Local and global well-posedness for a quadratic Schr\"odinger system on spheres and Zoll manifolds

TL;DR

This paper establishes local and global well-posedness results for a quadratic Schrödinger system on spheres and Zoll manifolds, using bilinear Strichartz estimates and Gagliardo-Nirenberg inequalities.

Contribution

It introduces bilinear Strichartz estimates for quadratic Schrödinger systems on Zoll manifolds and proves well-posedness results extending previous work.

Findings

Local well-posedness for s > 1/4 on 2D spheres and Zoll manifolds.

Global well-posedness for s ≥ 1 in dimensions 2 and 3.

Bilinear Strichartz estimates derived for specific spectral conditions.

Abstract

We consider the initial value problem (IVP) associated to a quadratic Schr\"odinger system \begin{equation*} \begin{cases} i \partial_{t} v \pm \Delta_{g} v - v = \epsilon_{1} u \bar{v}, & t \in \mathbb{R},\; x \in M, \\[2ex] i \sigma \partial_{t} u \pm \Delta_{g} u - \alpha u = \frac{\epsilon_{2}}{2} v^{2}, & \sigma > 0, \;\alpha \in \mathbb{R},\; \epsilon_{i} \in \mathbb{C}\, (i = 1, 2),\\[2ex] (v(0), u(0)) = (v_0, u_0), \end{cases} \end{equation*} posed on a $d$-dimensional sphere $ \mathbb{S}^{d}$ or a compact Zoll manifold $M$. Considering $\sigma=\frac{\theta}{\beta}$ with $\theta, \beta\in \{n^2:n\in\mathbb{Z}\}$ we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data $(v_0, u_0)\in H^s(M)\times H^s(M)$ whenever $s>\frac{1}{4}$ in the case $M = \mathbb{S}^{2}$ or a Zoll manifold, and $s > \frac{d - 2}{2}$ in the case $M = \mathbb{S}^{d}$ ($d \geq 3$) induced with the canonical metric. Moreover, in dimensions $2$ and $3$, we use a Gagliardo-Nirenberg type inequality to prove that the local solution can be extended globally in time whenever $s \geq 1$.

Equations

$$\{\begin{array}{ll}i{\partial }_{t}v\pm {\Delta }_{g}v-v={ϵ}_{1}u\bar{v},& t\in \mathbb{R},x\in M,\\ i\sigma {\partial }_{t}u\pm {\Delta }_{g}u-\alpha u=\frac{{ϵ}_2}{2}{v}^2,& \sigma >0,\alpha \in \mathbb{R},{ϵ}_i\in \mathbb{C}(i=1,2),\\ (v(0),u(0))=(v_0,u_0),& \end{array}$$

$$\mathcal{M}(t)={\int }_M|v(t){|}^2+2\sigma |u(t){|}^{2}dg=\mathcal{M}(0),$$

$$\mathcal{E}(t)={\int }_M\left(|\nabla v(t){|}_g^2+|\nabla u(t){|}_g^2+|v(t){|}^2+\alpha |u(t){|}^2+{ϵ}_1\mathrm{Re}(v^2(t)\overline{u(t)})\right)dg=\mathcal{E}(0),$$

$$\{\begin{array}{l}i{\partial }_{t}v+\Delta v-v={ϵ}_{1}u\bar{v}\\ i{\partial }_{t}u+\frac{1}{\sigma }\Delta u-\frac{\alpha }{\sigma }u=\frac{{ϵ}_2}{2\sigma }{v}^2\\ (v(0),u(0))=(v_0,u_0).\end{array}$$

$$\{\begin{array}{l}v(t):=V(t)v_0-i{ϵ}_1{\int }_0^{t}V(t-t^{\prime })\overline{v}(t^{\prime })u(t^{\prime })d{t}^{\prime },\\ u(t):=U_{\sigma }(t)u_0-i\frac{{ϵ}_2}{2\sigma }{\int }_0^t{U}_{\sigma }(t-t^{\prime })v^2(t^{\prime })d{t}^{\prime },\end{array}$$

$$P_{k}f=⟨e_k,f{⟩}_{L^2(M)}{e}_k.$$

$$\Vert u{\Vert }_{H^s(M)}^2:=\sum _{k\ge 0}⟨{\mu }_k{⟩}^s\Vert P_{k}u{\Vert }_{L^2(M)}^2\simeq \Vert (1-\Delta {)}^{\frac{s}{2}}u{\Vert }_{L^2(M)}^2.$$

$$\Vert u{\Vert }_{X_{-\Delta }^{s,b}(\mathbb{R}\times M)}^2=\sum _k\Vert ⟨\tau +{\mu }_k{⟩}^b⟨{\mu }_k{⟩}^{\frac{s}{2}}\widehat{P_{k}u}(\tau ){\Vert }_{L^2(\mathbb{R};L^2(M))}^2=\Vert S(-t)u(t,\cdot ){\Vert }_{H^b({\mathbb{R}}_t;H_x^s(M))}^2,$$

$$\begin{split}\|u\|^{2}_{X_{-\delta\Delta+\gamma}^{s,b}(\mathbb{R}\times M)}&=\sum_{k}\Big{\|}\langle\tau+\delta\mu_{k}+\gamma\rangle^{b}\langle\mu_{k}\rangle^{\frac{s}{2}}\widehat{P_{k}u}(\tau)\Big{\|}_{L^{2}(\mathbb{R};L^{2}(M))}^{2}\\ &=\sum_{k}\langle\mu_{k}\rangle^{s}\int_{\mathbb{R}}\langle\tau+\delta\mu_{k}+\gamma\rangle^{2b}\|\widehat{P_{k}u}(\tau)\|_{L^{2}(M)}^{2}d\tau\\ &=\left\|e^{it(-\delta\Delta+\gamma)}u(t,\cdot)\right\|_{H^{b}(\mathbb{R}_{t};H_{x}^{s}(M))}^{2}.\end{split}$$

$$\|u\|_{X_{\delta,\gamma}^{s,b}(I\times M)}:=\inf_{w\in X_{\delta,\gamma}^{s,b}(\mathbb{R}\times M)}\Big{\{}\|w\|_{X_{\delta,\gamma}^{s,b}(\mathbb{R}\times M)}:w\big{|}_{I}\,=u\Big{\}}.$$

$$|p_k-q_k|\le C_0,$$

$$\frac{1}{C}⟨\tau +p_k⟩\le ⟨\tau +q_k⟩\le C⟨\tau +q_k⟩.$$

$$\Vert \cdot {\Vert }_{X_{-\delta \Delta +\gamma }^{s,b}}\simeq \Vert \cdot {\Vert }_{X_{-\delta \Delta }^{s,b}}=:\Vert \cdot {\Vert }_{X_{\delta }^{s,b}}.$$

$$\Vert u_1\overline{u_2}{\Vert }_{X_{-\Delta +1}^{s,-b^{\prime }}}\le C\Vert u_1{\Vert }_{X_{-\frac{1}{\sigma }\Delta +\frac{\alpha }{\sigma }}^{s,b}}\Vert u_2{\Vert }_{X_{-\Delta +1}^{s,b}},$$

$$\Vert v_1{v}_2{\Vert }_{X_{-\frac{1}{\sigma }\Delta +\frac{\alpha }{\sigma }}^{s,-b^{\prime }}}\le C\Vert v_1{\Vert }_{X_{-\Delta +1}^{s,b}}\Vert v_2{\Vert }_{X_{-\Delta +1}^{s,b}},$$

$$\Vert u_1\overline{u_2}{\Vert }_{X^{s,-b^{\prime }}}\le C\Vert u_1{\Vert }_{X_{1/\sigma }^{s,b}}\Vert u_2{\Vert }_{X^{s,b}},$$

$$\Vert v_1{v}_2{\Vert }_{X_{1/\sigma }^{s,-b^{\prime }}}\le C\Vert v_1{\Vert }_{X^{s,b}}\Vert v_2{\Vert }_{X^{s,b}}$$

$$\Vert \widetilde{u_1}{u}_2{\Vert }_{L^2(\mathbb{R}\times M)}\le C(\min (N,L){)}^s\Vert u_1{\Vert }_{X_{1/\sigma }^{0,b}(\mathbb{R}\times M)}\Vert u_2{\Vert }_{X^{0,b}(\mathbb{R}\times M)},$$

$$\Vert e^{\pm i\frac{t}{\sigma }\Delta }{u}_0{e}^{it\Delta }{v}_0{\Vert }_{L^2((0,1{)}_t\times M)}\le C(\min (N,L){)}^s\Vert u_0{\Vert }_{L^2(M)}\Vert v_0{\Vert }_{L^2(M)},$$

$$A(k)=\{(n_1,n_2)\in {\mathbb{Z}}^2;|n_1|,|n_2|\le N,n_1^2+n_2^2=k\}.$$

$$|t\varphi ({\mathbb{S}}^1)\cap {\mathbb{Z}}^2|\le C(\varphi ,\epsilon )t^{\epsilon }.$$

$$\left|k^2+k-\sigma ({\ell }^2+\ell )-\sigma m\right|<\frac{\sigma }{2}$$

$$\sigma =\frac{\beta }{\theta }\text{ with}\beta ,\theta \in \{n^2:n\in \mathbb{N}\},$$

$$(\sqrt{\theta }(2k+1){)}^2-(\sqrt{\beta }(2\ell +1){)}^2=p,$$

$$♯\{(x,y)\in {\mathbb{Z}}^2:Q(x,y)\le N\}=\frac{2\pi }{\sqrt{D}}N+O(N^{\frac{131}{416}+}),$$

$$\Vert e^{it{\Delta }_{{\mathbb{T}}^{d-1}}}{u}_0{\Vert }_{L^p([0,1]\times {\mathbb{T}}^{d-1})}{\lesssim }_{\epsilon }{N}^{\epsilon }\Vert u_0{\Vert }_{L^2}$$

$$\Vert e^{it{\Delta }_{{\mathbb{T}}^{d-1}}}{u}_0{\Vert }_{L^p([0,1]\times {\mathbb{T}}^{d-1})}{\lesssim }_{\epsilon }{N}^{\frac{d-1}{2}-\frac{d+1}{p}+\epsilon }\Vert u_0{\Vert }_{L^2}$$

$$\Vert e^{\pm i\frac{t}{\sigma }{\Delta }_M}{u}_0{e}^{it{\Delta }_M}{v}_0{\Vert }_{L^p([0,1]\times M)}$$

$$s_0(d):=\{\begin{array}{ll}\frac{1}{4}& \text{mbox}ifd=2\\ \frac{1}{2}+\epsilon & \text{mbox}ifd=3\\ \frac{d-2}{2}& \text{mbox}ifd\ge 4.\end{array}$$

Full text

[TABLE]

posed on a $d$-dimensional compact Zoll manifold $M$. Considering $\sigma=\frac{\theta}{\beta}$ with $\theta,\beta\in\{n^{2}:n\in\mathbb{Z}\}$ we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data $(v_{0},u_{0})\in H^{s}(M)\times H^{s}(M)$ whenever $s>\frac{1}{4}$ when $d=2$ and $s>\frac{d-2}{2}$ when $d\geq 3$. Moreover, in dimensions $2$ and $3$, we use a Gagliardo-Nirenberg type inequality and conservation laws to prove that the local solution can be extended globally in time whenever $s\geq 1$.

