2D-defocusing nonlinear Schrödinger equation with random data on irrational tori

Abstract

We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.

Appendices

Appendix A: Time localization of $X^{s,b}$

In this section, we summarize several standard time localization facts for the $X^{s,b}$ space, and also briefly recall the proof of Lemma 3.5. The presentation mainly follows that from [6]. Here $\phi$ is a fixed time cut off function. There are several basic facts about the $X^{s,b}$ space that we can recall below. We have

$$\begin{aligned}&\Vert \phi (t/\delta )u\Vert _{X^{s,b}}\lesssim _{b} \Vert u\Vert _{X^{s,b}}, \quad 0< b<\frac{1}{2} \end{aligned}$$

(A.1)

$$\begin{aligned}&\Vert \phi (t/\delta )u\Vert _{X^{s,b}}\lesssim _{b} \delta ^{\frac{1-2b}{2}}\Vert u\Vert _{X^{s,b}} \quad \frac{1}{2}<b<1. \end{aligned}$$

(A.2)

Also, Hausdorff–Young inequality gives the following estimate which is useful in the interpolation

$$\begin{aligned} \Vert \phi (t)u\Vert _{L_{t,x}^{4}}\lesssim _{\epsilon } \Vert u\Vert _{X^{1/2,\frac{1}{4}+\epsilon }}, \end{aligned}$$

(A.3)

which can be compared to estimates (95), (96) on page 26 of [6].

In what follows, one should think $1\gg s_{p}\gg \epsilon >0$. We will only do proof for (3.12) in Lemma 3.5.

Via Strichartz estimate and interpolation of Hausdorff Young inequality, one can obtain

$$\begin{aligned} \Vert \phi (t)u\Vert _{L_{t,x}^{4}}\lesssim _{\epsilon }\Vert u\Vert _{X^{3\epsilon ,\frac{1}{2}-\epsilon }} \end{aligned}$$

(A.4)

(One may change the 3 in the above to any number larger than 2.) Similarly, for $p>4$, one can obtain

$$\begin{aligned} \Vert \phi (t)u\Vert _{L_{t,x}^{p}}\lesssim _{\epsilon }\Vert u\Vert _{X^{s_{p}+10\epsilon ,\epsilon }} \end{aligned}$$

(A.5)

There are the following two Hölder inequalities,

• (1)
  $$\begin{aligned} \Vert \phi (t/\delta )u\Vert _{L_{t,x}^{4}}\le \Vert \phi (t/\delta )u\Vert ^{\theta _{p}}_{L_{t,x}^{2}}\Vert \phi (t)u\Vert ^{1-\theta _{p}}_{L^{ p}_{t,x}}, \end{aligned}$$

  (A.6)

  where one has $\frac{1}{4}=\frac{\theta _{p}}{2}+\frac{1-\theta _{p}}{p}$. $\theta _{p}=\frac{s_{p}}{1+s_{p}}\ge \frac{1}{2}s_{p}$

• (2)
  $$\begin{aligned} \Vert \phi (t/\delta )u\Vert _{L_{t,x}^{2}}\le \delta ^{1/4}\Vert \phi (t)u\Vert _{L_{t,x}^{4}} \end{aligned}$$

  (A.7)

One derives

$$\begin{aligned} \begin{aligned}&\Vert \phi (t/\delta ) u\Vert _{L_{t,x}^{4}}\\&\quad \le \Vert \phi (t\delta )u\Vert _{L_{t,x}^{2}}^{\theta _{p}}\Vert \phi (t) u\Vert _{L_{t,x}^{p}}^{1-\theta _{p}}\\&\quad \le \Vert \phi (t\delta )u\Vert _{L_{t,x}^{4}}^{\theta _{p}}\delta ^{\theta _{p}/4} \Vert \phi (t)u\Vert _{L_{t,x}^{p}}^{1-\theta _{p}}\\&\quad \lesssim _{\epsilon }\Vert u\Vert _{X^{3\epsilon ,\frac{1}{2} -\epsilon }}^{\theta _{p}}\Vert u\Vert ^{1-\theta _{p}}_{X^{s_{p} +10\epsilon ,\frac{1}{2}-\epsilon }}\delta ^{\frac{\theta _{p}}{4}} \end{aligned} \end{aligned}$$

(A.8)

