Local and global analyticity for $\mu$-Camassa-Holm equations

TL;DR

This paper establishes local and global analyticity results for various $mbda$-Camassa-Holm equations, proving local solvability and global existence of solutions in the analytic category, which is a novel achievement for these equations.

Contribution

It provides the first global-in-time analytic solutions for several $mbda$-Camassa-Holm equations, including $mbda$CH, $mbda$DP, and higher-order variants, with lifespan estimates.

Findings

Proved unique local solvability of the Cauchy problems.

Established existence of global-in-time analytic solutions for certain $mbda$-Camassa-Holm equations.

Provided lifespan estimates for solutions.

Abstract

We solve Cauchy problems for some $\mu$-Camassa-Holm integro-partial differential equations in the analytic category. The equations to be considered are $\mu$CH of Khesin-Lenells-Misio\l{}ek, $\mu$DP of Lenells-Misio\l{}ek-Ti\u{g}lay, the higher-order $\mu$CH of Wang-Li-Qiao and the non-quasilinear version of Qu-Fu-Liu. We prove the unique local solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Moreover, we show the existence of a unique global-in-time analytic solution for $\mu$CH, $\mu$DP and the higher-order $\mu$CH. The present work is the first result of such a global nature for these equations. AMS subject classification: 35R09, 35A01, 35A10, 35G25

Equations

$$\mu (u_t)-u_{txx}=-2\mu (u)u_x+2u_x{u}_{xx}+u{u}_{xxx},x\in S^1=\mathbb{R}/\mathbb{Z},$$

$$u_t+u{u}_x+{\partial }_x(\mu -{\partial }_x^2{)}^{-1}\left[2\mu (u)u+\frac{1}{2}{u}_x^2\right]=0.$$

$$u_t-u_{txx}=-3u{u}_x+2u_x{u}_{xx}+u{u}_{xxx}$$

$$u_t+u{u}_x+{\partial }_x(1-{\partial }_x^2{)}^{-1}\left[u^2+\frac{1}{2}{u}_x^2\right]=0.$$

$$\mu (u_t)-u_{txx}=-2\mu (u)u_x+2u_x{u}_{xx}+u{u}_{xxx},x\in S^1=\mathbb{R}/\mathbb{Z},$$

$$A(\phi )=\mu (\phi )-{\phi }_{xx}.$$

$$u_t+u{u}_x+{\partial }_x{A}^{-1}\left[2\mu (u)u+\frac{1}{2}{u}_x^2\right]=0.$$

$$u_t+u{u}_x+{\partial }_x{A}^{-1}\left[3\mu (u)u\right]=0.$$

$$u_t+u{u}_x+{\partial }_x{B}^{-1}\left[2\mu (u)u+\frac{1}{2}{u}_x^2-3u_x{u}_{xxx}-\frac{7}{2}{u}_{xx}^2\right]=0,$$

$$B(\phi )=\mu (\phi )+(-{\partial }_x^2+{\partial }_x^4)\phi ,$$

$$u_t+u{u}_x+{\partial }_x{A}^{-2}\left[2\mu (u)u+\frac{1}{2}{u}_x^2-3u_x{u}_{xxx}-\frac{7}{2}{u}_{xx}^2\right]=0$$

$$u_t+2\mu (u)u{u}_x-\frac{1}{3}{u}_x^3+{\partial }_x{A}^{-1}\left[2{\mu }^2(u)u+\mu (u)u_x^2+\gamma u\right]+\frac{1}{3}\mu (u_x^3)=0$$

$$G^{\delta ,s}=\left\{\phi \in L^2(S^1);\Vert \phi {\Vert }_{\delta ,s}<\infty \right\},\Vert \phi {\Vert }_{\delta ,s}^2=\sum _{k\in \mathbb{Z}}⟨k{⟩}^{2s}{e}^{4\pi \delta |k|}|\hat{\phi }(k){|}^2,$$

$$H^s=G^{0,s},H^{\infty }={\cap }_{s\ge 0}{H}^s$$

$$\Vert \phi {\Vert }_2^2=\Vert {\Lambda }^2\phi {\Vert }_0^2=\Vert \phi {\Vert }_0^2+\frac{1}{2{\pi }^2}\Vert {\phi }^{\prime }{\Vert }_0^2+\frac{1}{(2\pi {)}^4}\Vert {\phi }^{\prime \prime }{\Vert }_0^2,$$

$$S(\delta )$$

$$A(\delta )$$

$$\Vert f{\Vert }_{(\sigma ,s)}^2=\sum _{j=0}^{\infty }\frac{e^{4\pi j\sigma }}{j{!}^2}\Vert f^{(j)}{\Vert }_s^2(e^{2\pi \sigma }<\delta ).$$

$$|\phi (z){|}^2$$

$$=\Vert \phi {\Vert }_{\delta ,s}^2\sum ⟨k{⟩}^{-2s}{e}^{-4\pi \delta |k|}{e}^{-4k\pi y}.$$

$$|\phi (x){|}^2\le {\left|\sum \hat{\phi }(k)e^{2k\pi ix}\right|}^2\le \sum ⟨k{⟩}^{2s}|\hat{\phi }(k){|}^2\times \sum ⟨k{⟩}^{-2s}|e^{2k\pi ix}{|}^2\le \Vert \phi {\Vert }_s^2\sum ⟨k{⟩}^{-2s}.$$

$$\hat{\phi }(k)={\int }_0^1\phi \left(x\pm {\delta }^{\prime }i\right)\exp \left(-2k\pi i\left[x\pm {\delta }^{\prime }i\right]\right)dx.$$

$$a_k={\int }_0^1\phi \left(x-\mathrm{sgn}(k){\delta }^{\prime }i\right)e^{-2k\pi ix}dx=e^{2\pi |k|{\delta }^{\prime }}\hat{\phi }(k),k\ne 0.$$

$$\Vert \phi \psi {\Vert }_{\delta ,s}\le c_s\Vert \phi {\Vert }_{\delta ,s}\Vert \psi {\Vert }_{\delta ,s},c_s=\left[2(1+s^{2s})\sum _{k=0}^{\infty }⟨k{⟩}^{-2s}\right].$$

$$\Vert {\phi }_x{\Vert }_{{\delta }^{\prime },s}$$

$$\Vert {\phi }_x{\Vert }_{\delta ,s}$$

$$\Vert \phi {\Vert }_{\delta ,s}^2$$

$$\Vert {\phi }_x{\Vert }_{{\delta }^{\prime },s}^2$$

$$=\sum _{k\in \mathbb{Z}}⟨k{⟩}^{2s}{e}^{4\pi \delta |k|}f(2\pi |k|)|\hat{\phi }(k){|}^2.$$

$$\mu (\phi )$$

$$A(\phi )$$

$$A^{-1}(\phi )$$

$$B(\phi )$$

$$B^{-1}(\phi )$$

$$\left|\mu (\phi )\right|\le \Vert \phi {\Vert }_{\delta ,s},$$

Full text

AMS subject classification: 35R09, 35A01, 35A10, 35G25

This work was partially supported by JSPS KAKENHI Grant Number 26400127.

1. 1 Function spaces and operators

2. 2 Local-in-time solutions

3. 2.1 Autonomous Ovsyannikov theorem

4. 2.2 $\mu$CH and $\mu$DP equations

5. 2.3 Higher-order $\mu$CH equation

6. 3 Global-in-time solutions

7. 3.1 Statement of the main results

8. 3.2 Regularity theorem by Kato-Masuda

9. 3.3 Pairing and estimates

10. 3.3.1 Estimate of $\langle v^{(j)},\partial_{x}^{j}(vv_{x})\rangle_{2}$

11. 3.3.2 Estimate of $\langle v^{(j)},\partial_{x}^{j+1}A^{-1}\left[\mu(v)v\right]\rangle_{2}$

12. 3.3.3 Estimate of $\langle v^{(j)},\partial_{x}^{j+1}A^{-1}(v_{x}^{2})\rangle_{2}$

13. 3.4 Analyticity in the space variable

14. 3.5 Analyticity in the space and time variables

15. 4 Global-in-time solutions: higher-order case

16. 4.1 Statement of the main results

17. 4.2 Estimates: higher-order case

18. 4.2.1 Estimate of $\langle v^{(j)},\partial_{x}^{j+1}B^{-1}\left[v_{x}v_{xxx}\right]\rangle_{2}$

19. 4.2.2 Estimate of $\langle v^{(j)},\partial_{x}^{j+1}B^{-1}\left[v_{xx}^{2}\right]\rangle_{2}$

Introduction

We consider a functional equation

[TABLE]

called the $\mu$-Camassa-Holm equation ($\mu$CH), and its variants in the complex-analytic or real-analytic category. Here $\mu(u)=\int_{S^{1}}u\,dx$. Multiplying by the inverse of $\mu-\partial_{x}^{2}$, we get an evolution equation

[TABLE]

It motivates one to consider Cauchy problems, not only in Sobolev spaces but also in spaces of analytic functions. In the latter case, the solutions are analytic in both $t$ and $x$. Recall that solutions to the KdV equation can be analytic in $x$ but not in $t$. This is because it is ‘not Kowalevskian’, which means the first-order derivative $u_{t}$ equals a quantity involving higher derivatives. Our evolution equation mentioned above is ‘Kowalevskian’ in a generalized sense due to the presence of the negative order pseudodifferential operator $(\mu-\partial_{x}^{2})^{-1}$.