M. Nogueira was supported by CAPES and CNPq, Brazil.

M. Panthee was partially supported by CNPq (308131/2017-7) and FAPESP (2016/25864-6) Brazil.

1. Introduction

In this work we are interested in addressing some well-posedness issues to the following initial value problem (IVP) associated to a system involving nonlinear Schrödinger (NLS) equations with quadratic nonlinearities

[TABLE]

where $v=v(t,x)$ and $u=u(t,x)$ are complex functions, $M$ is a compact Zoll manifold of dimension $d\geq 2$ and $\Delta_{g}$ is the Laplace-Beltrami operator.

The class of Zoll manifolds is defined as being that formed by all compact manifolds such that all geodesics are closed and possess the same period. In particular, this class contains all spheres as well as compact symmetric spaces of rank one (see [5], Chapter 4 for a rigorous exposition). The principal reason that motivated us to consider the system (1.1) posed on Zoll manifolds of dimension $d\geq 2$ is that the spectrum of the Laplacian consists of clusters of bounded width centered at the points $(k+\frac{Z_{0}}{4})^{2}$, $k\in\mathbb{N}$, where $Z_{0}$ is an integer which depends on the geometry of $M$ (see [42, 44, 45]). Note that the spectrum of the Laplacian on the sphere $\mathbb{S}^{d}$ is exactly of this form (see (5.11) below). This special feature of the spectrum allows to deduce some arithmetical properties which help us to understand the behavior of the Schrödinger type groups associated to (1.1) for some values of the parameter $\sigma>0$ and consequently to obtain bilinear estimates that are the main tools to get the well-posedness results (see Propositions 21 and 22 below).

The system of equations (1.1) appears in the study of non-linear optics, more specifically in studies related to the second harmonic generation (SHG) of type $I$, also known as Frequency Doubling, which is a nonlinear optical process discovered in the early 1960s. At that time, thanks to the invention of lasers, physicists have obtained a powerful source of coherent light, so that many of the non-linear optical effects, such as SHG, were demonstrated (see [46]). The functions $v$ and $u$ represent, respectively, the amplitudes of the envelopes of the first and second harmonics of an optical wave. For mathematical derivation of similar quadratic models posed on the whole space $\mathbb{R}^{n}$ and a detailed study of the associated Cauchy problems, we refer to the recent work [18]. The SHG system (1.1) posed on Riemannian manifold $M$ with the non-Euclidean metric $g$ describes the interaction of these harmonics in a medium in which the optical index is variable. In the system (1.1), there are four combinations of signs $(\pm,\pm)$ which are determined by the signs of dispersions/diffractions (temporal/spatial cases respectively). The constant $\sigma>0$ measures the dispersion/diffraction rates and plays an important role in the local and global theory. We observe that in order to establish the local well-posedness theory for (1.1) it is necessary that $\sigma$ assumes fractional values which is a contrast with the necessary assumption in the one-dimensional case. The parameter $\alpha>0$ is dimensionless, and in the one-dimensional physical model one needs to have $0<\alpha<1$ (see [43]).

If one considers $\epsilon_{j}\in\{1,-1\}$, $j=1,2$, for sufficiently regular solutions, the following quantities

[TABLE]

and

[TABLE]

where $\mathrm{Re}$ denotes the real part, are conserved. The quantities in (1.2) and (1.3) represent respectively the mass and energy of the system (1.1).

For simplicity of exposition we consider the combination $(+,+)$ of signs and use the notation $\Delta$ to represent $\Delta_{g}$ throughout this work. The study of the Schrödinger equations with quadratic non-linearities has attracted attention of several mathematicians over the past decades, see for instance [4, 34, 36] and references therein. As far as we know, in the literature, the system (1.1) has been studied considering $M=\mathbb{R}^{d}$, $(1\leq d\leq 5)$ in [46] where the focus is on the variational questions, and $M=\mathbb{T}$ in [1] where the focus is on the local well-posedness issues for $(u_{0},v_{0})\in H^{s}(\mathbb{T})\times H^{s}(\mathbb{T})$. More precisely, in [1] the local well-posedness results are obtained for given data with regularity $s\geq 0$ if $1/\sigma>0$ and $s>-1/2$ if $1/\sigma\neq 1$. In addition, the authors in [1] used mass conservation (1.2) and proved global well-posedness for $s\geq 0$. Also, we can cite [30] where the questions about scattering theory in $\mathbb{R}^{d},d\geq 3$ are addressed for a similar system to (1.1), and [31] where a study of well-posedness and blow-up is performed.

As mentioned earlier, if we choose $(+,+)$ signs and $\sigma>0$, the system (1.1) can be rewritten as

[TABLE]

The main objective of this work is in addressing the well-posedness issues for the system (1.4) posed on $d$-dimensional compact Zoll manifold $M$ with given data $(u_{0},v_{0})$ in a suitable Sobolev spaces $H^{s}(M)\times H^{s}(M)$. As far as we know, the results involving the well-posedness theory for (1.4) posed on compact manifolds in dimension $d\geq 2$ are not known.

To accomplish our objective, using Duhamel’s formula, we can consider the IVP (1.4) in the following equivalent integral formulation

[TABLE]

where $V(t)=e^{it(\Delta-1)}$ and $U_{\sigma}(t)=e^{it(\frac{1}{\sigma}\Delta-\frac{\alpha}{\sigma})}$ are the respective unitary groups associated with the linear problem. From now on, we consider the equivalent system (1.5) and use the contraction mapping argument in an appropriate space to get the required solution. Before announcing the main results on well-posedness theory, we introduce some definitions and function spaces on which we will be working. For convenience, let us start recapitulating some notions on Sobolev spaces in compact manifolds.

We denote by $\{e_{k}\}_{k\geq 0}\subset L^{2}(M,\mathbb{C})$ an orthonormal basis formed by eigenfunctions of $-\Delta$, with eigenvalues $\{\mu_{k}\}_{k\geq 0}$ and by $P_{k}:L^{2}(M)\rightarrow L^{2}(M)$ the orthogonal projection on $e_{k}$, given by

[TABLE]

In this way, we can define the Sobolev space $H^{s}(M)$ as being the completion of the space $C_{0}^{\infty}(M)$ with respect to the norm

[TABLE]

In what follows, we introduce a generalization of the spaces $X^{s,b}(\mathbb{R}\times\mathbb{T}^{d})$ previously introduced by Bourgain in [8] in the context of the NLS equation. The generalization in the context of the compact manifolds $M$ is due to Burq, Gérard and Tzvetkov in [13, 15, 16] where the authors studied the NLS equation. The definition uses the structure of the spectrum of $-\Delta$.

Definition 1**.**

Let $s,b\in\mathbb{R}$. The space $X^{s,b}(\mathbb{R}\times M)$ is the completion of the space $C_{0}^{\infty}(\mathbb{R}\times M)$ with respect to the norm

[TABLE]

where $S(-t):=e^{-it\Delta}$ and $\widehat{P_{k}u}(\tau)$ denotes the Fourier transform of the function $t\mapsto P_{k}u(t,\cdot)$.

Taking in consideration the spaces $X^{s,b}$ given by Definition 1, we introduce a family of spaces associated with the linear structure of (1.4). Our definition is appropriate to the modulation produced by the unitary groups $V(t)$ and $U_{\sigma}(t)$.

Definition 2**.**

Let $\delta,\gamma\in\mathbb{R}$ be fixed. Given $s,b\in\mathbb{R}$, we define the spaces $X^{s,b}_{\delta,\gamma}(\mathbb{R}\times M)$, as being the completion of the space $C_{0}^{\infty}(\mathbb{R}\times M)$ with respect to the norm

[TABLE]

In particular, if $\delta\neq 0$ and $\gamma=0$, we denote this space by $X^{s,b}_{\delta}$.

In order to establish the local theory, we need to define a local version of the spaces $X^{s,b}_{\delta,\gamma}(\mathbb{R}\times M)$ with respect to the variable $t\in I\subset\mathbb{R}$.

Definition 3**.**

Let $I\subset\mathbb{R}$ be a compact interval. We define the restriction space $X_{\delta,\gamma}^{s,b}(I\times M)$ equipped with the following norm

[TABLE]

Remark 4**.**

The definition (and the norm) of the spaces $X^{s,b}_{\delta,\gamma}$ clearly depend on the operator $P:=-\delta\Delta+\gamma$. However, if there is an operator $Q$ of the same order as $P$, having the same eigenfunctions and such that the eigenvalues $p_{k}$ and $q_{k}$ of $P$ and $Q$ respectively, obey the condition

[TABLE]

for some constant $C_{0}>0$, then one can easily show that there exists $C>0$ such that for all $k\in\mathbb{N}$ and $\tau\in\mathbb{R}$,

[TABLE]

Consequently, $X^{s,b}_{P}$ and $X^{s,b}_{Q}$ have equivalent norms.