Note that

$$\begin{aligned} \theta _{p}=\frac{s_{p}}{1+s_{p}}\ge \frac{1}{2}s_{p}. \end{aligned}$$

(A.9)

Thus, to summarize, when $s\ll 1$, and $\epsilon \ll s$, one has

$$\begin{aligned} \Vert \phi (t/\delta ) u\Vert _{L_{t,x}^{4}}\lesssim _{\epsilon } \Vert u\Vert _{X^{s+10\epsilon ,\frac{1}{2}-\epsilon }}\delta ^{\frac{s}{4}}, \end{aligned}$$

(A.10)

which, for convenience, can be written as

$$\begin{aligned} \Vert \phi (t/\delta )u\Vert _{L_{t,x}^{4}}\lesssim _{\epsilon }\Vert u\Vert _{X^{s, \frac{1}{2}-\epsilon }}\delta ^{s/8}. \end{aligned}$$

(A.11)

Localizing at frequency N, this gives Lemma 3.5 for balls B of radius N, which are centered at origin point. To prove general B centered at $n_{0}$, one simply observes

$$\begin{aligned} \sum _{n\in B}a_{n}e^{inx}e^{in^{2}t}e^{i\lambda t}=\sum _{|n-n_{0}|\le N}a_{n}e^{i(n-n_{0})(x+2n_{0})}e^{i(n-n_{0})^{2}t}e^{i\lambda t}e^{in_{0}x}e^{-in_{0}^{2}t} \end{aligned}$$

(A.12)

and the $L_{t,x}^{4}$ norm of a function is invariant under multiplying $e^{in_{0}x}e^{-in_{0}^{2}t}$ and doing space translation in x variable by $n_{0}$. This ends the proof.

Appendix B: Proof of Lemma 3.6, 3.7, 3.8

We briefly sketch the proof of those three Lemmata here.

We start with Lemma 3.8. Let $h(n,t), f_{i}(n,t)$ be space of $h,f_{i}$, and we will also short handedly write them as $h(n), f_{i}(n)$. We only prove

$$\begin{aligned}&\left| \int \phi (t/\delta )h{\mathcal {N}}_{2}(P_{N_{1}}f_{1}, P_{N_{1}}f_{2}, P_{N_{1}}f_{3})\right| \nonumber \\&\quad \lesssim (\delta ^{1/4}\Vert P_{N_{1}}h\Vert _{X_{0,1-b_{0}}}\Vert f_{1}\Vert _{X^{0,b_{0}}}\sup _{|n|\sim N_{1}}\Pi _{j\ne 1}\Vert f_{j}(n)e^{inx}\Vert _{X_{0,b_{0}}}). \end{aligned}$$

(B.1)

To see this, observe

$$\begin{aligned} \begin{aligned}&\left| \int \phi (t/\delta )h{\mathcal {N}}_{2}(P_{N_{1}}f_{1}, P_{N_{1}}f_{2}, P_{N_{1}}f_{3})\right| =\left| \sum _{|n|\sim N_{1}}\int \phi (t/\delta ) \bar{h}(n)f_{1}(n)\bar{f}_{2}(n)f_{3}(n)\right| \\&\quad \lesssim \Vert \phi (t/\delta )\Vert _{L_{t}^{2}}\Vert \phi (t)h(n)\Vert _{L_{t}^{2}}\Vert \phi (t)f_{1}\Vert _{L_{t}^{\infty }}\sup _{|n|\sim N_{1}}\Pi _{j\ne 1}\Vert f_{j}\Vert _{L^{\infty }_{t}}. \end{aligned} \end{aligned}$$

(B.2)

Now we have, (note that one only has one mode in all the estimates below)

$$\begin{aligned} \Vert \phi (t)h(n)\Vert _{L_{t}^{2}}\lesssim \Vert h(n)e^{inx}\Vert _{X^{0,1-b_{0}}}, \Vert f_{i}(n)\Vert _{L_{t}^{\infty }}\lesssim \Vert f_{i}(n)e^{inx}\Vert _{X^{0,b_{0}}} \end{aligned}$$

(B.3)

then, (B.1) will follow from (B.2) by Cauchy Schwarz.

