Asymptotic behavior for a class of derivative nonlinear Schrödinger systems

Abstract

We consider the initial value problem for systems of derivative nonlinear Schrödinger equations with nonlinearity of the critical power in one and two space dimensions. Li–Sunagawa and Sakoda–Sunagawa introduced a sufficient condition, which is weaker than the so-called null condition, for the small data global existence of solutions. In this paper, we investigate the asymptotic behavior of global solutions under a condition related to the above one.

Appendix: Improvement in the one space dimensional case

Our aim in this appendix is to show the following improved version of Theorem 2.1 for one dimensional case.

Theorem 7.1

Let$d=1$, $p=3$, and each$F_j$be a homogeneous polynomial of degree$p(=3)$in$(u,\,\partial _x u,\,\overline{u},\,\overline{\partial _x u})$. Letsbe a positive integer. Assume the conditions (a) and (b) are satisfied. For any positive constants$C_0$and$\delta$with$0<\delta <1/12$, there is a positive constant$\varepsilon _0$with the following property: If$u\in C\bigl ([0,\infty ); \mathcal {H}^{s+2, 1}(\mathbb {R})\bigr )$is a global solution to (1.1) satisfying (2.5), and if $0<\varepsilon \le \varepsilon _0$, then there exists a$\mathbb {C}^N$-valued function$\alpha ^+=(\alpha ^+_j(\xi ))_{j\in I_N}$on$\mathbb {R}$, satisfying

$$\begin{aligned} |\alpha ^+(\xi )| \le C\varepsilon \langle \xi \rangle ^{-(s+1)}, \quad \xi \in \mathbb {R}, \end{aligned}$$

(7.1)

as well as (2.9), such that we have (2.6) for$|\beta |\le s+1$, where$A^+=(A^+_j(t,\xi ))_{j\in I_N}$is the global solution to the profile system (2.7).

Remark 7.1

It is clear that the conclusion of Theorem 2.2 is also true under the assumptions in Theorem 7.1. $\square$

Let $d=1$ and each nonlinear term $F_j$ be a homogeneous polynomial of degree 3. We can show Theorem 7.1 by following the proof of Theorem 2.1 if we use the next lemma instead of Lemma 3.3:

Lemma 7.1

Letlbe a positive integer. Suppose that the condition$(\mathbf {a})$is satisfied. Then, for any multi-index$\beta$with$|\beta |\le l+1$, we have

$$\begin{aligned}&\left\| \mathcal {F}_{m_j}\mathcal {U}_{m_j}^{-1} \partial _x^\beta F_j(u,\partial _x u)-\frac{(im_j \xi )^\beta }{t} P_j(\xi , A)\right\| _{L^\infty } \nonumber \\&\qquad \le Ct^{-\frac{5}{4}}\left( \sum _{|\gamma |\le 1} \Vert \mathcal {J}_{{\varvec{M}}}(t)^\gamma u(t)\Vert _{H^{l+1}}\right) ^3. \end{aligned}$$

(7.2)

Compared to Lemma 3.3, the improvement here is that (7.2) holds not only for $|\beta |\le l$, but also for $|\beta |=l+1$. To prove this lemma, we use the technique from [26], which is applicable to the cubic nonlinearity.

Lemma 7.2

Let$m\in \mathbb {R}{\setminus }\{0\}$. Then we have

$$\begin{aligned} \left\| \mathcal {F}_m^{-1} \mathcal {U}_m(t)( \phi _1\phi _2\phi _3 )\right\| _{L^\infty (\mathbb {R})}\le C\Vert \phi _1\Vert _{L^\infty (\mathbb {R})}\Vert \phi _2\Vert _{L^2(\mathbb {R})}\Vert \phi _3\Vert _{L^2(\mathbb {R})}. \end{aligned}$$