In particular, the spaces $X^{s,b}_{-\delta\Delta+\gamma}$ and $X^{s,b}_{-\delta\Delta}$, have equivalent norms,

[TABLE]

Hence, according to the norm equivalence given by Remark 4, instead of $X^{s,b}_{\delta,\gamma}$ we can work in the $X^{s,b}_{\delta}$ and $X^{s,b}$ spaces. Note that, considering the system (1.5), to obtain the local well-posedness results using the contraction mapping argument, one needs to establish the following crucial bilinear estimates

[TABLE]

and

[TABLE]

for some $C>0$ and $(b,b^{\prime})\in\mathbb{R}^{2}$ satisfying $b+b^{\prime}<1$ and $0<b^{\prime}<1/2<b$.

In view of Remark 4, to obtain the above estimates, it is sufficient to prove that

[TABLE]

and

[TABLE]

hold for some $C>0$ and $(b,b^{\prime})\in\mathbb{R}^{2}$ satisfying $b+b^{\prime}<1$ and $0<b^{\prime}<1/2<b$.

To prove (1.9) and (1.10) we use a duality argument followed by dyadic decompositions on the functions $u_{j}$ and $v_{j}$, $j=1,2$. The crucial fact in this process is to analyse the decompositions of $u_{1}$ and $u_{2}$ when they are localized on the dyadic frequencies $\sim N$ and $\sim L$ respectively, with $N\ll L$ or $L\ll N$.

In view of the equivalence that will be given below in Lemma 20, we find that such estimates are closely related with the bilinear Strichartz estimates where discrepancy between frequencies is “controlled”   by $(\min(N,L))^{s}$, where $s>s_{0}(M)$. More precisely, we will show that

[TABLE]

with $\tilde{u_{1}}=u_{1}$ or $\overline{u_{1}}$, is equivalent to

[TABLE]

where $N,L$ are dyadic numbers on which $u_{0}$ and $v_{0}$ are spectrally localized respectively. In the next section we highlight some difficulties in the proof of the bilinear estimate (1.12).

2. Arithmetical properties of the spectrum and applications to bilinear Strichartz estimates

One of the useful properties of the Laplace-Beltrami operator on compact manifolds is that it has a discrete spectrum. In the case $M=\mathbb{T}^{2}$, this property as well as expansion in Fourier series is widely used. For instance, in Bourgain’s pioneer works [8, 9] the $L^{4}$–Strichartz estimate was reduced to counting the number of elements in the set

[TABLE]

It is well known that, $\sharp A(k)\lesssim N^{\epsilon}$ for any $\epsilon>0$. However, in (1.12) the product of Schrödinger semigroups with distinct spectra produces a different dispersion phenomena, especially due to the constant $\sigma>0$. Bilinear estimate to spectral projectors was proved by Burq, Gérard, Tzvetkov in [15] for the case of compact surfaces (see Proposition 15 below) and in [13, 16] for the higher dimensional case, viz., $\mathbb{S}^{d}$ $d\geq 2$ (see Proposition 16 below). In general, the knowledge of the spectrum is combined with some estimates from the analytic number theory. At this point, we highlight that, as in [8, 13, 15], estimates on the upper bound for the number of intersections of a closed curve with a lattice are strongly used. For example, an estimate for the number of intersection points of oval shaped closed curves with $\mathbb{Z}^{2}$, is following result due to Bombieri, Pila [7].

Theorem 5**.**

Suppose $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ analytic. Then, for all $\varepsilon>0$,

[TABLE]

Besides, it is proved that if $C$ is an affine plane algebraic curve of degree $d$ with integer coefficients, and if one takes a box with sides of length $N$, then $C$ can contain no more than $O_{d,\epsilon}(N^{\frac{1}{d}+\varepsilon})$ integer points within the box.

Nevertheless, in the case of the plane algebraic curves of the form $\{(x,y)\in\mathbb{R}^{2}:ax^{2}-by^{2}=m\}$ where $m\in\mathbb{Z}$ and $a,b\in\mathbb{R}$, Theorem 5 is not valid.

To describe better the situation which appears in our analysis, let us consider $M=\mathbb{S}^{2}$. Suppose that $N,L$ are natural numbers with $N\leq L$. Due to the fact that $Spec(-\Delta_{\mathbb{S}^{2}})=\{j(j+1)\}_{j\in\mathbb{N}}$, we need to prove that the number of solutions of the inequality

[TABLE]

where $N\leq k\leq 2N$ and $m\in\mathbb{Z}$ is $O(N^{\varepsilon})$. In general, we don’t know how to estimate the number of solutions to (2.1). However, if we choose the parameter $\sigma>0$ in the following form

[TABLE]

we obtain that (2.1) is reduced to the problem of find the number of solutions of the equation

[TABLE]

for some explicit values of $p\in\mathbb{Z}$. Thus, to consider this case, we will use Lemma 17 to estimate the number of solutions to (2.3). This is the point, that lead us to impose restriction (2.2) on the parameter $\sigma$. It is worth mentioning that:

1. $\bullet$

Using the Strichartz estimates proved by Burq, Gérard and Tzvetkov in [16], we may obtain local well posedness for the quadratic system (1.1) without any restriction on the parameter $\sigma>0$ for general compact manifolds of dimension $d\geq 2$ for initial data in $\mathbf{H}^{s}(M)$ with $s>\frac{d-1}{2}$, by using a procedure as in Nogueira and Panthee [37]. However, in our case, i.e., in the case of compact Zoll manifolds, the use of the Bourgain’s spaces improves the range of Sobolev index for $s>\frac{1}{4}$ for $d=2$ and $s>\frac{d-2}{2}$ for $d\geq 3$. In particular, in dimension 3, this allows us to extend the solutions (with initial data in $\mathbf{H}^{1}(M^{3})$ ) globally in time, which is not possible in the case of general $3$-dimensional compact manifolds in the case of the cubic nonlinear Schrödinger equation (see [16] p. 571). 2. $\bullet$

A result due to Huxley [32] asserts that if $Q(x,y)=ax^{2}+bxy+cy^{2}$ is a quadratic form with $a>0$ and $D:=4ac-b^{2}>0$, then

[TABLE]

when $N\rightarrow+\infty$. Observe that considering (2.1) and completing squares, we can suppose that the associated quadratic form is $Q(X,Y)=\frac{1}{4}X^{2}-\frac{\sigma}{4}Y^{2}$, where we performed the change of variables $X=2x+1$ and $Y=2y+1$. Thus, since $b=0$, we have $D=-\frac{\sigma}{4}<0$. So, the above estimate is not applicable in our case. However, this result is very important in the study of the local well-posedness for the NLS equation on the bidimensional irrational tori, see Demirbas [24]. 3. $\bullet$

The recent methods via the application of the Bourgain-Demeter [12, 23] decoupling theory seem promising to solve many questions related to questions of this nature. For example, let $u_{0}\in\mathbb{T}^{d-1}$ (flat torus), with $\mathrm{supp}(\widehat{u_{0}})\subset[-N,N]$. Then, for each $\varepsilon>0$ an application of the decoupling theory in [12] guarantees the full range of expected $L_{xt}^{p}$ Strichartz estimates

[TABLE]

if $2\leq p\leq\frac{2(d+1)}{d-1}$, and

[TABLE]

if $p\geq\frac{2(d+1)}{d-1}$. For more details we refer to the recent work of Demeter [22].

Recently, Fan et. al [26] used the decoupling type argument from [12] to obtain the bilinear Strichartz type estimate for irrational tori, recovering and generalizing the result of [21].

The decoupling theory methods are applied to obtain estimates for exponential sums whose phase function is associated with some curved surface, for example a truncated cone. However, we do not have idea how to use these methods to improve the estimates for our bilinear Strichartz estimates, which are of the form

[TABLE]

where $M$ is a Zoll manifold or the $d$-dimension sphere and $u_{0},v_{0}\in L^{2}(M)$, in the general case $\sigma>0$, i.e., without any restriction on $\sigma$.

Remark 6**.**

This sort of problems are also studied for algebraic plane curves of varying degrees (see [25, 41]).

3. Main Results

In this section, we state the main results on the well-posedness for the IVP (1.4) posed on the class of $d$-dimensional Zoll manifolds $M$ that we introduced in the beginning of the introduction. To simplify the notations, we define $\mathbf{H}^{s}(M):=H^{s}(M)\times H^{s}(M)$ and consider the IVP (1.4) with initial data $(u_{0},v_{0})\in\mathbf{H}^{s}(M)$.

Before stating the main results, let $d\geq 2$ and define

[TABLE]

The main result concerning the local well-posedness theory in $\mathbf{H}^{s}(M)$ for $s>s_{0}(d)$ is the following.

Theorem 7**.**

Let $(M,g)$ be a $d$-dimensional Zoll manifold and $\sigma=\frac{\beta}{\theta}$ with $\beta,\theta\in\{n^{2}:n\in\mathbb{N}\}$. For any $(v_{0},u_{0})\in\mathbf{H}^{s}(M)$, with $s>s_{0}(d)$ where $s_{0}(d)$ is defined in (3.1), there exist $T=T(\|(v_{0},u_{0})\|_{\mathbf{H}^{s}})>0$ and a unique solution $(v(t),u(t))$ of the IVP (1.4) on the interval $[0,T]$ such that, for some $b>\frac{1}{2}$

1. $(i)$

$(v,u)\in X_{T}^{s,b}(M)\times(X_{1/\sigma}^{s,b})_{T}(M)$, 2. $(ii)$

$(v,u)\in C([0,T],\mathbf{H}^{s}(M))$.

Moreover, for any $T^{\prime}\in(0,T)$ there exists an open ball $B((v_{0},u_{0}),r)\subset\mathbf{H}^{s}(M)$ such that the application

[TABLE]

is Lipschitz-continuous for some $r>0$.

Remark 8**.**

A natural question is whether the local result given by Theorem 7 is sharp. As far as we know, even for $d=2$, the local well-posedness for the cubic or quadratic nonlinearities for $s=1/4$ in classical Sobolev spaces $H^{s}(M)$ are open problems in the context of compact manifolds. However, using Besov spaces, Takaoka [40] proved that the local well-posedness can be achieved for $s=1/4$ for $M=\mathbb{S}^{2}$. The analogous question for the quadratic system (1.1) is being addressed in a work in progress by the first author in [38]. Ill-posedness results will be addressed elsewhere.