Because of the nonlocal nature of $(\mu-\partial_{x}^{2})^{-1}$, our considerations are always global in $x$. So we will work with the Sobolev space $H^{m}(S^{1})$ or $A(\delta)$, the space of analytic functions on $S^{1}\ni x$ which admits analytic continuation to $|y|<\delta$. On the other hand, we can work either locally or globally in $t$. Our local study will be given in Section 2 and Appendix. It is based on the Ovsyannikov theorem used in [1] and [8]. It is a kind of abstract Cauchy-Kowalevsky theorem about a scale of Banach spaces and enables us to obtain local-in-time solutions which are analytic in both $t$ and $x$. Our global study will be given in Sections 3 and 4. Since global-in-time solutions are known to exist in Sobolev spaces, what remains to be done here is to prove their analyticity. We carry out this task by using the method of [9] following [3]. In the final part of the proof, we quote a result in [12], which gives a useful criterion of real analyticity.

Now we explain some background and history. In the course of it, we will introduce some equations that will be studied in the present paper. All the equations mentioned below are integrable in some sense.

The original Camassa-Holm equation

[TABLE]

was introduced in [4] (shallow water wave) and in [7] (hereditary symmetries). It is known to be completely integrable and admits peaked soliton (peakon) solutions. The Cauchy problem for this equation can be formulated by introducing a pseudodifferential operator. Indeed, (0.1) can be written in the form

[TABLE]

Since (0.2) is Kowalevskian in a generalized sense, it is natural to solve this equation in the analytic setting as in the classical Cauchy-Kowalevsky theorem. In [1], the authors introduced a kind of Sobolev spaces with exponential weights consisting of holomorphic functions in a strip of the type $|y|<\text{const.}$ Since these spaces form a scale of Banach spaces, an Ovsyannikov type argument can be applicable. It leads to the unique solvability and an estimate of the lifespan of the solution in the periodic and non-periodic cases.

There are a lot of works about solutions of the Cauchy problem for (0.1) or (0.2) in Sobolev spaces. See the references in [1] and [20]. Local well-posedness and blowup mechanism are major topics. In [1], the local unique solvability in the analytic category was proved. Moreover, there is a result about the global-in-time solvability in [3]. Indeed, according to [3], if the initial value is in $H^{s}(\mathbb{R}),s>5/2$, and the McKean quantity $m_{0}=(1-\partial_{x}^{2})u_{0}$ does not change sign, then the Cauchy problem for (a generalization of) (0.2) has a unique global-in-time solution $u\in\mathcal{C}([0,\infty);H^{s}(\mathbb{R}))$. See [15] for the necessity and sufficiency of the no-change-of-sign condition. Moreover, in [3], it is proved that this solution is analytic in both $t$ and $x$ if the initial value is in the space of analytic functions mentioned above. In the present paper, we follow [1] for local theory and the analyticity part of [3] for global theory.

In [11], the $\mu$-version of (0.1), namely

[TABLE]

was introduced. The authors call this equation $\mu$HS (HS is for Hunter-Saxton), while it is called $\mu$CH in [14]. We have $\mu(u_{t})=0$, but we keep $\mu(u_{t})$ because this formulation facilitates later calculation. The interest of (0.3) lies, for example, in the fact that it describes evolution of rotators in liquid crystals with external magnetic field and self-interaction, and it is related to the diffeomorphism groups of the circle with a natural metric. Set

[TABLE]

Then it is invertible for a suitable choice of function spaces and commutes with $\partial_{x}$. The equation (0.3) can be written in the following form ([11, (5.1)]):

[TABLE]

In [11], the local well-posedness and the global existence in Sobolev spaces is demonstrated. In the global problem, the $\mu$-McKean quantity $(\mu-\partial_{x}^{2})u_{0}(x)$ is assumed to be free from change of sign.

There are similar $\mu$-equations. In [14, (5.3)], the following equation, called $\mu$DP (DP is for Degasperis-Procesi), was introduced:

[TABLE]

The local well-posedness in $H^{s}(S^{1}),s>3/2$, and the global existence in $H^{s}(S^{1}),s>3$, was proved in [14].

In [5], a family of higher-order Camassa-Holm equations depending on $k=2,3,\dots$ was introduced. It is related to diffeomorphisms of the unit circle. In [18], the $\mu$-version of the $k=2$ case

[TABLE]

was formulated. The local well-posedness and the global existence in $H^{s}(S^{1}),s>7/2$, was proved in [18]. Notice that the no-change-of-sign condition is not imposed in this work. In [19], a different version

[TABLE]

is studied. The global existence is proved there.

The modified $\mu$-Camassa-Holm equation (modified $\mu$CH) with non-quasilinear terms

[TABLE]

was introduced in [16, (3.2)] ($\gamma=0$) and [16, (2.7)], [17]. The global existence in Sobolev spaces remains open, as opposed to that of the other equations mentioned above, so it is not possible to show the global existence of analytic solutions by using the method in the present article. Because of this exceptional nature of the equation, we treat it in Appendix separately from the others. Notice that local theory for (0.8) is developed in a certain space of analytic functions as well as in Besov spaces in [17].

The outline of this article is as follows. In Section 1, we introduce some function spaces and operators and investigate their properties. In Section 2, we prove the local existence of analytic solutions of (0.4), (0.5), (0.6) and (0.7). These results are used in the proofs of the global existence theorems in Sections 3 and 4 about (0.4), (0.5) and (0.6). In Appendix, the local existence of analytic solutions of (0.8) is proved.

1. Function spaces and operators

In the present paper, $L^{2}(S^{1})$ consists of real-valued square-integrable functions on $S^{1}=\mathbb{R}/\mathbb{Z}$. We sometimes identify an element of it with a function on $\mathbb{R}$ with period 1. For a function on $S^{1}$, we set $\hat{\varphi}(k)=\int_{S^{1}}\varphi(x)e^{-2k\pi ix}\,dx$. We introduce a family of Hilbert spaces $G^{\delta,s}$ ($\delta\geq 0,s\geq 0$) by

[TABLE]

where $\langle k\rangle=(1+k^{2})^{1/2}$ (Japanese bracket). Notice that our definition is not exactly the same as that in [1, 2]. In particular, the base space is $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$ in [1, 2]. It is easy to see that we have continuous injections $G^{\delta,s}\to G^{\delta^{\prime},s^{\prime}}$ and $G^{\delta,s}\to G^{\delta,s^{\prime}}$ if $0\leq\delta^{\prime}<\delta,0\leq s^{\prime}<s$. Their norms are 1. We have a continuous injection $G^{\delta,s}\to G^{\delta^{\prime},s^{\prime}}$ under the weaker assumption $0\leq\delta^{\prime}<\delta$. We recover the usual Sobolev spaces

[TABLE]

and we set $\|\varphi\|_{s}=\|\varphi\|_{0,s}$. Notice that $\|\cdot\|_{0}$ is the $L^{2}$ norm. The corresponding inner product is denoted by $\langle\cdot,\cdot\rangle_{0}$. Set $\Lambda^{2}=1-(2\pi)^{-2}\partial_{x}^{2}$. Then we have

[TABLE]

because $\int_{S^{1}}\varphi\varphi^{\prime\prime}\,dx=-\int_{S^{1}}(\varphi^{\prime})^{2}\,dx$.

When $\delta>0$, set

[TABLE]

Remark 1*.*

If a function is analytic on $S^{1}$, then it belong to $A(\delta)$ for some $\delta>0$. See Proposition 4 below. Notice that an analogous statement does not hold true if $S^{1}$ is replaced with $\mathbb{R}$.

We identify an element of $A(\delta)$ with its analytic continuation. For $f\in A(\delta)$, set

[TABLE]

This norm will be used in the global theory. Do not confuse $\|\cdot\|_{(\sigma,s)}^{2}$ with $\|\cdot\|_{\delta,s}$.

Proposition 2**.**

For $\delta>0,s\geq 0$, the space $G^{\delta,s}$ is continuously embedded in $A(\delta)$.

Proof.

Assume $\varphi\in G^{\delta,s}$. The series $\varphi(z)=\sum\hat{\varphi}(k)e^{2k\pi iz}=\sum\hat{\varphi}(k)e^{-2k\pi y}e^{2k\pi ix}$ converges locally uniformly in $|y|<\delta$, because

[TABLE]

Therefore $G^{\delta,s}\subset A(\delta)$ and this embedding is continuous from $G^{\delta,s}$ to $A(\delta)$ with the topology (i) in Corollary 11. ∎

Remark 3*.*

Proposition 2 is a variant of the Sobolev embedding theorem: if $s>1/2$, then there is a continuous embedding $H^{s}=G^{0,s}\to\mathcal{C}^{0}(S^{1})$ as is proved by d

[TABLE]

Proposition 4**.**

If $\varphi$ is a real-analytic function on $S^{1}$, then there exists $\delta>0$ such that $\varphi\in G^{\delta,s}$ for any $s$. More precisely, if $\varphi\in A(\delta)$, then $\varphi\in G^{\delta^{\prime},s}$ for any $\delta^{\prime}\in]0,\delta[$ and any $s$.

Proof.

(This proposition is given in [1] without proof.) By the periodicity, contour deformation gives

[TABLE]

Set

[TABLE]

These are the Fourier coefficients of $\varphi(x\pm i\delta^{\prime})$ and we have $\sum_{k\in\mathbb{Z}\setminus\left\{0\right\}}\langle k\rangle^{2s}|a_{k}|^{2}<\infty$ for any $s$. Therefore the series $\sum_{k\in\mathbb{Z}}\langle k\rangle^{2s}e^{4\pi\delta^{\prime}|k|}|\hat{\varphi}(k)|^{2}$ converges. ∎

Proposition 5**.**

We have the following three estimates about products of functions:

(i) Assume $s>1/2,\delta\geq 0$. Then $G^{\delta,s}$ is closed under pointwise multiplication and we have

[TABLE]

(ii) There exists a positive constant $d_{s}$ such that we have $\|\varphi\psi\|_{0}\leq d_{s}\|\varphi\|_{0}\|\psi\|_{s}$ for any $\varphi\in H^{0}=L^{2}(S^{1})$ and any $\psi\in H^{s}\,(s\geq 1)$.

