\lambda +(a+b)\, (u_x,u_x)\} \partial _xu_{x} + (5a+c)\, (\partial _xu_{x},u_x) u_{x} \right] , \end{aligned}$$
(1.4)

where \(u:\mathbb {R}\times \mathbb {T}\rightarrow \mathbb {S}^2\subset \mathbb {R}^3\), \(\partial _x\) is the partial differential operator in x acting on \(\mathbb {R}^3\)-valued functions, \((\cdot ,\cdot )\) is the inner product in \(\mathbb {R}^3\), and \(\wedge \) is the exterior product in \(\mathbb {R}^3\). In particular, the \(\mathbb {S}^2\)-valued model (1.4) with \(3a-2b+c=0\) and \(\lambda =1\) models the continuum limit of the Heisenberg spin-chain systems with biquadratic exchange interactions ([12]), where each of abc is decided by two independent physical constants. Interestingly, the same equation can be derived from an equation modelling the motion of a vortex filament in an incompressible perfect fluid in \(\mathbb {R}^3\) by taking into account the elliptical deformation effect of the core due to the self-induced strain ([7, 8]).

For the Schrödinger flow equation (1.3) and the higher-dimensional generalization, almost all results on the existence of solutions have been established assuming essentially that (NJg) is a compact Kähler manifold; see, e.g., [2, 10, 11, 13, 14, 19] and references therein. Under the assumption, the classical energy method combined with geometric analysis works to show the local existence results. On the other hand, if (NJg) is a compact almost Hermitian manifold without the Kähler condition, then the classical energy method breaks down, since the so-called loss of derivatives occurs from the covariant derivative of the almost complex structure. However, Chihara in [3] overcame the difficulty using the geometric energy method combined with a kind of gauge transformation acting on the pullback bundle. Indeed, he established a local existence and uniqueness result for maps from a compact Riemannian manifold into a compact almost Hermitian manifold. After that, he and the author obtained similar results in [5, 16,17,18] for a third-order dispersive flow equation for maps from \(\mathbb {R}\) or \(\mathbb {T}\) into a compact almost Hermitian manifold.

In contrast, for our fourth-order dispersive flow equation (1.1), we face with the difficulty due to loss of derivatives even if (NJg) satisfies the Kähler condition, which is also the case for the \(\mathbb {S}^2\)-valued physical model (1.4). If the spatial domain is the real line \(\mathbb {R}\) instead of \(\mathbb {T}\), the difficulty can be overcome by making use of the local dispersive smoothing effect of the equation in some sense. Besides, there is much room for the solvable structure. Indeed, in [6], the local existence and the uniqueness of a solution to the problem on \(\mathbb {R}\) were established and extended to compact Kähler manifolds as N. Unfortunately, however, the local smoothing effect is absent in our problem since the spatial domain \(\mathbb {T}\) is compact. In other words, the method of the proof in [6] is not applicable to our problem. Thus, the obstruction coming from the loss of derivatives is expected to be avoided by finding out a kind of special good solvable structure of the equation.

The previous studies of (1.1) on \(\mathbb {T}\) are limited as follows: Guo et al. in [9] investigated the \(\mathbb {S}^2\)-valued physical model (1.4) with \(3a-2b+c=0\) and \(\lambda =1\) imposing an additional assumption \(c=0\). Under the assumption, (1.4) is completely integrable, and they made use of some conservation laws of (1.4) to show the local existence of a weak solution to the initial value problem, though the uniqueness was unsolved. Chihara in [4] investigated the fourth-order dispersive systems for \(\mathbb {C}^2\)-valued functions including a system which is reduced from (1.1) by the generalized Hasimoto transformation, and pointed out that the assumption that the sectional curvature of N is constant provides the solvable structure of the initial value problem. To the present author’s knowledge, although the insights seems to grasp the solvable structure of (1.1)–(1.2) essentially, it is nontrivial whether we can recover the solution to (1.1)–(1.2) from the solution to the reduced dispersive system.

Motivated by them, the present author tried to solve directly (1.1)–(1.2) imposing that the sectional curvature on N is constant, without using the generalized Hasimoto transformation. Recently, he succeeded to show in [18] the local existence of a unique solution to the initial value problem for the \(\mathbb {S}^2\)-valued model (1.4) without any assumption on \(a,b,c,\lambda \) (except for \(a\ne 0\)), where \(u_0\) is taken so that \(u_{0x}\in H^k(\mathbb {T};\mathbb {R}^3)\) with \(k\geqslant 6\). This is proved by the energy method based on the standard Sobolev norm for \(\mathbb {R}^3\)-valued functions, combined with a kind of gauge transformation.

The purpose of the present paper is to extend the results obtained in [18] for \(\mathbb {S}^2\)-valued model (1.4), that is, to establish the time-local existence and uniqueness theorem for (1.1)–(1.2) under the assumption that \(k\geqslant 6\) and the sectional curvature on (Ng) is constant. More precisely, our main results are stated as follows:

Theorem 1.1

Suppose that (NJg) is a compact Riemann surface, the sectional curvature of which is constant. Let k be an integer satisfying \(k\geqslant 6\). Then for any \(u_0\in C(\mathbb {T};N)\) satisfying \(u_{0x}\in H^k(\mathbb {T};TN)\), there exists \(T=T(\Vert u_{0x}\Vert _{H^4(\mathbb {T};TN)})>0\) such that the initial value problem (1.1)–(1.2) has a unique solution \(u\in C([-T,T]\times \mathbb {T};N)\) satisfying \(u_x\in C([-T,T];H^{k}(\mathbb {T};TN)). \)

Notation For \(\phi :\mathbb {T}\rightarrow N\), we denote by \(\Gamma (\phi ^{-1}TN)\) the set of all vector fields along \(\phi \). Let \(V\in \Gamma (\phi ^{-1}TN)\) and let m be a nonnegative integer. Then we say \(V\in H^m(\mathbb {T};TN)\) if

$$\begin{aligned} \Vert V\Vert _{H^m(\mathbb {T};TN)} := \sum _{\ell =0}^m \int _{\mathbb {T}} g\big (\nabla _x^{\ell }V(x), \nabla _x^{\ell }V(x)\big )\,\mathrm{d}x <\infty . \end{aligned}$$

In particular, if \(m=0\), we replace \(H^0(\mathbb {T};TN)\) with \(L^2(\mathbb {T};TN)\).

Remark 1.2

Precisely speaking, the existence time T of the solution in Theorem 1.1 depends on \(a,b,c,\lambda \), and the constant sectional curvature of (Ng) as well as \(\Vert u_{0x}\Vert _{H^4(\mathbb {T};TN)}\).

Remark 1.3

The local existence of the solution in Theorem 1.1 holds if \(k\geqslant 4\). The assumption \(k\geqslant 6\) comes from the requirement to show the uniqueness.

Remark 1.4

Let w be an isometric embedding of (Ng) into some Euclidean space \(\mathbb {R}^d\) so that N is considered as a submanifold of \(\mathbb {R}^d\). By the Gagliardo–Nirenberg inequality, it is found for \(u_0\) in Theorem 1.1 that \(u_{0x}\in H^k(\mathbb {T};TN)\) if and only if \((w{\circ }u_0)_x\in H^k(\mathbb {T};\mathbb {R}^d)\), where \(H^k(\mathbb {T};\mathbb {R}^d)\) denotes the standard kth-order Sobolev space for \(\mathbb {R}^d\)-valued functions on \(\mathbb {T}\). By the equivalence, Theorem 1.1 actually extends the results obtained in [18].

Remark 1.5

We can extend Theorem 1.1 to the case where (NJg) is a compact Kähler manifold with nonzero constant sectional curvature. Indeed, the argument using (2.12) and (3.28) in the proof can be replaced by that using (2.9) if the curvature is not zero. This seems a little bit artificial and the proof is not so different. Thus we do not pursue that.

Remark 1.6

It is unlikely that we can remove the assumption on the curvature of (Ng) in general. To check this, let (Ng) be a Riemann surface, the sectional curvature of which is not necessarily constant. In view of [4, Sect. 4], if we can construct a sufficiently smooth solution u to (1.1)–(1.2), then the following necessary condition:

$$\begin{aligned} \int _{\mathbb {T}} \frac{\partial }{\partial x} \left\{ S(u(t,x)) \right\} g(u_x(t,x),u_x(t,x)) \,\mathrm{d}x =0 \end{aligned}$$
(1.5)

is expected to be satisfied for all existence time, where S(u(tx)) denotes the sectional curvature of (Ng) at \(u(t,x)\in N\). This requires at least that the left-hand side of (1.5) is a conserved quantity in time. Even if (1.5) is true, the initial map \(u_0\) is required to satisfy

$$\begin{aligned} \int _{\mathbb {T}} {\frac{\partial }{\partial x}} \left\{ S(u_0(x)) \right\} g(u_{0x}(x),u_{0x}(x)) \,\mathrm{d}x =0. \end{aligned}$$
(1.6)

On the other hand, (1.5) and (1.6) are obviously satisfied if the sectional curvature of (Ng) is constant.

The idea of the proof of the local existence comes from the following formal observation. Suppose that u solves (1.1)–(1.2). If \(k\geqslant 4\), then \(\nabla _x^ku_x\) satisfies

$$\begin{aligned} \left( \nabla _t-a\,J_u\nabla _x^4-c_1\,P_1\nabla _x^2-c_2\,P_2\nabla _x\right) \nabla _x^ku_x&= \mathcal {O} \left( \sum _{m=0}^{k+2} |\nabla _x^mu_x|_g \right) , \end{aligned}$$
(1.7)

where \(|\cdot |_g=\left\{ g(\cdot ,\cdot )\right\} ^{1/2}\), \(c_1\) and \(c_2\) are real constants depending on abck and the sectional curvature on (Ng), and \(P_1\) and \(P_2\) are, respectively, defined by

$$\begin{aligned} P_1Y&= g(Y,u_x)J_uu_x, \quad P_2Y = g(\nabla _xu_x,u_x)J_uY \end{aligned}$$

for any \(Y\in \Gamma (u^{-1}TN)\). It is found that (1.7) leads to the classical energy estimate for \(\Vert \nabla _x^ku_x\Vert _{L^2(\mathbb {T};TN)}^2\) with loss of derivatives coming only from \(c_1\,P_1\nabla _x^2\) and \(c_2\,P_2\nabla _x\). Although the right-hand side of (1.7) includes \(\nabla _x^2(\nabla _x^ku_x)\) and \(\nabla _x(\nabla _x^ku_x)\), no loss of derivatives occurs thanks to the curvature condition and the Kähler condition on (NJg). To eliminate the loss of derivatives coming from \(c_1\,P_1\nabla _x^2\) and \(c_2\,P_2\nabla _x\), we introduce the so-called gauged function \(V_k\) defined by

$$\begin{aligned} V_k&= \nabla _x^ku_x -\frac{d_1}{2a}\, g\left( \nabla _x^{k-2}u_x,J_uu_x\right) J_uu_x + \frac{d_2}{8a}\, g(u_x,u_x)\nabla _x^{k-2}u_x, \end{aligned}$$
(1.8)

where \(d_1\) and \(d_2\) are constants which will be decided later and \(V_k\) is formally expressed as \(V_k=(I_d+\Phi _1\nabla _x^{-2}+\Phi _2\nabla _x^{-2})\nabla _x^ku_x\), where \(I_\mathrm{d}\) is the identity on \(\Gamma (u^{-1}TN)\) and

$$\begin{aligned} \Phi _1Y&= -\frac{d_1}{2a}\, g(Y,J_uu_x)J_uu_x, \quad \Phi _2 = \frac{d_2}{8a}\, g(u_x,u_x)Y \end{aligned}$$

for any \(Y\in \Gamma (u^{-1}TN)\). Noting that \(J_u\) commutes with \(\Phi _2\) and not with \(\Phi _1\), we obtain

$$\begin{aligned} \left[ a\,J_u\nabla _x^4, \Phi _1\nabla _x^{-2} \right] \nabla _x^ku_x&= \left( d_1\,P_1\nabla _x^2-d_1\,P_2\nabla _x\right) \nabla _x^ku_x +\text {harmless terms}, \end{aligned}$$
(1.9)
$$\begin{aligned} \left[ a\,J_u\nabla _x^4, \Phi _2\nabla _x^{-2} \right] \nabla _x^ku_x&= d_2\,P_2\nabla _x\nabla _x^ku_x + \text {harmless terms}. \end{aligned}$$
(1.10)

Therefore, if we set \(d_1=c_1\) and \(d_2=c_1+c_2\), the above two commutators eliminate \(c_1\,P_1\nabla _x^2+c_2\,P_2\nabla _x\) in the partial differential equation satisfied by \(V_k\), and hence the energy estimate for \(\Vert V_k\Vert _{L^2(\mathbb {T};TN)}^2\) works. The good choice of the above gauged function is inspired by [4].

The strategy for the proof of the local existence of a solution is as follows: First, we construct a family of fourth-order parabolic regularized solutions \(\left\{ u^{\varepsilon }\right\} _{\varepsilon \in (0,1]}\). Second, we obtain \(\varepsilon \)-independent uniform estimates for \(\Vert u_x^{\varepsilon }\Vert _{H^{k-1}(\mathbb {T};TN)}^2+\Vert V_k^{\varepsilon }\Vert _{L^2(\mathbb {T};TN)}^2\) and the lower bound \(T>0\) of existence time of \(\left\{ u^{\varepsilon }\right\} _{\varepsilon \in (0,1]}\), where \(V_k^{\varepsilon }\) is defined by (1.8) replacing u with \(u^{\varepsilon }\). Finally, the standard compactness argument concludes the existence of \(u\in C([0,T]\times \mathbb {T};N)\) so that \(u_x\in L^{\infty }(0,T;H^k(\mathbb {T};TN))\cap C([0,T];H^{k-1}(\mathbb {T};TN))\) and u solves (1.1)–(1.2). The two commutators (1.9) and (1.10) in the above formal observation will be generated essentially in the computation of the second and third terms of the right-hand side of (2.6). One can refer to [10, 11, 13] for tools of computation and [4,5,6] for the method of gauged energy employed in the proof.

The strategy for the proof of the uniqueness of the solution is stated as follows: Suppose that \(u, v\in C([0,T]\times \mathbb {T};N)\) are solutions to (1.1)–(1.2) satisfying \(u_x, v_x\in L^{\infty }(0,T;H^6(\mathbb {T};TN)) \cap C([0,T];H^{5}(\mathbb {T};TN)) \) with the same initial data \(u_0\). Their existence is ensured by the above local existence results. To estimate the difference between u and v, we regard u and v as the functions with values in some Euclidean space \(\mathbb {R}^d\). Indeed, letting w be an isometric embedding of (Ng) into \(\mathbb {R}^d\), we consider \(\mathbb {R}^d\)-valued functions defined as follows:

$$\begin{aligned} U&:=w{\circ } u, \quad V:=w{\circ } v, \quad Z:=U-V, \\ \mathcal {U}&:=\mathrm{d}w_u(\nabla _xu_x), \quad \mathcal {V}:=\mathrm{d}w_v(\nabla _xv_x), \quad \mathcal {W}:=\mathcal {U}-\mathcal {V}, \end{aligned}$$

where \(\mathrm{d}w_p:T_pN\rightarrow T_{w{\circ }p}\mathbb {R}^d\cong \mathbb {R}^d\) is the differential of w at \(p\in N\). To complete the proof of the uniqueness, it suffices to show \(Z=0\). First, as shown in (3.94), we obtain the classical energy estimate for \(\Vert Z\Vert _{L^2}^2+\Vert Z_x\Vert _{L^2}^2+\Vert \mathcal {W}\Vert _{L^2}^2\) with the loss of derivatives, where \(\Vert \cdot \Vert _{L^2}\) expresses the standard \(L^2\)-norm for \(\mathbb {R}^d\)-valued functions on \(\mathbb {T}\). The loss of derivatives has a similar form to that eliminated by the method of gauge transformation in the proof of the local existence of a solution. Observing the analogy, we can easily find \(\widetilde{\mathcal {W}}=\mathcal {W}+\widetilde{\Lambda }\) as a gauged function of \(\mathcal {W}\) so that the energy estimate for \(\Vert Z\Vert _{L^2}^2+\Vert Z_x\Vert _{L^2}^2+\Vert \widetilde{\mathcal {W}}\Vert _{L^2}^2\) can be closed. This shows \(Z=0\). The precise form of \(\widetilde{\Lambda }\) will be presented in (3.97).

In the proof of the uniqueness, we face with another difficulty, which does not appear in the proof of the local existence. On one hand, the proof of the local existence seems clear, thanks to the good matching between the geometric formulation of (1.1) and the geometric \(L^2\)-norm \(\Vert \cdot \Vert _{L^2(\mathbb {T};TN)}\). On the other hand, the proof of the uniqueness requires lengthier computations, due to the worse matching between the form of the equation satisfied by U and the standard \(L^2\)-norm \(\Vert \cdot \Vert _{L^2(\mathbb {T};\mathbb {R}^d)}\). More concretely, the most crucial part of the proof of the uniqueness is how to derive the energy estimate for \(\mathcal {W}\) of the form (3.94). To derive this, the partial differential equation satisfied by \(\mathcal {W}\) and the energy estimate in \(L^2(\mathbb {T};\mathbb {R}^d)\) are required. However, the analysis of the structure of lower order terms in the equation becomes complicated, since many terms related to the second fundamental form on N and the derivatives appear to describe the equation satisfied by U or V. As (1.1) is a higher-order equation than the Schrödinger flow equation or the third-order dispersive flow equation studied previously, the situation becomes worse. Fortunately, however, we can successfully formulate the Kähler condition and the curvature condition on (NJg) to be applicable to our problem, and demonstrate that only weak loss of derivatives is allowed to appear in the energy estimate for \(\Vert \mathcal {W}\Vert _{L^2(\mathbb {T};\mathbb {R}^d)}^2\). In addition, it is to be noted that we does not choose \(\partial _xZ_x\) but choose \(\mathcal {W}\) in the energy estimate. The choice also plays an important role (see, e.g., Lemma 3.1) in our proof, as well as the choice of \(\widetilde{\Lambda }\).

By the way, the geometric formulation of (1.1) was originally proposed by [15]. Independently, Anco and Myrzakulov in [1] derived the equation, named a fourth-order Schrödinger map equation, for \(u:\mathbb {R}\times \mathbb {R}\rightarrow N\) or \(u:\mathbb {R}\times \mathbb {T}\rightarrow N\) of the form:

$$\begin{aligned} -u_t&=J_{u}\nabla _x^3u_x +\frac{1}{2} \nabla _x\left\{ g(u_x,u_x)J_{u}u_x \right\} -\frac{1}{2} g(J_{u}u_x,\nabla _xu_x)u_x. \end{aligned}$$
(1.11)

Interestingly, if N is a Riemann surface, (1.11) is identical with (1.1) where \(a=-1\), \(b=-1\), \(c=-1/2\), and \(\lambda =0\). Therefore, we immediately find that Theorem 1.1 is valid for the initial value problem for (1.11).

The organization of the present paper is as follows: In Sect. 2, a time-local solution to (1.1)–(1.2) is constructed. In Sect. 3, the proof of Theorem 1.1 is completed.

Proof of the Existence of a Time-Local Solution

This section is devoted to the construction of a time-local solution to (1.1)–(1.2). More concretely, the goal of this section is to show the following.

Theorem 2.1

Suppose that the sectional curvature of (Ng) is constant. Let k be an integer satisfying \(k\geqslant 4\). Then for any \(u_0\in C(\mathbb {T};N)\) satisfying \(u_{0x}\in H^k(\mathbb {T};TN)\), there exists \(T=T(\Vert u_{0x}\Vert _{H^4(\mathbb {T};TN)})>0\) such that the initial value problem (1.1)–(1.2) has a solution \(u\in C([-T,T]\times \mathbb {T};N)\) satisfying \(u_x\in L^{\infty }(-T,T;H^{k}(\mathbb {T};TN)) \cap C([-T,T];H^{k-1}(\mathbb {T};TN)). \)

Proof of Theorem 2.1

Let \(k\geqslant 4\) be fixed. It suffices to solve the problem in the positive direction in time. We first assume that \(u_{0}\in C^{\infty }(\mathbb {T};N)\) and construct a local solution.

As a beginning, we consider the initial value problem of the form:

$$\begin{aligned}&u_t =\, (-\varepsilon + a\,J_u)\nabla _x^3u_x \nonumber \\&\ \quad \quad + \,b\, g(u_x,u_x)J_u\nabla _xu_x + c\,g(\nabla _xu_x,u_x)J_uu_x +\lambda \, J_u\nabla _xu_x \quad \text {in}\quad (0,\infty ){\times } \mathbb {T}, \end{aligned}$$
(2.1)
$$\begin{aligned}&u(0,x) = \,u_0(x) \quad \text {in}\quad \mathbb {T}, \end{aligned}$$
(2.2)

where \(\varepsilon \in (0,1]\) is a small positive parameter. Thanks to the added term \(-\varepsilon \,\nabla _x^3u_x\), (2.1) is a fourth-order quasilinear parabolic system and the initial value problem (2.1)–(2.2) has a unique local smooth solution which we will denote \(u^{\varepsilon }\).

Lemma 2.2

For each \(\varepsilon \in (0,1]\), there exists a positive constant \(T_{\varepsilon }\) depending on \(\varepsilon \) and \(\Vert u_{0x}\Vert _{H^4(\mathbb {T};TN)}\) such that the initial value problem (2.1)–(2.2) has a unique solution \(u^{\varepsilon }\in C^{\infty }([0,T_{\varepsilon }]\times \mathbb {T};N)\).

We can show Lemma 2.2 by the mix of a sixth-order parabolic regularization and a geometric classical energy method without the constant curvature condition on (Ng). The proof almost falls into the scope of that of [6, Lemma 3.1] by replacing \(\mathbb {R}\) with \(\mathbb {T}\) and by restricting to a compact Riemann surface as N. Although a slight modification is required in the proof, the difference is not essential and thus we omit the detail of the proof.

In the next step, letting \(\left\{ u^{\varepsilon }\right\} _{\varepsilon \in (0,1]}\) be a family of solutions to (2.1)–(2.2) constructed in Lemma 2.2, we obtain \(\varepsilon \)-independent energy estimates for \(\left\{ u^{\varepsilon }_x\right\} _{\varepsilon \in (0,1]}\). Precisely speaking, we obtain a uniform lower bound T of \(\left\{ T_{\varepsilon }\right\} _{\varepsilon \in (0,1]}\) and show that \(\left\{ u^{\varepsilon }_x\right\} _{\varepsilon \in (0,1]}\) is bounded in \(L^{\infty }(0,T;H^k(\mathbb {T};TN))\). However, the classical energy estimate for \(\Vert u^{\varepsilon }_x\Vert _{H^k(\mathbb {T};TN)}\) causes the loss of derivatives. To overcome this difficulty, we introduce a gauged function \(V^{\varepsilon }_k\) defined by

$$\begin{aligned} V^{\varepsilon }_k&= \nabla _x^ku_x^{\varepsilon } + \Lambda ^{\varepsilon } = \nabla _x^ku_x^{\varepsilon } + \Lambda ^{\varepsilon }_1 + \Lambda ^{\varepsilon }_2, \end{aligned}$$
(2.3)

where

$$\begin{aligned} \Lambda _1^{\varepsilon }&= -\frac{d_1}{2a}\, g\left( \nabla _x^{k-2}u_x^{\varepsilon },J_uu_x^{\varepsilon }\right) J_uu_x^{\varepsilon }, \quad \Lambda _2^{\varepsilon } = \frac{d_2}{8a}\, g\left( u_x^{\varepsilon },u_x^{\varepsilon }\right) \nabla _x^{k-2}u_x^{\varepsilon }, \end{aligned}$$

and \(d_1, d_2\in \mathbb {R}\) are real constants which will be decided later depending only on abck, and the constant sectional curvature of (Ng). Furthermore, we introduce the associated gauged energy \(N_k(u^{\varepsilon }(t))\) defined by

$$\begin{aligned} N_k(u^{\varepsilon }(t)) = \sqrt{ \big \Vert u_x^{\varepsilon }(t)\big \Vert _{H^{k-1}(\mathbb {T};TN)}^2 + \big \Vert V_k^{\varepsilon }(t)\big \Vert _{L^2(\mathbb {T};TN)}^2 }. \end{aligned}$$
(2.4)

We restrict the time interval on \([0,T_{\varepsilon }^{\star }]\), where \(T^{\star }_{\varepsilon }\) is defined by

$$\begin{aligned} T^{\star }_{\varepsilon } = \sup \left\{ T>0 \ | \ N_4\big (u^{\varepsilon }(t)\big )\le 2N_4(u_0) \quad \text {for all} \quad t\in [0,T] \right\} . \end{aligned}$$

By the Sobolev embedding, we immediately find that it holds that

$$\begin{aligned} \frac{1}{C}N_k\big (u^{\varepsilon }(t)\big ) \le \big \Vert u_x^{\varepsilon }(t)\big \Vert _{H^k(\mathbb {T};TN)} \le C\,N_k\big (u^{\varepsilon }(t)\big ) \quad \ \text {for any} \quad \ t\in [0,T_{\varepsilon }^{\star }] \end{aligned}$$
(2.5)

with \(C=C(\Vert u_{0x}\Vert _{H^4(\mathbb {T};TN)})>1\) being an \(\varepsilon \)-independent constant. We shall show that there exists a constant \(T=T(\Vert u_{0x}\Vert _{H^4(\mathbb {T};TN)})>0\) which is independent of \(\varepsilon \in (0,1]\) and k such that \(T^{\star }_{\varepsilon }\geqslant T\) uniformly in \(\varepsilon \in (0,1]\) and that \(\left\{ N_k(u^{\varepsilon })\right\} _{\varepsilon \in (0,1]}\) is bounded in \(L^{\infty }(0,T)\). If it is true, this together with (2.5) implies that \(\left\{ u_x^{\varepsilon }\right\} _{\varepsilon \in (0,1]}\) is bounded in \(L^{\infty }(0,T;H^k(\mathbb {T};TN))\).