Note that the system (1.4) has mass and energy conservation laws given respectively by (1.2) and (1.3). For $s\geq 1$ we use these conserved quantities together with a Gagliardo-Nirenberg inequality (see Proposition 30 below) to prove that the solutions given by Theorem 7 in the case $d=2,3$ are in fact global. This is the content of the following theorem.

Theorem 9**.**

Let $(M,g)$ be a $d$-dimensional Zoll manifold, $s\geq 1$ and $T^{\ast}>0$ be the maximal time of existence for the solution

[TABLE]

given by Theorem 7 when $d=2,3$. Then $T^{\ast}=+\infty$, that is, the solution is global in time.

This paper is organized as follows. In Section 4 we record some basic properties of the $X^{s,b}$ spaces. In Sections 5 and 6 we will derive the bilinear estimates that are crucial in our argument. The proofs of the main results are presented in Section 7 with subsection 7.1 devoted to supply the proof for Theorem 7, and subsection 7.2 for Theorem 9.

4. Basic properties of the Bourgain spaces

We begin exploring some basics properties of the Bourgain spaces $X^{s,b}$ and $X^{s,b}_{\delta}$. Here, $M$ denotes a general $d$-dimensional compact manifold.

Proposition 10**.**

The following properties are valid

1. (i)

For $s_{1}\leq s_{2}$ and $b_{1}\leq b_{2}$, one has $X^{s_{2},b_{2}}(\mathbb{R}\times M)\hookrightarrow X^{s_{1},b_{1}}(\mathbb{R}\times M)$. 2. (ii)

$X^{0,\frac{1}{6}}(\mathbb{R}\times M)\hookrightarrow L^{3}(\mathbb{R},L^{2}(M))$. 3. (iii)

If $b>\frac{1}{2}$, then the inclusion $X^{s,b}(\mathbb{R}\times M)\hookrightarrow C(\mathbb{R},H^{s}(M))$ holds.

Proof. The part $(i)$ follows directly from the Definition 1. The part $(ii)$ follows from the fact that $u\in X^{s,b}(\mathbb{R}\times M)$, if and only if, $S(-t)u(t,\cdot)\in H^{b}(\mathbb{R},H^{s}(M))$ and from the immersion $H^{1/6}(\mathbb{R})\hookrightarrow L^{3}(\mathbb{R})$, (see [2] p. 1866, for more details). The proof of $(iii)$, is given in [15], p. 196. $\Box$

Proposition 11**.**

Let $u\in C_{0}^{\infty}(\mathbb{R}\times M)$ and $(s,b)\in\mathbb{R}^{2}$. Considering the expression

[TABLE]

one has

[TABLE]

Moreover,

[TABLE]

Proof. The proof follows by applying orthogonal projection, Fubini’s theorem and Planchrel’s identity. $\Box$

We note that the results stated in Propositions 10 and 11 hold for the $X^{s,b}_{\delta}$ spaces as well.

4.1. Spectral projectors

In this section, we will introduce spectral projection operators and their properties which are used in the decomposition of functions. Also we record some estimates involving these operators that are used especially in the following section, where we derive some estimates involving the spectral localization of the functions related to the $X_{\delta,\gamma}^{s,b}$ spaces. In what follows, we use $N=2^{n},L=2^{\ell}$, $n,\ell\in\mathbb{N}$, to denote the dyadic integers. For clarity of exposition we consider $\delta=1$ and $\gamma=0$, the similar results hold for $\delta\neq 1$ and $\gamma\neq 0$ too.

Definition 12**.**

Given a dyadic integer $N$, we say that a function $u$ on $M$ is spectrally localized at frequency $N$ if

[TABLE]

Moreover, we denote $\Pi_{1}=\sum_{k:0\leq\langle\mu_{k}\rangle^{1/2}<2}P_{k}u$.

Let $u=u(t,x)\in C_{0}^{\infty}(\mathbb{R}\times M)$. Using a $L^{2}$ -decomposition of $u(t,\cdot)$, we can write

[TABLE]

Now, applying the Fourier inversion theorem to the function $t\mapsto\Pi_{N}u(t,\cdot)$, we obtain

[TABLE]

where

[TABLE]

According to this construction, we can identify two types of localization operators, the one with the spatial variable and the next with the time variable. More precisely, we have

1. $\bullet$

Localization with respect to the time variable:

[TABLE]

where $\Lambda_{L}:=\{\tau\in\mathbb{R};L\leq\langle\tau+\mu_{k}\rangle\leq 2L\}$. 2. $\bullet$

Localization with respect to the space variable:

[TABLE]

From the previous definitions, it is easy to see that the operators $\textbf{1}_{L\leq\langle\tau+\mu_{k}\rangle\leq 2L}$ and $\Pi_{N}$ commute with each other and can be applied in different orders if necessary.

Using the expression of the projector $\Pi_{NL}$ given in (4.5) and the definition of the $X^{s,b}$-norm given in Definition 1, we can prove the following basic estimates.

Lemma 13**.**

Let $s,b\in\mathbb{R}$. Then, there exists $C>0$ such that:

[TABLE]

[TABLE]

where the summation is taken over all dyadic values of $N$ and $L$.

Similar estimates also hold for $X^{s,b}_{\delta}$.

5. Bilinear estimates and applications

In this section, we derive bilinear interaction estimates for the quadratic nonlinear terms with respect to the semi-groups associated to the linear part of the system (1.4).

First, recall the work of Bourgain in [11] where a refinement of Strichartz estimates in the Euclidean case was derived. Considering $u_{0}$, $v_{0}$ localized in frequency on the sets $\{\xi\in\mathbb{R}^{d}:|\xi|\sim N_{1}\}$ and $\{\xi\in\mathbb{R}^{d}:|\xi|\lesssim N_{2}\}$ respectively, with $N_{2}\leq N_{1}$, the author in [11] proved the following estimate

[TABLE]

Notice that if $u_{0}=v_{0}$, $N_{1}=N_{2}$ and $d=2$, we can use the usual Littlewood-Paley decomposition on $\mathbb{R}^{d}$ to obtain the already known Strichartz estimate

[TABLE]

As we are working on a compact Zoll manifolds considering two different groups, we need to derive an analogue of (5.1) that fits in our context. Before entering to the details, we introduce the following more general definition and list the known results in the literature.

Definition 14**.**

Let $P$, $Q$ be differential operators on $M$ of order $n$, $m$ respectively. We say that the associated semi-groups $e^{itP},e^{itQ}$ satisfy a bilinear estimate of order $s_{0}:=s_{0}(M)\geq 0$ if, for any $u_{0}$ localized in a frequency $N_{1}$ and $v_{0}$ localized on a frequency $N_{2}$ one has that

[TABLE]

where $I\subset\mathbb{R}$ is a finite interval.

In what follows we list some works on the bilinear estimates (5.3) obtained in the context of compact Riemannian manifolds.

[TABLE]

With the above information in mind, as in the case of the single NLS equation, we plan to obtain the bilinear estimates for the (now mixed) semi-groups

[TABLE]

that describe the solution to the linear part

[TABLE]

associated to the IVP (1.4).

To better investigate the properties of the operators involved in the expressions of these semigroups, let us consider the operator

[TABLE]

If $(e_{k},\mu_{k})$ is an eigenpair of $-\Delta$, it is easy to check that $(e_{k},\delta\mu_{k}+\gamma)$ is an eigenpair of $-\Delta_{\delta}$. Considering the case where $\delta=\frac{1}{\sigma}$ and $\gamma=-\frac{\alpha}{\sigma}$, we have

[TABLE]

Thus, considering the product in $L^{2}$ with the semigroup generated by the choice of the parameters $\beta=1$ and $\gamma=-1$, we have

[TABLE]

This shows that the product of the mixed groups is not affected by the factors $e^{-it\frac{\alpha}{\sigma}},\;e^{-it}$, where $I\subset\mathbb{R}$.

5.1. Bilinear estimates on compact manifolds

We start with the following result that provides a bilinear estimate for localized functions on different regimes of frequency, in the case of compact manifolds.

Proposition 15**.**

([13] p. 261). Let $M$ be a compact smooth manifold without boundary of dimension $d$. Let $\chi\in C_{0}^{\infty}(\mathbb{R})$ and $s_{0}(d)$ defined in (3.1) . Given $A>0$, introduce the following approximated spectral projector, via functional calculus:

[TABLE]

Then, there is $C>0$ such that, for all $\lambda,\mu\geq 1$, and any $f,g\in L^{2}(M)$,

[TABLE]

Next, we use Proposition 15 to characterize the localization operator $\chi_{A}$.

1. $\bullet$

Choose $\chi\in C_{0}^{\infty}(\mathbb{R})$ be such that $\mathrm{supp}\chi\subset(0,2\delta_{0})$, $\chi\geq 0$ and $\chi=1$ in $[\delta_{0}/2,3\delta_{0}/2]$. Take $\sqrt{\mu_{k_{0}}}\in\mathrm{Spec}(\sqrt{-\Delta})$. As the spectrum of $\sqrt{-\Delta}$ is discrete, there exists $\delta_{0}>0$ such that

[TABLE]

Let $A_{0}=\sqrt{\mu_{k_{0}}}-\delta_{0}$, then $\chi(t-A_{0})>0$ if $t\in I_{0}$. Moreover, $\chi(t-A_{0})=1$ if

[TABLE]

Thus,

[TABLE]

Choose $\lambda=A_{0}$ and $\mu=\tilde{A_{0}}=(\sqrt{\mu_{\tilde{k}_{0}}}-\tilde{\delta}_{0})$, with $f=P_{k_{0}}$ and $g=P_{\tilde{k}_{0}}$. Then from (5.7) we have

[TABLE] 2. $\bullet$

Now, let $\chi\in C_{0}^{\infty}(\mathbb{R})$ with $\mathrm{supp}\chi\subset(0,B)$ ($B>0$), $\chi\geq 0$ and $\chi=1$ in $[\delta,B-\delta]$ for some $0<\delta<B/2$. Then, for $A>0$

[TABLE]

where $\chi(\sqrt{\mu_{k}}-A)>0$ only for the values of $k\in\mathbb{N}$ such that $\sqrt{\mu_{k}}\in(A,A+B)$ and $\chi(\sqrt{\mu_{k}}-A)=1$ if $\sqrt{\mu_{k}}\in[A+\delta,A+B-\delta]$. Therefore, we can write