(iii) There exists a constant $\gamma$ such that we have $\|\varphi\psi\|_{2}\leq\gamma\left(\|\varphi\|_{1}\|\psi\|_{2}+\|\varphi\|_{2}\|\psi\|_{1}\right)$ for any $\varphi,\psi\in H^{2}$.

Proof.

The proof of (i) is given in [2]. Although different formulations are used in [2] and the present article, the constant $c_{s}$ is the same. This is because $\widehat{\varphi\psi}(k)=\sum_{n}\hat{\varphi}(n)\hat{\psi}(k-n)$ holds in either situations. The difference of the base spaces are offset by that of conventions, namely the presence or the absence of the $1/(2\pi)$ factor in the definition of the Fourier coefficients. The other three estimates follow from the boundedness of $H^{1}\hookrightarrow C^{0}(S^{1})$ (and the Leibniz rule). ∎

Remark 6*.*

A better estimate than Proposition 5 (iii) can be found in [10], which implies that $\|\cdot\|_{1}$ can be replaced with $\|\cdot\|_{L^{\infty}}$. It is used in [18], but (iii) is good enough in the present paper.

Proposition 7**.**

([1, Lemma 2]) If $0\leq\delta^{\prime}<\delta,s\geq 0$ and $\varphi\in G^{\delta,s}$, then

[TABLE]

A derivative can be estimated in two ways: ‘larger $\delta$’ or ‘larger $s$’.

Proof.

The second inequality is easy to prove. We give a proof of the first in order to clarify that the assumption $\delta\leq 1$ in [1, 2] is superfluous. The present author thinks that the authors of [1, 2] wrote $\delta\leq 1$ not because they really needed it for the omitted proof but for the sole reason that they were interested only in $0<\delta\leq 1$.

Set $f(x)=x^{2}\exp[2(-\delta+\delta^{\prime})x],\,x\geq 0$. We have

[TABLE]

Since $0\leq f(x)\leq(e^{-1}/(\delta-\delta^{\prime}))^{2}$, we get $\|\varphi_{x}\|_{\delta^{\prime},s}^{2}\leq(e^{-1}/(\delta-\delta^{\prime}))^{2}\|\varphi\|_{\delta,s}^{2}$. ∎

We set $A(\varphi)=\mu(\varphi)-\varphi_{xx}$, $B(\varphi)=\mu(\varphi)+(-\partial_{x}^{2}+\partial_{x}^{4})\varphi$. For $\varphi=\sum_{k\in\mathbb{Z}}a_{k}e^{2k\pi ix}\in G^{\delta,s}$, we have

[TABLE]

It follows that $A$ is a bounded operator from $G^{\delta,s+2}$ to $G^{\delta,s}$. It is a bijection and its inverse $A^{-1}$ is a pseudodifferential operator of order $-2$. Therefore it is bounded from $G^{\delta,s}$ to $G^{\delta,s+2}$. On the other hand, $B^{-1}$ is a pseudodifferential operator of order $-4$. Notice that $A^{-1}$ and $B^{-1}$ commute with $\partial_{x}$.

The following proposition is easy to prove.

Proposition 8**.**

We have

[TABLE]

for $\varphi\in G^{\delta,s}$.

If $\varphi\in G^{\delta,s}$, then by Propositions 7 and 8, we have the following estimates of the ‘larger $\delta$, smaller $s$’ type:

[TABLE]

In these estimates, the left-hand sides can be replaced with $\|\partial_{x}A^{-1}(\varphi)\|_{\delta^{\prime},s+2}$ and $\|\partial_{x}B^{-1}(\varphi)\|_{\delta^{\prime},s+4}$ respectively, but (1.2) and (1.3) are good enough.

In later sections, we will use the following estimates repeatedly. Let $R>0$ and $u_{0}\in G^{\delta,s+1}$ be given. If $\|u_{j}-u_{0}\|_{\delta,s+1}<R$ for $u_{j}\in G^{\delta,s+1}\subset G^{\delta,s}\,(j=1,2)$, then we have

[TABLE]

Proposition 9**.**

(cf. [9, Lem 2.2]) If $f\in H^{\infty}$ satisfies $\|f\|_{(\sigma,s)}<\infty$ for some $s\geq 0$ and for any $\sigma$ with $e^{2\pi\sigma}<\delta$, then $f\in A(\delta)$.

Conversely, $f\in A(\delta)$ implies $\|f\|_{(\sigma,s)}<\infty$ for any $\sigma,s$ with $e^{2\pi\sigma}<\delta$ and $s\geq 0$.

Proof.

In the proof of the first part, we may assume $s=0$. Let

[TABLE]

Then $\|f^{(j)}\|_{0}\leq j!e^{-2\pi j\sigma}M$ and

[TABLE]

Since there exists $C>0$ such that $\|\varphi\|_{\mathcal{\mathcal{C}}^{0}}\leq C(\|\varphi\|_{0}+\|\varphi^{\prime}\|_{0})$ for any $\varphi\in H^{1}$ by the Sobolev embedding, we have

[TABLE]

for any $x$. Therefore $f(x+iy)=\sum_{j=0}^{\infty}f^{(j)}(x)(iy)^{j}/j!$ is holomorphic in $|y|<e^{2\pi\sigma}$ for any $\sigma$ with $e^{2\pi\sigma}<\delta$. Hence $f\in A(\delta)$.

Next, we show the second part. Assume $0<\delta^{\prime}<\delta$. It is enough to prove $\|f\|_{(\sigma,s)}^{2}<\infty$ for any $\sigma,s$ with $e^{2\pi\sigma}<\delta^{\prime}$. Let $S=\sup_{S(\delta^{\prime})}|f|<\infty$. Set $L_{\pm}=\left\{t\pm iy;\,0\leq t\leq 1\right\}$. If $0<|y|\leq\delta^{\prime}$, the periodicity of $f$ and Goursat’s formula yield

[TABLE]

We have $|f^{(j)}(x)|\leq\pi^{-1}j!S/y^{j+1}$. Then $y^{2(j+1)}\|f^{(j)}\|_{0}^{2}\leq\pi^{-2}j!^{2}S^{2}$ in $|y|\leq\delta^{\prime}$. Integrating in $y\in[0,\delta^{\prime}]$, we get

[TABLE]

It follows that

[TABLE]

For $s>0$, we can prove that

[TABLE]

where $e^{2\pi\sigma}<e^{2\pi\sigma^{\prime}}<\delta^{\prime}$. To see this, we may assume that $s$ is a positive integer in view of $\|f\|_{(\sigma,s_{1})}\leq\|f\|_{(\sigma,s_{2})}\,(s_{1}\leq s_{2})$. For simplicity, we explain the case of $s=1$ only. (The general case follows the same line of proof, the only additional tool being the binomial expansion of powers of the Japanese bracket.) We have

[TABLE]

where $I_{p}=\sum_{j=0}^{\infty}(j!)^{-2}e^{4\pi j\sigma}\sum_{k\in\mathbb{Z}}k^{2p}(2\pi k)^{2j}|\hat{f}(k)|^{2}\,(p=0,1)$. Obviously $I_{0}=\|f\|_{(\sigma,0)}^{2}\leq\|f\|_{(\sigma^{\prime},0)}^{2}$. On the other hand, setting $\ell=j+1$, we get

[TABLE]

The proof of the second part is over. ∎

Lemma 10**.**

If $\sigma<\sigma^{\prime}$, we have

[TABLE]

for any $s,s^{\prime}\geq 0$, where the constant depends on $s,s^{\prime},\sigma,\sigma^{\prime}$.

Proof.

This estimate follows from (1.8) immediately. ∎

Corollary 11**.**

The following four families of norms on $A(\delta)$ determine the same topology as a Fréchet space.

(i) $\sup_{z\in S(\delta^{\prime})}|f(z)|\,(0<\delta^{\prime}<\delta),\qquad$ (ii) $\|\cdot\|_{(\sigma,s)}\,(e^{2\pi\sigma}<\delta,s\geq 0),$

*(iii) $\|\cdot\|_{(\sigma,2)}\,(e^{2\pi\sigma}<\delta),\qquad$(iv) $\|\cdot\|_{(\sigma,0)}\,(e^{2\pi\sigma}<\delta)$.

With this topology, $A(\delta)$ is continuously embedded in $H^{\infty}(S^{1})=\cap_{s\geq 0}H^{s}(S^{1})$.*

Proof.

It is trivial that (ii) is stronger (not weaker) than (iv) and that (iii) is between (ii) and (iv). On the other hand, (1.8) implies (iv) is stronger than (ii).

The estimate (1.6) and the Taylor expansion imply that (iv) is stronger than (i). On the other hand, (1.7) implies (i) is stronger than (iv). ∎

Proposition 12**.**

(cf. [9, Lemma 2.4]) Let $f_{n}\in A(\delta)\,(n=0,1,2,\dots)$ be a sequence with $\|f_{n}\|_{(\sigma,s)}$ bounded, where $e^{2\pi\sigma}<\delta$, and assume $f_{\infty}\in A(\delta)$. If $f_{n}\to f_{\infty}$ in $H^{\infty}$ as $n\to\infty$, then $\|f_{n}\|_{(\sigma^{\prime},s)}\to f_{\infty}$ for each $\sigma^{\prime}<\sigma$.

Proof.