Having them in mind, let us focus on the uniform energy estimate for \(\left\{ N_k(u^{\varepsilon })\right\} _{\varepsilon \in (0,1]}\). We set \(u=u^{\varepsilon }\), \(V_k=V_k^{\varepsilon }\), \(\Lambda =\Lambda ^{\varepsilon }\), \(\Lambda _1=\Lambda _1^{\varepsilon }\), \(\Lambda _2=\Lambda _2^{\varepsilon }\), \(\Vert \cdot \Vert _{H^0(\mathbb {T};TN)}=\Vert \cdot \Vert _{L^2(\mathbb {T};TN)}=\Vert \cdot \Vert _{L^2}\), \(\Vert \cdot \Vert _{H^m(\mathbb {T};TN)}=\Vert \cdot \Vert _{H^m}\) for \(m=1,\ldots ,k\), and \(\sqrt{g(\cdot ,\cdot )}=|\cdot |_g\), for ease of notation. Since g is a Hermitian metric, \(g(J_uY_1,J_uY_2)=g(Y_1,Y_2)\) holds for any \(Y_1,Y_2\in \Gamma (u^{-1}TN)\). Since Riemann surfaces with Hermitian metric are Kähler manifolds, \(\nabla _xJ_u=J_u\nabla _x\) and \(\nabla _tJ_u=J_u\nabla _t\) hold. We denote the sectional curvature of (Ng) by S which is constant. Any positive constant which depends on a, b, c, \(\lambda \), k, S, \(\Vert u_{0x}\Vert _{H^4}\) and not on \(\varepsilon \in (0,1]\) will be denoted by the same C. Note that \(k\geqslant 4\) and the Sobolev embedding \(H^1(\mathbb {T})\subset C(\mathbb {T})\) yield \(\Vert \nabla _x^4u_x\Vert _{L^{\infty }(0,T_{\varepsilon }^{\star };L^2)}\le C\) and \(\Vert \nabla _x^mu_x\Vert _{L^{\infty }((0,T_{\varepsilon }^{\star })\times \mathbb {T})}\le C\) for \(m=0,1,\ldots ,3\). These properties will be used without any comment in this section.

We now investigate the energy estimate for \(\Vert V_k\Vert _{L^2}^2\). It follows that

$$\begin{aligned} \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t} \Vert V_k\Vert _{L^2}^2 =&\int _{\mathbb {T}} g(\nabla _tV_k,V_k) \mathrm{d}x \nonumber \\ =&\int _{\mathbb {T}} g\left( \nabla _t\left( \nabla _x^ku_x\right) ,V_k\right) \mathrm{d}x + \int _{\mathbb {T}} g(\nabla _t\Lambda ,V_k) \mathrm{d}x \nonumber \\ =&\int _{\mathbb {T}} g\left( \nabla _t\left( \nabla _x^ku_x\right) ,\nabla _x^ku_x\right) \mathrm{d}x + \int _{\mathbb {T}} g\left( \nabla _t\left( \nabla _x^ku_x\right) ,\Lambda \right) \mathrm{d}x \nonumber \\&+ \int _{\mathbb {T}} g(\nabla _t\Lambda ,V_k) \mathrm{d}x. \end{aligned}$$
(2.6)

To evaluate the right-hand side (denoted by RHS hereafter) of (2.6), we compute the partial differential equation satisfied by \(\nabla _x^ku_x\). Recalling that \(\nabla _xu_t=\nabla _tu_x\) and \((\nabla _x\nabla _t-\nabla _t\nabla _x)Y=R(u_x,u_t)Y\) for any \(Y\in \Gamma (u^{-1}TN)\), where \(R=R(\cdot ,\cdot )\) denotes the Riemann curvature tensor on (Ng), we have

$$\begin{aligned} \nabla _t\left( \nabla _x^ku_x\right)&= \nabla _x^{k+1}u_t + \sum _{m=0}^{k-1} \nabla _x^{k-1-m} \left\{ R(u_t,u_x)\nabla _x^mu_x \right\} =: \nabla _x^{k+1}u_t +Q . \end{aligned}$$
(2.7)

First, we use (2.1) to compute the second term of the RHS of the above equation, which becomes

$$\begin{aligned} Q&= -\varepsilon \, \sum _{m=0}^{k-1}\nabla _x^{k-1-m} \left\{ R(\nabla _x^3u_x,u_x)\nabla _x^mu_x \right\} \\&\quad +a\, \sum _{m=0}^{k-1}\nabla _x^{k-1-m} \left\{ R(J_u\nabla _x^3u_x,u_x)\nabla _x^mu_x \right\} \\&\quad +\lambda \, \sum _{m=0}^{k-1}\nabla _x^{k-1-m} \left\{ R(J_u\nabla _xu_x,u_x)\nabla _x^mu_x \right\} \\&\quad +b\, \sum _{m=0}^{k-1}\nabla _x^{k-1-m} \left\{ g(u_x,u_x)R(J_u\nabla _xu_x,u_x)\nabla _x^mu_x \right\} \\&\quad +c\, \sum _{m=0}^{k-1}\nabla _x^{k-1-m} \left\{ g(\nabla _xu_x,u_x)R(J_uu_x,u_x)\nabla _x^mu_x \right\} . \end{aligned}$$

Thus, by using the Sobolev embedding and the Gagliardo–Nirenberg inequality, we obtain

$$\begin{aligned} Q&= \varepsilon \, \mathcal {O} \left( |\nabla _x^{k+2}u_x|_g\right) +a\,Q_0 + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) , \end{aligned}$$
(2.8)

where

$$\begin{aligned} Q_0 = \sum _{m=0}^{k-1}\nabla _x^{k-1-m} \left\{ R\left( J_u\nabla _x^3u_x,u_x\right) \nabla _x^mu_x \right\} . \end{aligned}$$

Since S is the constant sectional curvature of (Ng),

$$\begin{aligned} R(Y_1,Y_2)Y_3 =S \left\{ g(Y_2,Y_3)Y_1 -g(Y_1,Y_3)Y_2 \right\} \end{aligned}$$
(2.9)

holds for any \(Y_1,Y_2,Y_3\in \Gamma (u^{-1}TN)\). Using this formula, \(Q_0\) is expressed as follows:

$$\begin{aligned} Q_0&= S\,\sum _{m=0}^{k-1} \nabla _x^{k-1-m} \left\{ g\left( \nabla _x^mu_x,u_x\right) J_u\nabla _x^3u_x -g\left( \nabla _x^mu_x, J_u\nabla _x^3u_x\right) u_x \right\} \nonumber \\&=S\,(Q_{0,1}+Q_{0,2}+Q_{0,3}), \end{aligned}$$
(2.10)

where

$$\begin{aligned} Q_{0,1}&= \nabla _x^{k-1} \left\{ g(u_x,u_x)J_u\nabla _x^3u_x -g\left( u_x,J_u\nabla _x^3u_x\right) u_x \right\} , \\ Q_{0,2}&=\nabla _x^{k-2} \left\{ g(\nabla _xu_x,u_x)J_u\nabla _x^3u_x -g\left( \nabla _xu_x,J_u\nabla _x^3u_x\right) u_x \right\} ,\\ Q_{0,3}&= \sum _{m=2}^{k-1} \nabla _x^{k-1-m} \left\{ g\left( \nabla _x^mu_x,u_x\right) J_u\nabla _x^3u_x -g\left( \nabla _x^mu_x, J_u\nabla _x^3u_x\right) u_x \right\} . \end{aligned}$$

For \(Q_{0,1}\), the product formula implies

$$\begin{aligned} Q_{0,1}&= \sum _{\mu +\nu =0}^{k-1} \frac{(k-1)!}{\mu !\nu !(k-1-\mu -\nu )!} \, g\left( \nabla _x^{\mu }u_x, \nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+2-\mu -\nu }u_x \nonumber \\&\quad - \sum _{\mu +\nu =0}^{k-1} \frac{(k-1)!}{\mu !\nu !(k-1-\mu -\nu )!} \, g\left( \nabla _x^{\mu }u_x, J_u\nabla _x^{\nu +3}u_x\right) \nabla _x^{k-1-\mu -\nu }u_x \nonumber \\&= g(u_x,u_x)J_u\nabla _x^{k+2}u_x + 2(k-1)g(\nabla _xu_x,u_x)J_u\nabla _x^{k+1}u_x \nonumber \\&\qquad \quad \qquad \qquad \quad \qquad \qquad \quad \qquad \qquad \quad \qquad -g\left( u_x,J_u\nabla _x^{k+2}u_x\right) u_x \nonumber \\&\quad -(k-1)g\left( \nabla _xu_x,J_u\nabla _x^{k+1}u_x\right) u_x -(k-1)g\left( u_x,J_u\nabla _x^{k+1}u_x\right) \nabla _xu_x \nonumber \\&\quad + \sum _{\mu +\nu =2}^{k-1} \frac{(k-1)!}{\mu !\nu !(k-1-\mu -\nu )!} \, g\left( \nabla _x^{\mu }u_x, \nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+2-\mu -\nu }u_x \nonumber \\&\quad - \sum _{\begin{array}{c} \mu +\nu =0,\\ \nu \le k-3 \end{array}}^{k-1} \frac{(k-1)!}{\mu !\nu !(k-1-\mu -\nu )!} \, g\left( \nabla _x^{\mu }u_x, J_u\nabla _x^{\nu +3}u_x\right) \nabla _x^{k-1-\mu -\nu }u_x \nonumber \\&= g(u_x,u_x)J_u\nabla _x^{k+2}u_x + 2(k-1)g(\nabla _xu_x,u_x)J_u\nabla _x^{k+1}u_x \nonumber \\&\quad -g\left( u_x,J_u\nabla _x^{k+2}u_x\right) u_x -(k-1)g\left( \nabla _xu_x,J_u\nabla _x^{k+1}u_x\right) u_x \nonumber \\&\quad -(k-1)g\left( u_x,J_u\nabla _x^{k+1}u_x\right) \nabla _xu_x + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.11)

Here it is to be emphasized that

$$\begin{aligned} g(Y, u_x)u_x+g(Y, J_uu_x)J_uu_x&=g(u_x,u_x)Y \end{aligned}$$
(2.12)

holds for any \(Y\in \Gamma (u^{-1}TN)\), since N is a two-dimensional real manifold. Using (2.12) with \(Y=J_u\nabla _x^{k+2}u_x\), we rewrite the third term of the RHS of (2.11) to have

$$\begin{aligned} -g\left( u_x,J_u\nabla _x^{k+2}u_x\right) u_x&= -g(u_x,u_x)J_u\nabla _x^{k+2}u_x +g\left( J_uu_x,J_u\nabla _x^{k+2}u_x\right) J_uu_x \nonumber \\&= -g(u_x,u_x)J_u\nabla _x^{k+2}u_x +g\left( \nabla _x^{k+2}u_x,u_x\right) J_uu_x. \end{aligned}$$
(2.13)

Substituting (2.13) into the RHS of (2.11), we obtain

$$\begin{aligned} Q_{0,1}&= 2(k-1)g(\nabla _xu_x,u_x)J_u\nabla _x^{k+1}u_x +g\left( \nabla _x^{k+2}u_x,u_x\right) J_uu_x \nonumber \\&\quad +(k-1)g\left( \nabla _x^{k+1}u_x, J_u\nabla _xu_x\right) u_x +(k-1)g\left( \nabla _x^{k+1}u_x, J_uu_x\right) \nabla _xu_x \nonumber \\&\quad + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.14)

For \(Q_{0,2}\), in the same way as that for \(Q_{0,1}\), we deduce

$$\begin{aligned} Q_{0,2}&= \sum _{\mu +\nu =0}^{k-2} \frac{(k-2)!}{\mu !\nu !(k-2-\mu -\nu )!} \, g\left( \nabla _x^{\mu +1}u_x, \nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+1-\mu -\nu }u_x \nonumber \\&\quad -\sum _{\mu +\nu =0}^{k-2} \frac{(k-2)!}{\mu !\nu !(k-2-\mu -\nu )!} \, g\left( \nabla _x^{\mu +1}u_x, J_u\nabla _x^{\nu +3}u_x\right) \nabla _x^{k-2-\mu -\nu }u_x \nonumber \\&= g(\nabla _xu_x,u_x)J_u\nabla _x^{k+1}u_x -g\left( \nabla _xu_x, J_u\nabla _x^{k+1}u_x\right) u_x \nonumber \\&\quad + \sum _{\mu +\nu =1}^{k-2} \frac{(k-2)!}{\mu !\nu !(k-2-\mu -\nu )!} \, g\left( \nabla _x^{\mu +1}u_x, \nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+1-\mu -\nu }u_x \nonumber \\&\quad -\sum _{\begin{array}{c} \mu +\nu =0,\\ \nu \le k-3 \end{array}}^{k-2} \frac{(k-2)!}{\mu !\nu !(k-2-\mu -\nu )!} \, g\left( \nabla _x^{\mu +1}u_x, J_u\nabla _x^{\nu +3}u_x\right) \nabla _x^{k-2-\mu -\nu }u_x \nonumber \\&= g(\nabla _xu_x,u_x)J_u\nabla _x^{k+1}u_x +g\left( \nabla _x^{k+1}u_x, J_u\nabla _xu_x\right) u_x + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.15)

For \(Q_{0,3}\), the Sobolev embedding and the Gagliardo–Nirenberg inequality imply

$$\begin{aligned} Q_{0,3}&=\mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.16)

Combining (2.8), (2.10), (2.14), (2.15), and (2.16), we obtain

$$\begin{aligned} Q&= \varepsilon \, \mathcal {O} \left( \big |\nabla _x^{k+2}u_x\big |_g \right) +aS\, g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x \nonumber \\&\quad +aS(2k-1)\,g\left( \nabla _xu_x,u_x\right) J_u\nabla _x\left( \nabla _x^ku_x\right) +aSk\,g\left( \nabla _x\left( \nabla _x^ku_x\right) ,J_u\nabla _xu_x\right) u_x \nonumber \\&\quad + aS(k-1)\, g\left( \nabla _x\left( \nabla _x^ku_x\right) , J_uu_x\right) \nabla _xu_x + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.17)

Second, we use (2.1) to compute the first term of the RHS of (2.7). A simple computation shows

$$\begin{aligned} \nabla _x^{k+1}u_t&= -\varepsilon \nabla _x^4\left( \nabla _x^ku_x\right) +a\,J_u\nabla _x^4\left( \nabla _x^ku_x\right) +\lambda \,J_u\nabla _x^2\left( \nabla _x^ku_x\right) +b\,Q_{1,1}+c\,Q_{1,2}, \end{aligned}$$
(2.18)

where

$$\begin{aligned} Q_{1,1}&= \nabla _x^{k+1}\left\{ g(u_x,u_x)J_u\nabla _xu_x \right\} \nonumber \\&= \sum _{\mu +\nu =0}^{k+1} \frac{(k+1)!}{\mu !\nu !(k+1-\mu -\nu )!} \, g\left( \nabla _x^{\mu }u_x,\nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+2-\mu -\nu }u_x \nonumber \\&= g(u_x,u_x)J_u\nabla _x^{k+2}u_x +2(k+1)g(\nabla _xu_x,u_x)J_u\nabla _x^{k+1}u_x \nonumber \\&\quad +2g\left( \nabla _x^{k+1}u_x,u_x\right) J_u\nabla _xu_x \nonumber \\&\quad +\sum _{\begin{array}{c} \mu +\nu =2, \\ \mu ,\nu \le k \end{array}}^{k+1} \frac{(k+1)!}{\mu !\nu !(k+1-\mu -\nu )!} \, g\left( \nabla _x^{\mu }u_x,\nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+2-\mu -\nu }u_x \nonumber \\&= \nabla _x\left\{ g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \right\} +2k\, g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \nonumber \\&\quad +2\,g\left( \nabla _x\left( \nabla _x^ku_x\right) , u_x\right) J_u\nabla _xu_x + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) , \end{aligned}$$
(2.19)

and

$$\begin{aligned} Q_{1,2}&= \nabla _x^{k+1} \left\{ g(\nabla _xu_x,u_x)J_uu_x \right\} \nonumber \\&= \sum _{\mu +\nu =0}^{k+1} \frac{(k+1)!}{\mu !\nu !(k+1-\mu -\nu )!} \,g\left( \nabla _x^{\mu +1}u_x, \nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+1-\mu -\nu }u_x \nonumber \\&= g(\nabla _xu_x,u_x)J_u\nabla _x^{k+1}u_x +g\left( \nabla _x^{k+2}u_x,u_x\right) J_uu_x \nonumber \\&\quad +(k+1)g\left( \nabla _x^{k+1}u_x,\nabla _xu_x\right) J_uu_x + g\left( \nabla _xu_x,\nabla _x^{k+1}u_x\right) J_uu_x \nonumber \\&\quad + (k+1)g\left( \nabla _x^{k+1}u_x,u_x\right) J_u\nabla _xu_x \nonumber \\&\quad + \sum _{\begin{array}{c} \mu +\nu =1, \\ \mu \le k-1, \\ \nu \le k \end{array}}^{k+1} \frac{(k+1)!}{\mu !\nu !(k+1-\mu -\nu )!} \,g\left( \nabla _x^{\mu +1}u_x, \nabla _x^{\nu }u_x\right) J_u\nabla _x^{k+1-\mu -\nu }u_x \nonumber \\&= g\left( \nabla _x^2\left( \nabla _x^ku_x\right) ,u_x\right) J_uu_x +g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \nonumber \\&\quad +(k+2)g\left( \nabla _x\left( \nabla _x^ku_x\right) ,\nabla _xu_x\right) J_uu_x +(k+1)g\left( \nabla _x\left( \nabla _x^ku_x\right) ,u_x\right) J_u\nabla _xu_x \nonumber \\&\quad + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.20)

By combining (2.17) and (2.18) with (2.19) and (2.20), we have

$$\begin{aligned} \nabla _t\left( \nabla _x^ku_x\right)&= -\varepsilon \nabla _x^4\left( \nabla _x^ku_x\right) + \varepsilon \, \mathcal {O} \left( \big |\nabla _x^{k+2}u_x\big |_g \right) \nonumber \\&\quad +a\,J_u\nabla _x^4\left( \nabla _x^ku_x\right) +\lambda \,J_u\nabla _x^2\left( \nabla _x^ku_x\right) +b\,\nabla _x \left\{ g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \right\} \nonumber \\&\quad +(aS+c)\,g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x \nonumber \\&\quad +\left\{ aS(2k-1)+2kb+c \right\} \,g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \nonumber \\&\quad + \left\{ 2b+(k+1)c \right\} \, g\left( \nabla _x\left( \nabla _x^ku_x\right) ,u_x\right) J_u\nabla _xu_x \nonumber \\&\quad + (k+2)c\, g\left( \nabla _x\left( \nabla _x^ku_x\right) ,\nabla _xu_x\right) J_uu_x \nonumber \\&\quad +aSk\,g\left( \nabla _x\left( \nabla _x^ku_x\right) ,J_u\nabla _xu_x\right) u_x \nonumber \\&\quad +aS(k-1)\,g\left( \nabla _x\left( \nabla _x^ku_x\right) , J_uu_x\right) \nabla _xu_x \nonumber \\&\quad +\mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.21)

Furthermore, we modify the expression of some terms including \(\nabla _x(\nabla _x^ku_x)\) to detect their essential structure. Let \(Y\in \Gamma (u^{-1}TN)\) be fixed. We first use (2.12) to obtain

$$\begin{aligned} g(u_x,u_x)J_uY&= g(J_uY,u_x)u_x+g(J_uY,J_uu_x)J_uu_x \nonumber \\&= g(Y,u_x)J_uu_x-g(Y,J_uu_x)u_x. \nonumber \end{aligned}$$

Taking the covariant derivative of both sides of the above with respect to x, we have

$$\begin{aligned} 2\,g(\nabla _xu_x,u_x)J_uY&= g(Y,\nabla _xu_x)J_uu_x + g(Y,u_x)J_u\nabla _xu_x \nonumber \\&\quad -g(Y,J_u\nabla _xu_x)u_x -g(Y,J_uu_x)\nabla _xu_x. \end{aligned}$$
(2.22)

We next introduce the following expression:

$$\begin{aligned} A_1Y&= g(Y,\nabla _xu_x)J_uu_x + g(Y,u_x)J_u\nabla _xu_x \nonumber \\&\quad +g(Y,J_u\nabla _xu_x)u_x +g(Y,J_uu_x)\nabla _xu_x, \nonumber \\ A_2Y&= g(Y,J_uu_x)\nabla _xu_x -g(Y,J_u\nabla _xu_x)u_x. \nonumber \end{aligned}$$

We find \({}^tA_1=A_1\) and \({}^tA_2=A_2\) in \(T_uN\). More precisely we can show the following.

Proposition 2.3

Let \(Y_1,Y_2\in \Gamma (u^{-1}TN)\). Then

$$\begin{aligned} g(A_iY_1,Y_2)&=g(Y_1,A_iY_2) \end{aligned}$$
(2.23)

holds for each \((t,x)\in [0,T_{\varepsilon }^{*}]\times \mathbb {T}\) with \(i=1,2\).

Proof of Proposition 2.3

If \(i=1\), then (2.23) immediately follows from the definition of \(A_1\). If \(i=2\), (2.23) follows from

$$\begin{aligned} \left\{ g(u_x,u_x)\right\} ^2 \left\{ g(A_2Y_1,Y_2)-g(Y_1,A_2Y_2) \right\} =0, \end{aligned}$$
(2.24)

since both sides of (2.23) vanish at the point (tx) with \(u_x(t,x)=0\). Indeed, we can show (2.24) by the following computations. We first write

$$\begin{aligned} g(u_x,u_x)A_2Y_1&= g(u_x,u_x) \left\{ g(Y_1,J_uu_x)\nabla _xu_x -g(Y_1,J_u\nabla _xu_x)u_x \right\} \nonumber \\&= g(g(u_x,u_x)Y_1,J_uu_x)\nabla _xu_x -g(g(u_x,u_x)Y_1,J_u\nabla _xu_x)u_x, \nonumber \end{aligned}$$

and we use (2.12) with \(Y=Y_1\) to obtain

$$\begin{aligned} g(u_x,u_x)A_2Y_1&=g(g(Y_1,u_x)u_x+g(Y_1,J_uu_x)J_uu_x,J_uu_x)\nabla _xu_x \nonumber \\&\quad -g(g(Y_1,u_x)u_x+g(Y_1,J_uu_x)J_uu_x,J_u\nabla _xu_x)u_x\nonumber \\&= g(u_x,u_x)g(Y_1,J_uu_x)\nabla _xu_x-g(u_x,J_u\nabla _xu_x)g(Y_1,u_x)u_x \nonumber \\&\quad -g(u_x,\nabla _xu_x)g(Y_1,J_uu_x)u_x. \nonumber \end{aligned}$$

This implies

$$\begin{aligned} \left\{ g(u_x,u_x)\right\} ^2g(A_2Y_1,Y_2)&= g(g(u_x,u_x)A_2Y_1, g(u_x,u_x)Y_2) \nonumber \\&= g(u_x,u_x)g(Y_1,J_uu_x)g(\nabla _xu_x,g(u_x,u_x)Y_2) \nonumber \\&\quad -g(u_x,J_u\nabla _xu_x)g(Y_1,u_x)g(u_x,g(u_x,u_x)Y_2) \nonumber \\&\quad -g(u_x,\nabla _xu_x)g(Y_1,J_uu_x)g(u_x,g(u_x,u_x)Y_2). \end{aligned}$$
(2.25)

Using (2.12) again with \(Y=Y_2\), we obtain

$$\begin{aligned} g(u_x,g(u_x,u_x)Y_2)&= g(u_x,u_x)g(Y_2,u_x), \nonumber \\ g(\nabla _xu_x,g(u_x,u_x)Y_2)&= g(\nabla _xu_x,u_x)g(Y_2,u_x) + g(\nabla _xu_x,J_uu_x)g(Y_2,J_uu_x). \nonumber \end{aligned}$$

Substituting these into (2.25), we have

$$\begin{aligned}&\left\{ g(u_x,u_x)\right\} ^2 g(A_2Y_1,Y_2) \nonumber \\&\quad = g(u_x,u_x)g(\nabla _xu_x,J_uu_x) \left\{ g(Y_1,J_uu_x)g(Y_2,J_uu_x) + g(Y_1,u_x)g(Y_2,u_x) \right\} . \nonumber \end{aligned}$$

As the form of the RHS is symmetric with respect to \(Y_1\) and \(Y_2\), we immediately conclude that the desired property (2.24) holds.

Using (2.22) and the definitions of \(A_1\) and \(A_2\), we have

$$\begin{aligned}&g(Y,J_uu_x)\nabla _xu_x \nonumber \\&\quad = \frac{1}{4} \biggl \{ g(Y,\nabla _xu_x)J_uu_x + g(Y,u_x)J_u\nabla _xu_x +g(Y,J_u\nabla _xu_x)u_x +g(Y,J_uu_x)\nabla _xu_x \biggr \} \nonumber \\&\qquad -\frac{1}{4} \biggl \{ g(Y,\nabla _xu_x)J_uu_x + g(Y,u_x)J_u\nabla _xu_x -g(Y,J_u\nabla _xu_x)u_x -g(Y,J_uu_x)\nabla _xu_x \biggr \} \nonumber \\&\qquad +\frac{1}{2} \biggl \{ g(Y,J_uu_x)\nabla _xu_x -g(Y,J_u\nabla _xu_x)u_x \biggr \} \nonumber \\&\quad = -\frac{1}{2}\, g(\nabla _xu_x,u_x)J_uY +\frac{1}{4}A_1Y +\frac{1}{2}A_2Y. \end{aligned}$$
(2.26)

In the same way, we have

$$\begin{aligned} g(Y,J_u\nabla _xu_x)u_x&= -\frac{1}{2}\, g(\nabla _xu_x,u_x)J_uY +\frac{1}{4}A_1Y -\frac{1}{2}A_2Y. \end{aligned}$$
(2.27)

Using \({}^{t}J_u=-J_u\) in \(T_uN\), (2.23), and (2.27), we deduce

$$\begin{aligned} g(Y,u_x)J_u\nabla _xu_x&= {}^{t}\left( g(\cdot ,J_u\nabla _xu_x)u_x \right) Y \nonumber \\&= -\frac{1}{2}\, g(\nabla _xu_x,u_x)\,{}^{t}J_uY +\frac{1}{4}\,{}^tA_1Y -\frac{1}{2}\,{}^tA_2Y \nonumber \\&= \frac{1}{2}\, g(\nabla _xu_x,u_x)J_uY +\frac{1}{4}A_1Y -\frac{1}{2}A_2Y, \end{aligned}$$
(2.28)

and

$$\begin{aligned} g(Y,\nabla _xu_x)J_uu_x&= {}^{t}\left( g(\cdot ,J_uu_x)\nabla _xu_x \right) Y \nonumber \\&= \frac{1}{2}\, g(\nabla _xu_x,u_x)J_uY +\frac{1}{4}A_1Y +\frac{1}{2}A_2Y. \end{aligned}$$
(2.29)

Applying (2.26), (2.27), (2.28), and (2.29) to the RHS of (2.21), we derive

$$\begin{aligned} \nabla _t\left( \nabla _x^ku_x\right)&= -\varepsilon \nabla _x^4\left( \nabla _x^ku_x\right) + \varepsilon \, \mathcal {O} \left( |\nabla _x^{k+2}u_x|_g \right) \nonumber \\&\quad +a\,J_u\nabla _x^4\left( \nabla _x^ku_x\right) +\lambda \,J_u\nabla _x^2\left( \nabla _x^ku_x\right) +b\,\nabla _x \left\{ g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \right\} \nonumber \\&\quad +c_1\,g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x +c_2 \,g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \nonumber \\&\quad +c_3\,A_1\nabla _x\left( \nabla _x^ku_x\right) +c_4\,A_2\nabla _x\left( \nabla _x^ku_x\right) \nonumber \\&\quad +\mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) , \end{aligned}$$
(2.30)

where \(c_1,\ldots ,c_4\) are constants given by abc, and S. More concretely,

$$\begin{aligned} c_1&=aS+c, \end{aligned}$$
(2.31)
$$\begin{aligned} c_2&=\left\{ aS(2k-1)+2kb+c \right\} + \frac{1}{2} \left\{ 2b+(k+1)c +(k+2)c-aSk-aS(k-1) \right\} \nonumber \\&= \left( k-\frac{1}{2}\right) aS +(2k+1)b +\left( k+\frac{5}{2}\right) c. \end{aligned}$$
(2.32)

We omit to describe the explicit form of \(c_3\) and \(c_4\), as they will not be used later.