Proof

Recall that $\mathcal {M}_m^{\pm 1}$ is the multiplication by $\psi _{\pm m}(t,x):=e^{\pm im\frac{x^2}{2t}}$. As $\mathcal {W}_m^{-1}=\mathcal {F}_m \mathcal {M}_m^{-1}\mathcal {F}_m^{-1}$ and $\Vert \mathcal {F}_m\psi _{-m}\Vert _{L^\infty }\le C\sqrt{t}$, we get

$$\begin{aligned} \Vert \mathcal {W}_m^{-1} \phi \Vert _{L^\infty }=C\Vert (\mathcal {F}_m \psi _{-m})*\phi \Vert _{L^\infty } \le C\Vert \mathcal {F}_m \psi _{-m}\Vert _{L^\infty }\Vert \phi \Vert _{L^1}\le C\sqrt{t}\Vert \phi \Vert _{L^1}. \end{aligned}$$

Writing $\mathcal {F}_m^{-1}\mathcal {U}_m=\mathcal {W}_m^{-1}\mathcal {D}^{-1}\mathcal {M}_m^{-1}$, we obtain

$$\begin{aligned} \left\| \mathcal {F}_m^{-1} \mathcal {U}_m(\phi _1\phi _2\phi _3\bigr )\right\| _{L^\infty }&\le Ct^{\frac{1}{2}}\left\| \mathcal {D}^{-1}\mathcal {M}_m^{-1}(\phi _1\phi _2\phi _3)\right\| _{L^1}\\&\le Ct^{-\frac{1}{2}}\left\| (\mathcal {D}^{-1}\phi _1)(\mathcal {D}^{-1}\phi _2)(\mathcal {D}^{-1} \phi _3)\right\| _{L^1}\\&\le Ct^{-\frac{1}{2}}\Vert \mathcal {D}^{-1}\phi _1\Vert _{L^\infty }\Vert \mathcal {D}^{-1}\phi _2\Vert _{L^2}\Vert \mathcal {D}^{-1}\phi _3\Vert _{L^2}\\&\le C\Vert \phi _1\Vert _{L^\infty }\Vert \phi _2\Vert _{L^2}\Vert \phi _3\Vert _{L^2}, \end{aligned}$$

which is the desired result. $\square$

Lemma 7.3

Let$m, \nu , \mu _1, \ldots , \mu _L \in \mathbb {R}{\setminus }\{0\}$, and let$H=H(Z)$be a homogeneous polynomial of degree 2 in$Z=(Z_j)_{j\in I_L}\in \mathbb {C}^L$. Suppose that we have

$$\begin{aligned} H(e^{i{\varvec{\mu }}\theta } Z)e^{i\nu \theta }=e^{im\theta } H(Z),\quad Z\in \mathbb {C}^L,\ \theta \in \mathbb {R}, \end{aligned}$$

where${\varvec{\mu }}={\mathrm {diag}\,}(\mu _1, \ldots , \mu _L)$. We put$B=\mathcal {F}_{{\varvec{\mu }}}\mathcal {U}_{{\varvec{\mu }}}^{-1} v$and$B_0=\mathcal {F}_{\nu }\mathcal {U}_{\nu }^{-1} w$. Then, for a$\mathbb {C}^L$-valued functionvand a$\mathbb {C}$-valued functionwon$\mathbb {R}$, we have

$$\begin{aligned}&\left\| \mathcal {F}_m\mathcal {U}_m(t)^{-1}\partial _x \bigl (H(v)w\bigr )-\frac{im\xi }{t}H(B)B_0\right\| _{L^\infty }\\&\quad \le Ct^{-\frac{5}{4}} \left\{ \sum _{|\gamma |\le 1} \bigl (\Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v(t)\Vert _{H^1}+\Vert \mathcal {J}_\nu ^\gamma w(t)\Vert _{L^2}\bigr )\right\} ^3. \end{aligned}$$

Proof

By the assumption, H(Z) consists of $Z_jZ_k$ with $\mu _j+\mu _k=m-\nu$. Hence we let $\mu _j+\mu _k=m-\nu$ in the sequel. By direct calculations, we have

$$\begin{aligned} \partial _x\bigl (v_jv_k w\bigr )=\frac{m}{\mu _k} \bigl (v_j (\partial _x v_k) w \bigr )+\frac{m}{it} \left\{ \mathcal {J}_m\bigl (v_jv_kw\bigr )-v_j (\mathcal {J}_{\mu _k} v_k)w\right\} . \end{aligned}$$