We may assume $f_{\infty}=0$. Let $M^{2}=\sup_{n}\|f_{n}\|_{(\sigma,s)}^{2}$. Then $j!^{-2}e^{4\pi j\sigma^{\prime}}\|f_{n}^{(j)}\|_{s}^{2}\leq Me^{4\pi(\sigma^{\prime}-\sigma)j}$. Since $\lim_{n\to\infty}\|f_{n}^{(j)}\|_{s}=0$ for each $j$ and $\sum_{j\geq 0}M^{2}e^{4\pi(\sigma^{\prime}-\sigma)j}<\infty$, we can apply (the sum version of ) Lebesgue’s dominated convergence theorem to $\|f_{n}\|_{(\sigma^{\prime},s)}^{2}=\sum_{j\geq 0}j!^{-2}e^{4\pi j\sigma^{\prime}}\|f_{n}^{(j)}\|_{s}^{2}\,(n=0,1,2,\dots)$. ∎

2. Local-in-time solutions

2.1. Autonomous Ovsyannikov theorem

We recall some basic facts about the autonomous Ovsyannikov theorem. Among many versions, we adopt the one in [1, 2]. Let $\left\{X_{\delta},\|\cdot\|_{\delta}\right\}_{0<\delta\leq 1}$ be a (decreasing) scale of Banach spaces, i.e. each $X_{\delta}$ is a Banach space and $X_{\delta}\subset X_{\delta^{\prime}},\|\cdot\|_{\delta^{\prime}}\leq\|\cdot\|_{\delta}$ for any $0<\delta^{\prime}<\delta\leq 1$. (For $s$ fixed, $\left\{G^{\delta,s},\|\cdot\|_{\delta,s}\right\}_{0<\delta\leq 1}$ is a scale of Banach spaces.) Assume that $F\colon X_{\delta}\to X_{\delta^{\prime}}$ is a mapping satisfying the following conditions.

(a) For any $u_{0}\in X_{1}$ and $R>0$, there exist $L=L(u_{0},R)>0,M=M(u_{0},R)>0$ such that we have

[TABLE]

if $0<\delta<1$ and

[TABLE]

if $0<\delta^{\prime}<\delta\leq 1$ and $u,v\in X_{\delta}$ satisfies $\|u-u_{0}\|_{\delta}<R,\|v-u_{0}\|_{\delta}<R$.

(b) If $u(t)$ is holomorphic on the disk $D(0,a(1-\delta))=\left\{t\in\mathbb{C}\colon|t|<a(1-\delta)\right\}$ with values in $X_{\delta}$ for $a>0,0<\delta<1$ satisfying $\sup_{|t|<a(1-\delta)}\|u(t)-u_{0}\|_{\delta}<R$, then the composite function $F(u(t))$ is a holomorphic function on $D(0,a(1-\delta))$ with values in $X_{\delta^{\prime}}$ for any $0<\delta^{\prime}<\delta$.

The autonomous Ovsyannikov theorem below is our main tool. For the proof, see [1].

Theorem 13**.**

Assume that the mapping $F$ satisfies the conditions (a) and (b). For any $u_{0}\in X_{1}$ and $R>0$, set

[TABLE]

Then, for any $\delta\in]0,1[$, the Cauchy problem

[TABLE]

has a unique holomorphic solution $u(t)$ in the disk $D(0,T(1-\delta))$ with values in $X_{\delta}$ satisfying

[TABLE]

2.2. $\mu$CH and $\mu$DP equations

First we consider the analytic Cauchy problem for the $\mu$CH equation (0.4), namely.

[TABLE]

Theorem 14**.**

Let $s>1/2$. If $u_{0}\in G^{1,s+1}$, then there exists a positive time $T=T(u_{0},s)$ such that for every $\delta\in]0,1[$, the Cauchy problem (2.5) has a unique solution which is a holomorphic function valued in $G^{\delta,s+1}$ in the disk $D(0,T(1-\delta))$. Furthermore, the analytic lifespan $T$ satisfies

[TABLE]

Proof.

Assume $\|u-u_{0}\|_{\delta,s+1}<R,\|v-u_{0}\|_{\delta,s+1}<R$. By Proposition 5 (i), the first inequality in Proposition 7 and (1.5),

[TABLE]

On the other hand, since we have $\mu(u)u-\mu(v)v=\mu(u-v)u+\mu(v)(u-v)$, Proposition 8 and (1.4) imply

[TABLE]

Therefore (1.2) yields

[TABLE]

Next, by Proposition 5 (i) and the second inequality in Proposition 7,

[TABLE]

Hence (1.2) gives

[TABLE]

Now set

[TABLE]

Then (2.6), (2.7) and (2.8) give the Lipschitz continuity of $F_{\mu}$:

[TABLE]

where $L=C\left(\|u_{0}\|_{1,s+1}+R\right),C=e^{-1}(c_{s+1}+4+4\pi^{2}c_{s})$.

Next we will derive an estimate of $\|F_{\mu}(u_{0})\|_{\delta,s+1}$. Since

[TABLE]

we have

[TABLE]

We set

[TABLE]

Because of Theorem 13, there exists a unique solution $u=u(t)$ to (2.5) which is a holomorphic mapping from $D(0,T(1-\delta))$ to $G^{\delta,s+1}$ and

[TABLE]

If we set $R=\|u_{0}\|_{1,s+1},$ we have

[TABLE]

∎

In Theorem 14, we assumed the initial value $u_{0}$ was in $G^{1,s+1}$. We can relax this assumption as in the following theorem.

Theorem 15**.**

If $u_{0}$ is a real-analytic function on $S^{1}$, then the Cauchy problem (2.5) has a holomorphic solution near $t=0$. More precisely, we have the following:

(i) There exists $\Delta>0$ such that $u_{0}\in G^{\Delta,s+1}\subset A(\Delta)$ for any $s$.

(ii) If $s>1/2$, there exists a positive time $T_{\Delta}=T(u_{0},s,\Delta)$ such that for every $d\in]0,1[$, the Cauchy problem (2.5) has a unique solution which is a holomorphic function valued in $G^{\Delta d,s+1}$ in the disk $D(0,T_{\Delta}(1-d))$. Furthermore, the analytic lifespan $T_{\Delta}$ satisfies

[TABLE]

when $\Delta$ is fixed.

Proof.

The first statement is nothing but Proposition 4.

Set $X_{d}=G^{\Delta d,s+1},\|\cdot\|_{d,s+1}^{(\Delta)}=\|\cdot\|_{\Delta d,s+1}$. Then $\left\{X_{d},\|\cdot\|_{d,s+1}^{(\Delta)}\right\}_{0<d\leq 1}$ is a (decreasing) scale of of Banach spaces and $u_{0}\in X_{1}$.

Assume $\|u-u_{0}\|_{d,s+1}^{(\Delta)}<R,\|v-u_{0}\|_{d,s+1}^{(\Delta)}<R$ and $0<d^{\prime}<d\leq 1$. Then (2.6), (2.7) and (2.8) give ($\delta=\Delta d,\delta^{\prime}=\Delta d^{\prime}$) the following counterpart of (2.10):

[TABLE]

where $L_{\Delta}=\Delta^{-1}C\left(\|u_{0}\|_{1,s+1}^{(\Delta)}+R\right)=C\left(\|u_{0}\|_{\Delta,s+1}+R\right),C=e^{-1}(c_{s+1}+4+4\pi^{2}c_{s})$. Simpler estimates give

[TABLE]

We set

[TABLE]

Because of Theorem 13, there exists a unique solution $u=u(t)$ to (2.5) which is a holomorphic mapping from $D(0,T(1-\delta))$ to $X_{d}=G^{\Delta d,s+1}$ and

[TABLE]

If we set $R=\|u_{0}\|_{1,s+1}^{(\Delta)}=\|u_{0}\|_{\Delta,s+1},$ we have

[TABLE]

∎

We can study the following Cauchy problem for the $\mu$DP equation (0.5) by using the same estimates (2.6) and (2.7).

[TABLE]

Theorem 16**.**

*If $u_{0}$ is a real-analytic function on $S^{1}$, then the Cauchy problem (2.12) has a holomorphic solution near $t=0$. More precisely, we have the following:

(i) There exists $\Delta>0$ such that $u_{0}\in G^{\Delta,s+1}\subset A(\Delta)$ for any $s$.

(ii) If $s>1/2$, there exists a positive time $T_{\Delta}=T(u_{0},s,\Delta)$ such that for every $d\in]0,1[$, the Cauchy problem (2.12) has a unique solution which is a holomorphic function valued in $G^{\Delta d,s+1}$ in the disk $D(0,T_{\Delta}(1-d))$. Furthermore, the analytic lifespan $T_{\Delta}$ satisfies*

[TABLE]

when $\Delta$ is fixed.

2.3. Higher-order $\mu$CH equation

We consider the analytic Cauchy problem for the higher-order $\mu$CH equation (0.6), namely.

[TABLE]

Theorem 17**.**

Let $s>1/2$. If $u_{0}\in G^{1,s+3}$, then there exists a positive time $T=T(u_{0},s)$ such that for every $\delta\in]0,1[$, the Cauchy problem (2.13) has a unique solution which is a holomorphic function valued in $G^{\delta,s+3}$ in the disk $D(0,T(1-\delta))$. Furthermore, the analytic lifespan $T$ satisfies

[TABLE]

Proof.