We are now in a position to evaluate the first term of the RHS of (2.6). Using (2.30), we have

$$\begin{aligned}&\int _{\mathbb {T}} g\left( \nabla _t\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\quad = -\varepsilon \int _{\mathbb {T}} g\left( \nabla _x^4\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x + \varepsilon \, \int _{\mathbb {T}} g\left( \mathcal {O} \left( |\nabla _x^{k+2}u_x|_g \right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\qquad +a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^4\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x +\lambda \, \int _{\mathbb {T}} g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\qquad +b\, \int _{\mathbb {T}} g\left( \nabla _x \left\{ g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \right\} , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\qquad +c_1\, \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x, \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\qquad +c_2 \,\int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\qquad +c_3\,\int _{\mathbb {T}} g\left( A_1\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x +c_4\,\int _{\mathbb {T}} g\left( A_2\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\qquad +\int _{\mathbb {T}} g\left( \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) , \nabla _x^ku_x\right) \mathrm{d}x. \nonumber \end{aligned}$$

We compute each term of the above separately. Integrating by parts, we obtain

$$\begin{aligned}&a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^4\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x = a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , \nabla _x^2\left( \nabla _x^ku_x\right) \right) \mathrm{d}x =0, \nonumber \\&\lambda \, \int _{\mathbb {T}} g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x = -\lambda \, \int _{\mathbb {T}} g\left( J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x\left( \nabla _x^ku_x\right) \right) \mathrm{d}x =0, \nonumber \\&b\, \int _{\mathbb {T}} g\left( \nabla _x \left\{ g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \right\} , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\quad = -b\, \int _{\mathbb {T}} g\left( g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x\left( \nabla _x^ku_x\right) \right) \mathrm{d}x =0. \nonumber \end{aligned}$$

Using the Cauchy–Schwartz inequality, we have

$$\begin{aligned} \int _{\mathbb {T}} g(\mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) , \nabla _x^ku_x)\mathrm{d}x \le C\Vert u_x\Vert _{H^k}\big \Vert \nabla _x^ku_x\big \Vert _{L^2} \le C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.33)

Using the integration by parts, the Young inequality \(AB\le A^2/2+B^2/2\) for any \(A,B\geqslant 0\), and \(\varepsilon \le 1\), we deduce

$$\begin{aligned}&-\varepsilon \int _{\mathbb {T}} g\left( \nabla _x^4\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x + \varepsilon \, \int _{\mathbb {T}} g\left( \mathcal {O} \left( |\nabla _x^{k+2}u_x|_g \right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\quad \le -\varepsilon \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) )\big \Vert _{L^2}^2 + \varepsilon \,C \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2} \big \Vert \nabla _x^ku_x\big \Vert _{L^2} \nonumber \\&\quad \left. \le -\varepsilon \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \right) \big \Vert _{L^2}^2 + \frac{\varepsilon }{2} \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 + \frac{\varepsilon \,C^2}{2} \big \Vert \nabla _x^ku_x\big \Vert _{L^2}^2 \nonumber \\&\quad \le - \frac{\varepsilon }{2} \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 + \frac{C^2}{2} \Vert u_x\Vert _{H^k}^2. \nonumber \end{aligned}$$

Integrating by parts and using (2.23), we have

$$\begin{aligned}&c_3\,\int _{\mathbb {T}} g\left( A_1\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x +c_4\,\int _{\mathbb {T}} g\left( A_2\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\quad = -\frac{c_3}{2} g\left( \nabla _x(A_1)\nabla _x^ku_x, \nabla _x^ku_x\right) \mathrm{d}x -\frac{c_4}{2} g\left( \nabla _x(A_2)\nabla _x^ku_x, \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\quad \le C\Vert u_x\Vert _{H^k}^2. \nonumber \end{aligned}$$

Combining them and noting that \(\Vert u_x\Vert _{H^k}\le CN_k(u)\) follows from (2.5), we derive

$$\begin{aligned}&\int _{\mathbb {T}} g\left( \nabla _t\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\quad \le -\frac{\varepsilon }{2}\ \big |\nabla _x^2\left( \nabla _x^ku_x\right) \ \big |_{L^2}^2 +c_1\, \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x, \nabla _x^ku_x\right) \mathrm{d}x \nonumber \\&\qquad +c_2 \,\int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \mathrm{d}x +C\,(N_k(u))^2. \end{aligned}$$
(2.34)

We next evaluate the second term of the RHS of (2.6). In the computation, it is to be noted that \(\Lambda = \mathcal {O}(|\nabla _x^{k-2}u_x|_g)\) and

$$\begin{aligned} \nabla _t\left( \nabla _x^ku_x\right) = -\varepsilon \,\nabla _x^4\left( \nabla _x^ku_x\right) + a\,J_u\nabla _x^4\left( \nabla _x^ku_x\right) + \mathcal {O}\left( \sum _{m=0}^{k+2} |\nabla _x^mu_x|_g \right) . \end{aligned}$$

By noting them and integrating by parts, we obtain

$$\begin{aligned}&\int _{\mathbb {T}} g\left( \nabla _t\left( \nabla _x^ku_x\right) , \Lambda \right) \mathrm{d}x \nonumber \\&\quad \le -\varepsilon \, \int _{\mathbb {T}} g\left( \nabla _x^4\left( \nabla _x^ku_x\right) , \Lambda \right) \mathrm{d}x + a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^4\left( \nabla _x^ku_x\right) , \Lambda \right) \mathrm{d}x + C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.35)

For the first term of the RHS of (2.35), by using \(\varepsilon \le 1\), the integration by parts, the Young inequality \(AB\le A^2/8+2B^2\) for any \(A,B\geqslant 0\), and \(\Lambda =\mathcal {O}(|\nabla _x^{k-2}u_x|_g)\), we have

$$\begin{aligned} -\varepsilon \, \int _{\mathbb {T}} g\left( \nabla _x^4\left( \nabla _x^ku_x\right) , \Lambda \right) \mathrm{d}x&= -\varepsilon \, \int _{\mathbb {T}} g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , \nabla _x^2(\Lambda )\right) \mathrm{d}x \nonumber \\&\le \varepsilon \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}\big \Vert \nabla _x^2(\Lambda ))\big \Vert _{L^2} \nonumber \\&\le \frac{\varepsilon }{8} \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 + 2\varepsilon \big \Vert \nabla _x^2(\Lambda ))\big \Vert _{L^2}^2 \nonumber \\&\le \frac{\varepsilon }{8} \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 +C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.36)

For the second term of the RHS of (2.35), we compute \(\nabla _x^2\Lambda \) as follows:

$$\begin{aligned} \nabla _x^2\Lambda&= -\frac{d_1}{2a}\nabla _x^2 \left\{ g\left( \nabla _x^{k-2}u_x,J_uu_x\right) J_uu_x \right\} + \frac{d_2}{8a} \nabla _x^2\left\{ g(u_x,u_x)\nabla _x^{k-2}u_x \right\} \nonumber \\&= -\frac{d_1}{2a} g\left( \nabla _x^ku_x,J_uu_x\right) J_uu_x + \frac{d_2}{8a} g(u_x,u_x)\nabla _x^ku_x \nonumber \\&\quad -\frac{d_1}{a} g\left( \nabla _x^{k-1}u_x,J_u\nabla _xu_x\right) J_uu_x -\frac{d_1}{a} g\left( \nabla _x^{k-1}u_x,J_uu_x\right) J_u\nabla _xu_x \nonumber \\&\quad +\frac{d_2}{2a} g(\nabla _xu_x,u_x)\nabla _x^{k-1}u_x + \mathcal {O} \left( \sum _{m=0}^{k-2} |\nabla _x^mu_x|_g \right) . \nonumber \end{aligned}$$

Thus, integrating by parts and substituting the above, we obtain

$$\begin{aligned}&a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^4\left( \nabla _x^ku_x\right) , \Lambda \right) \mathrm{d}x \nonumber \\&\quad = a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , \nabla _x^2\Lambda \right) \mathrm{d}x \nonumber \\&\quad = -\frac{d_1}{2}Q_{2,1} +\frac{d_2}{8}Q_{2,2} -d_1Q_{2,3}-d_1Q_{2,4} +\frac{d_2}{2}Q_{2,5} +Q_{2,6}, \end{aligned}$$
(2.37)

where

$$\begin{aligned} Q_{2,1}&= \int _{\mathbb {T}} g\left( \nabla _x^ku_x,J_uu_x\right) g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) ,J_uu_x\right) \,\mathrm{d}x, \nonumber \\ Q_{2,2}&= \int _{\mathbb {T}} g(u_x,u_x) g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \,\mathrm{d}x, \nonumber \\ Q_{2,3}&= \int _{\mathbb {T}} g\left( \nabla _x^{k-1}u_x,J_u\nabla _xu_x\right) g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) ,J_uu_x\right) \,\mathrm{d}x, \nonumber \\ Q_{2,4}&= \int _{\mathbb {T}} g\left( \nabla _x^{k-1}u_x,J_uu_x\right) g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) ,J_u\nabla _xu_x\right) \,\mathrm{d}x, \nonumber \\ Q_{2,5}&= \int _{\mathbb {T}} g(\nabla _xu_x,u_x) g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , \nabla _x^{k-1}u_x\right) \,\mathrm{d}x, \nonumber \\ Q_{2,6}&= \int _{\mathbb {T}} g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , \mathcal {O} \left( \sum _{m=0}^{k-2} |\nabla _x^mu_x|_g \right) \right) \,\mathrm{d}x. \nonumber \end{aligned}$$

We compute \(Q_{2,1},\ldots ,Q_{2,6}\) separately. Using integration by parts and the property of Hermitian metric g, we deduce

$$\begin{aligned} Q_{2,1}&= \int _{\mathbb {T}} g\left( \nabla _x^ku_x,J_uu_x\right) g\left( \nabla _x^2\left( \nabla _x^ku_x\right) ,u_x\right) \,\mathrm{d}x \nonumber \\&= \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x, \nabla _x^ku_x\right) \,\mathrm{d}x, \nonumber \\ Q_{2,2}&= \int _{\mathbb {T}} g\left( \nabla _x\left\{ g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \right\} , \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&\quad -2\,\int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&= -2\,\int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \,\mathrm{d}x, \nonumber \\ Q_{2,3}&= \int _{\mathbb {T}} g\left( \nabla _x^{k-1}u_x,J_u\nabla _xu_x\right) g\left( \nabla _x^2\left( \nabla _x^ku_x\right) ,u_x\right) \,\mathrm{d}x \nonumber \\&\le -\int _{\mathbb {T}} g\left( \nabla _x^{k}u_x,J_u\nabla _xu_x\right) g\left( \nabla _x\left( \nabla _x^ku_x\right) ,u_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2 \nonumber \\&= -\int _{\mathbb {T}} g\left( g\left( \nabla _x\left( \nabla _x^ku_x\right) , u_x\right) J_u\nabla _xu_x, \nabla _x^ku_x \right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2, \nonumber \\ Q_{2,4}&= \int _{\mathbb {T}} g\left( \nabla _x^{k-1}u_x,J_uu_x\right) g\left( \nabla _x^2\left( \nabla _x^ku_x\right) ,\nabla _xu_x\right) \,\mathrm{d}x \nonumber \\&\le -\int _{\mathbb {T}} g\left( \nabla _x^{k}u_x,J_uu_x\right) g\left( \nabla _x\left( \nabla _x^ku_x\right) ,\nabla _xu_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2 \nonumber \\&= -\int _{\mathbb {T}} g\left( g\left( \nabla _x\left( \nabla _x^ku_x\right) , \nabla _xu_x\right) J_uu_x, \nabla _x^ku_x \right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2, \nonumber \\ Q_{2,5}&\le -\int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x) J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x \right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2, \nonumber \\ Q_{2,6}&\le C\Vert u_x\Vert _{H^k}^2. \nonumber \end{aligned}$$

Applying them to (2.37), we obtain

$$\begin{aligned} a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^4\left( \nabla _x^ku_x\right) , \Lambda \right) \mathrm{d}x&\le -\frac{d_1}{2} \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x, \nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\quad -\frac{3d_2}{4} \int _{\mathbb {T}} g\left( g\left( \nabla _xu_x,u_x\right) J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&\quad +d_1\, \int _{\mathbb {T}} g\left( g\left( \nabla _x\left( \nabla _x^ku_x\right) , u_x\right) J_u\nabla _xu_x, \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&\quad +d_1\, \int _{\mathbb {T}} g\left( g\left( \nabla _x\left( \nabla _x^ku_x\right) , \nabla _xu_x\right) J_uu_x, \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&\quad +C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.38)

Here we rewrite the sum of the third and fourth terms of the RHS using (2.28) and (2.29), and use the integration by parts and (2.23) to find

$$\begin{aligned}&d_1\, \int _{\mathbb {T}} g\left( g\left( \nabla _x\left( \nabla _x^ku_x\right) , u_x\right) J_u\nabla _xu_x, \nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\quad +d_1\, \int _{\mathbb {T}} g\left( g\left( \nabla _x\left( \nabla _x^ku_x\right) , \nabla _xu_x\right) J_uu_x, \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&= d_1\,\int _{\mathbb {T}} g\left( g\left( \nabla _xu_x,u_x\right) J_u\nabla _x\left( \nabla _x^ku_x\right) ,\nabla _x^ku_x\right) \,\mathrm{d}x\nonumber \\&\quad + \frac{d_1}{2} \int _{\mathbb {T}} g\left( A_1\nabla _x\left( \nabla _x^ku_x\right) ,\nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\le d_1\,\int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) ,\nabla _x^ku_x\right) \,\mathrm{d}x + C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.39)

Combining (2.38) and (2.39), we have

$$\begin{aligned} a\, \int _{\mathbb {T}} g\left( J_u\nabla _x^4\left( \nabla _x^ku_x\right) , \Lambda \right) \mathrm{d}x&\le -\frac{d_1}{2} \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x, \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&\quad +\left( d_1-\frac{3d_2}{4}\right) \nonumber \\&\quad \times \int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x) J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\quad +C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.40)

Therefore, from (2.35), (2.36), and (2.40), it follows that

$$\begin{aligned}&\int _{\mathbb {T}} g\left( \nabla _t\left( \nabla _x^ku_x\right) , \Lambda \right) \,\mathrm{d}x \nonumber \\&\quad \le \frac{\varepsilon }{8} \big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 -\frac{d_1}{2} \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x, \nabla _x^ku_x \right) \,\mathrm{d}x \nonumber \\&\qquad +\left( d_1-\frac{3d_2}{4}\right) \int _{\mathbb {T}} g\left( g(\nabla _xu_x,u_x) J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x \right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.41)

We next evaluate the third term of the RHS of (2.6). For this purpose, we compute \(\nabla _t\Lambda \). Using the product formula and noting \(\nabla _tu_x=\nabla _xu_t =\mathcal {O} \left( \displaystyle \sum _{m=0}^4|\nabla _x^mu_x|_g \right) , \) we have

$$\begin{aligned} \nabla _t\Lambda&= -\frac{d_1}{2a} g\left( \nabla _t\nabla _x^{k-2}u_x,J_uu_x\right) J_uu_x -\frac{d_1}{2a} g\left( \nabla _x^{k-2}u_x,J_u\nabla _tu_x\right) J_uu_x \nonumber \\&\quad -\frac{d_1}{2a} g\left( \nabla _x^{k-2}u_x,J_uu_x\right) J_u\nabla _tu_x + \frac{d_2}{8a}g(u_x,u_x)\nabla _t\nabla _x^{k-2}u_x \nonumber \\&\quad +\frac{d_2}{4a}g\big (\nabla _xu_t,u_x\big )\nabla _x^{k-2}u_x \nonumber \\&= -\frac{d_1}{2a} g\left( \nabla _t\nabla _x^{k-2}u_x,J_uu_x\right) J_uu_x + \frac{d_2}{8a}g(u_x,u_x)\nabla _t\nabla _x^{k-2}u_x \nonumber \\&\quad + \mathcal {O} \left( |\nabla _x^{k-2}u_x|_g \sum _{m=0}^4|\nabla _x^mu_x|_g \right) . \nonumber \end{aligned}$$

Thus, we have

$$\begin{aligned} \int _{\mathbb {T}} g(\nabla _t\Lambda ,V_k)\,\mathrm{d}x&= Q_{3,1}+Q_{3,2}+Q_{3,2}, \nonumber \end{aligned}$$

where

$$\begin{aligned} Q_{3,1}&= -\frac{d_1}{2a} \int _{\mathbb {T}} g\left( g\left( \nabla _t\nabla _x^{k-2}u_x, J_uu_x\right) J_uu_x,V_k\right) \,\mathrm{d}x, \nonumber \\ Q_{3,2}&= \frac{d_2}{8a} \int _{\mathbb {T}} g\left( g(u_x,u_x)\nabla _t\nabla _x^{k-2}u_x,V_k\right) \,\mathrm{d}x, \nonumber \\ Q_{3,3}&= \int _{\mathbb {T}} g\left( \mathcal {O} \left( |\nabla _x^{k-2}u_x|_g \sum _{m=0}^4|\nabla _x^mu_x|_g \right) ,V_k\right) \,\mathrm{d}x. \nonumber \end{aligned}$$

For \(Q_{3,3}\), since \(k\geqslant 4\), we use the Sobolev embedding and the Cauchy–Schwartz inequality to obtain

$$\begin{aligned} Q_{3,3}&\le C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.42)

To obtain \(Q_{3,1}\) and \(Q_{3,2}\), we need to compute \(\nabla _t\nabla _x^{k-2}u_x\). Indeed, by the same computation as that used to obtain \(\nabla _t(\nabla _x^{k}u_x)\), we find

$$\begin{aligned} \nabla _t\nabla _x^{k-2}u_x&= -\varepsilon \nabla _x^4\left( \nabla _x^{k-2}u_x\right) +a\,J_u\nabla _x^4\left( \nabla _x^{k-2}u_x\right) + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) \nonumber \\&= \varepsilon \, \mathcal {O}\left( |\nabla _x^{k+2}u_x|_g\right) +a\,J_u\nabla _x^2\left( \nabla _x^{k}u_x\right) + \mathcal {O} \left( \sum _{m=0}^k |\nabla _x^mu_x|_g \right) . \end{aligned}$$
(2.43)

Applying (2.43), we deduce

$$\begin{aligned} Q_{3,1}&= -\frac{d_1}{2a} \int _{\mathbb {T}} g\left( g\left( \nabla _t\nabla _x^{k-2}u_x, J_uu_x\right) J_uu_x,\nabla _x^ku_x+\Lambda \right) \,\mathrm{d}x \nonumber \\&\le -\frac{d_1}{2a} \int _{\mathbb {T}} g\left( g\left( \nabla _t\nabla _x^{k-2}u_x, J_uu_x\right) J_uu_x,\nabla _x^ku_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2 \nonumber \\&\le \varepsilon \, \int _{\mathbb {T}} g\left( \mathcal {O}\left( |\nabla _x^{k+2}u_x|_g\right) ,\nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\quad -\frac{d_1}{2} \int _{\mathbb {T}} g\left( g\left( J_u\nabla _x^2\left( \nabla _x^ku_x\right) , J_uu_x\right) J_uu_x,\nabla _x^ku_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2 \nonumber \\&\le \frac{\varepsilon }{8}\big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 -\frac{d_1}{2} \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x,\nabla _x^ku_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.44)

In the same way, applying (2.43), we deduce

$$\begin{aligned} Q_{3,2}&= \frac{d_2}{8a} \int _{\mathbb {T}} g\left( g(u_x,u_x)\nabla _t\nabla _x^{k-2}u_x,\nabla _x^ku_x+\Lambda \right) \,\mathrm{d}x \nonumber \\&\le \frac{d_2}{8a} \int _{\mathbb {T}} g\left( g(u_x,u_x)\nabla _t\nabla _x^{k-2}u_x,\nabla _x^ku_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2 \nonumber \\&\le \varepsilon \, \int _{\mathbb {T}} g\left( \mathcal {O}(|\nabla _x^{k+2}u_x|_g),\nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\quad + \frac{d_2}{8} \int _{\mathbb {T}} g\left( g(u_x,u_x)J_u\nabla _x^2\left( \nabla _x^ku_x\right) ,\nabla _x^ku_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2 \nonumber \\&\le \frac{\varepsilon }{8}\big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 + \frac{d_2}{8} \int _{\mathbb {T}} g\left( \nabla _x\left\{ g(u_x,u_x)J_u\nabla _x\left( \nabla _x^ku_x\right) \right\} , \nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\quad -\frac{d_2}{4} \int _{\mathbb {T}} g\left( g\left( \nabla _xu_x,u_x\right) J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2 \nonumber \\&= \frac{\varepsilon }{8}\big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 -\frac{d_2}{4} \int _{\mathbb {T}} g\left( g\left( \nabla _xu_x,u_x\right) J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\quad +C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.45)

Combining (2.42), (2.44), and (2.45), we obtain

$$\begin{aligned}&\int _{\mathbb {T}} g\big (\nabla _t\Lambda , V_k\big )\,\mathrm{d}x \nonumber \\&\quad \le \frac{\varepsilon }{4}\big \Vert \nabla _x^2\left( \nabla _x^ku_x\right) \big \Vert _{L^2}^2 -\frac{d_1}{2} \int _{\mathbb {T}} g\left( g\left( \nabla _x^2\left( \nabla _x^ku_x\right) , u_x\right) J_uu_x,\nabla _x^ku_x\right) \,\mathrm{d}x \nonumber \\&\qquad -\frac{d_2}{4} \int _{\mathbb {T}} g\left( g\left( \nabla _xu_x,u_x\right) J_u\nabla _x\left( \nabla _x^ku_x\right) , \nabla _x^ku_x\right) \,\mathrm{d}x +C\Vert u_x\Vert _{H^k}^2. \end{aligned}$$
(2.46)

Consequently, combining the information (2.6), (2.34), (2.41), and (2.46), we derive

where \(c_1\) and \(c_2\) are given by (2.31) and (2.32). To cancel the second and third terms of the RHS of the above equation, we set \(d_1\) and \(d_2\) as follows:

$$\begin{aligned} d_1&=c_1=aS+c, \nonumber \\ d_2&=c_2+d_1 = \left( k+\frac{1}{2}\right) aS +(2k+1)b +\left( k+\frac{7}{2}\right) c. \nonumber \end{aligned}$$

Therefore, using \(\Vert u_x\Vert _{H^k}\le CN_k(u)\), we conclude that

(2.47)

holds on \([0,T_{\varepsilon }^{\star }]\)

Let us now go back to the original purpose to derive the uniform estimate for \(\left\{ N_k(u^{\varepsilon })\right\} _{\varepsilon \in (0,1]}\). To achieve this, it remains to consider the energy estimate for \(\big \Vert u_x^{\varepsilon }\big \Vert _{H^{k-1}}^2\). However, using the integration by parts, the Sobolev embedding, and the Cauchy–Schwartz inequality repeatedly, we can easily show that

(2.48)

Therefore, from (2.47) and (2.48), we conclude that there exists a positive constant C depending on \(a,b,c,k,\lambda , S, \Vert u_{0x}\Vert _{H^4}\) and not on \(\varepsilon \) such that

on the time interval \([0,T_{\varepsilon }^{\star }]\). This implies \((N_k(u^{\varepsilon }(t)))^2 \le (N_k(u_0))^2 e^{Ct} \) for any \(t\in [0,T_{\varepsilon }^{\star }]\). Thus, by the definition of \(T_{\varepsilon }^{\star }\), it holds that

$$\begin{aligned} 4(N_4(u_0))^2&= \left( N_4(u^{\varepsilon }(T_{\varepsilon }^{\star }))\right) ^2 \le (N_4(u_0))^2 e^{C_4T_{\varepsilon }^{\star }} \nonumber \end{aligned}$$

with \(C_4>0\) which depends on \(a,b,c,\lambda , S, \Vert u_{0x}\Vert _{H^4}\) and not on \(\varepsilon \). This shows \(e^{C_4T_{\varepsilon }^{\star }}\geqslant 4\) and hence \(T_{\varepsilon }^{\star }\geqslant \log 4/C_4\) holds. Therefore, if we set \(T=\log 4/C_4\), it follows that \(T_{\varepsilon }^{\star }\geqslant T\) for any \(\varepsilon \in (0,1]\) and \(\left\{ N_k(u^{\varepsilon }) \right\} _{\varepsilon \in (0,1]}\) is bounded in \(L^{\infty }(0,T)\).

As stated before, this shows that \(\left\{ u_x\right\} _{\varepsilon \in (0,1]}\) is bounded in \(L^{\infty }(0,T;H^k(\mathbb {T};TN))\). Hence, the standard compactness argument and the compactness of N show the existence of a map \(u\in C([0,T]\times \mathbb {T};N)\) and a subsequence \(\left\{ u^{\varepsilon (j)}\right\} _{j=1}^{\infty }\) of \(\left\{ u^{\varepsilon }\right\} _{\varepsilon \in (0,1]}\) that satisfy

$$\begin{aligned}&u_x^{\varepsilon (j)}\rightarrow u_x \quad&\text {in} \quad&C([0,T];H^{k-1}(\mathbb {T};TN)), \nonumber \\&u_x^{\varepsilon (j)}\rightarrow u_x \quad&\text {in} \quad&L^{\infty }(0,T;H^{k}(\mathbb {T};TN)) \quad \text {weakly star} \nonumber \end{aligned}$$

as \(j\rightarrow \infty \), and this u is smooth and solves (1.1)–(1.2).