We turn to Lemma 3.6. We start with (3.14) to (3.17). Estimates (3.14), (3.16) follows from (3.11), (3.12) via Hölder inequality. We point out that the naive loss will be $N_{1}^{C\epsilon }$ rather than $\max (N_{2},N_{3})^{C\epsilon }$, but this can be handled by a standard $L^{2}$ orthogonality argument, See, for example,[2, 6] for more details. Now we show how to derive (3.17) from (3.16). We shall see that (3.15) can be derived similarly form (3.14).

Recall that we used the notation

$$\begin{aligned} f_{i}(x,t)=\sum _{n}f_{i}(n,t)e^{inx}, \quad i=1,2.3 \end{aligned}$$

(B.4)

i.e. $f_{i}(n,t)$ is the space Fourier transform. For the sake of convenience, we denote $f_{i}(n,t)$ with $f_{i}(n)$. Similarly, we wrtie $h=\sum h(n,t)e^{inx}$.

Given (3.16), in order to derive (3.17), we need to further prove

• If $N_{1}\sim N_{2}$

  $$\begin{aligned}&\sum _{n_{1}\sim |N_{1}|, n_{3}\sim N_{3}}\left| \int \phi (t/\delta \bar{h}(n_{1})f_{1}(n_{1})\bar{f}_{2}(n_{3})f_{3}(n_{3})\right| \nonumber \\&\quad \lesssim \delta ^{1/10}(\max (N_{2},N_{3}))^{C\epsilon _{0}} \Vert h\Vert _{X^{0,1-b_{0}}}\prod _{i}\Vert P_{N_{i}}f_{i}\Vert _{X^{0,b_{0}}} \end{aligned}$$

  (B.5)

• If $N_{2}\sim N_{3}$

  $$\begin{aligned}&\sum _{n_{1}\sim N_{1}, n_{2}\sim N_{2}}\left| \int \phi (t/\delta )\bar{h}(n_{1})f_{1}(n_{1})\bar{f}_{2}(n_{2})f_{3}(n_{2})\right| \nonumber \\&\quad \lesssim \delta ^{1/10}(\max (N_{2},N_{3}))^{C\epsilon _{0}} \Vert h\Vert _{X^{0,1-b_{0}}}\prod _{i}\Vert P_{N_{i}}f_{i}\Vert _{X^{0,b_{0}}} \end{aligned}$$

  (B.6)

• If $N_{1}\sim N_{2}\sim N_{3}$,

  $$\begin{aligned}&\int |\phi (t/\delta )h{\mathcal {N}}_{2}(P_{N_{1}}f_{1}, P_{N_{1}}f_{2}, P_{N_{1}}f_{3})|\nonumber \\&\quad \lesssim \delta ^{1/10}(\max (N_{2},N_{3}))^{C\epsilon _{0}} \Vert h\Vert _{X_{0,1-b_{0}}}\prod _{i}\Vert P_{N_{i}}f_{i}\Vert _{X^{0,b_{0}}} \end{aligned}$$

  (B.7)

Estimate (B.7) follows from Lemma 3.8. The proof of estimates (B.5) and (B.6) are similar, and we only work on (B.5). Note that the integration on the left side is only in t. One has, (by Sobolev embedding in the t variable if necessary), that

$$\begin{aligned} \begin{aligned}&\Vert h(n_{1})\Vert _{L_{t}^{2}}\lesssim \Vert h(n_{1})e^{in_{1}x}\Vert _{X^{0,1-b_{0}}},\\&\Vert f(n_{1})\Vert _{L_{t}^{\infty }}\lesssim \Vert f(n_{1})e^{in_{1}x}\Vert _{X^{0,b_{0}}},\\&\Vert f_{i}(n_{3})\Vert _{L_{t}^{\infty }}\lesssim \Vert f_{i}(n_{3})e^{in_{3}x}\Vert _{X^{0,b_{0}}}. \end{aligned} \end{aligned}$$

(B.8)

Then the desired estimates follow from a Hölder inequality in t and Cauchy Schwarz inequality in $n_{1},n_{3}$. Estimates (3.18), (3.20), (3.21) are of similar flavor. We prove (3.18) and leave the rest to the interested readers. Estimate (3.18) follows from the following four estimates.