Assume $u_{0}\in G^{\delta,s+3}$ and $\|u-u_{0}\|_{\delta,s+3}<R,\|v-u_{0}\|_{\delta,s+3}<R$. We follow the proofs of (2.6),(2.7) and (2.8) with (1.3) instead of (1.2) to obtain

[TABLE]

Next we study the difference associated with $\partial_{x}B^{-1}(u_{x}u_{xxx})$. Since $u_{x}u_{xxx}-v_{x}v_{xxx}=(u_{x}-v_{x})u_{xxx}+v_{x}(u_{xxx}-v_{xxx})$, we have

[TABLE]

Since

[TABLE]

we have

[TABLE]

Therefore by (1.3)

[TABLE]

Next we study the difference associated with $\partial_{x}B^{-1}(u_{xx}^{2})$. We have $u_{xx}^{2}-v_{xx}^{2}=(u_{xx}+v_{xx})(u_{xx}-v_{xx})$ and

[TABLE]

Since

[TABLE]

we have

[TABLE]

and

[TABLE]

Now we set

[TABLE]

Then by (2.14), (2.15), (2.16), (2.17) and (2.18), we obtain

[TABLE]

We need an estimate of $G(u_{0})$. By using

[TABLE]

we obtain

[TABLE]

We set

[TABLE]

Because of Theorem 13, there exists a unique solution $u=u(t)$ to (2.13) which is a holomorphic mapping from $D(0,T(1-\delta))$ to $G^{\delta,s+3}$ and

[TABLE]

If we set $R=\|u_{0}\|_{1,s+3}$, we have

[TABLE]

∎

Similarly, we have the following theorem about the other higher order $\mu$CH (0.7).

Theorem 18**.**

Let $s>1/2$. If $u_{0}\in G^{1,s+3}$, then there exists a positive time $T=T(u_{0},s)$ such that for every $\delta\in(0,1)$, the Cauchy problem

[TABLE]

has a unique solution which is a holomorphic function valued in $G^{\delta,s+3}$ in the disk $D(0,T(1-\delta))$. Furthermore, the analytic lifespan $T$ satisfies

[TABLE]

Proof.

The proof is almost the same as for Theorem 17. Indeed, $A^{-2}$ has the same properties as those of $B^{-1}$. ∎

Global-in-time analytic solutions of (2.5), (2.12), (2.13) and (0.7) will be studied in the following sections. The proofs will rely on known results about the global existence in Sobolev spaces. On the other hand, that kind of existence for the non-quasilinear equation (0.8) is unknown. Therefore, the argument given below is not valid for it. The local theory of (0.8) will be given in Appendix.

3. Global-in-time solutions

3.1. Statement of the main results

We recall known results about global-in-time solutions to $\mu$CH and $\mu$DP in Sobolev spaces. First, as for $\mu$CH, we have

Theorem 19**.**

([11, Theorem 5.1, 5.5]) Let $s>5/2$. Assume that $u_{0}\in H^{s}(S^{1})$ has non-zero mean and satisfies the condition

[TABLE]

*Then (2.5) has a unique global-in-time solution in $\mathcal{C}(\mathbb{R},H^{s}(S^{1}))\cap\mathcal{C}^{1}(\mathbb{R},H^{s-1}(S^{1}))$.

Moreover, local well-posedness (in particular, uniqueness) holds.*

The quantity $(\mu-\partial_{x}^{2})u_{0}$ is the $\mu$-version of the McKean quantity $(1-\partial_{x}^{2})u_{0}$ ([15]). There is an analogous result about $\mu$DP.

Theorem 20**.**

([14, Theorem 5,1, 5.4]) Let $s>3$. Assume that $u_{0}\in H^{s}(S^{1})$ has non-zero mean and satisfies the condition

[TABLE]

*Then (2.12) has a unique global-in-time solution in $\mathcal{C}(\mathbb{R},H^{s}(S^{1}))\cap\mathcal{C}^{1}(\mathbb{R},H^{s-1}(S^{1}))$.

Moreover, local well-posedness holds.*

We will discuss global-in-time analytic solutions. These solutions are analytic in both the time and space variables. Notice that in the KdV and other cases treated in [9], solutions are analytic in the space variable only. This is due to the absence of Cauchy-Kowalevsky type theorems. Our main result about the $\mu$-Camassa-Holm equation is the following:

Theorem 21**.**

Assume that a real-analytic function $u_{0}$ on $S^{1}$ has non-zero mean and satisfies the condition

[TABLE]

Then the Cauchy problem (2.5) has a unique solution $u\in\mathcal{C}^{\omega}(\mathbb{R}_{t}\times S_{x}^{1})$.

We have the following estimate of the radius of analyticity. Let $u_{0}\in A(r_{0})$. Fix $\sigma_{0}<(\log r_{0})/(2\pi)$ and set

[TABLE]

Then, for any fixed $T>0$, we have $u(\cdot,t)\in A(e^{2\pi\sigma(t)})$ for $t\in[-T,T]$.

There is an analogue about the $\mu$DP equation which reads as follows:

Theorem 22**.**

Assume that a real-analytic function $u_{0}$ on $S^{1}$ has non-zero mean and satisfies the condition

[TABLE]

Then the Cauchy problem (2.12) has a unique solution $u\in\mathcal{C}^{\omega}(\mathbb{R}_{t}\times S_{x}^{1})$.

We have the following estimate of the radius of analyticity. Let $u_{0}\in A(r_{0})$. Fix $\sigma_{0}<(\log r_{0})/(2\pi)$ and set

[TABLE]

Then, for any fixed $T>0$, we have $u(\cdot,t)\in A(e^{2\pi\sigma^{\prime}(t)})$ for $t\in[-T,T]$.

The proof of Theorem 21 will be given in the later subsections. First we will prove the analyticity in $x$ and establish the lower bound of $\sigma(t)$. Next we will establish the analyticity in $(t,x)$. The proof of Theorem 22 is essentially contained in that of Theorem 21. We can employ Remark 26 instead of Proposition 25.

3.2. Regularity theorem by Kato-Masuda

In [9], the authors used their theory of Liapnov families to prove a regularity result about the KdV and other equations. Later, it was applied to a generalized Camassa-Holm equation in [3]. Here we recall the abstract theorem in [9] in a weaker, more concrete form. It is good enough for our purpose. The inequality (3.1) is a special case of that in [9]. Notice that $v$ and $u(t,\cdot)$ are elements of $\mathcal{O}$ in (3.1) and (3.2). In applications, making a suitable choice of the subset $\mathcal{O}$ of $Z$ is essential.

Theorem 23**.**

Let $X$ and $Z$ be Banach spaces. Assume that $Z$ is a dense subspace of $X$. Let $\mathcal{O}$ be an open subset of $Z$ and $F$ be a continuous mapping from $Z$ to $X$. Let $\left\{\Phi_{s};-\infty<s<\infty\right\}$ be a family of real-valued functions on $Z$ satisfying

(a) The Fréchet partial derivative of $\Phi_{s}(v)$ in $v\in Z$ exists not only in $\mathcal{L}(Z;\mathbb{R})$ but also in $\mathcal{L}(X;\mathbb{R})$. It is denoted by $D\Phi_{s}(v)$. This statement makes sense because $\mathcal{L}(X;\mathbb{R})\subset\mathcal{L}(Z;\mathbb{R})$ by the canonical identification. [(a) follows from (b) below.]

(b) The Fréchet derivative of $\Phi_{s}(v)$ in $(s,v)$ exists not only in $\mathcal{L}(\mathbb{R}\times Z;\mathbb{R})$ but also in $\mathcal{L}(\mathbb{R}\times X;\mathbb{R})$ and is continuous from $\mathbb{R}\times Z$ to $\mathcal{L}(\mathbb{R}\times X;\mathbb{R})$. This statement makes sense because $\mathcal{L}(\mathbb{R}\times X;\mathbb{R})\subset\mathcal{L}(\mathbb{R}\times Z;\mathbb{R})$ by the canonical identification.

(c) There exist positive constants $K$ and $L$ such that

[TABLE]

holds for any $v\in\mathcal{O}.$ Here $\langle\cdot,\cdot\rangle$ (no subscript) is the pairing of $X$ and $\mathcal{L}(X;\mathbb{R})$.

Let $u\in\mathcal{C}([0,T];\,\mathcal{O})\cap\mathcal{C}^{1}([0,T];\,X)$ be the solution to the Cauchy problem

[TABLE]

Moreover, for a fixed constant $s_{0}$, set

[TABLE]

for $t\in[0,T]$. Then we have

[TABLE]

Roughly speaking, this theorem means that the regularity of $u(t)$ for $t\in[0,T]$ follows from that of $u_{0}$. Later we will use this when $X=H^{m+2},\,Z=H^{m+5}$ and $\Phi_{s}$ is related to some variant of the Sobolev norms.

We can extend Theorem 23 to $t\leq 0$.

Corollary 24**.**

Let $u\in\mathcal{C}([-T,T];\,\mathcal{O})\cap\mathcal{C}^{1}([-T,T];\,X)$ be the solution to the Cauchy problem (3.2). Extend the definition of $\rho$ and $\sigma$ sot that $r(t)=r(|t|),s(t)=s(|t|)$ by

[TABLE]

for $t\in[-T,T]$. Then we have

[TABLE]

Proof.