Finally, in the general case where \(u_0\in C(\mathbb {T};N)\) and \(u_{0x}\in H^k(\mathbb {T};TN)\), we modify the above argument as follows: We consider a sequence \(\left\{ u_{0}^i\right\} _{i=1}^{\infty }\subset C^{\infty }(\mathbb {T};N)\) such that

$$\begin{aligned} u_{0x}^i \rightarrow u_{0x} \quad \text {in} \quad H^{k}(\mathbb {T};TN) \end{aligned}$$
(2.49)

as \(i\rightarrow \infty \). There exist \(T_i=T(\Vert u_{0x}^i\Vert _{H^4})>0\) and \(u^i\in C^{\infty }([0,T_i]\times \mathbb {T};N)\) which satisfies (1.1) and \(u^i(0,x)=u^i(x)\) for each \(i=1,2,\ldots \), since \(u_0^i\in C^{\infty }(\mathbb {T};N)\). Recalling the above argument, it is not difficult to show the estimate \(T_i^{\star }\geqslant \log 4/C_{4,i}\), where

$$\begin{aligned} T^{\star }_{i} = \sup \left\{ T>0 \ | \ N_4\left( u^{i}(t)\right) \le 2N_4\left( u_{0}^i\right) \quad \text {for all} \quad t\in [0,T] \right\} , \end{aligned}$$

and \(C_{4,i}>0\) depends on \(a,b,c,\lambda ,S, \big \Vert u_{0x}^i\big \Vert _{H^4}\). Note that \(C_{4,i}\) depends on \(\big \Vert u_{0x}^i\big \Vert _{H^4}\) continuously. This together with (2.49) shows that there exists \(C_4^{\prime }>0\) depending on \(a,b,c,\lambda , S, \Vert u_{0x}\Vert _{H^4}\) and not on i such that \(T_i^{\star }\geqslant \log 4/C_{4}^{\prime }\) for sufficiently large i. Therefore, if we set \(T=\log 4/C_4^{\prime }\), there exists a sufficiently large \(i_0\in \mathbb {N}\) such that \(T_{i}^{\star }\geqslant T\) for any \(i\geqslant i_0\) and \(\left\{ N_k(u^{i}) \right\} _{i=i_0}^{\infty }\) is bounded in \(L^{\infty }(0,T)\). Therefore, by applying the compactness argument again, we can construct the desired solution to (1.1)–(1.2). This completes the proof. \(\square \)

Proof of Theorem 1.1

The goal of this section is to complete the proof of Theorem 1.1. Throughout this section, it is assumed that \(k\geqslant 6\).

Proof of Theorem 1.1

Since \(k\geqslant 6\geqslant 4\), Theorem 2.1 established in Sect. 2 guarantees the existence of \(T=T(\Vert u_{0x}\Vert _{H^4(\mathbb {T};TN)})>0\) and a map \(u\in C([0,T]\times \mathbb {T};N)\) so that \(u_x\in L^{\infty }(0,T;H^k(\mathbb {T};TN)) \cap C([0,T];H^{k-1}(\mathbb {T};TN))\) and u solves (1.1)-(1.2) on the time interval [0, T]. In what follows, we shall concentrate on the proof of the uniqueness of the solution. Once the uniqueness is established, we can easily prove the time continuity of \(\nabla _x^ku_x\) in \(L^2\) by the standard argument, which implies \(u_x\in C([0,T];H^k(\mathbb {T};TN))\). In this way, the proof of Theorem 1.1 is completed.

Let uv be solutions constructed in Theorem 2.1. Then u and v solve (1.1)–(1.2) and satisfy \(u_x, v_x \in L^{\infty }(0,T;H^6(\mathbb {T};TN)) \cap C([0,T];H^{5}(\mathbb {T};TN))\). We shall show \(u=v\). For this purpose, fix w as an isometric embedding of (Ng) into some Euclidean space \(\mathbb {R}^d\) so that N is considered as a submanifold of \(\mathbb {R}^d\). We set \(U=w{\circ }u\), \(V=w{\circ }v\), \(Z=U-V\), \(\mathcal {U}=\mathrm{d}w_u(\nabla _xu_x)\), \(\mathcal {V}=\mathrm{d}w_v(\nabla _xv_x)\), and \(\mathcal {W}=\mathcal {U}-\mathcal {V}\). To prove \(u=v\), it suffices to show \(Z=0\). The proof of \(Z=0\) consists of the following four steps:

  1. 1.

    Notations and tools of computations used below.

  2. 2.

    Analysis of the partial differential equation satisfied by \(\mathcal {U}\).

  3. 3.

    Classical energy estimates for \(\Vert \mathcal {W}\Vert _{L^2(\mathbb {T};\mathbb {R}^d)}\) with the loss of derivatives.

  4. 4.

    Energy estimates for \(\Vert \widetilde{\mathcal {W}}\Vert _{L^2(\mathbb {T};\mathbb {R}^d)}\) (defined later) to eliminate the loss of derivatives.

1. Notations and tools of computations used below

We state some notations and gather tools of computations which will be used below.

The inner product and the norm in \(\mathbb {R}^d\) are expressed as \((\cdot ,\cdot )\) and \(|\cdot |\), respectively. The inner product and the norm in \(L^2\) for \(\mathbb {R}^d\)-valued functions on \(\mathbb {T}\) are expressed as \(\left\langle \cdot \right\rangle \) and \(\Vert \cdot \Vert _{L^2}\), respectively. That is, for \(\phi , \psi :\mathbb {T}\rightarrow \mathbb {R}^d\), \(\left\langle \phi ,\psi \right\rangle \) and \(\Vert \phi \Vert _{L^2}\) are given by \( \left\langle \phi ,\psi \right\rangle = \int _{\mathbb {T}} (\phi (x),\psi (x)) \,\mathrm{d}x\) and \(\Vert \phi \Vert _{L^2} =\sqrt{\left\langle \phi ,\phi \right\rangle } \), respectively.

Let \(p\in N\) be a fixed point. We consider the orthogonal decomposition \( \mathbb {R}^d = \mathrm{d}w_p(T_pN) \oplus \left( \mathrm{d}w_p(T_pN) \right) ^{\perp } \), where \(\mathrm{d}w_p:T_pN\rightarrow T_{w{\circ }p}\mathbb {R}^d\cong \mathbb {R}^d\) is the differential of \(w:N\rightarrow \mathbb {R}^d\) at \(p\in N\) and \(\left( \mathrm{d}w_p(T_pN) \right) ^{\perp }\) is the orthogonal complement of \(\mathrm{d}w_p(T_pN)\) in \(\mathbb {R}^d\). We denote the orthogonal projection mapping of \(\mathbb {R}^d\) onto \(\mathrm{d}w_p(T_pN)\) by \(P(w{\circ }p)\) and define \(N(w{\circ }p)\) by \(N(w{\circ }p)=I_\mathrm{d}-P(w{\circ }p)\), where \(I_\mathrm{d}\) is the identity mapping on \(\mathbb {R}^d\). Moreover, we define \(J(w{\circ }p)\) as an action on \(\mathbb {R}^d\) by first projecting onto \(\mathrm{d}w_p(T_pN)\) and then applying the complex structure at \(p\in N\). More precisely, we define \(J(w{\circ }p)\) by

$$\begin{aligned} J(w{\circ }p)&= \mathrm{d}w_p\circ J_{p}\circ \mathrm{d}w_{w{\circ }p}^{-1}\circ P(w{\circ }p). \end{aligned}$$
(3.1)

We can extend \(P(\cdot )\), \(N(\cdot )\), and \(J(\cdot )\) to a smooth linear operator on \(\mathbb {R}^d\) so that P(q), N(q), and J(q) make sense for all \(q\in \mathbb {R}^d\) following the argument in, e.g., [14, pp. 17]. Although J(q) is not skew-symmetric and the square is not the minus identity in general, similar properties hold if q is restricted to w(N). Indeed, from the definitions of \(P(\omega {\circ }p)\) and \(J(\omega {\circ }p)\), it follows that

$$\begin{aligned} (P(w{\circ }p)Y_1,Y_2)&=(Y_1,P(w{\circ }p)Y_2), \end{aligned}$$
(3.2)
$$\begin{aligned} (J(w{\circ }p)Y_1,Y_2)&=-(Y_1,J(w{\circ }p)Y_2), \end{aligned}$$
(3.3)
$$\begin{aligned} \left( J(w{\circ }p) \right) ^2Y_3&= -P(w{\circ }p)Y_3, \end{aligned}$$
(3.4)

for any \(p\in N\) and \(Y_1,Y_2\in \mathbb {R}^d\).

Let \(Y\in \Gamma (u^{-1}TN)\) be fixed. For \((t,x)\in [0,T]\times \mathbb {T}\), let \(\left\{ \nu _3, \ldots , \nu _d\right\} \) denote a smooth local orthonormal frame field for the normal bundle \((dw(TN))^{\perp }\) near \(U(t,x)=w{\circ }u(t,x)\in w(N)\). Recalling that \(\mathrm{d}w_u(\nabla _xY)\) is the \(\mathrm{d}w_u(T_uN)\) component of \(\partial _x(\mathrm{d}w_u(Y))\), we obtain

$$\begin{aligned} \mathrm{d}w_u(\nabla _xY)&= \partial _x \left( \mathrm{d}w_u(Y) \right) - \sum _{k=3}^d \left( \partial _x(\mathrm{d}w_u(Y)), \nu _k(U)\right) \nu _k(U) \nonumber \\&= \partial _x\left( \mathrm{d}w_u(Y) \right) + \sum _{k=3}^d (\mathrm{d}w_u(Y), \partial _x\left( \nu _k(U)\right) )\nu _k(U) \nonumber \\&= \partial _x\left( \mathrm{d}w_u(Y) \right) + \sum _{k=3}^d (\mathrm{d}w_u(Y), D_k(U)U_x)\nu _k(U) \nonumber \\&= \partial _x\left( \mathrm{d}w_u(Y) \right) + A(U)(\mathrm{d}w_u(Y),U_x), \end{aligned}$$
(3.5)

where \(D_k={\text {grad}} \nu _k\) for \(k=3,\dots ,d\) and \(A(q)(\cdot ,\cdot )= \displaystyle \sum _{k=3}^d (\cdot , D_k(q)\cdot )\nu _k(q)\) is the second fundamental form at \(q\in w(N)\). In the same way, only by replacing x with t, we obtain

$$\begin{aligned} \mathrm{d}w_u(\nabla _tY)&= \partial _t\left( \mathrm{d}w_u(Y) \right) + \sum _{k=3}^d \big (\mathrm{d}w_u(Y), D_k(U)U_t\big )\nu _k(U). \end{aligned}$$
(3.6)

The Sobolev embedding and the Gagliardo–Nirenberg inequality lead to the equivalence between \(U_x, V_x\in L^{\infty }(0,T;H^6(\mathbb {T};\mathbb {R}^d))\) and \(u_x,v_x\in L^{\infty }(0,T;H^6(\mathbb {T};TN))\). In particular, from the Sobolev embedding \(H^1(\mathbb {T})\) into \(C(\mathbb {T})\), it follows that \(\partial _x^kU_x, \partial _x^kV_x\in L^{\infty }((0,T)\times \mathbb {T};\mathbb {R}^d)\) for \(k=0,1,\ldots ,5\), which will be used below without any comments.

We next observe some properties related to \(\nu _k\) and \(D_k\) for \(k=3\ldots ,d\).

Lemma 3.1

For each \((t,x)\in [0,T]\times \mathbb {T}\), the following properties hold:

$$\begin{aligned} J(U)\nu _k(U)&=0, \end{aligned}$$
(3.7)
$$\begin{aligned} (\nu _k(U), \mathcal {W})&= -(\nu _k(U)-\nu _k(V), \mathcal {V}) =\mathcal {O}(|Z|), \end{aligned}$$
(3.8)
$$\begin{aligned} (\nu _k(U),\partial _x\mathcal {W})&= -(D_k(U)U_x,\mathcal {W}) -(D_k(U)Z_x,\mathcal {V}) + \mathcal {O}(|Z|), \end{aligned}$$
(3.9)
$$\begin{aligned} (\nu _k(U),\partial _x^2\mathcal {W})&= -2\,(D_k(U)U_x,\partial _x\mathcal {W}) + \mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|), \end{aligned}$$
(3.10)
$$\begin{aligned} (D_k(U)Y_1,Y_2)&=(Y_1,D_k(U)Y_2) \ \ \ \text {for any } Y_1,Y_2:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d. \end{aligned}$$
(3.11)

Remark 3.2

In particular, in view of (3.8), we find (see the argument to show (3.66) in the third step) that the term including \(\partial _x^2\mathcal {W}\) or \(\partial _x\mathcal {W}\) can be handled as a harmless term if the vector part is described by \(\nu _k(U)\) with some \(k=3,\ldots ,d\). This is related to the reason why we choose \(\mathrm{d}w_u(\nabla _xu_x)-\mathrm{d}w_v(\nabla _xv_x)\) as \(\mathcal {W}\).

Proof of Lemma 3.1

First, (3.7) is a direct consequence of the definition of J(U) and the orthogonality \(\nu _k(U)\perp \mathrm{d}w_u(T_uN)\). Next, in view of \((\nu _k(U),\mathcal {U})=(\nu _k(V),\mathcal {V})=0\), we have

$$\begin{aligned} \big (\nu _k(U),\mathcal {W}\big ) = (\nu _k(U),\mathcal {U}-\mathcal {V}) = -(\nu _k(U),\mathcal {V}) = -(\nu _k(U)-\nu _k(V),\mathcal {V}), \nonumber \end{aligned}$$

which shows (3.8). Moreover, by taking the derivative of (3.8) with respect to x, we have

$$\begin{aligned}&(\nu _k(U),\partial _x\mathcal {W}) \nonumber \\&\quad = \partial _x\left\{ (\nu _k(U),\mathcal {W}) \right\} -(\partial _x\left\{ \nu _k(U) \right\} ,\mathcal {W}) \nonumber \\&\quad = -(\partial _x\left\{ \nu _k(U)-\nu _k(V) \right\} , \mathcal {V}) -(\nu _k(U)-\nu _k(V),\partial _x\mathcal {V}) -(D_k(U)U_x,\mathcal {W}) \nonumber \\&\quad = -(D_k(U)U_x-D_k(V)V_x, \mathcal {V}) -(\nu _k(U)-\nu _k(V),\partial _x\mathcal {V}) -(D_k(U)U_x,\mathcal {W}) \nonumber \\&\quad = -(D_k(U)Z_x-(D_k(U)-D_k(V))V_x, \mathcal {V}) -(\nu _k(U)-\nu _k(V),\partial _x\mathcal {V})\nonumber \\&\qquad -(D_k(U)U_x,\mathcal {W}) \nonumber \\&\quad =-(D_k(U)U_x,\mathcal {W})-(D_k(U)Z_x, \mathcal {V}) +\mathcal {O}(|Z|), \nonumber \end{aligned}$$

which shows (3.9). We obtain (3.10), by taking the derivative of (3.9) with respect to x. We omit the proof of (3.11), since it has been proved in [16, pp. 912].

The following lemma comes from the Kähler condition on (NJg).

Lemma 3.3

(i) For any \(Y\in \Gamma (u^{-1}TN)\),

$$\begin{aligned} \partial _x(J(U))\mathrm{d}w_u(Y)&= \sum _{k=3}^d \left( \mathrm{d}w_u(Y), J(U)D_k(U)U_x \right) \nu _k(U). \end{aligned}$$
(3.12)

(ii) For any \(Y:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\),

$$\begin{aligned} \partial _x(J(U))Y&= \sum _{k=3}^d \left( Y,J(U)D_k(U)U_x\right) \nu _k(U) -\sum _{k=3}^d \big (Y,\nu _k(U)\big )J(U)D_k(U)U_x. \end{aligned}$$
(3.13)

Remark 3.4

Using (3.13) combined with (3.8), we can handle the term \(\partial _x(J(U))\partial _xW\) as a harmless term in the energy estimate for \(\mathcal {W}\) in \(L^2\).

Proof of Lemma 3.3

First we show (i). For \(Y\in \Gamma (u^{-1}TN)\), the Kähler condition on (NJg) implies \(\nabla _xJ_uY=J_u\nabla _xY\). Hence, it holds that

$$\begin{aligned} \mathrm{d}w_u(\nabla _xJ_uY)=\mathrm{d}w_u(J_u\nabla _xY). \end{aligned}$$
(3.14)

From (3.5) and (3.7), the RHS of (3.14) satisfies

$$\begin{aligned} \mathrm{d}w_u(J_u\nabla _xY)&=J(U)\mathrm{d}w_u(\nabla _xY) = J(U) \partial _x(\mathrm{d}w_u(Y)). \end{aligned}$$
(3.15)

On the other hand, from (3.5), the left-hand side of (3.14) satisfies

$$\begin{aligned} \mathrm{d}w_u(\nabla _xJ_uY)&= \partial _x\left\{ \mathrm{d}w_u(J_uY)\right\} + \sum _{k=3}^d \left( \mathrm{d}w_u(J_uY), D_k(U)U_x \right) \nu _k(U) \nonumber \\&= \partial _x\left\{ J(U)\mathrm{d}w_u(Y)\right\} + \sum _{k=3}^d \left( J(U)\mathrm{d}w_u(Y), D_k(U)U_x \right) \nu _k(U) \nonumber \\&= \partial _x(J(U))\mathrm{d}w_u(Y) + J(U)\partial _x(\mathrm{d}w_u(Y)) \nonumber \\&\quad + \sum _{k=3}^d \left( J(U)\mathrm{d}w_u(Y), D_k(U)U_x \right) \nu _k(U). \end{aligned}$$
(3.16)

By substituting (3.15) and (3.16) into (3.14) and using (3.3), we have

$$\begin{aligned} \partial _x(J(U))\mathrm{d}w_u(Y)&= -\sum _{k=3}^d \left( J(U)\mathrm{d}w_u(Y), D_k(U)U_x \right) \nu _k(U) \nonumber \\&= \sum _{k=3}^d \left( \mathrm{d}w_u(Y), J(U)D_k(U)U_x \right) \nu _k(U), \nonumber \end{aligned}$$

which shows (3.12). Next we show (ii). Decomposing \(Y=P(U)Y+N(U)Y\), where \(P(U)Y\in dw(T_uN)\) and \(N(U)Y\in (dw(T_uN))^{\perp }\) for each (tx), we have

$$\begin{aligned} \partial _x(J(U))Y&=\partial _x(J(U))P(U)Y + \partial _x(J(U))N(U)Y. \end{aligned}$$
(3.17)

Using (3.12) and noting that N(U)Y is perpendicular to \(J(U)D_k(U)U_x\), we find that the first term of the RHS of (3.17) satisfies

$$\begin{aligned} \partial _x(J(U))P(U)Y&= \sum _{k=3}^d \left( P(U)Y, J(U)D_k(U)U_x \right) \nu _k(U) \nonumber \\&= \sum _{k=3}^d \left( Y, J(U)D_k(U)U_x \right) \nu _k(U). \end{aligned}$$
(3.18)

Moreover, since

$$\begin{aligned} \partial _x(J(U))\nu _k(U) = \partial _x(J(U)\nu _k(U)) -J(U)\partial _x(\nu _k(U)) = -J(U)D_k(U)U_x \end{aligned}$$
(3.19)

follows from (3.7), the second term of the RHS of (3.17) satisfies

$$\begin{aligned} \partial _x(J(U))N(U)Y&= \sum _{k=3}^d(Y,\nu _k(U))\partial _x(J(U))\nu _k(U) \nonumber \\&= -\sum _{k=3}^d(Y,\nu _k(U)) J(U)D_k(U)U_x. \end{aligned}$$
(3.20)

Substituting (3.18) and (3.20) into (3.17), we obtain (3.13).

As in the proof of Theorem 2.1, we denote the sectional curvature on (Ng) by S which is supposed to be a constant. Recall that the Riemann curvature tensor R is expressed as

$$\begin{aligned} R(Y_1,Y_2)Y_3&= S\left\{ g(Y_2,Y_3)Y_1 - g(Y_1,Y_3)Y_2 \right\} \end{aligned}$$
(3.21)

for any \(Y_1,Y_2,Y_3\in \Gamma (u^{-1}TN)\). The next lemma comes from (3.21).

Lemma 3.5

For any \(Y_1,Y_2,Y_3\in \Gamma (u^{-1}TN)\),

$$\begin{aligned}&\mathrm{d}w_u \left( R(Y_1,Y_2)Y_3 \right) = \sum _k \left( \mathrm{d}w_u(Y_3), D_k(U)\mathrm{d}w_u(Y_2) \right) P(U)D_k(U)\mathrm{d}w_u(Y_1) \nonumber \\&\quad \phantom {\mathrm{d}w_u \left( R(Y_1,Y_2)Y_3 \right) } \qquad - \sum _k \left( \mathrm{d}w_u(Y_3), D_k(U)\mathrm{d}w_u(Y_1) \right) P(U)D_k(U)\mathrm{d}w_u(Y_2), \end{aligned}$$
(3.22)
$$\begin{aligned}&\sum _k \left( \mathrm{d}w_u(Y_3), D_k(U)\mathrm{d}w_u(Y_2) \right) P(U)D_k(U)\mathrm{d}w_u(Y_1) \nonumber \\&\qquad - \sum _k \left( \mathrm{d}w_u(Y_3), D_k(U)\mathrm{d}w_u(Y_1) \right) P(U)D_k(U)\mathrm{d}w_u(Y_2) \nonumber \\&\quad = S\,\left\{ (\mathrm{d}w_u(Y_3), dw_u(Y_2))\mathrm{d}w_u(Y_1) -(\mathrm{d}w_u(Y_3), dw_u(Y_1))\mathrm{d}w_u(Y_2) \right\} . \end{aligned}$$
(3.23)

Proof of Lemma 3.5

We can realize that (3.22) is a kind of the expression of the Gauss–Codazzi formula in Riemannian geometry. Fix \((t,x)\in [0,T]\times \mathbb {T}\). We take a two-parameterized smooth map \(\gamma =\gamma (s,\sigma ):(-\delta ,\delta )\times (-\delta ,\delta )\rightarrow N\) with sufficiently small \(\delta >0\), and a \(Y_4\in \Gamma (\gamma ^{-1}TN)\) so that \(\gamma (0,0)=u(t,x)\), \(\gamma _s(0,0)=Y_1(t,x)\), \(\gamma _\sigma (0,0)=Y_2(t,x)\), and \(Y_4(0,0)=Y_3(t,x)\). Since \(R(\gamma _s,\gamma _\sigma )Y_4=\nabla _s\nabla _\sigma Y_4-\nabla _\sigma \nabla _sY_4\), we deduce

$$\begin{aligned} \mathrm{d}w_{\gamma } \left( R(\gamma _s,\gamma _\sigma )Y_4 \right)&= \mathrm{d}w_{\gamma } \left( \nabla _s\nabla _\sigma Y_4 \right) - \mathrm{d}w_{\gamma } \left( \nabla _\sigma \nabla _sY_4 \right) \nonumber \\&= P(w{\circ }\gamma ) \partial _s\left( \mathrm{d}w_{\gamma }(\nabla _\sigma Y_4) \right) - P(w{\circ }\gamma ) \partial _\sigma \left( \mathrm{d}w_{\gamma }(\nabla _sY_4) \right) \nonumber \\&= P(w{\circ }\gamma ) \left\{ \partial _s\left( \mathrm{d}w_{\gamma }(\nabla _\sigma Y_4) \right) - \partial _\sigma \left( \mathrm{d}w_{\gamma }(\nabla _sY_4) \right) \right\} . \end{aligned}$$
(3.24)

Similarly to (3.5) and (3.6), the definition of the covariant derivatives yields

$$\begin{aligned} \partial _s\left( \mathrm{d}w_{\gamma }(\nabla _\sigma Y_4) \right)&= \partial _s\left( \partial _\sigma (\mathrm{d}w_{\gamma }(Y_4))\right. \nonumber \\&\quad \left. + \sum _{k=3}^d \left( \mathrm{d}w_{\gamma }(Y_4), D_k(w{\circ }\gamma )\partial _\sigma (w{\circ }\gamma ) \right) \nu _k(w{\circ }\gamma ) \right) \nonumber \\&= \partial _s\partial _\sigma (\mathrm{d}w_{\gamma }(Y_4)) \nonumber \\&\quad + \sum _{k=3}^d \partial _s \left\{ \left( \mathrm{d}w_{\gamma }(Y_4), D_k(w{\circ }\gamma )\partial _\sigma (w{\circ }\gamma ) \right) \right\} \nu _k(w{\circ }\gamma ) \nonumber \\&\quad + \sum _{k=3}^d \left( \mathrm{d}w_{\gamma }(Y_4), D_k(w{\circ }\gamma )\partial _\sigma (w{\circ }\gamma ) \right) D_k(w{\circ }\gamma ) \partial _s(w{\circ }\gamma ), \end{aligned}$$
(3.25)

and

$$\begin{aligned} \partial _\sigma \left( \mathrm{d}w_{\gamma }(\nabla _sY_4) \right)&= \partial _\sigma \partial _s(\mathrm{d}w_{\gamma }(Y_4)) \nonumber \\&\quad + \sum _{k=3}^d \partial _\sigma \left\{ \left( \mathrm{d}w_{\gamma }(Y_4), D_k(w{\circ }\gamma )\partial _s(w{\circ }\gamma ) \right) \right\} \nu _k(w{\circ }\gamma ) \nonumber \\&\quad + \sum _{k=3}^d \left( \mathrm{d}w_{\gamma }(Y_4), D_k(w{\circ }\gamma )\partial _s(w{\circ }\gamma ) \right) D_k(w{\circ }\gamma ) \partial _\sigma (w{\circ }\gamma ). \end{aligned}$$
(3.26)

By substituting (3.25) and (3.26) into (3.24) and by noting \(P(w{\circ }\gamma )\nu _k(w{\circ }\gamma )=0\), we have

$$\begin{aligned} \mathrm{d}w_{\gamma } \left( R(\gamma _s,\gamma _\sigma )Y_4 \right) =&\sum _{k=3}^d \left( \mathrm{d}w_{\gamma }(Y_4), D_k(w{\circ }\gamma ) \partial _\sigma (w{\circ }\gamma ) \right) \\&\times P(w{\circ }\gamma ) D_k(w{\circ }\gamma ) \partial _s(w{\circ }\gamma ) \nonumber \\&- \sum _{k=3}^d \left( \mathrm{d}w_{\gamma }(Y_4), D_k(w{\circ }\gamma )\partial _s(w{\circ }\gamma ) \right) \\&\times P(w{\circ }\gamma ) D_k(w{\circ }\gamma ) \partial _\sigma (w{\circ }\gamma ). \nonumber \end{aligned}$$

Thus, by taking the limit \((s,\sigma )\rightarrow (0,0)\), we obtain

$$\begin{aligned} \mathrm{d}w_{u} \left( R(Y_1,Y_2)Y_3 \right)&= \sum _{k=3}^d \left( \mathrm{d}w_{u}(Y_3), D_k(w{\circ }u)\mathrm{d}w_u(Y_2) \right) P(w{\circ }u) D_k(w{\circ }u) \mathrm{d}w_u(Y_1) \nonumber \\&\quad - \sum _{k=3}^d \left( \mathrm{d}w_{u}(Y_3), D_k(w{\circ }u)\mathrm{d}w_u(Y_1) \right) P(w{\circ }u) D_k(w{\circ }u) \mathrm{d}w_u(Y_2) \nonumber \end{aligned}$$

for each (tx). This implies (3.22). Noting that \(w:(N,g)\rightarrow (\mathbb {R}^d,(\cdot ,\cdot ))\) is isometric, we observe that (3.23) follows from (3.21) and (3.22).