[TABLE]

Observe that $\mathrm{Spec}(\sqrt{-\Delta})\cap(A,A+B)=\{\sqrt{\mu_{k_{\ell}}}\}_{\ell=1}^{n}$, where $\sqrt{\mu_{k_{\ell}}}\leq\sqrt{\mu_{k_{\ell+1}}}$ for $1\leq\ell\leq n-1$. Therefore, if $0<\delta<\min\{|\sqrt{\mu_{k_{1}}}-A|,|\sqrt{\mu_{k_{n}}}-(A+B)|\}$, the second sum on the right hand side of (5.9) vanishes. Consequently, the spectral projectors for which (5.7) is true are of the form

[TABLE]

5.2. Bilinear estimates in spheres

We begin this section with a brief revision on the concept of Spherical Harmonics. For a complete introduction on the subject, see [3, 20]. Let $P=P(x_{0},x_{1},...,x_{d})$ be a homogeneous polynomial of degree $k$ in $\mathbb{R}^{d+1}$. If $P$ is harmonic, that is, $\sum_{i}\frac{\partial^{2}P}{\partial x_{i}^{2}}=0$ then it can be shown that the restriction $H:=P\mid_{\mathbb{S}^{d}}$ satisfies $\Delta_{\mathbb{S}^{d}}H+\mu_{k}H=0$, where

[TABLE]

denote the eigenvalues of $-\Delta_{\mathbb{S}^{d}}$. These functions are called spherical harmonics of degree $k$, which will be denoted by $H_{k}$.

Next, we state a result that provides bilinear spectral estimates in $d$-dimensional spheres, $d\geq 2$.

Proposition 16**.**

([13, 15]). Let $H_{k},\widetilde{H}_{\ell}$ be spherical harmonics of degrees $k,\ell\geq 1$ on $\mathbb{S}^{d}$ ($d\geq 2$). Then,

[TABLE]

where $s_{0}(d)$ is defined in (3.1).

5.3. From the spectral estimates to the evolution estimates

In this subsection, we will use the spectral bilinear estimates (5.7) and (5.12) to infer evolution bilinear estimates involving the interaction of the semigroups given in (5.6) by

[TABLE]

We emphasize here that the choice of the $\pm$ signs in (5.13) is due to the presence of complex conjugate in the expressions we are going to deal with and has no relation to the choice of $(+,+)$ signs that we made in the introduction. Notice that, on the sphere $(\mathbb{S}^{d},can)$ we can take advantage of the precise knowledge of the spectrum of the operator $-\Delta_{\mathbb{S}^{d}}$ (see (5.11)). After obtaining the bilinear estimates in the case of the spheres, we will extend those results to the case of Zoll manifolds. Since the localization of the eigenvalues of the operator $-\Delta_{Zoll}$ is also well understood (see Proposition 21 below), we will extend the estimate in this case by introducing a suitable abstract perturbation of the Laplacian to reduce to an analysis that is already carried out in the case of the sphere $\mathbb{S}^{d}$.

The following lemma will be important in our argument which is proved using the estimate for the number of divisors of a natural number.

Lemma 17**.**

For every $\varepsilon>0$, there is $C_{\varepsilon}>0$ such that, given $m\in\mathbb{Z}\setminus\{0\}$ and a positive integer $N$,

[TABLE]

Proof.

The proof for the case $x^{2}+y^{2}=m$ is given in [15] p. 207. Thus, it remains to prove the hyperbolic case $x^{2}-y^{2}=m$.

First, we consider the case when $m>0$. Since $N\leq x\leq 2N$, we have

[TABLE]

and consequently $y\in[0,2N]$. Let $\tilde{x}=x-y$ and $\tilde{y}=x+y$. Note that $\tilde{x}\tilde{y}=m$ is possible only if $\tilde{x}>0$. Hence, it suffices to estimate

[TABLE]

The number of divisors of a natural number $n$, denoted by $d(n)$, satisfies $d(n)=O(n^{\epsilon})$. Thus, one obtains $A_{m}\lesssim d(8N^{2})=O(N^{\varepsilon})$.

Next, we consider the case when $m<0$. In this case, we write

[TABLE]

where $m^{\prime}:=-m>0$. Then, $\sqrt{N^{2}+m^{\prime}}\leq y\leq\sqrt{4N^{2}+m^{\prime}}$ and so, $y\in I_{m^{\prime}}:=\left[\sqrt{N^{2}+m^{\prime}},\sqrt{4N^{2}+m^{\prime}}\right]$.

If $m^{\prime}<3N^{4}$ then $y\in[N,4N^{2}]$ and we may reduce to the same analysis of $A_{m}$.

If $m^{\prime}\geq 3N^{4}$, then we simply observe that $y$ takes values in $I_{m^{\prime}}$, which contains at most one integer, since $|I_{m^{\prime}}|<1$. The proof of the lemma is completed taking into account that if we fix $y$ then $x$ can not take more than one value. ∎

Also, we will use the following result involving Fourier series.

Lemma 18**.**

([13] p. 289). Let $A\subset{\mathbb{R}}$ be a countable set. Then, for every $T>0$ there is $C_{T}>0$ such that for every sequence $(a_{\lambda})$ indexed by $A$, one has

[TABLE]

In what follows, we use the Lemmas 17 and 18 and the estimate (5.12) to derive the bilinear Strichartz estimate on spheres.

Proposition 19**.**

(Bilinear Strichartz estimate on $\mathbb{S}^{d}$). Let $d\geq 2$ and $s>s_{0}(d)$. Consider the semi-groups given by (5.13) with $\sigma=\frac{\beta}{\theta}$ where $\beta,\theta\in\{n^{2}:n\in\mathbb{N}\}$. Then, there is $C>0$ such that for any $u_{0},v_{0}\in L^{2}(\mathbb{S}^{d})$ satisfying the spectral localization conditions $u_{0}^{N}=u_{0}$, $v_{0}^{L}=v_{0}$, one has

[TABLE]

Proof. Initially consider $d=2$. Let us consider the “ - ” sign, the proof for the  “+” sign will follow in a similar way.

Using the series expansion associated with the semi-groups (5.13) and the fact that $u_{0}^{N}=u_{0}$, $v_{0}^{L}=v_{0}$, we can write

[TABLE]

Applying the Fubini’s theorem, we get

[TABLE]

Now, from Lemma 18, we obtain

[TABLE]

Let

[TABLE]

Applying the Cauchy-Schwarz inequality in the sum of $(k,\ell)$ and the triangle inequality for the $L^{2}(\mathbb{S}^{2})$-norm, we obtain

[TABLE]

By Proposition 16, it follows that

[TABLE]

On the one hand, considering $\sigma:=\frac{\beta}{\theta}$ with $\beta,\theta\in\{n^{2}:n\in\mathbb{N}\}$ and $\mu_{j}=j^{2}+j$ ($j\in\mathbb{N}$), we can write

[TABLE]

where $k_{\theta}:=\sqrt{\theta}(2k+1)$ and $\ell_{\beta}:=\sqrt{\beta}(2\ell+1)$. Notice that the condition on $\sigma$ is of technical character, since we must complete the square to obtain the expression of the set given in (5.20) and also because we need to obtain an upper bound for its cardinality using Lemma 17. Furthermore, we have

[TABLE]

where

[TABLE]

$N_{\theta}:=\sqrt{\theta}(N+1)$ and $L_{\beta}:=\sqrt{\beta}(L+1)$.

Let $\beta^{\prime}\in([-2\beta,2\beta]\cap\mathbb{Z})$ then, using Lemma 17 uniformly with respect to $m$, we obtain that the number of the pairs $(\tilde{k},\tilde{\ell})$, satisfying

[TABLE]

is bounded by $O_{\epsilon}(\min(N,L)^{\epsilon})$. Since there are $4\beta+1$ possibilities for $\beta^{\prime}$, we have

[TABLE]

From (5.19), (5.21) and (5.23) it follows that

[TABLE]

On the other hand, according to (5.18), we have $\Lambda^{NL}(m)=\emptyset$ if

[TABLE]

Thus, to bound the summations in the RHS of (5.24) we need to consider

[TABLE]

But,

[TABLE]

Therefore, the summations on the right side of (5.24) can be bounded by

[TABLE]

That is,

[TABLE]

By taking $\epsilon:=2s-1/2$ we finish the proof for $d=2$.

For $d\geq 3$, we follow the same lines of the proof given for $d=2$. First, exploiting the knowledge of the spectrum of $-\Delta_{\mathbb{S}^{d}}$ (see (5.11)), we need to analyse the cardinality of the set

[TABLE]

with $\mu_{k}=k(k+d-1)$ and $\mu_{\ell}=\ell(\ell+d-1)$.

Analogously to what was done previously, we must find a bound for the number of solutions of the inequality

[TABLE]

where $k\in[N/2,N]$ and $\ell\in[L/2,L]$. Using Lemma 17 and the spectral bilinear estimate given by Lemma 16 for $d\geq 3$, we obtain

[TABLE]

for all $s>s_{0}(d)$ where $s_{0}(d)$ is defined in (3.1). $\Box$

The following lemma provides a reformulation of the bilinear Strichartz estimates in terms of bilinear estimates involving the mixed spaces $X^{s,b}_{\delta}$ and $X^{s,b}$.

Lemma 20**.**

Let $s>0$ and $\sigma>0$ be as in the statement of the Proposition 19. The following statements are equivalent

1. $(i)$

For any $u_{0},v_{0}\in L^{2}(M)$ satisfying $u_{0}^{N}=u_{0}$ and $v_{0}^{L}=v_{0}$, one has

[TABLE] 2. $(ii)$

For all $b>1/2$ and any functions $u_{1}\in X_{1/\sigma}^{0,b}(\mathbb{R}\times M)$ and $u_{2}\in X^{0,b}(\mathbb{R}\times M)$ satisfying $u_{1}^{N}=u_{1}$ and $u_{2}^{L}=u_{2}$, one has

[TABLE]

where $\tilde{u_{1}}=\overline{u_{1}}$ or $\tilde{u_{1}}=u_{1}$.