For $t\leq 0$, set $t^{\prime}=-t,\,\tilde{u}(t^{\prime})=u(-t^{\prime})=u(t)$. Then $\tilde{u}$ satisfies $d\tilde{u}/dt^{\prime}=-F(\tilde{u}),\,\tilde{u}(0,x)=u_{0}(x)$. Notice that $-F$ satisfies (3.1). By Theorem 23, we have $\Phi_{s(t^{\prime})}(\tilde{u}(t^{\prime}))\leq r(t^{\prime})$ for $t^{\prime}\in[0,T]$. It means $\Phi_{s(t)}(u(t))\leq r(t)$ for $t\in[-T,0]$. ∎

3.3. Pairing and estimates

Recall the norm $\|\cdot\|_{(\sigma,s)}$ in Section 1. When $s=2$, we have

[TABLE]

It is approximated by the finite sum

[TABLE]

Set $\Psi_{j}(v)=\frac{1}{2}\|v^{(j)}\|_{2}^{2}=\frac{1}{2}\|\Lambda^{2}v^{(j)}\|_{0}^{2}=\frac{1}{2}\int_{S^{1}}\left(\Lambda^{2}v^{(j)}\right)^{2}\,dx$ for $j=0,1,\dots,m$. We introduce

[TABLE]

Later $\left\{\Phi_{\sigma,m}\right\}$ will play the role of $\left\{\Phi_{s}\right\}$ in Corollary 24. Assume $w\in H^{m+2},v\in H^{m+5}.$ Let $D\Psi_{j}$ be the Fréchet derivative of $\Psi_{j}$ and $\langle w,D\Psi_{j}(v)\rangle$ be the pairing of $w\in H^{m+2}$ and $D\Psi_{j}(v)\in\left(H^{m+2}\right)^{*}\subset\left(H^{m+5}\right)^{*}$. Indeed, $D\Psi_{j}(v)$ is an element of $\left(H^{m+2}\right)^{*}$, because

[TABLE]

where $\langle\cdot,\cdot\rangle_{2}$ is the $H^{2}$ inner product. Recall

[TABLE]

It is easy to see that $F_{\mu}$ is continuous from $H^{m+5}$ to $H^{m+2}$.

Proposition 25**.**

We have

[TABLE]

for $v\in H^{m+5}$, where $D\Phi_{\sigma,m}$ is the Fréchet derivative of $\Phi_{\sigma,m}$.

Proof.

We have

[TABLE]

By (3.4), we have

[TABLE]

By using (3.6), (3.7) and the estimates (3.21), (3.22) and (3.25) below, we obtain (3.5). In the following subsections, we will calculate the three terms in (3.7) separately. ∎

Remark 26*.*

The $\mu$DP equation can be studied by using the following estimate. Set

[TABLE]

Then we have

[TABLE]

3.3.1. Estimate of $\langle v^{(j)},\partial_{x}^{j}(vv_{x})\rangle_{2}$

We have

[TABLE]

Because of (1.1), we have

[TABLE]

where $I_{i}=\langle v^{(j+i)},\partial_{x}^{i}(vv^{(j+1)})\rangle_{0}$. Set $w=v^{(j)}$ for brevity. Then $I_{i}=\langle w^{(i)},\partial_{x}^{i}(vw_{x})\rangle_{0}$. Integrating $\partial_{x}(vw^{2}/2)=vww_{x}+v_{x}w^{2}/2$, we obtain

[TABLE]

Therefore the Schwarz inequality and Proposition 5 (ii) imply

[TABLE]

Next, we have $I_{1}=\int_{S^{1}}w_{x}\partial_{x}(vw_{x})\,dx=-\int_{S^{1}}vw_{x}w_{xx}\,dx$ by integration by parts. Integrating $\partial_{x}(vw_{x}^{2}/2)=vw_{x}w_{xx}+v_{x}w_{x}^{2}/2$, we obtain

[TABLE]

Therefore we get, by the Schwarz inequality, Proposition 5 (ii) and Proposition 7,

[TABLE]

The estimate of $I_{2}$ is a bit complicated. We divide it into a sum of three terms as in

[TABLE]

where

[TABLE]

We have

[TABLE]

Integrating $\partial_{x}(vw_{xx}^{2}/2)=vw_{xx}w_{xxx}+v_{x}w_{xx}^{2}/2$, we obtain

[TABLE]

Hence

[TABLE]

The three inequalities (3.14), (3.15) and (3.16) give

[TABLE]

Combining (3.11), (3.12), (3.13) and (3.17), we obtain

[TABLE]

and

[TABLE]

Next we calculate $\sum_{j=1}^{m}j!^{-2}e^{4\pi\sigma j}Q_{j}$. Recall

[TABLE]

By Proposition 5 (iii), we have

[TABLE]

Combining this estimate with the Schwarz inequality, we get

[TABLE]

Now we set $b_{k}=k!^{-1}e^{2\pi\sigma k}\|v^{(k)}\|_{2}\,(k=0,1,\dots,j)$. Then we have

[TABLE]

Set $B=\left(\sum_{j=0}^{m}b_{j}^{2}\right)^{1/2},\widetilde{B}=\left(\sum_{j=1}^{m}jb_{j}^{2}\right)^{1/2}$. Notice that $B^{2}=\|v\|_{(\sigma,2,m)}^{2}=2\Phi_{\sigma,m}(v),\,\widetilde{B}^{2}=(2\pi)^{-1}\partial_{\sigma}\Phi_{\sigma,m}(v)$. The repeated use of the Schwarz inequality gives ([9, Lemma 3.1])

[TABLE]

Recalling $Q_{0}=0$, we obtain

[TABLE]

The inequalities (3.19) and (3.20) give, together with (3.8),

[TABLE]

3.3.2. Estimate of $\langle v^{(j)},\partial_{x}^{j+1}A^{-1}\left[\mu(v)v\right]\rangle_{2}$

We consider the second term of the right-hand side of (3.7). By (1.1), we have

[TABLE]

where $J_{i}=\langle v^{(j+i)},A^{-1}v^{(j+i+1)}\rangle_{0}\,(i=0,1,2)$. Set $w=v^{(j)}$ for brevity. We have

[TABLE]

Since $A^{-1}\partial_{x}^{i}:H^{0}\to H^{0}\,(i=0,1,2)$ are bounded operators whose norms are equal to $1$, we have

[TABLE]

Therefore we get

[TABLE]

and

[TABLE]

3.3.3. Estimate of $\langle v^{(j)},\partial_{x}^{j+1}A^{-1}(v_{x}^{2})\rangle_{2}$

We consider the third term of the right-hand side of (3.7).

First, assume $j=0$. Since the norm of $\partial_{x}A^{-1}\colon H^{1}\to H^{2}$ does not exceed 1, we have

[TABLE]

Next we assume $j\geq 1$. We have

[TABLE]

This is similar to $Q_{j}$ in (3.10). Since the norm of $\partial_{x}^{2}A^{-1}\colon H^{2}\to H^{2}$ is 1, this operator can be neglected in estimating $\langle v^{(j)},(\partial_{x}^{2}A^{-1})(v^{(\ell)}v^{(j-\ell+1)})\rangle_{2}$. Moreover we have

[TABLE]

We can follow (3.20) for $j\geq 1$ and employ (3.23) for $j=0$ (Recall $Q_{0}=0$). Finally we obtain

[TABLE]

It completes the proof of Proposition 25.

3.4. Analyticity in the space variable

In this subsection, we prove a part of Theorem 21. We assume that $u_{0}\in A(r_{0})$ is as in Theorem 21. Then we can apply Theorem 19 with arbitrarily large $s$. We will prove the analyticity of $u(t)$ in $x$ for each fixed $t$.

Proposition 27**.**

Let $T>0$ be given. We have $u(t)\in A(e^{2\pi\sigma(t)})\subset A(e^{2\pi\sigma(T)})$ for $t\in[-T,T]$, where the function $\sigma(t)$ is defined in Theorem 21.

Proof.

Theorem 19 implies $u(t)\in H^{\infty}$. Set

[TABLE]

Then $u(t)\in\mathcal{O}$ for $t\in[-T,T]$. Proposition 25 implies that

[TABLE]

holds for any $v\in\mathcal{O}.$ Fix $\sigma_{0}$ with $e^{2\pi\sigma_{0}}<r_{0}$. Then $u_{0}\in A(r_{0})$ implies $\|u_{0}\|_{(\sigma_{0},2)}<\infty$ by Proposition 9.

Set

[TABLE]

These functions correspond to $r(t)$ and $s(t)$ in Corollary 24. Similarly, set

[TABLE]

We have $\rho_{m}(t)\leq\rho_{m+1}(t)\leq\rho(t)$, $\rho_{m}(t)\to\rho(t)\,(\text{as }m\to\infty)$ and $\sigma_{m}(t)\geq\sigma_{m+1}(t)\geq\sigma(t)$, $\sigma_{m}(t)\to\sigma(t)\,(\text{as }m\to\infty)$. We can apply Corollary 24 and obtain

[TABLE]

By Fatou’s Lemma,

[TABLE]

Therefore $u(t)\in A(e^{2\pi\sigma(t)})\subset A(e^{2\pi\sigma(T)})$ for $t\in[-T,T]$ by Proposition 9. ∎

Proposition 28**.**

The mapping $[-T,T]\to A(e^{2\pi\sigma(T)}),\,t\mapsto u(t)$ is continuous.

Proof.

Let $\left\{t_{n}\right\}\subset[-T,T]$ be a sequence converging to $t_{\infty}\in[-T,T]$. We have $u(t_{n})\to u(t_{\infty})$ in $H^{\infty}$. On the other hand, $\|u(t_{n})\|_{(\sigma(T),2)}\,(n\geq 0)$ is bounded since

[TABLE]

Proposition 12 implies that $u(t_{n})$ converges to $u(t_{\infty})$ with respect to $\|\cdot\|_{(\sigma^{\prime},2)}\,(\sigma^{\prime}<\sigma(T))$. By Corollary 11, this means convergence in $A(e^{2\pi\sigma(T)})$. ∎

3.5. Analyticity in the space and time variables

We continue the proof of Theorem 21. In the previous subsection, we have established the analyticity in the space variable. Here, we will prove the analyticity in the space and time variables. By convention, a real-analytic function on a closed interval is real-analytic on some open neighborhood of the closed interval.

Proposition 29**.**

Under the situation of Theorem 21, for any $T>0$, there exists $\delta_{T}>0$ such that we have $u\in\mathcal{C}^{\omega}([-T,T],A(\delta_{T}))$.

Proof.