The following properties come from the assumption that N is a two-dimensional real manifold. Noting that \(\left\{ \frac{U_x}{|U_x|}, \frac{J(U)U_x}{|U_x|}, \nu _3(U), \ldots , \nu _d(U) \right\} \) forms an orthonormal basis of \(\mathbb {R}^d\) for each (tx) where \(U_x(t,x)\ne 0\), we observe that

$$\begin{aligned} |U_x|^2Y = (Y,U_x)U_x + (Y,J(U)U_x)J(U)U_x + \sum _{k=3}^d \left( |U_x|^2Y,\nu _k(U)\right) \nu _k(U) \end{aligned}$$
(3.27)

holds for every (tx). Note also that (3.27) is valid for (tx) where \(U_x(t,x)=0\), as each of the both sides of (3.27) vanishes. Using (3.27) with J(U)Y instead of Y, we have

$$\begin{aligned} |U_x|^2J(U)Y&= (J(U)Y,U_x)U_x + (J(U)Y,J(U)U_x)J(U)U_x \nonumber \\&\quad + \sum _{k=3}^d \left( |U_x|^2J(U)Y,\nu _k(U)\right) \nu _k(U) \nonumber \\&= -(Y,J(U)U_x)U_x +(Y,U_x)J(U)U_x. \end{aligned}$$
(3.28)

Moreover, we introduce \(T_2(U), \ldots , T_5(U):[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\) defined by the following.

Definition 3.6

For any \(Y:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\),

$$\begin{aligned} T_2(U)Y&= \frac{1}{2}|U_x|^2J(U)Y, \end{aligned}$$
(3.29)
$$\begin{aligned} T_3(U)Y&= \frac{1}{2} \biggl \{ (Y,\partial _xU_x)J(U)U_x +(Y,U_x)J(U)\partial _xU_x \nonumber \\&\qquad \quad + (Y,J(U)\partial _xU_x)U_x + (Y,J(U)U_x)\partial _xU_x \biggr \}, \end{aligned}$$
(3.30)
$$\begin{aligned} T_4(U)Y&= \left( Y,\partial _xU_x+\sum _{k=3}^d(U_x,D_k(U)U_x)\nu _k(U) \right) J(U)U_x\nonumber \\&\qquad \quad - (Y,U_x)J(U)\partial _xU_x, \end{aligned}$$
(3.31)
$$\begin{aligned} T_5(U)Y&= \frac{1}{2} \biggl \{ (Y,\partial _xU_x)J(U)U_x + (Y,U_x)J(U)\partial _xU_x \nonumber \\&\qquad \quad - (Y,J(U)\partial _xU_x)U_x - (Y,J(U)U_x)\partial _xU_x \biggr \}. \end{aligned}$$
(3.32)

We use (3.27) or (3.28) to show the following.

Lemma 3.7

For any \(Y,Y_1,Y_2:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\), it follows that

$$\begin{aligned} T_2(U)Y&= \frac{1}{2} \left\{ (Y,U_x)J(U)U_x -(Y,J(U)U_x)U_x \right\} , \end{aligned}$$
(3.33)
$$\begin{aligned} \partial _x(T_2(U))Y&= (\partial _xU_x,U_x)J(U)Y + \frac{1}{2} |U_x|^2 \partial _x(J(U))Y, \end{aligned}$$
(3.34)
$$\begin{aligned} \partial _x(T_2(U))Y&= T_5(U)Y -\frac{1}{2} (Y,U_x)\sum _{k=3}^d(J(U)U_x,D_k(U)U_x)\nu _k(U) \nonumber \\&\quad +\frac{1}{2} \sum _{k=3}^d(Y,\nu _k(U))(J(U)U_x,D_k(U)U_x)U_x, \end{aligned}$$
(3.35)
$$\begin{aligned} (T_3(U)Y_1, Y_2)&= (Y_1,T_3(U)Y_2), \end{aligned}$$
(3.36)
$$\begin{aligned} (T_4(U)Y_1, Y_2)&= (Y_1,T_4(U)Y_2). \end{aligned}$$
(3.37)

Proof of Lemma 3.7

First, (3.33) is a direct consequence of (3.28). Second, (3.34) follows from the substitution of (3.29) into \(\partial _x(T_2(U))Y = \partial _x\left\{ T_2(U)Y \right\} -T_2(U)\partial _xY\). Third, by substituting (3.33) into \(\partial _x(T_2(U))Y=\partial _x\left\{ T_2(U)Y\right\} -T_2(U)\partial _xY\) and using (3.32), we have

$$\begin{aligned} \partial _x(T_2(U))Y&= T_5(U)Y + \frac{1}{2} (Y,U_x)\partial _x(J(U))U_x -\frac{1}{2} (Y,\partial _x(J(U))U_x)U_x. \end{aligned}$$
(3.38)

Recall here that (3.13) yields \(\partial _x(J(U))U_x = \displaystyle \sum _{k=3}^d (U_x,J(U)D_k(U)U_x)\nu _k(U)\). Substituting this into the RHS of (3.38), we obtain (3.35). Next, in view of (3.30), it is immediate to see that (3.36) holds. Finally we show (3.37). The proof is reduced to that of (2.23) with \(i=2\). Noting that there exists \(\Xi _i\in \Gamma (u^{-1}TN)\) such that \(\mathrm{d}w_u(\Xi _i)=P(U)Y_i\) for each \(i=1,2\), we have

$$\begin{aligned} T_4(U)Y_1&= (Y_1,\mathrm{d}w_u(\nabla _xu_x))\mathrm{d}w_u(J_uu_x) -(Y_1,\mathrm{d}w_u(u_x))dw_u(J_u\nabla _xu_x) \nonumber \\&= (P(U)Y_1,\mathrm{d}w_u(\nabla _xu_x))\mathrm{d}w_u(J_uu_x) -(P(U)Y_1,\mathrm{d}w_u(u_x))dw_u(J_u\nabla _xu_x) \nonumber \\&= (\mathrm{d}w_u(\Xi _1),\mathrm{d}w_u(\nabla _xu_x))\mathrm{d}w_u(J_uu_x) -(\mathrm{d}w_u(\Xi _1),\mathrm{d}w_u(u_x))dw_u(J_u\nabla _xu_x). \nonumber \end{aligned}$$

Since w is an isometric, this shows

$$\begin{aligned} T_4(U)Y_1 =\mathrm{d}w_u\left\{ g(\Xi _1,\nabla _xu_x)J_uu_x -g(\Xi _1,u_x)J_u\nabla _xu_x \right\} , \end{aligned}$$

and thus we obtain

$$\begin{aligned} (T_4(U)Y_1,Y_2)&= (\mathrm{d}w_u\left\{ g(\Xi _1,\nabla _xu_x)J_uu_x -g(\Xi _1,u_x)J_u\nabla _xu_x \right\} ,P(U)Y_2) \nonumber \\&= (\mathrm{d}w_u\left\{ g(\Xi _1,\nabla _xu_x)J_uu_x -g(\Xi _1,u_x)J_u\nabla _xu_x \right\} ,\mathrm{d}w_u(\Xi _2)) \nonumber \\&= g(g(\Xi _1,\nabla _xu_x)J_uu_x -g(\Xi _1,u_x)J_u\nabla _xu_x, \Xi _2) \nonumber \\&= g(A_2\Xi _1,\Xi _2). \nonumber \end{aligned}$$

Since \(g(A_2\Xi _1,\Xi _2)=g(\Xi _1,A_2\Xi _2)\) follows from (2.23), we conclude that (3.37) holds.

In what follows, for simplicity, we sometimes write dw instead of \(\mathrm{d}w_u\) or \(dw_v\) and use \(\displaystyle \sum _{k}\) and \(\displaystyle \sum _{k,\ell }\) instead of \(\displaystyle \sum _{k=3}^{d}\) and \(\displaystyle \sum _{k,\ell =3}^{d}\), respectively. Any confusion will not occur.

2. Analysis of the partial differential equation satisfied by \(\mathcal {U}\).

We compute the PDE satisfied by \(\mathcal {U}\).

First, we start by the computation of the PDE satisfied by U. Since u satisfies (1.1),

$$\begin{aligned} U_t&= dw(u_t) \nonumber \\&= a\,dw(\nabla _xJ_u\nabla _x^2u_x) + \lambda \,dw(J_u\nabla _xu_x) \nonumber \\&\quad + b\,dw(g(u_x,u_x)J_u\nabla _xu_x) + c\,dw(g(\nabla _xu_x,u_x)J_uu_x) \nonumber \\&= a\,dw(\nabla _xJ_u\nabla _x^2u_x) + \lambda \,J(U)dw(\nabla _xu_x) \nonumber \\&\quad + b\, (dw(u_x),dw(u_x))J(U)dw(\nabla _xu_x) + c\, (dw(\nabla _xu_x),dw(u_x))J(U)dw(u_x) \nonumber \\&= a\,dw\big (\nabla _xJ_u\nabla _x^2u_x\big ) + \left\{ \lambda +b\,|U_x|^2\right\} J(U)\mathcal {U}+ c\, (\mathcal {U},U_x)J(U)U_x. \end{aligned}$$
(3.39)

Using (3.5) and (3.7) repeatedly, we have

$$\begin{aligned} dw\big (\nabla _x^2u_x\big )&= \partial _x\left( dw(\nabla _xu_x) \right) + \sum _{\ell } (dw(\nabla _xu_x), D_{\ell }(U)U_x)\nu _{\ell }(U) \nonumber \\&= \partial _x\mathcal {U}+ \sum _{\ell } (\mathcal {U}, D_{\ell }(U)U_x)\nu _{\ell }(U), \end{aligned}$$
(3.40)
$$\begin{aligned} J(U)dw(\nabla _x^2u_x)&= J(U)\partial _x\mathcal {U}+ \sum _{\ell } (\mathcal {U}, D_{\ell }(U)U_x)J(U)\nu _{\ell }(U) = J(U)\partial _x\mathcal {U}, \end{aligned}$$
(3.41)
$$\begin{aligned} dw(\nabla _xJ_u\nabla _x^2u_x)&= \partial _x \left( dw(J_u\nabla _x^2u_x) \right) + \sum _k (dw(J_u\nabla _x^2u_x), D_k(U)U_x)\nu _k(U) \nonumber \\&= \partial _x\left( J(U)dw(\nabla _x^2u_x) \right) \nonumber \\&\quad + \sum _k (J(U)dw(\nabla _x^2u_x), D_k(U)U_x)\nu _k(U) \nonumber \\&= \partial _x\left( J(U)\partial _x\mathcal {U}\right) + \sum _k (J(U)\partial _x\mathcal {U}, D_k(U)U_x)\nu _k(U). \end{aligned}$$
(3.42)

From (3.39) and (3.42), we have

$$\begin{aligned} U_t&= a\,\partial _x\left( J(U)\partial _x\mathcal {U}\right) + a\, \sum _k (J(U)\partial _x\mathcal {U}, D_k(U)U_x)\nu _k(U) \nonumber \\&\quad + \mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.43)

Next, we compute the PDE satisfied by \(\mathcal {U}\). From (1.1), (3.21), and (3.6), it follows that

$$\begin{aligned} \partial _t\mathcal {U}&= \partial _t(dw(\nabla _xu_x)) \nonumber \\&= dw(\nabla _t\nabla _xu_x) - \sum _{k} (dw(\nabla _xu_x), D_k(U)U_t)\nu _k(U) \nonumber \\&= dw\big (\nabla _x^2u_t+R(u_t,u_x)u_x\big ) - \sum _{k} (\mathcal {U}, D_k(U)U_t)\nu _k(U) \nonumber \\&= dw\big (\nabla _x^2u_t+S\left\{ g(u_x,u_x)u_t-g(u_t,u_x)u_x\right\} \big ) - \sum _{k} (\mathcal {U}, D_k(U)U_t)\nu _k(U) \nonumber \\&= dw\big (\nabla _x^2u_t\big ) +S\left\{ (dw(u_x),dw(u_x))dw(u_t) -(dw(u_t),dw(u_x))dw(u_x) \right\} \nonumber \\&\quad - \sum _{k} (\mathcal {U}, D_k(U)U_t)\nu _k(U) \nonumber \\&= dw\big (\nabla _x^2u_t\big ) + S|U_x|^2U_t -S(U_x,U_t)U_x - \sum _{k} (\mathcal {U}, D_k(U)U_t)\nu _k(U) \nonumber \\&=:I+II+III+IV. \end{aligned}$$
(3.44)

We compute II, III, IV, and I in order.

Applying (3.43), we have

$$\begin{aligned} II&= aS\,\partial _x\left\{ |U_x|^2J(U)\partial _x\mathcal {U}\right\} -2aS\,(\partial _xU_x,U_x)J(U)\partial _x\mathcal {U}\nonumber \\&\quad +aS\,|U_x|^2\sum _k (J(U)\partial _x\mathcal {U}, D_k(U)U_x)\nu _k(U) + \mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.45)

In the same way, by noting \((U_x,\nu _k(U))=0\) and (3.2), we obtain

$$\begin{aligned} III&= -aS\, \left( U_x, \partial _x\left( J(U)\partial _x\mathcal {U}\right) \right) U_x + \mathcal {O}(|U|+|U_x|+|\mathcal {U}|) \nonumber \\&= -aS\, \big (J(U)\partial _x^2\mathcal {U},U_x\big )U_x -aS\, (\partial _x(J(U))\partial _x\mathcal {U},U_x)U_x +\mathcal {O}(|U|+|U_x|+|\mathcal {U}|) \nonumber \\&= aS\, \big (\partial _x^2\mathcal {U},J(U)U_x\big )U_x -aS\, (\partial _x(J(U))\partial _x\mathcal {U},U_x)U_x +\mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.46)

Furthermore, by substituting (3.28) with \(Y=\partial _x^2\mathcal {U}\) into the first term of the RHS of (3.46),

$$\begin{aligned} III&= aS\, (\partial _x^2\mathcal {U},U_x)J(U)U_x -aS\, |U_x|^2J(U)\partial _x^2\mathcal {U}-aS\, (\partial _x(J(U))\partial _x\mathcal {U},U_x)U_x \nonumber \\&\quad +\mathcal {O}(|U|+|U_x|+|\mathcal {U}|) \nonumber \\&= aS\, (\partial _x^2\mathcal {U},U_x)J(U)U_x -aS\, \partial _x \left\{ |U_x|^2J(U)\partial _x\mathcal {U}\right\} \nonumber \\&\quad +2aS\, (\partial _xU_x,U_x)J(U)\partial _x\mathcal {U}+aS\, |U_x|^2 \partial _x(J(U))\partial _x\mathcal {U}\nonumber \\&\quad -aS\, (\partial _x(J(U))\partial _x\mathcal {U},U_x)U_x +\mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.47)

In the same way, by applying (3.43),

$$\begin{aligned} IV&= -a\, \sum _k \left( \mathcal {U}, D_k(U)\partial _x\left( J(U)\partial _x\mathcal {U}\right) \right) \nu _k(U) \nonumber \\&\quad -a\, \sum _k \left( \mathcal {U}, D_k(U)\sum _{\ell } (J(U)\partial _x\mathcal {U}, D_{\ell }(U)U_x)\nu _{\ell }(U) \right) \nu _k(U) \nonumber \\&\quad + \mathcal {O}(|U|+|U_x|+|\mathcal {U}|) \nonumber \\&= -a\, \sum _k \left( \mathcal {U}, D_k(U)\partial _x\left( J(U)\partial _x\mathcal {U}\right) \right) \nu _k(U) \nonumber \\&\quad -a\, \sum _{k,\ell } (\mathcal {U}, D_k(U)\nu _{\ell }(U)) (J(U)\partial _x\mathcal {U}, D_{\ell }(U)U_x) \nu _k(U) \nonumber \\&\quad + \mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.48)

Let us now move on to the computation of I. We start with

$$\begin{aligned} I&=dw\big (\nabla _x^2u_t\big ) \nonumber \\&= a\, dw\big (\nabla _ x^4J_u\nabla _xu_x\big ) + \lambda \, dw(\nabla _x^2J_u\nabla _xu_x) \nonumber \\&\quad + b\,dw \left( \nabla _x^2\left\{ g(u_x,u_x)J_u\nabla _xu_x \right\} \right) + c\,dw \left( \nabla _x^2\left\{ g(\nabla _xu_x,u_x)J_uu_x \right\} \right) \nonumber \\&=: I_1+I_2+I_3+I_4. \end{aligned}$$
(3.49)

We compute \(I_2\), \(I_3\), \(I_4\), and \(I_1\) in order.

For \(I_2\), we have

$$\begin{aligned} I_2&= \lambda \, dw\left( \nabla _x(\nabla _xJ_u\nabla _xu_x)\right) \nonumber \\&= \lambda \, \partial _x\left( dw(\nabla _xJ_u\nabla _xu_x) \right) + \lambda \, \sum _{\ell } \left( dw(\nabla _xJ_u\nabla _xu_x), D_{\ell }(U)U_x \right) \nu _{\ell }(U). \nonumber \end{aligned}$$

Since

$$\begin{aligned} dw(\nabla _xJ_u\nabla _xu_x)&= \partial _x\left( dw(J_u\nabla _xu_x) \right) + \sum _k \left( dw(J_u\nabla _xu_x), D_k(U)U_x \right) \nu _k(U) \nonumber \\&= \partial _x\left( J(U)\mathcal {U}\right) + \sum _k \left( J(U)\mathcal {U}, D_k(U)U_x \right) \nu _k(U) \nonumber \\&= J(U)\partial _x\mathcal {U}+ \partial _x(J(U))\mathcal {U}+ \sum _k \left( J(U)\mathcal {U}, D_k(U)U_x \right) \nu _k(U), \end{aligned}$$
(3.50)

we obtain

$$\begin{aligned} I_2&= \lambda \, \partial _x\left\{ J(U)\partial _x\mathcal {U}\right\} + \lambda \, \partial _x(J(U))\partial _x\mathcal {U}+ 2\lambda \, \sum _k \left( J(U)\partial _x\mathcal {U}, D_k(U)U_x \right) \nu _k(U) \nonumber \\&\quad +\mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.51)

For \(I_3\), we have

$$\begin{aligned} I_3&= b\,dw\left\{ g(u_x,u_x)\nabla _x^2J_u\nabla _xu_x \right\} + b\,dw\left\{ 2\nabla _x(g(u_x,u_x))\nabla _xJ_u\nabla _xu_x \right\} \nonumber \\&\quad + b\,dw\left\{ \nabla _x^2(g(u_x,u_x))J_u\nabla _xu_x \right\} \nonumber \\&= b\,g(u_x,u_x) dw\big (\nabla _x^2J_u\nabla _xu_x\big ) + 4b\, g(\nabla _xu_x,u_x)dw(\nabla _xJ_u\nabla _xu_x) \nonumber \\&\quad + 2b\, g\big (\nabla _x^2u_x,u_x\big )dw(J_u\nabla _xu_x) + 2b\, g(\nabla _xu_x,\nabla _xu_x)dw(J_u\nabla _xu_x) \nonumber \\&= b\, |U_x|^2dw\big (\nabla _x^2J_u\nabla _xu_x\big ) + 4b\, (dw(\nabla _xu_x),U_x)dw(\nabla _xJ_u\nabla _xu_x) \nonumber \\&\quad + 2b\, (dw(\nabla _x^2u_x),U_x)dw(J_u\nabla _xu_x) + 2b\, |dw(\nabla _xu_x)|^2dw(J_u\nabla _xu_x) \nonumber \\&= b\, |U_x|^2dw\big (\nabla _x^2J_u\nabla _xu_x\big ) + 4b\, (\mathcal {U},U_x)dw(\nabla _xJ_u\nabla _xu_x) \nonumber \\&\quad + 2b\, (dw(\nabla _x^2u_x),U_x)J(U)\mathcal {U}+ 2b\, |\mathcal {U}|^2J(U)\mathcal {U}. \nonumber \end{aligned}$$

Here, we recall the Kähler condition on (NJg) to observe \(dw(\nabla _x^2J_u\nabla _xu_x)=dw(\nabla _xJ_u\nabla _x^2u_x)\). Hence, substituting (3.42), (3.50), and (3.40), we deduce

$$\begin{aligned} I_3&= b\,|U_x|^2 \partial _x\left\{ J(U)\partial _x\mathcal {U}\right\} +b\,|U_x|^2 \sum _k (J(U)\partial _x\mathcal {U}, D_k(U)U_x)\nu _k(U) \nonumber \\&\quad + 4b\, (\mathcal {U},U_x)J(U)\partial _x\mathcal {U}+ 2b\, (\partial _x\mathcal {U},U_x)J(U)\mathcal {U}+\mathcal {O}(|U|+|U_x|+|\mathcal {U}|) \nonumber \\&= b\, \partial _x \left\{ |U_x|^2 J(U)\partial _x\mathcal {U}\right\} -2b\, (\partial _xU_x,U_x) J(U)\partial _x\mathcal {U}\nonumber \\&\quad +b\,|U_x|^2 \sum _k (J(U)\partial _x\mathcal {U}, D_k(U)U_x)\nu _k(U) \nonumber \\&\quad + 4b\, (\mathcal {U},U_x)J(U)\partial _x\mathcal {U}+ 2b\, (\partial _x\mathcal {U},U_x)J(U)\mathcal {U}+\mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.52)

Furthermore, by noting \(\mathcal {U}= dw(\nabla _xu_x) = \partial _xU_x + \displaystyle \sum _k (U_x,D_k(U)U_x)\nu _k(U)\), we obtain

$$\begin{aligned} (\mathcal {U},U_x)&= (\partial _xU_x,U_x) + \sum _k(U_x,D_k(U)U_x)(\nu _k(U),U_x) = (\partial _xU_x,U_x), \end{aligned}$$
(3.53)
$$\begin{aligned} J(U)\mathcal {U}&= J(U)\left( \partial _xU_x + \sum _k (U_x,D_k(U)U_x)\nu _k(U) \right) = J(U)\partial _xU_x. \end{aligned}$$
(3.54)

Combining the information (3.52), (3.53), and (3.54), we obtain

$$\begin{aligned} I_3&= b\, \partial _x \left\{ |U_x|^2 J(U)\partial _x\mathcal {U}\right\} +2b\, (\partial _xU_x,U_x) J(U)\partial _x\mathcal {U}+ 2b\, (\partial _x\mathcal {U},U_x)J(U)\partial _xU_x \nonumber \\&\quad +b\,|U_x|^2 \sum _k (J(U)\partial _x\mathcal {U}, D_k(U)U_x)\nu _k(U) +\mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.55)

For \(I_4\), we have

$$\begin{aligned} I_4&= c\,dw \left( g\big (\nabla _x^3u_x,u_x\big )J_uu_x \right) + 3c\,dw \left( g(\nabla _x^2u_x,\nabla _xu_x)J_uu_x \right) \nonumber \\&\quad + 2c\,dw \left( g\big (\nabla _x^2u_x,u_x\big )J_u\nabla _xu_x \right) + 2c\,dw \left( g(\nabla _xu_x,\nabla _xu_x)J_u\nabla _xu_x \right) \nonumber \\&\quad + c\,dw \left( g(\nabla _xu_x,u_x)J_u\nabla _x^2u_x. \right) \nonumber \\&= c\, \big (dw\big (\nabla _x^3u_x\big ),U_x\big )J(U)U_x + 3c\, (dw(\nabla _x^2u_x),\mathcal {U})J(U)U_x \nonumber \\&\quad + 2c\, \big (dw\big (\nabla _x^2u_x\big ),U_x\big )J(U)\mathcal {U}+ 2c\, |\mathcal {U}|^2J(U)\mathcal {U}+ c\, (\mathcal {U},U_x)J(U)dw(\nabla _x^2u_x). \nonumber \end{aligned}$$

From (3.40), it follows that

$$\begin{aligned} dw\big (\nabla _x^3u_x\big )&= \partial _x\left\{ dw(\nabla _x^2u_x) \right\} + \sum _{\ell } \left( dw\big (\nabla _x^2u_x\big ),D_{\ell }(U)U_x \right) \nu _{\ell }(U) \nonumber \\&= \partial _x^2\mathcal {U}+ 2\sum _{k} \left( \partial _x\mathcal {U}, D_k(U)U_x \right) \nu _k(U) + \mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \nonumber \end{aligned}$$

This together with \((\nu _k(U),U_x)=0\) yields

$$\begin{aligned} \big (dw\big (\nabla _x^3u_x\big ),U_x\big )&= (\partial _x^2\mathcal {U},U_x) + \mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.56)

Using (3.40), (3.53), (3.54), and (3.56), we obtain

$$\begin{aligned} I_4&= c\, \big (\partial _x^2\mathcal {U},U_x\big )J(U)U_x + 3c\, (\partial _x\mathcal {U},\mathcal {U})J(U)U_x \nonumber \\&\quad + 2c\, (\partial _x\mathcal {U},U_x)J(U)\mathcal {U}+ c\, (\mathcal {U},U_x)J(U)\partial _x\mathcal {U}+ \mathcal {O}(|U|+|U_x|+|\mathcal {U}|) \nonumber \\&= c\, \big (\partial _x^2\mathcal {U},U_x\big )J(U)U_x + 3c\, (\partial _x\mathcal {U},\partial _xU_x)J(U)U_x \nonumber \\&\quad + 2c\, (\partial _x\mathcal {U},U_x)J(U)\partial _xU_x + c\, (\partial _xU_x,U_x)J(U)\partial _x\mathcal {U}+ \mathcal {O}(|U|+|U_x|+|\mathcal {U}|). \end{aligned}$$
(3.57)

For \(I_1=a\,dw(\nabla _x\nabla _xJ_u\nabla _x^2\nabla _xu_x)\), we start with

$$\begin{aligned}&dw\big (\nabla _x\nabla _xJ_u\nabla _x^2\nabla _xu_x\big ) \nonumber \\&= \partial _x\left\{ dw\big (\nabla _xJ_u\nabla _x^2\nabla _xu_x\big ) \right\} + \sum _{k} \left( dw\big (\nabla _xJ_u\nabla _x^2\nabla _xu_x\big ), D_k(U)U_x \right) \nu _k(U), \end{aligned}$$
(3.58)

and

$$\begin{aligned}&dw\big (\nabla _xJ_u\nabla _x^2\nabla _xu_x\big ) \nonumber \\&= \partial _x\left\{ dw\big (J_u\nabla _x^2\nabla _xu_x\big ) \right\} + \sum _{\ell } \left( dw\big (J_u\nabla _x^2\nabla _xu_x\big ), D_{\ell }(U)U_x \right) \nu _{\ell }(U). \end{aligned}$$
(3.59)