Proof. The proof follows with simple modification of Lemma 2.3 in [15] using the semigroups $S_{1/\sigma}(\pm t)$, $S(t)$ and properties of the spaces $X^{s,b}_{1/\sigma}$, see also Proposition 4.3 in [13]. $\Box$

Now, we prove that the bilinear Strichartz estimate stated in Proposition 19 can be extended to the class of the $d$-dimensional Zoll manifolds. For this purpose, as we have already said, the following result due to Colin de Verdière [19] and Guillemin [27] on the localization of the eigenvalues of the Laplacian on these manifolds (which we denote by $-\Delta_{Zoll}$ just to make it more explicit) plays crucial role. It is worth mentioning that this localization property holds in higher dimensions and not only in the case of surfaces $(d=2)$ (see for example [42, 44, 45]).

Proposition 21**.**

If the geodesics of $M$ are $2\pi$-periodic111This renormalization is done by applying a dilation in the Riemannian metric $g$, that is, given an appropriate scalar $\lambda\in(0,+\infty)$ we consider the metric $\lambda g$ instead of $g$. , there are $Z_{0}\in\mathbb{N}$ and $E>0$ such that the spectrum of $-\Delta_{Zoll}$ is contained in $\bigcup_{k=1}^{+\infty}I_{k}$, where

[TABLE]

Proposition 22**.**

(Bilinear evolution estimates on Zoll manifolds). Let $M$ be a $d$-dimensional Zoll manifold and $s>s_{0}(d)$ be given. Consider the semi-groups given by (5.13) with $\sigma=\frac{\beta}{\theta}$ where $\beta,\theta\in\{n^{2}:n\in\mathbb{N}\}$. Then, there is $C>0$ such that for any $u_{0},v_{0}\in L^{2}(M)$ satisfying the spectral localization conditions $u_{0}^{N}=u_{0}$, $v_{0}^{L}=v_{0}$, one has

[TABLE]

Proof. Denote by $(e_{n})_{n}$ the sequence of eigenfunctions of $-\frac{1}{\sigma}\Delta_{Zoll}$ associated to the eigenvalues $\lambda_{n}^{\sigma}$ (counting its multiplicity). Using the Proposition 21, it follows that $\mathrm{Spec}(-\frac{1}{\sigma}\Delta_{Zoll})\subset\bigcup_{k=1}^{+\infty}I^{\sigma}_{k}$, where $I^{\sigma}_{k}:=\frac{1}{\sigma}I_{k}$. Observing the expressions of the $I_{k}$ intervals, it can be inferred that they are not necessarily disjoint. In this way, it is easy to see that, if we take

[TABLE]

we have $I^{\sigma}_{m}\cap I^{\sigma}_{\ell}=\emptyset$, whenever $m\neq l$ and $m,\ell\geq K_{0}$.

Now, set $N_{0}$ as being the smallest natural number such that

[TABLE]

In this case, since the sequence of eigenvalues is non-decreasing, we have, for all $n\geq N_{0}$, that $\lambda_{\sigma}$ belongs to only one interval of the form $I_{k}^{\sigma}$ (with $k\geq K_{0}$). Take

[TABLE]

Define an abstract perturbation of the Laplacian on $L^{2}(M)$ by

[TABLE]

According to Lemma 20, to prove (5.29), it suffices to prove that (5.28) is valid for $s>s_{0}(M)=1/4$ and $X^{0,b}_{-\frac{1}{\sigma}\Delta_{Zoll}}$, $X^{0,b}_{-\Delta_{Zoll}}$. But, according to Remark 4,

[TABLE]

So, using the fact that $(ii)$ implies $(i)$ in the Lemma 20, it is sufficient to prove that (5.27) is valid with $s>1/4$ and

[TABLE]

Replacing $P_{k}$ in (5.16) by222Note that this is a type of spectral projector for which the estimate (15) is valid, and the same is true if we consider $\Pi_{I_{k}^{\sigma}}$. $\Pi_{I_{k}}=\sum_{n\in\mathbb{N}:\lambda_{n}\in\mathrm{Spec}(-\Delta_{Zoll}^{\bigstar})\cap I_{k}}P_{n}$ and the set in (5.25) by

[TABLE]

with $\mu_{k}=(k+Z_{0}/4)^{2}$ and $\mu_{\ell}=(\ell+Z_{0}/4)^{2}$ and proceeding analogously as in the case of $\mathbb{S}^{d}$, we need to find a bound for the number of solutions of the inequality

[TABLE]

Thus, the proof given in Proposition 19 can be applied to the pair $(\pm\frac{1}{\sigma}\Delta_{Zoll}^{\bigstar},\Delta_{Zoll}^{\bigstar})$ as well. $\Box$

Remark 23**.**

In view of the result obtained in the Proposition 22, the equivalence of the estimates given in the Lemma 20 can be used for the case of the Zoll manifolds whenever $s>s_{0}(d)$.

6. Bilinear estimates for quadratic interactions

In this section, we will use the estimates obtained in the Propositions 19 and 22 and the equivalence given by Lemma 20 to obtain the bilinear estimates involving the quadratic interactions of the system (1.4). More precisely, we prove the following result.

Proposition 24**.**

(Bilinear estimates). Let $M$ be a $d$-dimensional Zoll manifold, $s>s_{0}(d)$ and $\sigma=\frac{\beta}{\theta}$ with $\beta,\theta\in\{n^{2}:n\in\mathbb{N}\}$. Then there exist $(b,b^{\prime})\in\mathbb{R}^{2}$ satisfying $0<b^{\prime}<1/2<b$, $b+b^{\prime}<1$ and $C>0$ such that

[TABLE]

and

[TABLE]

Also, there exist $(b_{1},b^{\prime}_{1})\in\mathbb{R}^{2}$ satisfying $0<b^{\prime}_{1}<1/2<b_{1}$, $b_{1}+b_{1}^{\prime}<1$ and $C>0$ such that

[TABLE]

and

[TABLE]

To prove Proposition 24, we need a series of basic results that we derive in the following subsection.

6.1. Auxiliary results

We begin by proving a result which provides an estimate for the product of three functions in $M$ localized on three frequency intervals $I_{k}$, such that one of the intervals is very dislocated in relation to the others.

As shown in [29] page 1203, in the case of $M=\mathbb{T}^{d}$ or $\mathbb{S}^{d}$, we have

[TABLE]

if $\mu_{k_{0}}>\mu_{k_{1}}+\mu_{k_{2}}+\mu_{k_{3}}$, where $\tilde{e_{k_{j}}}=e_{k_{j}}$ or $\overline{e_{k_{j}}}$. Therefore, in these cases the lemma that we will announce below is not necessary. This kind of property is often used when decompositions of function $u$ over $\mathbb{S}^{d}$ with respect to the spherical harmonics of degree $k$ are considered, that is, decompositions of the type

[TABLE]

for details see [39] page 145 in the case of $\mathbb{S}^{4}$ and [35] page 819 in the case of $\mathbb{S}^{3}$. To deal with the case of Zoll manifolds (where the cancellation property may fail), it is necessary to use the following result.

Lemma 25**.**

Let $M$ be a compact $d$-dimensional Riemannian manifold. If there exist $C>1$ such that $C(\mu_{k_{1}}+\mu_{k_{2}})\leq\mu_{k_{0}},$ then for all $p>0$ there exists $C_{p}>0$ such that, for all $w_{j}\in L^{2}(M)$, $j=0,1,2$,

[TABLE]

Proof. The proof consists in reducing the estimate of Theorem 4.2 of [29] to the case of the product of three projections. More precisely in [29], (see [15] p. 198 for a different proof in the case $d=2$), it has been proved that if $C(\mu_{k_{1}}+\mu_{k_{2}}+\mu_{k_{3}})\leq\mu_{k_{0}}$, then for all $\mathbb{N}\ni p>0$ there exists $C_{p}>0$ such that, for any $w_{j}\in L^{2}(M)$, $j=0,1,2,3$,

[TABLE]

Thus, if we take $k_{3}=0$ in (6.5), we have $\mu_{k_{3}}=\mu_{0}=0$ (first eigenvalue of $-\Delta$ with eigenfunction $e_{0}=1$). Now, if $w_{3}=e_{0}=1\in L^{2}(M)$, we have

[TABLE]

Thus, if $C(\mu_{k_{1}}+\mu_{k_{2}})\leq\mu_{k_{0}}$, one obtains

[TABLE]

as required. $\Box$

The estimate obtained in the Lemma 25 will be crucial to establish estimates for an integral involving the product of three functions localized in different frequency regimes under the condition $N_{0}\geq C(N_{1}+N_{2})$.

Remark 26**.**

From now on, to simplify the notation, we will use $\langle\mu_{k}\rangle\sim N_{j}^{2}$ for $j=0,1,2$ to denote that $N_{j}\leq\langle\mu_{k}\rangle^{\frac{1}{2}}<2N_{j}$ and $\langle\tau+\widetilde{\mu_{k}}\rangle\sim L_{j}$ to indicate that $L_{j}\leq\langle\tau+\widetilde{\mu_{k}}\rangle<2L_{j}$, where $\widetilde{\mu_{k}}:=\frac{\mu_{k}}{\sigma}$.

Lemma 27**.**

Let $M$ be a compact $d$-dimensional Riemannian manifold. Consider the expression

[TABLE]

where

[TABLE]

with

[TABLE]

Let $s>0$. Then, there exist $C_{1}>0$ and $p>0$ such that if $N_{0}\geq C(N_{1}+N_{2})$, we have

[TABLE]

where $s+6-p<0$.