Since

$$\begin{aligned} (e^{i\mu _j\theta } v_j)(e^{i\mu _k\theta }\partial _x v_k)(e^{i\nu \theta }w)=e^{im\theta }\bigl (v_j(\partial _x v_k)w\bigr ), \end{aligned}$$

we can use Lemma 3.6 to obtain

$$\begin{aligned}&\left\| \mathcal {F}_m\mathcal {U}_m^{-1}\bigl (v_j (\partial _x v_k) w \bigr )-\frac{i\mu _k\xi }{t}B_jB_k B_0\right\| _{L^\infty }\\&\quad \le Ct^{-\frac{5}{4}} \left\{ \sum _{|\gamma |\le 1} \bigl (\Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v(t)\Vert _{H^1}+\Vert \mathcal {J}_\nu ^\gamma w(t)\Vert _{L^2}\bigr )\right\} ^3. \end{aligned}$$

As $\mathcal {J}_m=(it/m)\mathcal {M}_m\partial _x\mathcal {M}_m^{-1}$, we have

$$\begin{aligned} \mathcal {J}_m\bigl (v_jv_kw\bigr ) =\frac{\mu _j}{m}(\mathcal {J}_{\mu _j}v_j)v_k w+\frac{\mu _k}{m}v_j(\mathcal {J}_{\mu _k}v_k)w +\frac{\nu }{m}v_jv_k(\mathcal {J}_\nu w). \end{aligned}$$

Lemma 7.2 yields

$$\begin{aligned}&\left\| \mathcal {F}_m\mathcal {U}_m^{-1}\left\{ \mathcal {J}_m\bigl (v_jv_kw\bigr )-v_j (\mathcal {J}_{\mu _k} v_k)w\right\} \right\| _{L^\infty }\\&\quad \le C\Vert v\Vert _{L^\infty }\left( \sum _{|\gamma |\le 1}\Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v\Vert _{L^2}\right) \left( \sum _{|\gamma |\le 1} \Vert \mathcal {J}_\nu ^\gamma w\Vert _{L^2}\right) \\&\quad \le Ct^{-\frac{1}{2}}\left( \sum _{|\gamma |\le 1}\Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v\Vert _{L^2}\right) ^2\left( \sum _{|\gamma |\le 1} \Vert \mathcal {J}_\nu ^\gamma w\Vert _{L^2}\right) , \end{aligned}$$

as the identity $\partial _x(\mathcal {M}_{\mu _j}^{-1}\phi )=-(i\mu _j/t)\mathcal {J}_{\mu _j}\phi$ implies

$$\begin{aligned} \Vert \phi \Vert _{L^\infty }&=\Vert \mathcal {M}_{\mu _j}^{-1} \phi \Vert _{L^\infty } \le \Vert \mathcal {M}_{\mu _j}^{-1}\phi \Vert _{L^2}^{\frac{1}{2}}\Vert \partial _x(\mathcal {M}_{\mu _j}^{-1}\phi )\Vert _{L^2}^{\frac{1}{2}} \le \sqrt{\frac{|\mu _j|}{t}}\Vert \phi \Vert _{L^2}^{\frac{1}{2}}\Vert \mathcal {J}_{\mu _j}\phi \Vert _{L^2}^{\frac{1}{2}}. \end{aligned}$$

To sum up, we obtain

$$\begin{aligned}&\left\| \mathcal {F}_m\mathcal {U}_m^{-1}\partial _x(v_j v_k w \bigr )-\frac{im\xi }{t}B_jB_k B_0\right\| \\&\qquad \le Ct^{-\frac{5}{4}} \left\{ \sum _{|\gamma |\le 1} \bigl (\Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v(t)\Vert _{H^1}+\Vert \mathcal {J}_\nu ^\gamma w(t)\Vert _{L^2}\bigr )\right\} ^3, \end{aligned}$$

which implies the desired result. $\square$

The main ingredient in Lemma 7.3 is that we do not need $\Vert \mathcal {J}_\nu w\Vert _{H^1}$ in the estimate.

Let G, v, B be as in Sect. 3.2. Then Corollary 3.1 is replaced by the following.