From (3.3), (3.13) with \(Y=\partial _x\mathcal {U}\), the Kähler condition on (NJg), and (3.42), it follows that

$$\begin{aligned} dw(J_u\nabla _x^2\nabla _xu_x)&= J(U)\partial _x^2\mathcal {U}+ \partial _x(J(U))\partial _x\mathcal {U}+ \sum _k (J(U)\partial _x\mathcal {U}, D_k(U)U_x)\nu _k(U) \nonumber \\&= J(U)\partial _x^2\mathcal {U}+ \sum _k \left( \partial _x\mathcal {U},J(U)D_k(U)U_x\right) \nu _k(U) \nonumber \\&\quad -\sum _k (\partial _x\mathcal {U},\nu _k(U))J(U)D_k(U)U_x \nonumber \\&\quad - \sum _k (\partial _x\mathcal {U}, J(U)D_k(U)U_x)\nu _k(U) \nonumber \\&= J(U)\partial _x^2\mathcal {U}-\sum _k (\partial _x\mathcal {U},\nu _k(U))J(U)D_k(U)U_x. \nonumber \end{aligned}$$

Here note that \((\mathcal {U}, \nu _k(U))=0\) holds. By taking the derivative of both sides with respect to x, we obtain

$$\begin{aligned} (\partial _x\mathcal {U},\nu _k(U)) = -(\mathcal {U},\partial _x(\nu _k(U))) = -(\mathcal {U}, D_k(U)U_x). \nonumber \end{aligned}$$

Using this, we obtain

$$\begin{aligned} dw(J_u\nabla _x^2\nabla _xu_x)&= J(U)\partial _x^2\mathcal {U}+\sum _k (\mathcal {U},D_k(U)U_x)J(U)D_k(U)U_x. \end{aligned}$$
(3.60)

Furthermore, by substituting (3.60) into (3.59), we have

$$\begin{aligned}&dw\big (\nabla _xJ_u\nabla _x^2\nabla _xu_x\big ) \nonumber \\&= \partial _x\left\{ J(U)\partial _x^2\mathcal {U}+\sum _n (\mathcal {U},D_n(U)U_x)J(U)D_n(U)U_x \right\} \nonumber \\&\quad + \sum _{\ell } \left( J(U)\partial _x^2\mathcal {U}+\sum _n (\mathcal {U},D_n(U)U_x)J(U)D_n(U)U_x, D_{\ell }(U)U_x \right) \nu _{\ell }(U) \nonumber \\&= \partial _x\left\{ J(U)\partial _x^2\mathcal {U}\right\} - \sum _{\ell } \left( \partial _x^2\mathcal {U}, J(U)D_{\ell }(U)U_x \right) \nu _{\ell }(U) \nonumber \\&\quad + \sum _{n} (\partial _x\mathcal {U}, D_n(U)U_x)J(U)D_n(U)U_x \nonumber \\&\quad +\sum _{n} \left( \mathcal {U}, \partial _x \left\{ D_n(U)U_x \right\} \right) J(U)D_n(U)U_x \nonumber \\&\quad + \sum _{n} (\mathcal {U}, D_n(U)U_x) \partial _x\left\{ J(U)D_n(U)U_x \right\} \nonumber \\&\quad + \sum _{\ell ,n} (\mathcal {U},D_n(U)U_x) \left( J(U)D_n(U)U_x,D_{\ell }(U)U_x \right) \nu _{\ell }(U). \end{aligned}$$
(3.61)

Therefore, by substituting (3.61) into (3.58) and using \(\partial _x^2U_x=\partial _x\mathcal {U}+\mathcal {O}(|U|+|U_x|+|\mathcal {U}|)\), we deduce

$$\begin{aligned} I_1&= a\,\partial _x^2\left\{ J(U)\partial _x^2\mathcal {U}\right\} - a\,\sum _{\ell } \left( \partial _x^3\mathcal {U}, J(U)D_{\ell }(U)U_x \right) \nu _{\ell }(U) \nonumber \\&\quad - a\,\sum _{\ell } \left( \partial _x^2\mathcal {U}, \partial _x\left\{ J(U)D_{\ell }(U)U_x \right\} \right) \nu _{\ell }(U) \nonumber \\&\quad - a\,\sum _{\ell } \left( \partial _x^2\mathcal {U}, J(U)D_{\ell }(U)U_x \right) D_{\ell }(U)U_x \nonumber \\&\quad + a\,\sum _{n} (\partial _x^2\mathcal {U}, D_n(U)U_x)J(U)D_n(U)U_x \nonumber \\&\quad + a\,\sum _{n} (\partial _x\mathcal {U}, \partial _x\left\{ D_n(U)U_x\right\} )J(U)D_n(U)U_x \nonumber \\&\quad + a\,\sum _{n} (\partial _x\mathcal {U}, D_n(U)U_x)\partial _x\left\{ J(U)D_n(U)U_x\right\} \nonumber \\&\quad +a\,\sum _{n} \left( \partial _x\mathcal {U}, \partial _x \left\{ D_n(U)U_x \right\} \right) J(U)D_n(U)U_x \nonumber \\&\quad + a\,\sum _{n} \left( \mathcal {U}, D_n(U)\partial _x^2U_x \right) J(U)D_n(U)U_x \nonumber \\&\quad + a\,\sum _{n} (\partial _x\mathcal {U}, D_n(U)U_x) \partial _x\left\{ J(U)D_n(U)U_x \right\} \nonumber \\&\quad + a\,\sum _{n} (\mathcal {U}, D_n(U)U_x) J(U) D_n(U)\partial _x^2U_x \nonumber \\&\quad + a\,\sum _{\ell ,n} (\partial _x\mathcal {U},D_n(U)U_x) \left( J(U)D_n(U)U_x,D_{\ell }(U)U_x \right) \nu _{\ell }(U) \nonumber \\&\quad + a\,\sum _{k} \left( \partial _x\left\{ J(U)\partial _x^2\mathcal {U}\right\} , D_k(U)U_x \right) \nu _k(U) \nonumber \\&\quad - a\,\sum _{k,\ell } \left( \partial _x^2\mathcal {U}, J(U)D_{\ell }(U)U_x \right) \left( \nu _{\ell }(U), D_k(U)U_x \right) \nu _k(U) \nonumber \\&\quad + a\,\sum _{k,n} (\partial _x\mathcal {U}, D_n(U)U_x) \left( J(U)D_n(U)U_x, D_k(U)U_x \right) \nu _k(U) \nonumber \\&\quad + \mathcal {O} (|U|,|U_x|,|\mathcal {U}|) \nonumber \\&= a\,\partial _x^2\left\{ J(U)\partial _x^2\mathcal {U}\right\} - 2a\, F_1\big (\partial _x^3\mathcal {U}\big ) - a\,F_2\big (\partial _x^2\mathcal {U}\big ) + a\,F_3\big (\partial _x^2\mathcal {U}\big ) \nonumber \\&\quad + 2a\,F_4(\partial _x\mathcal {U}) + 2a\,F_5(\partial _x\mathcal {U}) +a\,F_6(\partial _x\mathcal {U}) + a\,F_7(\partial _x\mathcal {U}) \nonumber \\&\quad + \sum _k \mathcal {O} \left( |U|+|U_x|+|\mathcal {U}|+|\partial _x\mathcal {U}|+|\partial _x^2\mathcal {U}| \right) \nu _k(U) \nonumber \\&\quad + \mathcal {O} (|U|+|U_x|+|\mathcal {U}|), \end{aligned}$$
(3.62)

where, for any \(Y:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\),

$$\begin{aligned} F_1(Y)&=\sum _{k} \left( Y, J(U)D_{k}(U)U_x \right) \nu _{k}(U), \nonumber \\ F_2(Y)&=\sum _{k} \left( Y, J(U)D_{k}(U)U_x \right) D_{k}(U)U_x, \nonumber \\ F_3(Y)&=\sum _{k} (Y, D_k(U)U_x)J(U)D_k(U)U_x, \nonumber \\ F_4(Y)&=\sum _{k} (Y, \partial _x\left\{ D_k(U)U_x\right\} )J(U)D_k(U)U_x, \nonumber \\ F_5(Y)&=\sum _{k} (Y, D_k(U)U_x)\partial _x\left\{ J(U)D_k(U)U_x\right\} , \nonumber \\ F_6(Y)&=\sum _{k} \left( \mathcal {U}, D_k(U)Y \right) J(U)D_k(U)U_x, \nonumber \\ F_7(Y)&=\sum _{k} (\mathcal {U}, D_k(U)U_x) J(U) D_k(U)Y. \nonumber \end{aligned}$$

Combining (3.45), (3.47), (3.48), (3.51), (3.55), (3.57), and (3.62), we derive

$$\begin{aligned} \partial _x\mathcal {U}&= I_1+I_2+I_3+I_4+II+III+IV \nonumber \\&= a\,\partial _x^2\left\{ J(U)\partial _x^2\mathcal {U}\right\} - 2a\,F_1\left( \partial _x^3\mathcal {U}\right) + \lambda \, \partial _x\left\{ J(U)\partial _x\mathcal {U}\right\} \nonumber \\&\quad + (b+aS-aS)\, \partial _x \left\{ |U_x|^2 J(U)\partial _x\mathcal {U}\right\} + (c+aS)\, \left( \partial _x^2\mathcal {U},U_x\right) J(U)U_x \nonumber \\&\quad - a\,F_2\left( \partial _x^2\mathcal {U}\right) + a\,F_3\left( \partial _x^2\mathcal {U}\right) + 2a\,F_4 \left( \partial _x\mathcal {U}\right) + 2a\,F_5\left( \partial _x\mathcal {U}\right) \nonumber \\&\quad +a\,F_6(\partial _x\mathcal {U}) + a\,F_7(\partial _x\mathcal {U}) +(2b+c-2aS+2aS)\, (\partial _xU_x,U_x) J(U)\partial _x\mathcal {U}\nonumber \\&\quad + (2b+2c)\, (\partial _x\mathcal {U},U_x)J(U)\partial _xU_x + 3c\, (\partial _x\mathcal {U},\partial _xU_x)J(U)U_x \nonumber \\&\quad -aS\, \left( \partial _x(J(U))\partial _x\mathcal {U},U_x\right) U_x +aS\, |U_x|^2 \partial _x(J(U))\partial _x\mathcal {U}\nonumber \\&\quad +\lambda \, \partial _x(J(U))\partial _x\mathcal {U}+ r\left( U,U_x,\mathcal {U},\partial _x\mathcal {U},\partial _x^2\mathcal {U}\right) + \mathcal {O} (|U|+|U_x|+|\mathcal {U}|), \end{aligned}$$
(3.63)

where

$$\begin{aligned} r\left( U,U_x,\mathcal {U},\partial _x\mathcal {U},\partial _x^2\mathcal {U}\right) = \sum _k \mathcal {O} \left( |U|+|U_x|+|\mathcal {U}|+|\partial _x\mathcal {U}|+|\partial _x^2\mathcal {U}| \right) \nu _k(U). \nonumber \end{aligned}$$

3. Classical energy estimates for \(\Vert \mathcal {W}\Vert _{L^2(\mathbb {T};\mathbb {R}^d)}\) with the loss of derivatives

We compute \(\partial _t\mathcal {W}=\partial _t\mathcal {U}-\partial _t\mathcal {V}\) and next evaluate the classical energy estimate for \(\mathcal {W}\) in \(L^2\). Obviously, \(\mathcal {V}\) also satisfies (3.63) replacing \(\mathcal {U}\) with \(\mathcal {V}\). Hence, by using the mean value formula, we obtain

$$\begin{aligned} \partial _t\mathcal {W}&= a\,\partial _x^2\left\{ J(U)\partial _x^2\mathcal {W}\right\} - 2a\,F_1\left( \partial _x^3\mathcal {W}\right) + \lambda \, \partial _x\left\{ J(U)\partial _x\mathcal {W}\right\} \nonumber \\&\quad + b\, \partial _x \left\{ |U_x|^2 J(U)\partial _x\mathcal {W}\right\} + (c+aS)\, \left( \partial _x^2\mathcal {W},U_x\right) J(U)U_x \nonumber \\&\quad - a\,F_2\left( \partial _x^2\mathcal {W}\right) + a\,F_3\left( \partial _x^2\mathcal {W}\right) + 2a\,F_4(\partial _x\mathcal {W}) + 2a\,F_5(\partial _x\mathcal {W}) \nonumber \\&\quad +a\,F_6(\partial _x\mathcal {W}) + a\,F_7(\partial _x\mathcal {W}) +(2b+c)\, (\partial _xU_x,U_x) J(U)\partial _x\mathcal {W}\nonumber \\&\quad + (2b+2c)\, (\partial _x\mathcal {W},U_x)J(U)\partial _xU_x + 3c\, (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x \nonumber \\&\quad -aS\, \left( \partial _x(J(U))\partial _x\mathcal {W},U_x\right) U_x +aS\, |U_x|^2 \partial _x(J(U))\partial _x\mathcal {W}+\lambda \, \partial _x(J(U))\partial _x\mathcal {W}\nonumber \\&\quad + r\left( U,U_x,\mathcal {U},\partial _x\mathcal {U},\partial _x^2\mathcal {U}\right) - r\left( V,V_x,\mathcal {V},\partial _x\mathcal {V},\partial _x^2\mathcal {V}\right) \nonumber \\&\quad + \mathcal {O} (|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.64)

Note that \(F_1(\cdot ),\ldots ,F_7(\cdot )\) should be expressed globally, not by using local orthonormal frame. It is possible by using the second fundamental form on w(N) and the derivatives, or by following the argument in [18] to use the partition of unity on w(N). However, for simplicity and for better understanding, we will continue to use the local expression without loss of generality.

We now move on to the classical energy estimate for \(\Vert \mathcal {W}\Vert _{L^2}^2\). Since \(k\geqslant 6\), \(\mathcal {W}\in L^{\infty }(0,T;H^5)\cap C([0,T];H^4)\cap C^1([0,T];L^2)\). This together with (3.64) implies

(3.65)

Let us compute the RHS of the above term by term. First, by integrating by parts, it is immediate to observe that

Next, by the Cauchy–Schwartz inequality, it holds that

for some \(C>0\). Hereafter, various positive constants depending on \(\Vert u_x\Vert _{L^{\infty }(0,T;H^6)}\) and \(\Vert v_x\Vert _{L^{\infty }(0,T;H^6)}\) will be denoted by the same C without any comments. Besides, we use the notation D(t) so that the square is defined by

$$\begin{aligned}D(t)^2=\Vert Z\Vert _{L^2}^2 +\Vert Z_x\Vert _{L^2}^2 +\Vert \mathcal {W}\Vert _{L^2}^2. \end{aligned}$$

Next, by noting that

$$\begin{aligned}&r(U,U_x,\mathcal {U},\partial _x\mathcal {U},\partial _x^2\mathcal {U}) - r(V,V_x,\mathcal {V},\partial _x\mathcal {V},\partial _x^2\mathcal {V}) \nonumber \\&= \sum _k \mathcal {O} \left( |Z|+|Z_x|+|\mathcal {W}|+|\partial _x\mathcal {W}|+|\partial _x^2\mathcal {W}| \right) \nu _k(U) \nonumber \\&\quad +\sum _k \mathcal {O} \left( |U|+|U_x|+|\mathcal {U}|+|\partial _x\mathcal {U}|+|\partial _x^2\mathcal {U}| \right) (\nu _k(U)-\nu _k(V)), \nonumber \end{aligned}$$

we use (3.8) obtained in Lemma 3.1, \(\partial _xZ_x=\mathcal {W}+\mathcal {O}(|Z|+|Z_x|)\), and the integration by parts, to obtain

$$\begin{aligned}&\left\langle r(U,U_x,\mathcal {U},\partial _x\mathcal {U},\partial _x^2\mathcal {U}) - r(V,V_x,\mathcal {V},\partial _x\mathcal {V},\partial _x^2\mathcal {V}), \mathcal {W} \right\rangle \nonumber \\&\le \left\langle \sum _k \mathcal {O} \left( |Z|+|Z_x|+|\mathcal {W}|+|\partial _x\mathcal {W}|+|\partial _x^2\mathcal {W}| \right) \nu _k(U), \mathcal {W} \right\rangle +C\,D(t)^2 \nonumber \\&= \int _{\mathbb {T}} \sum _k \mathcal {O} \left( |Z|+|Z_x|+|\mathcal {W}|+|\partial _x\mathcal {W}|+|\partial _x^2\mathcal {W}| \right) \mathcal {O}(|Z|)\,\mathrm{d}x +C\,D(t)^2 \nonumber \\&= \int _{\mathbb {T}} \sum _k \mathcal {O} \left( |Z|+|Z_x|+|\mathcal {W}| \right) \mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|)\,\mathrm{d}x +C\,D(t)^2 \nonumber \\&\le C\,D(t)^2. \end{aligned}$$
(3.66)

In the next computation, the Kähler condition on (NJg) plays the crucial roles. Indeed, we apply (3.13) with \(Y=\partial _x\mathcal {W}\) and use (3.9) to obtain

$$\begin{aligned} \partial _x(J(U))\partial _x\mathcal {W}=&\sum _k \left( \partial _x\mathcal {W},J(U)D_k(U)U_x\right) \nu _k(U) \nonumber \\&-\sum _k (\partial _x\mathcal {W},\nu _k(U))J(U)D_k(U)U_x \nonumber \\ =&\sum _k \left( \partial _x\mathcal {W},J(U)D_k(U)U_x\right) \nu _k(U) +\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.67)

Using (3.67) and (3.8), we obtain

$$\begin{aligned} (\partial _x(J(U))\partial _x\mathcal {W},U_x) =&\sum _k \left( \partial _x\mathcal {W},J(U)D_k(U)U_x\right) (\nu _k(U), U_x)\nonumber \\&+\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|) \nonumber \\ =&\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|), \end{aligned}$$
(3.68)
$$\begin{aligned} (\partial _x(J(U))\partial _x\mathcal {W},\mathcal {W}) =&\sum _k \left( \partial _x\mathcal {W},J(U)D_k(U)U_x\right) (\nu _k(U), \mathcal {W})\nonumber \\&+\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|) \nonumber \\ =&\sum _k \left( \partial _x\mathcal {W},J(U)D_k(U)U_x\right) \mathcal {O}(|Z|) + \mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.69)

Thus, by using (3.68) and the Cauchy–Schwartz inequality, we have

$$\begin{aligned} -aS\, \left\langle \left( \partial _x(J(U))\partial _x\mathcal {W},U_x\right) U_x, \mathcal {W} \right\rangle&\le C\,D(t)^2. \nonumber \end{aligned}$$

In the same manner, by using (3.69) and the Cauchy–Schwartz inequality, together with the integration by parts, we deduce

$$\begin{aligned} aS\, \left\langle |U_x|^2 \partial _x(J(U))\partial _x\mathcal {W}, \mathcal {W} \right\rangle +\lambda \, \left\langle \partial _x(J(U))\partial _x\mathcal {W}, \mathcal {W} \right\rangle&\le C\,D(t)^2. \nonumber \end{aligned}$$

Combining them, we obtain

(3.70)

where

In what follows, we need to compute more carefully. Let us consider \(R_1\). We start by integrating by parts to obtain

$$\begin{aligned} R_1&= 2a\, \left\langle \sum _{k} \left( \partial _x^2\mathcal {W}, \partial _x\left\{ J(U)D_{k}(U)U_x\right\} \right) \nu _{k}(U) , \mathcal {W} \right\rangle \nonumber \\&\quad + 2a\, \left\langle \sum _{k} \left( \partial _x^2\mathcal {W}, J(U)D_{k}(U)U_x \right) D_k(U)U_x , \mathcal {W} \right\rangle \nonumber \\&\quad + 2a\, \left\langle \sum _{k} \left( \partial _x^2\mathcal {W}, J(U)D_{k}(U)U_x \right) \nu _{k}(U) , \partial _x\mathcal {W} \right\rangle . \nonumber \end{aligned}$$

By applying (3.8) to the first term of the RHS of the above and by applying (3.9) to the third term of the RHS of the above, we have

Here, by integrating by parts, the first and third terms are bounded by \(C \,D(t)^2 \). Therefore, we obtain

$$\begin{aligned} R_1&\le - 2a\, \left\langle \sum _{k} \left( \partial _x^2\mathcal {W}, J(U)D_{k}(U)U_x \right) D_k(U)Z_x , \mathcal {V} \right\rangle +C\,D(t)^2. \nonumber \end{aligned}$$

Furthermore, by using integration by parts, \(\partial _xZ_x=\mathcal {W}+\mathcal {O}(|Z|+|Z_x|)\), (3.3), and (3.11), we deduce

(3.71)

where

$$\begin{aligned} R_{11}&=2a\, \left\langle \sum _{k} \left( J(U)\mathcal {W}, D_{k}(U)U_x \right) P(U)D_k(U)\mathcal {U}, \partial _x\mathcal {W} \right\rangle , \nonumber \\ R_{12}&=2a\, \left\langle \sum _{k} \left( J(U)\mathcal {W}, D_{k}(U)U_x \right) N(U)D_k(U)\mathcal {U}, \partial _x\mathcal {W} \right\rangle . \nonumber \end{aligned}$$

For \(R_{12}\), recall (3.9) to observe

$$\begin{aligned}(N(U)D_k(U)\mathcal {U}, \partial _x\mathcal {W})= & {} \sum _{\ell } (D_k(U)\mathcal {U},\nu _{\ell }(U))(\nu _{\ell }(U),\partial _x\mathcal {W}) \\= & {} \mathcal {O} \left( |Z|+|Z_x|+|\mathcal {W}| \right) . \end{aligned}$$

This shows \(R_{12}\le C\,D(t)^2\). For \(R_{11}\), by using (3.2) and (3.11), we obtain

$$\begin{aligned} R_{11} =2a\, \left\langle \sum _{k} \left( J(U)\mathcal {W}, D_{k}(U)U_x \right) D_k(U)P(U)\partial _x\mathcal {W}, \mathcal {U} \right\rangle . \end{aligned}$$

Since \((N(U)D_k(U)P(U)\partial _x\mathcal {W}, \mathcal {U})=0\), we have

$$\begin{aligned} R_{11} =2a\, \left\langle \sum _{k} \left( J(U)\mathcal {W}, D_{k}(U)U_x \right) P(U)D_k(U)P(U)\partial _x\mathcal {W}, \mathcal {U} \right\rangle . \end{aligned}$$

Applying (3.23) to Lemma 3.5 with \(\mathrm{d}w_u(Y_1)=P(U)\partial _x\mathcal {W}\), \(\mathrm{d}w_u(Y_2)=U_x\), and \(\mathrm{d}w_u(Y_3)=J(U)\mathcal {W}\), we obtain

$$\begin{aligned} R_{11}&= 2a\, \left\langle \sum _{k} \left( J(U)\mathcal {W}, D_{k}(U)P(U)\partial _x\mathcal {W}\right) P(U)D_k(U)U_x, \mathcal {U} \right\rangle \nonumber \\&\quad +2aS\, \left\langle (J(U)\mathcal {W}, U_x)P(U)\partial _x\mathcal {W}, \mathcal {U} \right\rangle -2aS \left\langle (J(U)\mathcal {W}, P(U)\partial _x\mathcal {W})U_x, \mathcal {U} \right\rangle \nonumber \\&=:R_{111} +R_{112}+R_{113}. \nonumber \end{aligned}$$

Here we recall (3.9) to obtain

$$\begin{aligned} N(U)\partial _x\mathcal {W}=\sum _k (\partial _x\mathcal {W},\nu _k(U))\nu _k(U) = \mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.72)

This implies \(P(U)\partial _x\mathcal {W}=\partial _x\mathcal {W}+\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|).\) Using this (3.2), (3.3) and \(P(U)\mathcal {U}=\mathcal {U}\), we obtain

$$\begin{aligned} R_{111}&= -2a\, \left\langle \sum _{k} \left( \mathcal {W}, J(U)D_{k}(U)P(U)\partial _x\mathcal {W}\right) D_k(U)U_x, P(U)\mathcal {U} \right\rangle \nonumber \\&\le -2a\, \left\langle \sum _{k} \left( \mathcal {W}, J(U)D_{k}(U)\partial _x\mathcal {W}\right) D_k(U)U_x, \mathcal {U} \right\rangle +C\,D(t)^2 \nonumber \\&= -2a\, \left\langle \sum _{k} (\mathcal {U}, D_k(U)U_x) J(U)D_{k}(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle +C\,D(t)^2. \nonumber \end{aligned}$$

In the same way, using (3.2), (3.3), and \( \mathcal {U}=\partial _xU_x+ \sum _{\ell }(U_x,D_{\ell }(U)U_x)\nu _{\ell }(U)\), we obtain

$$\begin{aligned} R_{112}&\le -2aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x,\mathcal {W} \right\rangle +C\,D(t)^2, \nonumber \\ R_{113}&\le 2aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle +C\,D(t)^2. \nonumber \end{aligned}$$

Combining them, we obtain

$$\begin{aligned} R_1&=R_{111}+R_{112}+R_{113}+R_{12}+C\,D(t)^2 \nonumber \\&\le -2a\, \left\langle \sum _{k} (\mathcal {U}, D_k(U)U_x) J(U)D_{k}(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad -2aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x,\mathcal {W} \right\rangle + 2aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \nonumber \\&\quad +C\,D(t)^2. \end{aligned}$$
(3.73)

The first term of the RHS of (3.73) is cancelled with the same term appearing from the computation of \(R_6+R_7\).