Proof. Inserting the terms given in (6.7) for $j=0,1,2$ in (6.6), we obtain

[TABLE]

[TABLE]

where

[TABLE]

Denote

[TABLE]

Let $\delta$ be the Dirac measure. As in [15] p. 200, we can rewrite

[TABLE]

By triangular inequality for integrals, we get

[TABLE]

where $D$ is equipped with the measure $d\mu:=\delta(\tau_{1}-\tau_{2}-\tau_{0})d\tau_{0}d\tau_{1}d\tau_{2}$, and

[TABLE]

By Lemma 25, and noting that $\mu_{k_{0}}^{-p}\leq 2^{p}N_{0}^{-p}$ from (6.11), we obtain

[TABLE]

Let

[TABLE]

Performing the change of variables $(\tau_{0},\tau_{1})\mapsto(\tau_{0}^{\prime},\tau_{1}^{\prime}+\tau_{0}^{\prime})$ one may rewrite (6.13) as

[TABLE]

with,

[TABLE]

where we used the fact that for $f$ measurable and $x_{0}\in\mathbb{R}$, we have

[TABLE]

Using the Cauchy-Schwarz inequality with respect to $\tau_{0}^{\prime}$ we obtain

[TABLE]

Now, applying the Cauchy-Schwarz inequality with respect to $\tau_{1}^{\prime}$ we get

[TABLE]

where

[TABLE]

Since

[TABLE]

one obtains

[TABLE]

Thus, from (6.12) and (6.16), we have

[TABLE]

In the last line of (6.17) we used the Weyl’s Law for eigenvalues of Laplace operator in compact manifolds to find an estimate for the number of elements of the set $\Theta(N)$, that is,

[TABLE]

Suppose $N_{0}>C(N_{1}+N_{2})$. Thus, from (6.17), we obtain

[TABLE]

By Lemma 25, we can choose $p>0$ such that $s+6-p<0$. Thus,

[TABLE]

as desired. $\Box$

Since we will be dealing with dyadic sums involving different regimes on dyadic variables, the following lemma will be useful, whose proof can be found in [13], page 282.

Lemma 28**.**

(Discrete Schur lemma). For all $\gamma>0$ and all $\theta>0$ there is $C>0$ such that if $(c_{N})$ and $(d_{N^{\prime}})$ are two sequences of non-negative numbers indexed by dyadic integers, then

[TABLE]

6.2. Proof of bilinear estimates

We are now ready to provide proofs of the bilinear estimates related to quadratic interactions stated in Proposition 24.

Proof of Proposition 24. Let us prove first (6.1). By density, we can assume that $u_{j}\in C_{0}^{\infty}(\mathbb{R}\times M)$ ($j=1,2$). As in [15], we introduce the functions

[TABLE]

[TABLE]

Using Proposition 11, we can write

[TABLE]

and

[TABLE]

with

[TABLE]

Using the duality relation between $X^{s,-b^{\prime}}$ and $X^{-s,b^{\prime}}\approx(X^{s,-b^{\prime}})^{\ast}$ and (4.1), it follows that to prove (6.1), it suffices to show that

[TABLE]

where $w_{0}\in L^{2}(\mathbb{R}\times M)$ is arbitrary, and

[TABLE]

Now, we move to estimate the term $I$ in (6.20). Let $N_{0},N_{1},N_{2}$ and $L_{0},L_{1},L_{2}$ be the dyadic integers, that is, $N_{j}=2^{n_{j}}$ $(n_{j}\in\mathbb{N})$, $L_{j}=2^{\ell_{j}}$ $(\ell_{j}\in\mathbb{N})$, $j=0,1,2$. We define $N:=(N_{0},N_{1},N_{2})$, $L:=(L_{0},L_{1},L_{2})$, and use this notation throughout this proof. The sum $\Sigma_{N}$ denotes the summation over all possible dyadic values of $N_{0},N_{1},N_{2}$. Similar convention will be adopted for the sum over $L$.

Note that, we can decompose333See subsection 4.1 for more details on these spectral decompositions. the functions $u_{j}$ is a such way that

[TABLE]

with respective components

[TABLE]

where $\widetilde{\mu_{k}}=\frac{\mu_{k}}{\sigma}$ and $\tilde{b}=b$ or $b^{\prime}$.

Inserting this decomposition in the LHS of (6.20), one obtains

[TABLE]

where

[TABLE]

with

[TABLE]

[TABLE]

and

[TABLE]

Applying the Fubini’s theorem, we obtain from (6.21) that

[TABLE]

Inserting (6.27) in (6.21), and taking the $L^{2}(\mathbb{R}\times M)$-norm, we get

[TABLE]

where,

[TABLE]

Now, summing on the dyadic variables $L_{j}$ and $N_{j}$, and using the Plancherel’s theorem in the time variable, we obtain

[TABLE]

Considering the coefficients $c_{j}^{N_{j}L_{j}}$, we need to estimate the components of $u_{j}$ given in (6.21) in the norms of the $X^{s,b}$ and $X^{s,b}_{1/\sigma}$ spaces. For this, we consider an appropriate selection of real parameters $a_{1},b_{1},a_{2},b_{2}$, to use the estimate (4.8) of the Lemma 13 and then (6.28) to obtain

[TABLE]

[TABLE]

[TABLE]

By means of the Lemma 27, using (6.22), it is convenient to write

[TABLE]

where in $I_{1}$ the summation in $N$ is restricted to $N_{0}\geq C(N_{1}+N_{2})$ and all other possibilities are in $I_{2}$.

To bound $I_{1}$, note that the frequency regime over $N=(N_{0},N_{1},N_{2})$ is such that $N_{0}\geq C(N_{1}+N_{2})$ so that we can apply the estimate (6.9) of Lemma 27 with $p>s+6$. As $b>\frac{1}{2}$ and $b^{\prime}>0$, we can perform a sum of geometric series444By this we mean that the sums in question are of the form $\sum_{k=0}^{\infty}\left(\frac{1}{2^{\gamma}}\right)^{k}<\infty$, since $\gamma>0$. in all dyadic variables to conclude that

[TABLE]

Having the estimate (6.35) at hand, it remains to bound

[TABLE]

In this case, since the frequency regime satisfies the condition $N_{0}\leq C(N_{1}+N_{2})$, we cannot apply Lemma 27. To overcome this obstacle, we consider the cases $N_{1}\geq N_{2}$ and $N_{1}\leq N_{2}$ separately and find appropriate estimates in each case.

By symmetry, we just consider the case where $N_{1}\geq N_{2}$. In this case, $N_{0}\leq 2CN_{1}$. Once this frequency regime has been set, our next step will be to bound the term $I^{NL}$ in (6.36) in two different ways. The first way is to bound $I^{NL}$ using Cauchy-Schwarz inequality which behaves better with respect to the localization on $N$. The other way is to bound $I^{NL}$ using Hölder’s inequality which behaves better with respect to the localization on $L$. Interpolating these bounds, we obtain the required estimate. In the sequel, we describe this process in detail.

An use of the Cauchy-Schwarz inequality with respect to $N$ in (6.23), yields

[TABLE]

Applying (6.28) directly, it follows that

[TABLE]

Now, using the bilinear estimate (5.28) with $s^{\prime}>s_{0}(M)$, we have

[TABLE]

Considering (6.32) and (6.33) with the values $a_{i}=0$, $b_{i}=\frac{1}{2}+\epsilon_{0}\mbox{ }(i=1,2)$ and $s_{1}=s$, (where $\epsilon_{0}>0$ and $b>1/2$ will be chosen later in a suitable way), we get

[TABLE]

and

[TABLE]

We note here that the value of $s_{2}$ in (6.41) will be chosen so that $s_{2}=1$ if $s\geq 1$ and $s_{2}=s$ if $0<s<1$. Replacing the estimates (6.38) and (6.39) in (6.37), we obtain

[TABLE]

Now, we make a better estimate with respect to localization over $L$. In fact, by applying Hölder’s inequality in (6.23), we get

[TABLE]

By using the Sobolev embedding $H^{\frac{d}{2}+\frac{1}{10}}(M)\hookrightarrow L^{\infty}(M)$ in (6.43), we obtain

[TABLE]

Thus, from (6.43) we have

[TABLE]

Using part $(ii)$ of Lemma 10, we obtain $\|f\|_{L^{3}(\mathbb{R};L^{2}(M))}\lesssim\|f\|_{Y}$ where $Y=X_{1/\sigma}^{0,\frac{1}{6}}(\mathbb{R}\times M)$ or $X^{0,\frac{1}{6}}(\mathbb{R}\times M)$. Thus, using the identities involving the coefficients $c_{j}^{L_{j}N_{j}}$ given in (6.31), (6.32) and (6.33), it follows that

[TABLE]

An interpolation between (6.42) and (6.46) implies that for every $\theta\in(0,1)$

[TABLE]

Now, we consider a fixed value of the parameter $s$ such that $s^{\prime}<s$, where $0<s_{0}(M)<s^{\prime}<1$ and make an analysis dividing in two different cases.

Case 1. $s\in(0,1)$. In this case $s_{2}=s$ and (6.47) reduces to

[TABLE]

In order to make the exponent of $N_{2}$ negative (necessary for convergence), we choose $\theta$ so that $0<\theta<\frac{s-s^{\prime}}{\frac{d}{2}+\frac{1}{10}-s^{\prime}}<1$. Now, we choose $\epsilon_{0}$ such that $0<\epsilon_{0}<\frac{\theta}{9-3\theta}$. Hence, if we define $b^{\prime}:=\frac{1}{2}-2\epsilon_{0}$, this ensures that $b^{\prime}>\frac{\theta}{6}+(\frac{1}{2}+\epsilon_{0})\theta$. Moreover, one has $b^{\prime}>0\Longleftrightarrow\epsilon_{0}<\frac{1}{4}$, which is a consequence of $\frac{\theta}{9-3\theta}\leq\frac{1}{4}\Longleftrightarrow\theta\leq\frac{9}{7}=1+\frac{2}{7}$. Finally, choose $b:=\frac{1}{2}+\frac{3}{2}\epsilon_{0}$ and notice that with this choice of parameters, the basic conditions for $(b,b^{\prime})$ are verified, i.e., $0<b^{\prime}<\frac{1}{2}<b$ and $b+b^{\prime}<1$.

Case 2. $s\in[1,+\infty)$. In this case $s_{2}=1$ and (6.47) is reduced to

[TABLE]

Observe that

[TABLE]

As $0<s^{\prime}<1$, the choice of $\theta$ to obtain the exponent of $N_{2}$ negative is

[TABLE]

Therefore, at this point, we can choose the same values of $\epsilon_{0},b^{\prime}$ and $b$ of the Case 1.