Corollary 7.1

For a positive integerl, we have

$$\begin{aligned}&\left\| \mathcal {F}_m \mathcal {U}_m^{-1}\partial _x^{l+1} G\bigl (v(t)\bigr )-\frac{(im\xi )^{l+1}}{t} G\bigl (B(t)\bigr )\right\| _{L^\infty (\mathbb {R})}\\&\qquad \le Ct^{-\frac{5}{4}}\left( \sum _{|\gamma |\le 1} \Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v(t) \Vert _{H^l(\mathbb {R})} \right) ^3. \end{aligned}$$

Proof

Recall the definition of $G^{[l]}$ in Sect. 3.2, and observe that terms involving $Z^{(l)}$ in $G^{[l]}\bigl ((Z^{(\gamma )})_{0\le \gamma \le l})$ are $\sum _{j=1}^L \partial _{Z_j} G(Z) Z_j^{(l)}$. We put

$$\begin{aligned} \widetilde{G}^{[l]}\bigl ((Z^{(\gamma )})_{0\le \gamma \le l-1}\bigr ): =G^{[l]}\bigl ((Z^{(\gamma )})_{0\le \gamma \le l}\bigr )-\sum _{j=1}^L \frac{\partial G}{\partial Z_j}(Z) Z_j^{(l)} \end{aligned}$$

Then, since we have

$$\begin{aligned} \widetilde{G}^{[l]}\bigl ((e^{i{\varvec{\mu }}\theta }Z^{(\gamma )})_{0\le \gamma \le l-1}\bigr )=e^{im\theta } \widetilde{G}^{[l]}\bigl ((Z^{(\gamma )})_{0\le \gamma \le l-1}\bigr ), \end{aligned}$$

we can use Corollary 3.1 to get

$$\begin{aligned}&\left\| \mathcal {F}_m\mathcal {U}_m^{-1}\partial _x \widetilde{G}^{[l]}\bigl ((v^{(\gamma )})_{0\le \gamma \le l-1}\bigr ) -\frac{im\xi }{t}\widetilde{G}^{[l]}\left( \bigl ( (i{\varvec{\mu }})^{|\gamma |}\xi ^\gamma B\bigr )_{0\le \gamma \le l-1} \right) \right\| _{L^\infty }\\&\qquad \le Ct^{-\frac{5}{4}}\left( \sum _{|\gamma |\le 1} \Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v(t)\Vert _{H^l}\right) ^3. \end{aligned}$$

Using Lemma 7.3 with $H=\partial _{Z_j}G$ and $w=v_j^{(l)}$, we find

$$\begin{aligned}&\left\| \mathcal {F}_m\mathcal {U}_m^{-1}\partial _x \sum _{j=1}^L \frac{\partial G}{\partial Z_j}(v)v_j^{(l)}-\frac{im\xi }{t}\sum _{j=1}^L \frac{\partial G}{\partial Z_j}(B)(im_j\xi )^l B_j \right\| _{L^\infty }\\&\qquad \le Ct^{-\frac{5}{4}}\left( \sum _{|\gamma |\le 1} \Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v(t)\Vert _{H^l}\right) ^3. \end{aligned}$$

Therefore we have

$$\begin{aligned}&\left\| \mathcal {F}_m\mathcal {U}_m^{-1}\partial _x G^{[l]}\bigl ((v^{(\gamma )})_{0\le \gamma \le l}\bigr ) -\frac{im\xi }{t}G^{[l]}\left( \bigl ( (i{\varvec{\mu }})^{|\gamma |}\xi ^\gamma B\bigr )_{0\le \gamma \le l}\right) \right\| _{L^\infty }\\&\qquad \le Ct^{-\frac{5}{4}}\left( \sum _{|\gamma |\le 1} \Vert \mathcal {J}_{{\varvec{\mu }}}^\gamma v(t)\Vert _{H^l}\right) ^3, \end{aligned}$$

which leads to the desired result thanks to (3.13) and Lemma 3.5. $\square$

Using this corollary, we immediately obtain Lemma 7.1, and then Theorem 7.1, in a similar fashion to Remark 5.1.