We compute \(R_6+R_7 = a\,\left\langle F_6(\partial _x\mathcal {W}),\mathcal {W} \right\rangle +a\,\left\langle F_7(\partial _x\mathcal {W}),\mathcal {W} \right\rangle \). By noting \(J(U)=J(U)P(U)\) and applying (3.23) with \(\mathrm{d}w_u(Y_1)=U_x\), \(\mathrm{d}w_u(Y_2)=P(U)\partial _x\mathcal {W}\), and \(\mathrm{d}w_u(Y_3)=\mathcal {U}\), we obtain

$$\begin{aligned} F_6(\partial _x\mathcal {W})&= \sum _{k} \left( \mathcal {U}, D_k(U)\partial _x\mathcal {W}\right) J(U)D_k(U)U_x \nonumber \\&= J(U)\sum _{k} \left( \mathcal {U}, D_k(U)\partial _x\mathcal {W}\right) P(U)D_k(U)U_x \nonumber \\&= J(U)\sum _{k} \left( \mathcal {U}, D_k(U)P(U)\partial _x\mathcal {W}\right) P(U)D_k(U)U_x \nonumber \\&\quad + J(U)\sum _{k} \left( \mathcal {U}, D_k(U)N(U)\partial _x\mathcal {W}\right) P(U)D_k(U)U_x \nonumber \\&= J(U)\sum _{k} \left( \mathcal {U}, D_k(U)U_x \right) P(U)D_k(U)P(U)\partial _x\mathcal {W}\nonumber \\&\quad +S\,J(U) \left\{ (\mathcal {U},P(U)\partial _x\mathcal {W})U_x -(\mathcal {U},U_x)P(U)\partial _x\mathcal {W}\right\} \nonumber \\&\quad + J(U)\sum _{k} \left( \mathcal {U}, D_k(U)N(U)\partial _x\mathcal {W}\right) P(U)D_k(U)U_x. \nonumber \end{aligned}$$

Furthermore, we use \(J(U)=J(U)P(U)\) and (3.72) to obtain

$$\begin{aligned} F_6(\partial _x\mathcal {W})&= \sum _{k} \left( \mathcal {U}, D_k(U)U_x \right) J(U)D_k(U)P(U)\partial _x\mathcal {W}\nonumber \\&\quad +S\,(\partial _x\mathcal {W},\mathcal {U})J(U)U_x -S\,(\mathcal {U},U_x)J(U)\partial _x\mathcal {W}+\mathcal {O} (|Z|+|Z_x|+|\mathcal {W}) \nonumber \\&= \sum _{k} \left( \mathcal {U}, D_k(U)U_x \right) J(U)D_k(U)\partial _x\mathcal {W}\nonumber \\&\quad +S\,(\partial _x\mathcal {W},\partial _xU_x)J(U)U_x -S\,(\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}\nonumber \\&\quad +\mathcal {O} (|Z|+|Z_x|+|\mathcal {W}) \nonumber \\&= F_7(\partial _x\mathcal {W}) +S\,(\partial _x\mathcal {W},\partial _xU_x)J(U)U_x -S\,(\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}\nonumber \\&\quad +\mathcal {O} (|Z|+|Z_x|+|\mathcal {W}). \nonumber \end{aligned}$$

Hence, we obtain

$$\begin{aligned} R_6 + R_7&\le 2a\, \left\langle \sum _{k} \left( \mathcal {U}, D_k(U)U_x \right) J(U)D_k(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \nonumber \\&\quad +aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x ,\mathcal {W} \right\rangle \nonumber \\&\quad -aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle +C\,D(t)^2. \end{aligned}$$
(3.74)

Combining (3.73) and (3.74) and using (3.11), we have

$$\begin{aligned}&R_1 + R_6 + R_7 \nonumber \\&\le -aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x ,\mathcal {W} \right\rangle +aS\, \left\langle (U_x,\partial _xU_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle +C\,D(t)^2. \end{aligned}$$
(3.75)

Next, we compute \(R_2+R_3=-a\left\langle F_2(\partial _x^2\mathcal {W}),\mathcal {W} \right\rangle +a\left\langle F_3(\partial _x^2\mathcal {W}),\mathcal {W} \right\rangle \). As in deriving (3.73), we use (3.3), (3.11), and (3.23) with \(\mathrm{d}w_u(Y_1)=U_x\), \(\mathrm{d}w_u(Y_2)=J(U)\partial _x^2\mathcal {W}\), and \(\mathrm{d}w_u(Y_3)=U_x\) to deduce

$$\begin{aligned} F_2\left( \partial _x^2\mathcal {W}\right)&= \sum _k \left( \partial _x^2\mathcal {W},J(U)D_k(U)U_x\right) D_k(U)U_x\nonumber \\&= \sum _k \left( \partial _x^2\mathcal {W},J(U)D_k(U)U_x\right) N(U)D_k(U)U_x \nonumber \\&\quad + \sum _k \left( \partial _x^2\mathcal {W},J(U)D_k(U)U_x\right) P(U)D_k(U)U_x \nonumber \\&= \sum _k \left( \partial _x^2\mathcal {W},J(U)D_k(U)U_x\right) N(U)D_k(U)U_x \nonumber \\&\quad - \sum _k \left( U_x, D_k(U)J(U)\partial _x^2\mathcal {W}\right) P(U)D_k(U)U_x \nonumber \\&= \sum _k \left( \partial _x^2\mathcal {W},J(U)D_k(U)U_x\right) N(U)D_k(U)U_x \nonumber \\&\quad - \sum _k (U_x, D_k(U)U_x)P(U)D_k(U)J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad - S\, \left\{ \left( U_x,J(U)\partial _x^2\mathcal {W}\right) U_x -(U_x,U_x)J(U)\partial _x^2\mathcal {W}\right\} \nonumber \\&= -\sum _k (U_x, D_k(U)U_x)D_k(U)J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +S\,\left( \partial _x^2\mathcal {W},J(U)U_x\right) U_x +S\,|U_x|^2J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +\sum _k (U_x, D_k(U)U_x)N(U)D_k(U)J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad + \sum _k \left( \partial _x^2\mathcal {W},J(U)D_k(U)U_x\right) N(U)D_k(U)U_x \nonumber \\&= -\sum _k (U_x, D_k(U)U_x)D_k(U)J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +S\,\left( \partial _x^2\mathcal {W},J(U)U_x\right) U_x +S\,|U_x|^2J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +\sum _{\ell } \mathcal {O} \left( |\partial _x^2\mathcal {W}| \right) \nu _{\ell }(U), \end{aligned}$$
(3.76)

and in the same way we use (3.3), (3.11), and (3.23) with \(\mathrm{d}w_u(Y_1)=U_x\), \(\mathrm{d}w_u(Y_2)=\partial _x^2\mathcal {W}\), and \(\mathrm{d}w_u(Y_3)=U_x\) to deduce

$$\begin{aligned} F_3\left( \partial _x^2\mathcal {W}\right)&= \sum _{k} \left( \partial _x^2\mathcal {W}, D_k(U)U_x\right) J(U)D_k(U)U_x\nonumber \\&= J(U)\sum _{k} \left( U_x, D_k(U)\partial _x^2\mathcal {W}\right) P(U)D_k(U)U_x \nonumber \\&= J(U)\sum _{k} \left( U_x, D_k(U)N(U)\partial _x^2\mathcal {W}\right) P(U)D_k(U)U_x \nonumber \\&\quad + J(U)\sum _{k} \left( U_x, D_k(U)P(U)\partial _x^2\mathcal {W}\right) P(U)D_k(U)U_x\nonumber \\&= J(U)\sum _{k} \left( U_x, D_k(U)N(U)\partial _x^2\mathcal {W}\right) P(U)D_k(U)U_x \nonumber \\&\quad + J(U)\sum _{k} (U_x, D_k(U)U_x)P(U)D_k(U)P(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +S\,J(U) \left\{ \left( U_x, P(U)\partial _x^2\mathcal {W}\right) U_x-(U_x,U_x)P(U)\partial _x^2\mathcal {W}\right\} \nonumber \\&= \sum _{k} \left( U_x, D_k(U)N(U)\partial _x^2\mathcal {W}\right) J(U)D_k(U)U_x \nonumber \\&\quad + \sum _{k} (U_x, D_k(U)U_x)J(U)D_k(U)P(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +S\,\left( \partial _x^2\mathcal {W},U_x\right) J(U)U_x -S\,|U_x|^2J(U)\partial _x^2\mathcal {W}\nonumber \\&= \sum _{k} (U_x, D_k(U)U_x)J(U)D_k(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +S\,\left( \partial _x^2\mathcal {W},U_x\right) J(U)U_x -S\,|U_x|^2J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad + \sum _{k} \left( U_x, D_k(U)N(U)\partial _x^2\mathcal {W}\right) J(U)D_k(U)U_x \nonumber \\&\quad - \sum _{k} (U_x, D_k(U)U_x)J(U)D_k(U)N(U)\partial _x^2\mathcal {W}. \end{aligned}$$
(3.77)

Here, we use (3.10) to obtain

$$\begin{aligned} N(U)\partial _x^2\mathcal {W}&= \sum _{\ell } \left( \partial _x^2\mathcal {W},\nu _{\ell }(U)\right) \nu _{\ell }(U)\nonumber \\&= -2\sum _{\ell } \left( \partial _x\mathcal {W}, D_{\ell }(U)U_x\right) \nu _{\ell }(U) +\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.78)

By substituting (3.78) into the fourth and fifth terms of the RHS of (3.77), we have

$$\begin{aligned} F_3(\partial _x^2\mathcal {W})&= \sum _{k} (U_x, D_k(U)U_x)J(U)D_k(U)\partial _x^2\mathcal {W}\nonumber \\&\quad +S\,\left( \partial _x^2\mathcal {W},U_x\right) J(U)U_x -S\,|U_x|^2J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad -2 \sum _{k,\ell } \left( \partial _x\mathcal {W}, D_{\ell }(U)U_x\right) \left( U_x, D_k(U)\nu _{\ell }(U)\right) J(U)D_k(U)U_x \nonumber \\&\quad +2 \sum _{k,\ell } \left( \partial _x\mathcal {W}, D_{\ell }(U)U_x\right) (U_x, D_k(U)U_x)J(U)D_k(U)\nu _{\ell }(U) \nonumber \\&\quad + \mathcal {O} (|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.79)

Thus, from (3.76), (3.79), and (3.28), it follows that

$$\begin{aligned}&-F_2\left( \partial _x^2\mathcal {W}\right) +F_3\left( \partial _x^2\mathcal {W}\right) \nonumber \\&= \sum _k (U_x, D_k(U)U_x)D_k(U)J(U)\partial _x^2\mathcal {W}+ \sum _{k} (U_x, D_k(U)U_x)J(U)D_k(U)\partial _x^2\mathcal {W}\nonumber \\&\quad -S\,\left( \partial _x^2\mathcal {W},J(U)U_x\right) U_x +S\,\left( \partial _x^2\mathcal {W},U_x\right) J(U)U_x -2S\,|U_x|^2J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad -2 \sum _{k,\ell } \left( \partial _x\mathcal {W}, D_{\ell }(U)U_x\right) (U_x, D_k(U)\nu _{\ell }(U))J(U)D_k(U)U_x \nonumber \\&\quad +2 \sum _{k,\ell } \left( \partial _x\mathcal {W}, D_{\ell }(U)U_x\right) (U_x, D_k(U)U_x)J(U)D_k(U)\nu _{\ell }(U) \nonumber \\&\quad +\sum _{\ell } \mathcal {O} \left( |\partial _x^2\mathcal {W}| \right) \nu _{\ell }(U) + \mathcal {O} (|Z|+|Z_x|+|\mathcal {W}|) \nonumber \\&= \sum _k (U_x, D_k(U)U_x)(D_k(U)J(U)+J(U)D_k(U))\partial _x^2\mathcal {W}\nonumber \\&\quad -S\,|U_x|^2J(U)\partial _x^2\mathcal {W}\nonumber \\&\quad -2 \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)\nu _{\ell }(U))J(U)D_k(U)U_x \nonumber \\&\quad +2 \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)U_x)J(U)D_k(U)\nu _{\ell }(U) \nonumber \\&\quad +\sum _{\ell } \mathcal {O} \left( |\partial _x^2\mathcal {W}| \right) \nu _{\ell }(U) + \mathcal {O} (|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.80)

Therefore, from (3.80) and

$$\begin{aligned} |U_x|^2J(U)\partial _x^2\mathcal {W}= & {} \partial _x \left\{ |U_x|^2J(U)\partial _x\mathcal {W}\right\} -2(\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}\nonumber \\&-|U_x|^2\partial _x(J(U))\partial _x\mathcal {W}, \end{aligned}$$

we observe that \(R_2+R_3 =-a\left\langle F_2(\partial _x^2\mathcal {W}),\mathcal {W} \right\rangle +a\left\langle F_3(\partial _x^2\mathcal {W}),\mathcal {W} \right\rangle \) is evaluated as follows:

$$\begin{aligned} R_2 + R_3&\le a\, \left\langle \partial _x\biggl \{ \sum _k (U_x, D_k(U)U_x)(D_k(U)J(U)+J(U)D_k(U))\partial _x\mathcal {W}\biggr \}, \mathcal {W}\right\rangle \nonumber \\&\quad -a\, \left\langle \partial _x\biggl \{ \sum _k (U_x, D_k(U)U_x)(D_k(U)J(U)+J(U)D_k(U)) \biggr \} \partial _x\mathcal {W}, \mathcal {W}\right\rangle \nonumber \\&\quad -aS\, \left\langle \partial _x\biggl \{ |U_x|^2J(U)\partial _x\mathcal {W}\biggr \}, \mathcal {W} \right\rangle +2aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \nonumber \\&\quad +aS\, \left\langle |U_x|^2\partial _x(J(U))\partial _x\mathcal {W},\mathcal {W} \right\rangle \nonumber \\&\quad -2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)\nu _{\ell }(U))J(U)D_k(U)U_x,\mathcal {W} \right\rangle \nonumber \\&\quad +2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)U_x)J(U)D_k(U)\nu _{\ell }(U), \mathcal {W} \right\rangle \nonumber \\&\quad + \left\langle \sum _{\ell } \mathcal {O} \left( |\partial _x^2\mathcal {W}|\right) \nu _{\ell }(U), \mathcal {W} \right\rangle \nonumber \\&\quad +C\,D(t)^2. \end{aligned}$$
(3.81)

Note here that

$$\begin{aligned} ((J(U)D_k(U)+D_k(U)J(U))Y_1,Y_2) = -(Y_1, (J(U)D_k(U)+D_k(U)J(U))Y_2) \end{aligned}$$

holds for any \(Y_1,Y_2:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\). This implies that the first term of the RHS of (3.81) vanishes. Indeed, the integration by parts yields

$$\begin{aligned}&a\,\left\langle \partial _x\biggl \{ \sum _k (U_x, D_k(U)U_x)(D_k(U)J(U)+J(U)D_k(U))\partial _x\mathcal {W}\biggr \}, \mathcal {W}\right\rangle \nonumber \\&= -a\,\left\langle \sum _k (U_x, D_k(U)U_x)(D_k(U)J(U)+J(U)D_k(U))\partial _x\mathcal {W}, \partial _x\mathcal {W}\right\rangle&=0. \nonumber \end{aligned}$$

In addition, the third term of the RHS of (3.81) vanishes by integrating by parts. Besides, due to the presence of N(U), we can bound the eighth term of the RHS of (3.81) by \(C\,D(t)^2\) using the same argument to show (3.66). Consequently, we derive

$$\begin{aligned} R_2+R_3&\le -a\, \left\langle \partial _x\biggl \{ \sum _k (U_x, D_k(U)U_x)(D_k(U)J(U)+J(U)D_k(U)) \biggr \} \partial _x\mathcal {W}, \mathcal {W}\right\rangle \nonumber \\&\quad +2aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \nonumber \\&\quad -2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)\nu _{\ell }(U))J(U)D_k(U)U_x,\mathcal {W} \right\rangle \nonumber \\&\quad +2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)U_x)J(U)D_k(U)\nu _{\ell }(U), \mathcal {W} \right\rangle \nonumber \\&\quad + C\,D(t)^2. \end{aligned}$$
(3.82)

The third and fourth terms and the bad part of the first term of the RHS of (3.82) will be cancelled with the same term appearing in the computation of \(R_4+R_5\).

Let us next compute \(R_4+R_5\). To begin with, we introduce \(T_1(U)\) which is defined by

$$\begin{aligned} T_1(U)Y&= \sum _k (Y,D_k(U)U_x)J(U)D_k(U)U_x \nonumber \end{aligned}$$

for any \(Y:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\). Substituting this with \(Y=\partial _x^2\mathcal {W}\) into the RHS of \(\partial _x(T_1(U))\partial _x\mathcal {W}=\partial _x\left\{ T_1(U)\partial _x\mathcal {W}\right\} -T_1(U)\partial _x^2\mathcal {W}\), we can write

$$\begin{aligned} R_4 + R_5&= 2a\, \left\langle \partial _x(T_1(U))\partial _x\mathcal {W},\mathcal {W} \right\rangle . \end{aligned}$$
(3.83)

On the other hand, using (3.23) with \(\mathrm{d}w_u(Y_1)=U_x\), \(\mathrm{d}w_u(Y_2)=P(U)Y\), and \(\mathrm{d}w_u(Y_3)=U_x\), we find that \(T_1(U)Y\) has another expression as follows:

$$\begin{aligned} T_1(U)Y&= J(U)\sum _k (Y,D_k(U)U_x)P(U)D_k(U)U_x \nonumber \\&= J(U)\sum _k (P(U)Y,D_k(U)U_x)P(U)D_k(U)U_x \nonumber \\&\quad + J(U)\sum _k (N(U)Y,D_k(U)U_x)P(U)D_k(U)U_x \nonumber \\&= J(U)\sum _k (U_x, D_k(U)P(U)Y)P(U)D_k(U)U_x \nonumber \\&\quad + J(U)\sum _k (N(U)Y,D_k(U)U_x)P(U)D_k(U)U_x \nonumber \\&= J(U)\sum _k (U_x, D_k(U)U_x)P(U)D_k(U)P(U)Y \nonumber \\&\quad + S\,J(U) \left\{ (U_x,P(U)Y)U_x -|U_x|^2P(U)Y \right\} \nonumber \\&\quad + J(U)\sum _k (N(U)Y,D_k(U)U_x)P(U)D_k(U)U_x \nonumber \\&= \sum _k (U_x, D_k(U)U_x)J(U)D_k(U)P(U)Y \nonumber \\&\quad + S\,(U_x,P(U)Y)J(U)U_x -S|U_x|^2J(U)Y \nonumber \\&\quad + \sum _k (N(U)Y,D_k(U)U_x)J(U)D_k(U)U_x \nonumber \\&= \sum _k (U_x, D_k(U)U_x)J(U)D_k(U)Y \nonumber \\&\quad -\sum _k (U_x, D_k(U)U_x)J(U)D_k(U)N(U)Y \nonumber \\&\quad + S\,(Y,U_x)J(U)U_x -S|U_x|^2J(U)Y \nonumber \\&\quad + \sum _k (N(U)Y,D_k(U)U_x)J(U)D_k(U)U_x \nonumber \end{aligned}$$

for any \(Y:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\). If we adopt this formulation, we have

$$\begin{aligned} \partial _x(T_1(U))Y&= \partial _x\left\{ \sum _k (U_x, D_k(U)U_x)J(U)D_k(U) \right\} Y \nonumber \\&\quad -\partial _x\left\{ \sum _k (U_x, D_k(U)U_x)J(U)D_k(U)\right\} N(U)Y \nonumber \\&\quad -\sum _k (U_x, D_k(U)U_x)J(U)D_k(U)\partial _x(N(U))Y \nonumber \\&\quad +S\,(Y,\partial _xU_x)J(U)U_x +S\,(Y,U_x)\partial _x(J(U))U_x +S\,(Y,U_x)J(U)\partial _xU_x \nonumber \\&\quad -2S(\partial _xU_x,U_x)J(U)Y-S|U_x|^2\partial _x(J(U))Y \nonumber \\&\quad + \sum _k (\partial _x(N(U))Y,D_k(U)U_x)J(U)D_k(U)U_x \nonumber \\&\quad + \sum _k (N(U)Y,\partial _x\left\{ D_k(U)U_x\right\} )J(U)D_k(U)U_x \nonumber \\&\quad + \sum _k (N(U)Y,D_k(U)U_x)\partial _x\left\{ J(U)D_k(U)U_x\right\} . \end{aligned}$$
(3.84)

By substituting (3.84) into (3.83), we have

$$\begin{aligned} R_4 + R_5&= 2a\, \left\langle \partial _x\biggl \{ \sum _k (U_x, D_k(U)U_x)J(U)D_k(U) \biggr \} \partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad -2a\, \left\langle \partial _x \biggl \{ \sum _k (U_x, D_k(U)U_x)J(U)D_k(U) \biggr \} N(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad -2a\, \left\langle \sum _k (U_x, D_k(U)U_x)J(U)D_k(U)\partial _x(N(U))\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad +2aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x, \mathcal {W} \right\rangle +2aS\, \left\langle (\partial _x\mathcal {W},U_x)\partial _x(J(U))U_x, \mathcal {W} \right\rangle \nonumber \\&\quad +2aS\, \left\langle (\partial _x\mathcal {W},U_x)J(U)\partial _xU_x, \mathcal {W} \right\rangle -4aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad -2aS\, \left\langle |U_x|^2\partial _x(J(U))\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad + 2a\, \left\langle \sum _k (\partial _x(N(U))\partial _x\mathcal {W},D_k(U)U_x)J(U)D_k(U)U_x,\mathcal {W} \right\rangle \nonumber \\&\quad + 2a\, \left\langle \sum _k (N(U)\partial _x\mathcal {W},\partial _x\left\{ D_k(U)U_x\right\} )J(U)D_k(U)U_x,\mathcal {W} \right\rangle \nonumber \\&\quad + 2a\, \left\langle \sum _k (N(U)\partial _x\mathcal {W},D_k(U)U_x)\partial _x\left\{ J(U)D_k(U)U_x\right\} , \mathcal {W} \right\rangle . \end{aligned}$$
(3.85)

The second, tenth, and eleventh terms of the RHS of (3.85) are bounded by \(C\,D(t)^2\), in view of (3.72). For the fifth term of the RHS of (3.85), we use (3.13) and \((U_x,\nu _k(U))=0\) to obtain

$$\begin{aligned} \partial _x(J(U))U_x&= \sum _k \left( U_x,J(U)D_k(U)U_x\right) \nu _k(U), \end{aligned}$$
(3.86)

which combined with (3.8) implies \((\partial _x(J(U))U_x,\mathcal {W})=\mathcal {O}(|Z|)\). Therefore, using integration by parts, the fifth term of the RHS of (3.85) is bounded by \(C\,D(t)^2\). In the same way, we use (3.13), (3.8), and (3.9) to obtain

$$\begin{aligned}&(\partial _x(J(U))\partial _x\mathcal {W}, \mathcal {W}) \nonumber \\&= \sum _k \left( \partial _x\mathcal {W},J(U)D_k(U)U_x\right) (\nu _k(U),\mathcal {W}) -\sum _k (\partial _x\mathcal {W},\nu _k(U))(J(U)D_k(U)U_x,\mathcal {W}) \nonumber \\&= \sum _k \left( \partial _x\mathcal {W},J(U)D_k(U)U_x\right) \mathcal {O}(|Z|) + \mathcal {O} \left( (|Z|+|Z_x|+|\mathcal {W}|)|\mathcal {W}|\right) . \nonumber \end{aligned}$$

Thus the integration by parts shows that the eighth term of the RHS of (3.85) is bounded by \(C\,D(t)^2\). For, the third and ninth terms of the RHS of (3.85), in view of (3.72), we have

$$\begin{aligned} \partial _x(N(U))\partial _x\mathcal {W}&= \sum _{\ell } (\partial _x\mathcal {W},D_{\ell }(U)U_x)\nu _{\ell }(U) + \sum _{\ell } (\partial _x\mathcal {W},\nu _{\ell }(U))D_{\ell }(U)U_x \nonumber \\&= \sum _{\ell } (\partial _x\mathcal {W},D_{\ell }(U)U_x)\nu _{\ell }(U) + \mathcal {O} (|Z|+|Z_x|+|\mathcal {W}|), \nonumber \end{aligned}$$

which implies

$$\begin{aligned}&-2a\, \left\langle \sum _k (U_x, D_k(U)U_x)J(U)D_k(U)\partial _x(N(U))\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad \le -2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)U_x) J(U)D_k(U)\nu _{\ell }(U), \mathcal {W} \right\rangle + C\,D(t)^2, \nonumber \\ \end{aligned}$$

and

$$\begin{aligned}&2a\, \left\langle \sum _k (\partial _x(N(U))\partial _x\mathcal {W},D_k(U)U_x)J(U)D_k(U)U_x,\mathcal {W} \right\rangle \nonumber \\&\quad \le 2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (\nu _{\ell }(U),D_k(U)U_x)J(U)D_k(U)U_x,\mathcal {W} \right\rangle +C\,D(t)^2. \nonumber \end{aligned}$$

Combining them, we derive

$$\begin{aligned} R_4 + R_5&\le 2a\, \left\langle \partial _x\biggl \{ \sum _k (U_x, D_k(U)U_x)J(U)D_k(U) \biggr \} \partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad -2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (U_x, D_k(U)U_x) J(U)D_k(U)\nu _{\ell }(U), \mathcal {W} \right\rangle \nonumber \\&\quad + 2a\, \left\langle \sum _{k,\ell } (\partial _x\mathcal {W}, D_{\ell }(U)U_x) (\nu _{\ell }(U),D_k(U)U_x)J(U)D_k(U)U_x,\mathcal {W} \right\rangle \nonumber \\&\quad +2aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x, \mathcal {W} \right\rangle +2aS\, \left\langle (\partial _x\mathcal {W},U_x)J(U)\partial _xU_x, \mathcal {W} \right\rangle \nonumber \\&\quad -4aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle + C\,D(t)^2. \end{aligned}$$
(3.87)

Therefore, by (3.82) and (3.87), we obtain

$$\begin{aligned}&R_2+R_3+R_4+R_5 \nonumber \\&\quad \le a\, \left\langle \partial _x\biggl \{ \sum _k (U_x, D_k(U)U_x)(J(U)D_k(U)-D_k(U)J(U)) \biggr \} \partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\qquad +2aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x, \mathcal {W} \right\rangle +2aS\, \left\langle (\partial _x\mathcal {W},U_x)J(U)\partial _xU_x, \mathcal {W} \right\rangle \nonumber \\&\qquad -2aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle + C\,D(t)^2. \end{aligned}$$
(3.88)

Note here that

$$\begin{aligned}&\left( \sum _k(\mathcal {U},D_k(U)U_x)(J(U)D_k(U)-D_k(U)J(U))Y_1,Y_2 \right) \nonumber \\&\quad = \left( Y_1, \sum _k(\mathcal {U},D_k(U)U_x)(J(U)D_k(U)-D_k(U)J(U))Y_2 \right) \end{aligned}$$
(3.89)

holds for any \(Y_1,Y_2:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\). This follows immediately from (3.3) and (3.11). By taking the derivative of both sides of (3.89) in x, we obtain

$$\begin{aligned}&\left( \partial _x\left\{ \sum _k(\mathcal {U},D_k(U)U_x)(J(U)D_k(U)-D_k(U)J(U)) \right\} Y_1,Y_2 \right) \nonumber \\&\quad = \left( Y_1, \partial _x\left\{ \sum _k(\mathcal {U},D_k(U)U_x)(J(U)D_k(U)-D_k(U)J(U)) \right\} Y_2 \right) \nonumber \end{aligned}$$

holds for any \(Y_1,Y_2:[0,T]\times \mathbb {T}\rightarrow \mathbb {R}^d\). Hence, by integrating by parts, we observe that the first term of the RHS of (3.88) is bounded by \(C\,D(t)^2\). Therefore, we obtain

$$\begin{aligned}&R_2+R_3+R_4+R_5 \nonumber \\&\quad \le 2aS\, \left\langle (\partial _x\mathcal {W},\partial _xU_x)J(U)U_x, \mathcal {W} \right\rangle +2aS\, \left\langle (\partial _x\mathcal {W},U_x)J(U)\partial _xU_x,\mathcal {W} \right\rangle \nonumber \\&\qquad -2aS\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle + C\,D(t)^2. \end{aligned}$$
(3.90)