Hence, after the suitable choice of the parameters in (6.48) and (6.49), as $b^{\prime}<b$, we have that in any case, there are $\gamma_{1},\cdots,\gamma_{4}>0$ such that

[TABLE]

Next, using (6.50), it follows from (6.36) that

[TABLE]

The summations involving $L_{0},L_{1},L_{2},N_{2}$ in (6.50) can be performed via convergence of geometric series. Now, we indicate the coefficients associated with the dyadic variables $N_{1}$, $N_{0}$ and $N_{2}$ by

[TABLE]

Applying the Cauchy-Schwarz inequality on the variables $L_{j}$ ($j=0,1,2$) and using the relations given in (6.52), we obtain

[TABLE]

Applying the Cauchy-Schwarz inequality on the summation involving $N_{2}$ and using the relation (6.30), it follows from (6.53) that

[TABLE]

Now, from Lemma 28, it follows that

[TABLE]

Thus, replacing (6.55) in (6.54), we get

[TABLE]

Combining the estimates (6.35) and (6.56) we obtain the required bilinear estimates (6.1) and (6.2).

The proof of the bilinear estimate (6.3) is similar to the proof of the estimate (6.1) modulo some modifications. Let us highlight the main modifications here.

By using the duality relation between $X_{1/\sigma}^{s,-b_{1}^{\prime}}$ and $X_{1/\sigma}^{-s,b_{1}^{\prime}}\approx(X_{1/\sigma}^{s,-b_{1}^{\prime}})^{\ast}$ we see that to prove (6.3), it is necessary to prove that

[TABLE]

where $u_{0}\in X_{1/\sigma}^{-s,b_{1}^{\prime}}(\mathbb{R}\times M)$ is arbitrary. In this case, as in (6.23), it is necessary to work with

[TABLE]

Notice that the analogue of (6.21) is given by

[TABLE]

where we do not have the presence of conjugates. Using the bilinear estimate (5.28) with $\sigma=1$, we obtain

[TABLE]

In the same way, we can obtain the estimate (6.45). The other estimates can be obtained analogously. $\Box$

7. Local and Global Theory

In this section we provide proofs of the main local and global well-posedness results of this work.

7.1. Local Theory

We begin by recording some basic estimates that will be important in order to prove the well posedness result. Recall the class of the spaces $X_{\delta,\gamma}^{s,b}$ introduced in the Definition 1 and the restriction spaces given in Definition 3.

Proposition 29**.**

(Linear estimates in the spaces $X^{s,b}_{\delta}$). Let $M$ be a compact Riemannian manifold, $b,s>0$, $v_{0}\in H^{s}(M)$ and $\psi\in C_{0}^{\infty}(\mathbb{R})$ be such that $\psi=1$ in $[-1,1]$. Let $\psi_{T}:=\psi(\frac{\cdot}{T})$, then $\psi_{T}=1$ in $[-T,T]$. Under these assumptions, we have

[TABLE]

Let $0<b^{\prime}<\frac{1}{2}$ and $0<b<1-b^{\prime}$. Then, for all $F\in X_{\delta}^{s,-b^{\prime}}(M)$,

[TABLE]

when $0<T\leq 1$.

Now, we are in position to supply a proof to the result stated in Theorem 7 that provides the local well-posedness for the IVP (1.4) posed on $d$-dimensional Zoll manifolds.

Proof of Theorem 7. Let $M$ be a Zoll manifold of dimension $d\geq 2$, $(v_{0},u_{0})\in\mathbf{H}^{s}(M)$ with $s>s_{0}(d)$ and $\sigma=\frac{\beta}{\theta}$ with $\beta,\theta\in\{n^{2}:n\in\mathbb{N}\}$. Let $R>0$ and $T>0$ be constants to be suitably chosen later. Consider the spaces $\mathcal{X}^{s,b}_{T}:=X^{s,b}_{T}\times(X_{1/\sigma}^{s,b})_{T}$ endowed with norm

[TABLE]

and a closed ball of radius $R>0$ in $\mathcal{X}^{s,b}_{T}$ given by

[TABLE]

For $v,u\in B_{T}^{R}$ define the operators

[TABLE]

For appropriate choices of the constants $R>0$ and $T>0$, we will show that the application $(\Phi_{1},\Phi_{2})$ is a contraction.

Applying the linear estimates (7.1) and (7.2), the nonlinear estimates (6.1), (6.2) and (6.3), (6.4) in (7.3), using the definition of the $\mathcal{X}^{s,b}_{T}$ norm and noting that $u,v\in B_{T}^{R}$, one obtains that

[TABLE]

Considering $\Phi(u,v):=(\Phi_{1}(u,v),\Phi_{2}(u,v))$, we see from (7.4) that

[TABLE]

Choosing $R:=2c_{0}\|(v_{0},u_{0})\|_{\mathbf{H}^{s}(M)}$, we obtain

[TABLE]

Thus, if we choose 0<T<\Big{(}\frac{1}{2c_{1}R}\Big{)}^{\frac{1}{1-b-b^{\prime}}}, it is easy to see from (7.6) that $\Phi$ maps $B^{R}_{T}$ into itself.

Now, writing $u\overline{v}-\tilde{u}\overline{\tilde{v}}=(u-\tilde{u})\overline{v}+(\overline{v}-\overline{\tilde{v}})\tilde{u}$, with the similar calculations as above, we easily get

[TABLE]

Thus, choosing $T=T(\|(v_{0},u_{0})\|_{\mathbf{H}^{s}})>0$, in such a way that

[TABLE]

we get $\Phi:B_{T}^{R}\longrightarrow B_{T}^{R}$ is a contraction. Hence using the Banach fixed point theorem, we conclude that, there is a unique $(v,u)\in B_{T}^{R}$ which solves the integral system (1.5) for $t\in[0,T]$, with $T\leq 1$.

As $b>\frac{1}{2}$, we have the embedding $\mathcal{X}_{T}^{s,b}\hookrightarrow C([0,T],\mathbf{H}^{s})$. Hence, for $T\in(0,1)$ and $t\in(0,T)$,

[TABLE]

To end the proof of the theorem, we will show that

[TABLE]

is Lipschitz continuous for some $r>0$.

Given $T^{\prime}<T$ take $r>0$ such that

[TABLE]

where $\gamma(b,b^{\prime}):=\frac{1}{1-b-b^{\prime}}>0$ and $C>0$ is given in (7.7). Given $(\tilde{v_{0}},\tilde{u_{0}})\in\mathbf{H}^{s}(M)$, consider the respective solution $(\tilde{v},\tilde{u})\in\mathbf{H}^{s}(M)$, which exists on $[0,\tilde{T}]$, where $\tilde{T}=\tilde{T}(\|(\tilde{v_{0}},\tilde{u_{0}})\|)>0$, satisfies $T^{\prime}<\tilde{T}$. Then, both the solutions are well defined in $(0,T^{\prime})$.

Thus, similarly to what was done to obtain the estimate (7.5), we can get

[TABLE]

As $C(T^{\prime})^{1-b-b^{\prime}}R^{2}<1$,

[TABLE]

Now, using the embedding $\mathcal{X}_{T^{\prime}}^{s,b}\hookrightarrow C([0,T^{\prime}],\mathbf{H}^{s})$, we get

[TABLE]

which proves the assertion. $\Box$

7.2. Global Theory

In this subsection, we will use the Gagliardo-Nirenberg inequality on compact Riemannian manifolds to extend the local solution for the IVP (1.4) obtained in Theorem 7 to the global one in dimensions 2 and 3. We start with the following result.

Proposition 30**.**

Let $M$ be a compact Riemannian manifold of dimension $d\geq 2$ and $1<p\leq 2$. If $1\leq q<r<p^{\ast}=\frac{dp}{d-p}$ and $\theta=\theta(p,q,r)=\frac{dp(r-q)}{r(q(p-d)+dp)}\in(0,1]$, then

[TABLE]

where

[TABLE]

Proof. See [17], p. 854. $\Box$

Remark 31**.**

The explicit value of $A_{opt}$ is known if $q>p$ and $r=\frac{p(q-1)}{p-1}$, see [17] p. 853.

Now, we prove the global well-posedness result state in Theorem 9.

Proof of Theorem 9. First, consider $s=1$. Let $d=2,3$. If we choose $p=2=q$, the Gagliardo-Nirenberg inequality (7.8) yields

[TABLE]

where $\theta_{r}=\frac{d}{2}-\frac{d}{r}$, $2<r<\infty$ if $d=2$ and $2<r<\frac{2d}{d-2}$ if $d\geq 3$. Now, using (1.3), we obtain

[TABLE]

Hence

[TABLE]

Using the Cauchy-Schwarz inequality, we obtain

[TABLE]

From (1.2), we conclude that

[TABLE]

Using (7.10) with $r=4$, we get

[TABLE]

If $d=2,3$, we can bound (7.11) by

[TABLE]

From (7.12), we obtain

[TABLE]

Observe that

[TABLE]

Denote

[TABLE]

Thus,

[TABLE]

By Young’s inequality, one obtains

[TABLE]

That is,

[TABLE]

Now we use an a priori estimate given by (7.13) combined with a standard argument to prove that the solution $(v,u)\in C([0,T^{\ast});\mathbf{H}^{1}(M))$ is in fact global in $\mathbf{H}^{1}$ for dimensions $2$ and $3$, that is, we have $T^{\ast}=+\infty$.

Now, we consider $s>1$. From (6.2) and (6.4), we obtain

[TABLE]

Consider $\Phi(u,v):=(\Phi_{1}(u,v),\Phi_{2}(u,v))$. Then, from (7.4), we get

[TABLE]

If $(v,u)\in B^{R}_{T}$ we must choose $T$ so that

[TABLE]

that is,

[TABLE]

By the continuous dependence of $H^{1}(M)$, it follows that

[TABLE]

Thus, we have

[TABLE]

such that the solution exists on $[0,T]$, and

[TABLE]

Observe that $T$ depends only on the $\mathbf{H}^{1}$ norm of the initial data. Moreover, by (7.13), there exists a fixed constant $C=C(\sigma,A_{opt},B,\alpha,d)>0$ such that

[TABLE]

Thus, $T$ is bounded from below by a positive constant, and we can iterate this argument to extend the solution $(v,u)$ on any time interval $[0,T]$ for $s>1$. $\Box$

Acknowledgment

The authors would like to thank the anonymous referee for the careful reading of the manuscript and many constructive comments and suggestions that considerably improved the presentation.