We have $u_{0}\in G^{\Delta,s+1}$ for any $\Delta<r_{0}$. Let $s>1/2$. By Theorem 15, there exists $\widetilde{T}>0$ such that the Cauchy problem (2.5) has a unique solution $\tilde{u}\in\mathcal{C}^{\omega}(|t|\leq\widetilde{T}(1-d),G^{\Delta d,s+1})$ for $0<d<1$. We have $\tilde{u}=u$ by the local uniqueness, where $u$ is the solution in Theorem 19. Set $d=1/2,\,\widehat{T}=\widetilde{T}/2$. Then $u=\tilde{u}\in\mathcal{C}^{\omega}(|t|\leq\widehat{T},G^{\Delta/2,s+1})$. By Proposition 2, a convergent series in $G^{\Delta/2,s+1}$ is convergent in $A(\Delta/2)$. We have $u\in\mathcal{C}^{\omega}(|t|\leq\widehat{T},A(\Delta/2))$.

We have shown that $u$ is analytic in $t$ at least locally. Our next step is to show that $u$ is analytic in $t$ globally. Set

[TABLE]

We prove $T^{*}=\infty$ by contradiction. Assume $T^{*}<\infty$. By Proposition 27, $u(T^{*})$ is well-defined and there exists $\delta^{*}>0$ such that

[TABLE]

By Theorem 15 (with $t$ replaced with $t-T^{*}$), there exists $\varepsilon>0$ and $\hat{u}\in\mathcal{C}^{\omega}([T^{*}-\varepsilon,T^{*}+\varepsilon],G^{\delta^{\prime}/2,s+1})$ such that

[TABLE]

By the local uniqueness, we have $\hat{u}=u$. Namely, $\hat{u}$ is an extension of $u$ up to $|t|\leq T^{*}+\varepsilon$ (valued in $G^{\delta^{\prime}/2,s+1}\subset A(\delta^{\prime}/2)$). Therefore $T^{*}+\varepsilon\in S$. This is a contradiction. ∎

Proposition 30**.**

Under the situation of Theorem 21, the Cauchy problem (2.5) has a unique solution $u\in\mathcal{C}^{\omega}(\mathbb{R}_{t}\times S_{x}^{1})$.

Proof.

The uniqueness in $H^{\infty}$ implies the uniqueness in the real-analytic category.

Let $T$ be fixed. For $r>0$ sufficiently small, we have

[TABLE]

The integral is performed in $A(\delta_{T})$ and converges with respect to $\|\cdot\|_{(\sigma,0)}\,(e^{2\pi\sigma}<\delta_{T})$. By Cauchy’s estimate, there exists $C_{0}>1/r$ such that

[TABLE]

Therefore we have

[TABLE]

and there exists $C>0$ such that

[TABLE]

Set $\Delta=\partial^{2}/\partial x^{2}+\partial^{2}/\partial t^{2}$. The binomial expansion of $\Delta^{\ell}\,(\ell=0,1,2,\dots)$ and (3.28) yield

[TABLE]

This estimates implies the real-analyticity of $u$ due to Theorem 31 by Komatsu given below. ∎

In the last step of the proof of Proposition 30, we have used the following theorem.

Theorem 31**.**

*([12]) Let $\Omega$ be a domain in $\mathbb{R}^{n}$ and let $P=P(\partial/\partial x_{1},\dots,\partial/\partial x_{n})$ be an elliptic partial differential operator of order $m$ with constant coefficients. Then, for a function $f\in L_{\text{loc}}^{2}(\Omega)$ to be analytic in $\Omega$, it is (necessary and) sufficient that

1. for every $\ell\in\mathbb{Z}_{+}$, $P^{\ell}f$ (in the sense of distributions) belongs to $L_{\text{loc}}^{2}(\Omega)$, and that
2. for every compact subset $K\subset\Omega$, there exist positive constants $M$ and $A$ such that*

[TABLE]

A better known result in this direction is [13], in which $P=P(x,\partial/\partial x_{1},\dots,\partial/\partial x_{n})$, $x=(x_{1,},\dots,x_{n})$, is an elliptic operator of order $m$ with analytic coefficients and the right-hand side of (3.29) is replaced with $M^{\ell+1}(m\ell)!$. We can employ the result of [13] instead of Theorem 31.

4. Global-in-time solutions: higher-order case

In this section we consider (2.13). Global-in-time solutions in Sobolev spaces have been studied in [18]. Notice that the non-zero mean and the no-change-of-sign conditions are not imposed.

4.1. Statement of the main results

Theorem 32**.**

([18, Theorem 2.1, 3.5]) Let $u_{0}\in H^{s}(S^{1})$, $s>7/2$. Then the Cauchy problem (2.13) has a unique global solution $u$ in the space $\mathcal{C}(\mathbb{R},H^{s})\cap\mathcal{C}^{1}(\mathbb{R},H^{s-1})$. Moreover, local uniqueness holds.

Remark 33*.*

In [18], the authors solve the Cauchy problem for $t>0$ only. Since the equation is invariant under $(t,u)\mapsto(-t,-u)$, the result for $t<0$ follows immediately.

There is an analogous result about (2.24).

Theorem 34**.**

([19, Theorem 2.1, 3.2]) Let $u_{0}\in H^{s}(S^{1})$, $s>7/2$. Then the Cauchy problem (2.24) has a unique global solution $u$ in the space $\mathcal{C}(\mathbb{R},H^{s})\cap\mathcal{C}^{1}(\mathbb{R},H^{s-1})$. Moreover, local uniqueness holds.

Our main result about the higher order $\mu$CH is the following theorem.

Theorem 35**.**

If $u_{0}$ is a real-analytic function on $S^{1}$, then the Cauchy problem (2.13) has a unique solution $u\in\mathcal{C}^{\omega}(\mathbb{R}_{t}\times S_{x}^{1})$.

We have the following estimate of the radius of analyticity. Assume $u_{0}\in A(r_{0})$. Fix $\sigma_{0}<(\log r_{0})/(2\pi)$ and set

[TABLE]

Then, for any fixed $T>0$, we have $u(\cdot,t)\in A(\tilde{\sigma}(t))$ for $t\in[-T,T]$.

Proof.

Theorem 32 implies $u(t)\in H^{\infty}$ if $u_{0}\in H^{\infty}$. Set

[TABLE]

Then the proof is almost the same as that of Theorem 21 and follows from Proposition 37 below. It is an analogue of Proposition 25. Notice that $\|v\|_{2}$ in (3.5) has been replaced with $\|v\|_{4}$. ∎

There is an analogous result about (2.24).

Theorem 36**.**

If $u_{0}$ is a real-analytic function on $S^{1}$, then the Cauchy problem (2.24) has a unique solution $u\in\mathcal{C}^{\omega}(\mathbb{R}_{t}\times S_{x}^{1})$. The estimate of the radius of analyticity is the same as in Theorem 35.

The proof of Theorem 36 is almost the same as that of Theorem 35. One has only to replace $B^{-1}$ with $A^{-2}$. The rest of this section is devoted to the proof of Theorem 35. It is enough to prove Proposition 37 below.

Proposition 37**.**

We have

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are given in Theorem 35.

Proof.

Recall (2.19), namely

[TABLE]

We have

[TABLE]

By (3.4), we have

[TABLE]

We compare it with (3.7). We encountered $\langle v^{(j)},\partial_{x}^{j}(vv_{x})\rangle_{2}$ in (3.7). The following two terms $-2\langle v^{(j)},\partial_{x}^{j+1}B^{-1}\left[\mu(v)v\right]\rangle_{2}$ and $-\frac{1}{2}\langle v^{(j)},\partial_{x}^{j+1}B^{-1}(v_{x}^{2})\rangle_{2}$ have better estimates than $-2\langle v^{(j)},\partial_{x}^{j+1}A^{-1}\left[\mu(v)v\right]\rangle_{2}$ and $-\frac{1}{2}\langle v^{(j)},\partial_{x}^{j+1}A^{-1}(v_{x}^{2})\rangle_{2}$. Therefore we can employ (3.21) and analogues of (3.22) and (3.25). Here we replace $\|v\|_{2}$ with $(\|v\|_{2}\leq)\|v\|_{4}$. Our remaining task is to estimate $3\langle v^{(j)},\partial_{x}^{j+1}B^{-1}\left[v_{x}v_{xxx}\right]\rangle_{2}$ and $\frac{7}{2}\langle v^{(j)},\partial_{x}^{j+1}B^{-1}(v_{xx}^{2})\rangle_{2}$. The results will be given as (4.8) and (4.13) in the following subsection. ∎

4.2. Estimates: higher-order case

4.2.1. Estimate of $\langle v^{(j)},\partial_{x}^{j+1}B^{-1}\left[v_{x}v_{xxx}\right]\rangle_{2}$

First we assume $j\geq 3$. Since

[TABLE]

This is better than (3.10). Indeed, $\partial_{x}^{4}B^{-1}\colon H^{2}\to H^{2}$ is as good as $\partial_{x}^{2}A^{-1}\colon H^{2}\to H^{2}$ and the binomial coefficients have become smaller. We follow (3.20) with $j\geq 3$ instead of $j\geq 0$ and get

[TABLE]

Next we consider the case $j=0$. We have

[TABLE]

and, by Proposition 5 (ii),

[TABLE]

These two inequalities yield

[TABLE]

We have used $\|v\|_{3}\leq\|v\|_{4}$, because $\|v\|_{4}$ will inevitably appear later. We deal with it by modifying the definition of $\mathcal{O}$, so that $\|v\|_{3}$ and $\|v\|_{4}$ are bounded there. Proposition 25 should be modified accordingly.