Combining the information (3.70), (3.75), and (3.90), we derive

(3.91)

Furthermore, we rewrite the third and fourth terms of the RHS of (3.91) recalling Definition 3.6 and Lemma 3.7. From Definition 3.6, it follows that

$$\begin{aligned}&(\partial _x\mathcal {W},U_x)J(U)\partial _xU_x \nonumber \\&\quad = \frac{1}{2} \left( T_3(U)-T_4(U)+T_5(U) \right) \partial _x\mathcal {W}+ \frac{1}{2} \sum _k (U_x,D_k(U)U_x) (\partial _x\mathcal {W},\nu _k(U))J(U)U_x. \nonumber \end{aligned}$$

Using (3.34) and (3.35) with \(Y=\partial _x\mathcal {W}\), we obtain

$$\begin{aligned} T_5(U)\partial _x\mathcal {W}&= (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}+\frac{1}{2} |U_x|^2\partial _x(J(U))\partial _x\mathcal {W}\nonumber \\&\quad +\frac{1}{2}(\partial _x\mathcal {W},U_x) \sum _k (J(U)U_x,D_k(U)U_x)\nu _k(U) \nonumber \\&\quad -\frac{1}{2} \sum _k (\partial _x\mathcal {W},\nu _k(U)) (J(U)U_x,D_k(U)U_x)U_x. \nonumber \end{aligned}$$

Substituting this and using (3.9), we obtain

$$\begin{aligned}&(\partial _x\mathcal {W},U_x)J(U)\partial _xU_x \nonumber \\&\quad = \frac{1}{2}(\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}+ \frac{1}{2} \left( T_3(U)-T_4(U) \right) \partial _x\mathcal {W}\nonumber \\&\qquad + \frac{1}{4} |U_x|^2\partial _x(J(U))\partial _x\mathcal {W}+\frac{1}{4}(\partial _x\mathcal {W},U_x) \sum _k (J(U)U_x,D_k(U)U_x)\nu _k(U) \nonumber \\&\qquad +\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.92)

In the same way, by using Definition 3.6 and Lemma 3.7, we obtain

$$\begin{aligned}&(\partial _x\mathcal {W},\partial _xU_x)J(U)U_x \nonumber \\&\quad = \frac{1}{2}(\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}+ \frac{1}{2} \left( T_3(U)+T_4(U) \right) \partial _x\mathcal {W}\nonumber \\&\qquad + \frac{1}{4} |U_x|^2\partial _x(J(U))\partial _x\mathcal {W}+\frac{1}{4}(\partial _x\mathcal {W},U_x) \sum _k (J(U)U_x,D_k(U)U_x)\nu _k(U) \nonumber \\&\qquad +\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|). \end{aligned}$$
(3.93)

Thanks to (3.36) and (3.37), we can easily show \(\left\langle T_i(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \le C\,D(t)^2\) with \(i=3,4\), by integrating by parts. Besides, it is now immediate to observe that

$$\begin{aligned}&\left\langle |U_x|^2\partial _x(J(U))\partial _x\mathcal {W}, \mathcal {W} \right\rangle \le C\,D(t)^2,\nonumber \\&\quad \left\langle (\partial _x\mathcal {W},U_x) \sum _k (J(U)U_x,D_k(U)U_x)\nu _k(U) \mathcal {W} \right\rangle \le C\,D(t)^2 \nonumber \end{aligned}$$

by the argument using (3.13) and (3.8). Therefore, we substitute (3.92) and (3.93) into (3.91) to derive

(3.94)

Even if we use the integration parts, the first and second terms of the RHS of (3.94) cannot be bounded by \(C\,D(t)^2\). Fortunately, however, we will find in the next step that the two terms can be eliminated essentially by introducing a gauged function.

4. Energy estimates for \(\Vert \widetilde{\mathcal {W}}\Vert _{L^2(\mathbb {T};\mathbb {R}^d)}\) to eliminate the loss of derivatives

We introduce the function \(\widetilde{\mathcal {W}}\) which is defined by

$$\begin{aligned} \widetilde{\mathcal {W}}&= \mathcal {W}+\widetilde{\Lambda }, \end{aligned}$$
(3.95)

where

$$\begin{aligned} \widetilde{\Lambda }&= -\frac{e_1}{2a}(Z,J(U)U_x)J(U)U_x + \frac{e_2}{8a}|U_x|^2Z, \end{aligned}$$
(3.96)
$$\begin{aligned} e_1&=aS+c, \quad e_2=e_1+\frac{aS+6b+7c}{2}. \end{aligned}$$
(3.97)

Moreover, we introduce the energy \(\widetilde{D}(t)\), the square of which is defined by

$$\begin{aligned} \widetilde{D}(t)^2&= \Vert Z(t)\Vert _{L^2}^2 +\Vert Z_x(t)\Vert _{L^2}^2 +\Vert \widetilde{\mathcal {W}}(t)\Vert _{L^2}^2. \end{aligned}$$
(3.98)

Since u and v satisfy the same initial value, \(\widetilde{D}(0)=0\) holds. We shall show that there exists a positive constant C such that

(3.99)

for all \(t\in (0,T)\). If it is true, (3.99) together with \(\widetilde{D}(0)=0\) shows \(\widetilde{D}(t)\equiv 0\). This implies \(Z=0\).

In the proof of (3.99), by integrating by parts repeatedly, it is not difficult to obtain the following estimate permitting the loss of derivatives of order one:

(3.100)

Having them in mind, we hereafter concentrate on how to derive the estimate of the form:

(3.101)

For this purpose, we begin with

(3.102)

The first term of the RHS of (3.102) has already been investigated to satisfy (3.94). Hence, we compute the second and third terms of the RHS of (3.102) below. Observing \(\widetilde{\Lambda }=\mathcal {O}(|Z|)\), we obtain \(\widetilde{\mathcal {W}}=\mathcal {W}+\mathcal {O}(|Z|)\), \(\partial _x\widetilde{\mathcal {W}}=\partial _x\mathcal {W}+\mathcal {O}(|Z|+|Z_x|)\), and \(\partial _x^2\widetilde{\mathcal {W}}=\partial _x^2\mathcal {W}+\mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|)\), which will be often used without comments.

We start the computation of \(\left\langle \partial _t\widetilde{\Lambda },\widetilde{\mathcal {W}} \right\rangle \) by investigating \(\partial _t\widetilde{\Lambda }\). A simple computation shows

$$\begin{aligned} \partial _t\widetilde{\Lambda }&= -\frac{e_1}{2a}(Z_t,J(U)U_x)J(U)U_x + \frac{e_2}{8a}|U_x|^2Z_t +\mathcal {O}(|Z). \end{aligned}$$
(3.103)

Recalling (3.43), we obtain

$$\begin{aligned} Z_t&= a\,\partial _x\left( J(U)\partial _x\mathcal {W}\right) + a\, \sum _k (J(U)\partial _x\mathcal {W}, D_k(U)U_x)\nu _k(U) + \mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|) \end{aligned}$$
(3.104)
$$\begin{aligned}&= a\,J(U)\partial _x^2\mathcal {W}+a\,\partial _x(J(U))\partial _x\mathcal {W}+ a\, \sum _k (J(U)\partial _x\mathcal {W}, D_k(U)U_x)\nu _k(U) \nonumber \\&\quad + \mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|). \end{aligned}$$
(3.105)

Using (3.105), we obtain

$$\begin{aligned}&-\frac{e_1}{2a}(Z_t,J(U)U_x)J(U)U_x \nonumber \\&\quad = -\frac{e_1}{2}(J(U)\partial _x^2\mathcal {W},J(U)U_x)J(U)U_x -\frac{e_1}{2}(\partial _x(J(U))\partial _x\mathcal {W},J(U)U_x)J(U)U_x \nonumber \\&\qquad -\frac{e_1}{2} \sum _k (J(U)\partial _x\mathcal {W}, D_k(U)U_x)(\nu _k(U),J(U)U_x)J(U)U_x \\&\qquad + \mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|). \nonumber \end{aligned}$$

The third term of the RHS vanishes, since \((\nu _k(U),J(U)U_x)=0\). By noting (3.67), we observe that the second term of the RHS is \(\mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|)\). Thus, we have

$$\begin{aligned} -\frac{e_1}{2a}(Z_t,J(U)U_x)J(U)U_x&= -\frac{e_1}{2}(\partial _x^2\mathcal {W},U_x)J(U)U_x +\mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|). \end{aligned}$$
(3.106)

On the other hand, by using (3.104), we obtain

$$\begin{aligned} \frac{e_2}{8a}|U_x|^2Z_t&= \frac{e_2}{8}|U_x|^2 \partial _x\left( J(U)\partial _x\mathcal {W}\right) + \frac{e_2}{8}|U_x|^2 \sum _k (J(U)\partial _x\mathcal {W}, D_k(U)U_x)\nu _k(U) \nonumber \\&\quad +\mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|) \nonumber \\&= \frac{e_2}{8} \partial _x\left\{ |U_x|^2 J(U)\partial _x\mathcal {W}\right\} - \frac{e_2}{4} (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}\nonumber \\&\quad + \sum _k \mathcal {O} \left( |\partial _x\mathcal {W}| \right) \nu _k(U) +\mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|). \end{aligned}$$
(3.107)

By substituting (3.106) and (3.107) into (3.103), we obtain

$$\begin{aligned} \partial _t\widetilde{\Lambda }&= -\frac{e_1}{2}(\partial _x^2\mathcal {W},U_x)J(U)U_x + \frac{e_2}{8} \partial _x\left\{ |U_x|^2 J(U)\partial _x\mathcal {W}\right\} \\&\quad - \frac{e_2}{4} (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}\nonumber \\&\quad + \sum _k \mathcal {O} \left( |\partial _x\mathcal {W}| \right) \nu _k(U) +\mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|). \nonumber \end{aligned}$$

This shows that

$$\begin{aligned} \left\langle \partial _t\widetilde{\Lambda },\widetilde{\mathcal {W}} \right\rangle&= -\frac{e_1}{2} \left\langle (\partial _x^2\mathcal {W},U_x)J(U)U_x,\mathcal {W}+\mathcal {O}(|Z|) \right\rangle \nonumber \\&\quad +\frac{e_2}{8} \left\langle \partial _x\left\{ |U_x|^2 J(U)\partial _x\mathcal {W}\right\} ,\mathcal {W}+\mathcal {O}(|Z|) \right\rangle \nonumber \\&\quad -\frac{e_2}{4} \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W}+\mathcal {O}(|Z|) \right\rangle \nonumber \\&\quad + \left\langle \sum _k \mathcal {O} \left( |\partial _x\mathcal {W}| \right) \nu _k(U) , \mathcal {W}+\mathcal {O}(|Z|) \right\rangle \nonumber \\&\quad +\left\langle \mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|),\mathcal {W}+\mathcal {O}(|Z|) \right\rangle \nonumber \\&\le -\frac{e_1}{2} \left\langle (\partial _x^2\mathcal {W},U_x)J(U)U_x,\mathcal {W} \right\rangle +\frac{e_2}{8} \left\langle \partial _x\left\{ |U_x|^2 J(U)\partial _x\mathcal {W}\right\} ,\mathcal {W} \right\rangle \nonumber \\&\quad -\frac{e_2}{4} \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \nonumber \\&\quad + \left\langle \sum _k \mathcal {O} \left( |\partial _x\mathcal {W}| \right) \nu _k(U), \mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&\le -\frac{e_1}{2} \left\langle (\partial _x^2\mathcal {W},U_x)J(U)U_x,\mathcal {W} \right\rangle -\frac{e_2}{4} \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \nonumber \\&\quad +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.108)

We next compute \(\left\langle \partial _t\mathcal {W},\widetilde{\Lambda } \right\rangle \). Using (3.64), we obtain

$$\begin{aligned} \partial _t\mathcal {W}&=a\,\partial _x^2\left\{ J(U)\partial _x^2\mathcal {W}\right\} - 2a\, \sum _{k} \left( \partial _x^3\mathcal {W}, J(U)D_{k}(U)U_x \right) \nu _{k}(U) \nonumber \\&\quad + \mathcal {O}\big (|Z|+|Z_x|+|\mathcal {W}|+|\partial _x\mathcal {W}|+|\partial _x^2\mathcal {W}|\big ). \nonumber \end{aligned}$$

Using this and by noting \(\widetilde{\Lambda }=\mathcal {O}(|Z|)\), we integrate by parts to obtain

$$\begin{aligned} \left\langle \partial _t\mathcal {W},\widetilde{\Lambda } \right\rangle&\le R_8+R_9 +C\,\widetilde{D}(t)^2, \end{aligned}$$
(3.109)

where

$$\begin{aligned} R_8&=a\, \left\langle \partial _x^2\left\{ J(U)\partial _x^2\mathcal {W}\right\} ,\widetilde{\Lambda } \right\rangle , \nonumber \\ R_9&= -2a\, \left\langle \sum _{k} \left( \partial _x^3\mathcal {W}, J(U)D_{k}(U)U_x \right) \nu _{k}(U),\widetilde{\Lambda } \right\rangle . \nonumber \end{aligned}$$

For \(R_9\), noting \(\widetilde{\Lambda }=\mathcal {O}(|Z|)\), we use the integration by parts and \((\nu _k(U),J(U)U_x)=0\) to obtain

$$\begin{aligned} R_9&\le -2a\,(-1)^3 \left\langle \sum _{k} \left( \mathcal {W}, J(U)D_{k}(U)U_x \right) \nu _{k}(U),\partial _x^3\widetilde{\Lambda } \right\rangle + C\,\widetilde{D}(t)^2 \nonumber \\&\le 2a\, \left\langle \sum _{k} \left( \mathcal {W}, J(U)D_{k}(U)U_x \right) \nu _{k}(U), -\frac{e_1}{2a}\big (\partial _x^3Z,J(U)U_x\big )J(U)U_x + \frac{e_2}{8a}|U_x|^2\partial _x^3Z \right\rangle \nonumber \\&\quad + C\,\widetilde{D}(t)^2 \nonumber \\&=\frac{e_2}{4} \left\langle \sum _{k} \left( \mathcal {W}, J(U)D_{k}(U)U_x \right) \nu _{k}(U), |U_x|^2\partial _x^3Z \right\rangle + C\,\widetilde{D}(t)^2. \nonumber \end{aligned}$$

Furthermore, since \(\partial _x^3Z=\partial _x^2Z_x = \partial _x\mathcal {W}+ \mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|) =\partial _x\mathcal {W}+\mathcal {O}(|Z|+|Z_x|+|\widetilde{\mathcal {W}}|)\) and \((\nu _k(U),\partial _x\mathcal {W})=\mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|)\), we have

$$\begin{aligned} R_9&\le \frac{e_2}{4} \left\langle \sum _{k} \left( \mathcal {W}, J(U)D_{k}(U)U_x \right) \nu _{k}(U), |U_x|^2\partial _x\mathcal {W} \right\rangle + C\,\widetilde{D}(t)^2 \le C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.110)

For \(R_8\), we begin with

$$\begin{aligned} R_8&= -\frac{e_1}{2}\, \left\langle \partial _x^2\left\{ J(U)\partial _x^2\mathcal {W}\right\} , (Z,J(U)U_x)J(U)U_x \right\rangle \\&\quad + \frac{e_2}{8}\, \left\langle \partial _x^2\left\{ J(U)\partial _x^2\mathcal {W}\right\} , |U_x|^2Z \right\rangle \nonumber \\&=:R_{81}+R_{82}. \nonumber \end{aligned}$$

The integration by parts implies

$$\begin{aligned} R_{81}&= -\frac{e_1}{2}\, \left\langle J(U)\partial _x^2\mathcal {W}, \partial _x^2\left\{ (Z,J(U)U_x)J(U)U_x\right\} \right\rangle \nonumber \\&\le -\frac{e_1}{2}\, \left\langle J(U)\partial _x^2\mathcal {W}, (\partial _xZ_x,J(U)U_x)J(U)U_x \right\rangle \nonumber \\&\quad -e_1\, \left\langle J(U)\partial _x^2\mathcal {W}, (Z_x,\partial _x\left\{ J(U)U_x\right\} )J(U)U_x \right\rangle \nonumber \\&\quad -e_1\, \left\langle J(U)\partial _x^2\mathcal {W}, (Z_x,J(U)U_x)\partial _x\left\{ J(U)U_x\right\} \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&\le -\frac{e_1}{2}\, \left\langle J(U)\partial _x^2\mathcal {W}, (\partial _xZ_x,J(U)U_x)J(U)U_x \right\rangle \nonumber \\&\quad +e_1\, \left\langle J(U)\partial _x\mathcal {W}, (\partial _xZ_x,\partial _x\left\{ J(U)U_x\right\} )J(U)U_x \right\rangle \nonumber \\&\quad +e_1\, \left\langle J(U)\partial _x\mathcal {W}, (\partial _xZ_x,J(U)U_x)\partial _x\left\{ J(U)U_x\right\} \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&=:R_{83}+R_{84}+R_{85}+C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.111)

Since \(\mathcal {U}=\partial _xU_x+\displaystyle \sum _{k}(U_x,D_k(U)U_x)\nu _k(U)\) and \(\mathcal {V}=\partial _xV_x+\displaystyle \sum _{k}(V_x,D_k(V)V_x)\nu _k(V)\), we obtain

$$\begin{aligned}\partial _xZ_x=\mathcal {W}-\sum _{k}(Z_x,D_k(U)U_x)\nu _k(U) -\sum _k(V_x,D_k(U)Z_x)\nu _k(U) + \mathcal {O}(|Z|), \end{aligned}$$

and thus \( (\partial _xZ_x, J(U)U_x) = (\mathcal {W}, J(U)U_x) +\mathcal {O}(|Z|)\). Substituting this, using (3.3) and (3.4), and integrating by parts, we have

$$\begin{aligned} R_{83}&= -\frac{e_1}{2}\, \left\langle \partial _x^2\mathcal {W}, (\partial _xZ_x,J(U)U_x)U_x \right\rangle \nonumber \\&\le -\frac{e_1}{2}\, \left\langle \partial _x^2\mathcal {W}, (\mathcal {W},J(U)U_x)U_x \right\rangle -\frac{e_1}{2}\, \left\langle \partial _x^2\mathcal {W}, \mathcal {O}(|Z|) \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&\le -\frac{e_1}{2}\, \left\langle (\partial _x^2\mathcal {W}, U_x)J(U)U_x, \mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.112)

From (3.86), it follows that

$$\begin{aligned} \partial _x\left\{ J(U)U_x\right\} = J(U)\partial _xU_x+\sum _k(U_x,J(U)D_k(U)U_x)\nu _k(U). \nonumber \end{aligned}$$

Using this, \(\partial _xZ_x=\mathcal {W}+\mathcal {O}(|Z|+|Z_x|)\), and (3.8), we obtain

$$\begin{aligned}&(\partial _xZ_x, \partial _x\left\{ J(U)U_x\right\} ) \nonumber \\&= (\partial _xZ_x, J(U)\partial _xU_x) +\sum _k(U_x,J(U)D_k(U)U_x)(\nu _k(U),\partial _xZ_x) + \mathcal {O}(|Z|+|Z_x|) \nonumber \\&= (\mathcal {W}, J(U)\partial _xU_x) +\sum _k(U_x,J(U)D_k(U)U_x)(\nu _k(U),\mathcal {W}) + \mathcal {O}(|Z|+|Z_x|) \nonumber \\&=(\mathcal {W}, J(U)\partial _xU_x)+ \mathcal {O}(|Z|+|Z_x|). \nonumber \end{aligned}$$

This implies

$$\begin{aligned} R_{84}&= e_1\, \left\langle \partial _x\mathcal {W}, (\partial _xZ_x,\partial _x\left\{ J(U)U_x\right\} )U_x \right\rangle \nonumber \\&\le e_1\, \left\langle \partial _x\mathcal {W}, (\mathcal {W}, J(U)\partial _xU_x)U_x \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&= e_1\, \left\langle (\partial _x\mathcal {W}, U_x) J(U)\partial _xU_x, \mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.113)

In the same way, we use (3.4) to obtain

$$\begin{aligned}&(J(U)\partial _x\mathcal {W}, \partial _x\left\{ J(U)U_x\right\} ) \nonumber \\&\quad = (J(U)\partial _x\mathcal {W}, J(U)\partial _xU_x) + \sum _k(U_x,J(U)D_k(U)U_x) (J(U)\partial _x\mathcal {W}, \nu _k(U)) \nonumber \\&\quad = (\partial _x\mathcal {W}, P(U)\partial _xU_x) \nonumber \\&\quad = (\partial _x\mathcal {W}, \partial _xU_x+\sum _k(U_x, D_k(U)U_x)\nu _k(U)) \nonumber \\&\quad = (\partial _x\mathcal {W},\partial _xU_x) + \mathcal {O}(|Z|+|Z_x|+|\mathcal {W}|). \nonumber \end{aligned}$$

Substituting this, we obtain

$$\begin{aligned} R_{85}&=e_1\, \left\langle J(U)\partial _x\mathcal {W}, (\partial _xZ_x,J(U)U_x)\partial _x\left\{ J(U)U_x\right\} \right\rangle \nonumber \\&\le e_1\, \left\langle \partial _x\mathcal {W}, (\partial _xZ_x,J(U)U_x)\partial _xU_x \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&\le e_1\, \left\langle \partial _x\mathcal {W}, (\mathcal {W},J(U)U_x)\partial _xU_x \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&= e_1\, \left\langle (\partial _x\mathcal {W}, \partial _xU_x)J(U)U_x,\mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.114)

Substituting (3.112), (3.113), and (3.114) into (3.111), we observe that \(R_{81}=R_{83}+R_{84}+R_{85}+C\,\widetilde{D}(t)^2\) is bounded as follows:

$$\begin{aligned} R_{81}&\le -\frac{e_1}{2}\, \left\langle \big (\partial _x^2\mathcal {W}, U_x\big )J(U)U_x, \mathcal {W} \right\rangle + e_1\, \left\langle (\partial _x\mathcal {W}, U_x) J(U)\partial _xU_x, \mathcal {W} \right\rangle \nonumber \\&\quad + e_1\, \left\langle (\partial _x\mathcal {W}, \partial _xU_x)J(U)U_x,\mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2. \nonumber \end{aligned}$$

Furthermore, by applying (3.92) and (3.93) to the second and third terms of the RHS of the above, and by using \(\left\langle T_3(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle \le C\,\widetilde{D}(t)^2\) again, we deduce

$$\begin{aligned} R_{81}&\le -\frac{e_1}{2}\, \left\langle (\partial _x^2\mathcal {W}, U_x)J(U)U_x, \mathcal {W} \right\rangle + e_1\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad + e_1\, \left\langle T_3(U)\partial _x\mathcal {W},\mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&\le -\frac{e_1}{2}\, \left\langle (\partial _x^2\mathcal {W}, U_x)J(U)U_x, \mathcal {W} \right\rangle + e_1\, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.115)

For \(R_{82}\), the integration by parts and the same argument as above lead to

$$\begin{aligned} R_{82}&= \frac{e_2}{8} \left\langle J(U)\partial _x^2\mathcal {W}, \partial _x^2\left\{ |U_x|^2Z\right\} \right\rangle \nonumber \\&= \frac{e_2}{8} \left\langle J(U)\partial _x^2\mathcal {W}, |U_x|^2\partial _xZ_x+4(\partial _xU_x,U_x)Z_x+\mathcal {O}(|Z|) \right\rangle \nonumber \\&\le \frac{e_2}{8} \left\langle |U_x|^2J(U)\partial _x^2\mathcal {W}, \mathcal {W} \right\rangle + \frac{e_2}{2} \left\langle J(U)\partial _x^2\mathcal {W},(\partial _xU_x,U_x)Z_x \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&\le \frac{e_2}{8} \left\langle \partial _x \left\{ |U_x|^2J(U)\partial _x\mathcal {W}\right\} , \mathcal {W} \right\rangle - \frac{e_2}{4} \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad - \frac{e_2}{8} \left\langle |U_x|^2\partial _x(J(U))\partial _x\mathcal {W}, \mathcal {W} \right\rangle - \frac{e_2}{2} \left\langle J(U)\partial _x\mathcal {W},(\partial _xU_x,U_x)\mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2 \nonumber \\&\le - \frac{3e_2}{4} \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.116)

Therefore, from (3.115) and (3.116), it follows that \(R_8=R_{81}+R_{82}\) is bounded as follows:

$$\begin{aligned} R_8&\le -\frac{e_1}{2}\, \left\langle (\partial _x^2\mathcal {W}, U_x)J(U)U_x, \mathcal {W} \right\rangle + \left( e_1-\frac{3e_2}{4}\right) \, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle \nonumber \\&\quad +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.117)

Consequently, by substituting (3.110) and (3.117) into (3.109), we have

$$\begin{aligned} \left\langle \partial _t\mathcal {W}, \widetilde{\Lambda } \right\rangle&\le -\frac{e_1}{2}\, \left\langle (\partial _x^2\mathcal {W}, U_x)J(U)U_x, \mathcal {W} \right\rangle \nonumber \\&\quad + \left( e_1-\frac{3e_2}{4}\right) \, \left\langle (\partial _xU_x,U_x)J(U)\partial _x\mathcal {W}, \mathcal {W} \right\rangle +C\,\widetilde{D}(t)^2. \end{aligned}$$
(3.118)

Combining the information (3.102), (3.94), (3.108), (3.118), and (3.97), we conclude

which is the desired result (3.101). \(\square \)

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Acknowledgements

The author would like to thank Hiroyuki Chihara for valuable comments and encouragement. Thanks to his comments in [4], the proof of Theorems 1.1 and 2.1 is improved to be comprehensible. This work was supported by JSPS Grant-in-Aid for Young Scientists (B) #24740090.

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Correspondence to Eiji Onodera.

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Onodera, E. A Fourth-Order Dispersive Flow Equation for Closed Curves on Compact Riemann Surfaces. J Geom Anal 27, 3339–3403 (2017). https://doi.org/10.1007/s12220-017-9808-1

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Keywords

  • Dispersive flow
  • Geometric analysis
  • Local existence and uniqueness
  • Loss of derivatives
  • Energy method
  • Gauge transformation
  • Constant sectional curvature

Mathematics Subject Classification

  • Primary 53C44
  • Secondary 35Q35
  • 35Q55
  • 35G61