Next assume $j=1$. We have

[TABLE]

and, by Proposition 5 (ii) again,

[TABLE]

Therefore

[TABLE]

Next we assume $j=2$. We have

[TABLE]

and, by Proposition 5 (ii),

[TABLE]

Therefore

[TABLE]

By (4.4), (4.5), (4.6) and (4.7), we obtain

[TABLE]

4.2.2. Estimate of $\langle v^{(j)},\partial_{x}^{j+1}B^{-1}\left[v_{xx}^{2}\right]\rangle_{2}$

First we assume $j\geq 3$. Since

[TABLE]

This is better than (3.10). We follow (3.20) with $j\geq 3$ instead of $j\geq 0$ and get

[TABLE]

Next we consider the case $j=0$. We have

[TABLE]

and, by Proposition 5 (ii),

[TABLE]

These two inequalities yield

[TABLE]

Next assume $j=1$. We have

[TABLE]

and, by Proposition 5 (ii),

[TABLE]

Therefore

[TABLE]

Next assume $j=2$. We have

[TABLE]

and

[TABLE]

Therefore

[TABLE]

By using (4.9), (4.10), (4.11) and (4.12), we obtain

[TABLE]

Appendix: Local study of the non-quasilinear modified $\mu$CH equation

The difficulty of (0.8) lies in the presence of the non-quasilinear terms $-\frac{1}{3}u_{x}^{3}$ and $-\frac{1}{3}\mu(u_{x}^{3})$. In [2], the authors employed the power series method to deal with the non-quasilinear term $au^{k-2}u_{x}^{3}$ of the $k$-$abc$-equation. In the present paper, we overcome the difficulty of non-quasilinearity by a $\mu$-version of a classical trick used in the proof of the Cauchy-Kowalevsky theorem ([6]) following [8] and [17]. We set $v=u_{x}$ and differentiate (0.8) in $x$. It can be proved that (0.8) is equivalent to the following quasilinear modified $\mu$CH system:

[TABLE]

Of course, this trick works for the the $k$-$abc$-equation as well. We can prove unique solvability of the Cauchy problem for (4.14) and (0.8).

The Cauchy problem (0.8) for the non-quasilinear modified $\mu$CH equation can be written in the following form:

[TABLE]

We introduce the system below.

[TABLE]

Theorem 38**.**

The Cauchy problems (4.15) and (4.16) are equivalent to each other if $v_{0}(x)=\partial_{x}u_{0}(x)$. In particular, (4.16) implies $v=\partial_{x}u=u_{x}$ if $v_{0}(x)=\partial_{x}u_{0}(x)$.

Proof.

By differentiation with respect to $x$, we get the second equation in (4.16) from (4.15).

To show the converse, differentiate both sides of the first equation of (4.16) in $x$. By comparing it with the second equation, we get

[TABLE]

It is enough to prove that $w_{t}+\partial_{x}A^{-1}[a(t)w]=0$ and $w(0,x)=0$ imply $w=0$, where $a(t)$ is a continuous function in $t$. Set $w=\sum_{k\in\mathbb{Z}}w_{k}(t)e^{2k\pi ix}$. Then we have

[TABLE]

Since $w_{k}(0)=0$, we have $w_{k}(t)=0$ ($k\in\mathbb{Z}$) for any $t$. It implies $w=0$ and $v=u_{x}.$ ∎

Remark 39*.*

The trick of setting $v=u_{x}$ has been used in [8] (non-$\mu$ equation) and [17] in a different function space rather formally, i.e. without discussion corresponding to Theorem 38. In [8], this trick is applied to a quasilinear equation.

Theorem 40**.**

If $u_{0}$ and $v_{0}$ are real-analytic functions on $S^{1}$, then the Cauchy problem (4.16) has a holomorphic solution near $t=0$. More precisely, we have the following:

(i) There exists $\Delta>0$ such that $u_{0},v_{0}\in G^{\Delta,s+1}\subset A(\Delta)$ for any $s$.

(ii) If $s>1/2$, there exists a positive time $T_{\Delta}=T(u_{0},v_{0},s,\Delta)$ such that for every $d\in(0,1)$, the Cauchy problem (4.16) has a unique solution which is a holomorphic function valued in $\oplus^{2}G^{\Delta d,s+1}$ in the disk $D(0,T_{\Delta}(1-d))$. Furthermore, if $\gamma=0$, the analytic lifespan $T_{\Delta}$ satisfies

[TABLE]

when $\Delta$ is fixed. On the other hand, if $\gamma\neq 0$, we have the asymptotic behavior

[TABLE]

Proof.

We give a detailed proof assuming $\Delta=1$. The general case can be proved in the same way as for Theorem 15. The norm on $\oplus^{2}G^{\delta,s+1}$ is defined by $\|(u,v)\|_{\delta,s+1}=\|u\|_{\delta,s+1}+\|v\|_{\delta,s+1}$. We use the same notation $\|\cdot\|_{\delta,s+1}$ for $G^{\delta,s+1}$ and $\oplus^{2}G^{\delta,s+1}$. Assume $\|(u,v)-(u_{0},v_{0})\|_{\delta,s+1}<R,\|(u^{\prime},v^{\prime})-(u_{0},v_{0})\|_{\delta,s+1}<R$. Set $R_{s+1}=R+\max\left(\|u_{0}\|_{1,s+1},\|v_{0}\|_{1,s+1}\right)$. Assume $0<\delta^{\prime}<\delta\leq 1$. Then by (1.4), we have

[TABLE]

where $w$ is any of $u,v,u^{\prime},v^{\prime}$. Let us consider differences concerning $\mu(u)uv$ and $\mu(u)(uv)_{x}$. Since $uv-u^{\prime}v^{\prime}=u(v-v^{\prime})+(u-u^{\prime})v^{\prime}$, we get

[TABLE]

Combining (4.18) with Proposition 7, we get

[TABLE]

On the other hand, we have

[TABLE]

Combining

[TABLE]

with (4.17) and (4.18), we obtain

[TABLE]

Next, combining

[TABLE]

with (4.19) and (4.20), we obtain

[TABLE]

The next step is to consider $-v^{3}/3$, $-(v^{3})_{x}/3$ and $\mu(v^{3})$. The factorization $v^{3}-v^{\prime 3}=(v^{2}+vv^{\prime}+v^{\prime 2})(v-v^{\prime})$ implies

[TABLE]

Here we have

[TABLE]

Therefore

[TABLE]

and it immediately gives

[TABLE]

Combining (4.23) with Proposition 7, we get

[TABLE]

An immediate consequence of (4.24) is

[TABLE]

The last step is to consider the images of $\mu^{2}(u)u,\mu^{2}(u)v,\mu(u)v^{2}$ and $\mu(u)(v^{2})_{x}$ under $\partial_{x}A^{-1}$. Since

[TABLE]

we have, by $c_{s}\leq c_{s+1}$, $\|\cdot\|_{\delta,s}\leq\|\cdot\|_{\delta,s+1}$ and $R_{s}\leq R_{s+1}$,

[TABLE]

To derive the last one, we have used the second inequality of Proposition 7. We employ (1.2) to obtain

[TABLE]

It is much easier to obtain

[TABLE]

Set

[TABLE]

Then by (4.21), (4.23), (4.27), (4.29), (4.26) and (4.31), we have

[TABLE]

By (4.22), (4.25), (4.28), (4.30) and (4.32), we have

[TABLE]

We have obtained two inequalities of the Lipschitz type.

Next we estimate $F_{j}(u_{0},v_{0})\,(j=1,2)$. We have

[TABLE]

Since $X^{2}Y\leq(X+Y)^{3}/3,X^{3}\leq(X+Y)^{3}$ for $X,Y\geq 0$, we get

[TABLE]

Similarly we have

[TABLE]

Therefore

[TABLE]

∎

We set $R=\|(u_{0},v_{0})\|_{1,s+1}$. If $\gamma=0$, the constants corresponding to $L$ and $M$ in (2.1) and (2.2) are of degrees 2 and 3 respectively. Therefore $T$ equals a constant multiple of $\|(u_{0},v_{0})\|_{1,s+1}^{-2}$ if $\gamma=0$. If $\gamma\neq 0$, we have to consider two cases separately: large or small initial values. When the initial values are large, larger order terms are dominant and $T$ is approximated by a constant multiple of $\|(u_{0},v_{0})\|_{1,s+1}^{-2}$, while $T$ approaches a constant as the initial values approach 0. Theorem 38 allows us to get a result about the original Cauchy problem (4.15). We assume $u_{0}\in G^{\Delta,s+2}$ so that $v_{0}=\partial_{x}u_{0}$ belongs to $G^{\Delta,s+1}$.

Theorem 41**.**

If $u_{0}$ is a real-analytic function on $S^{1}$, then the Cauchy problem (4.15) has a holomorphic solution near $t=0$. More precisely, we have the following:

(i) There exists $\Delta>0$ such that $u_{0}\in G^{\Delta,s+2}$ for any $s$.

(ii) If $s>1/2$, there exists a positive time $T_{\Delta}=T(u_{0},s,\Delta)$ such that for every $d\in(0,1)$, the Cauchy problem (4.15) has a unique solution which is a holomorphic function valued in $G^{\Delta d,s+2}$ in the disk $D(0,T_{\Delta}(1-d))$. Furthermore, if $\gamma=0$, the analytic lifespan $T_{\Delta}$ satisfies

[TABLE]

when $\Delta$ is fixed. On the other hand, if $\gamma\neq 0$, we have the asymptotic behavior

[TABLE]

Remark 42*.*

In [17], the author solved (4.15) in a space of analytic functions following [8]. What is new in the present paper is precise estimates of the lifespan.

Acknowledgment

This work was partially supported by JSPS KAKENHI Grant Number 26400127.