• $$\begin{aligned}&\left| \int \psi (t)\bar{h} P_{N_{1}}f_{1}P_{N_{2}}\bar{f}_{2}P_{N_{3}}f_{3}\right| \nonumber \\&\quad \lesssim (\max (N_{2},N_{3}))^{C\epsilon _{0}}|h\Vert _{X^{0,1/3}}\Vert \sup _{J} \Vert P_{J}f_{1}\Vert _{L_{t,x}^\infty }\Vert f_{2}\Vert _{X^{0,1/3}}\Vert f_{3}\Vert _{X^{0,1/3}}, \end{aligned}$$

  (B.9)

• If $N_{1}\sim N_{2}$,

  $$\begin{aligned}&\sum _{n_{1}\sim |N_{1}|, n_{3}\sim N_{3}}\left| \int \psi (t)\overline{h(n_{1})}f_{1}(n_{1})\bar{f}_{2}(n_{3}) f_{3}(n_{3})\right| \nonumber \\&\quad \lesssim \Vert P_{N_{1}}f_{1}\Vert _{X^{0,b_{0}}}\Vert P_{N_{1}}f_{2}\Vert _{X^{0,1/3}} \Vert P_{N_{3}}f_{3}\Vert _{X^{0,1/3}}\Vert P_{N_{3}}h\Vert _{X^{0,1/3}}, \end{aligned}$$

  (B.10)

• If $N_{2}\sim N_{3}$,

  $$\begin{aligned}&\sum _{n_{1}\sim N_{1}, n_{2}\sim N_{2}}\left| \int \psi (t)\overline{{h}(n_{1})}f_{1}(n_{1})\bar{f}_{2}(n_{2})f_{3}(n_{3})\right| \nonumber \\&\quad \lesssim \Vert P_{N_{1}}f\Vert _{X^{0,b_{0}}}\Vert P_{N_{2}}f_{2}\Vert _{X^{0,1/3}} \Vert P_{N_{2}}f_{3}\Vert _{X^{0,1/3}}\Vert P_{N_{1}}h\Vert _{X^{0,1/3}} \end{aligned}$$

  (B.11)

• If $N_{1}\sim N_{2}\sim N_{3}$

  $$\begin{aligned}&\left| \int \psi (t)\bar{h}{\mathcal {N}}_{2}(P_{N_{1}}f_{1}P_{N_{2}}\bar{f}_{2}P_{N_{3}}f_{3})\right| \nonumber \\&\quad \lesssim \min (\Vert P_{N_{1}}h\Vert _{X^{0,1-b_{0}}}\Vert f_{i}\Vert _{X^{0,b_{0}}}\sup _{|n|\sim N_{1}}\prod _{j\ne i}\Vert f_{j}(n)e^{inx}\Vert _{X^{0,b_{0}}} \end{aligned}$$

  (B.12)

Again estimate (B.12) follows from Lemma 3.8. We will only prove estimate (B.9), (B.10). The proof of (B.11) is similar to that for (B.10).

We start with (B.9). We may only consider the case $N_{2}\ge N_{3}$, as the case $N_{2}\le N_{3}$ can be proved similarly.

We may further only consider the case $N_{1}\gg N_{2}$, otherwise one may replace $P_{J}$ by $P_{<N_{1}}$. Observe that (using $L^{2}$ orthogonality),

$$\begin{aligned}&\int \psi (t)\bar{h} P_{N_{1}}f_{1}P_{N_{2}}\bar{f}_{2}P_{N_{3}}f_{3}\nonumber \\&\quad =\sum _{J} \int \psi (t)\bar{h} P_{J}f_{1}P_{N_{2}}\bar{f}_{2}P_{N_{3}}f_{3}\nonumber \\&\quad = \sum _{J}\int \psi (t)P_{J}\bar{h}P_{J}f_{1}P_{N_{2}}\bar{f}P_{N_{3}}f_{3}. \end{aligned}$$

(B.13)

For each J, we may estimate as follows,

$$\begin{aligned}&\left| \int \int \psi (t)P_{J}\bar{h}P_{J}f_{1}P_{N_{2}}\bar{f}P_{N_{3}}f_{3}\right| \nonumber \\&\quad \lesssim \Vert \phi (t)P_{J}h\Vert _{L_{L_{t,x}^{3}}}\Vert \phi (t)P_{J}f_{1} \Vert _{L_{t,x}^{\infty }}\Vert \phi (t)P_{N_{2}}f_{2}\Vert _{L_{t,x}^{3}} \Vert \phi (t)P_{N_{3}}f_{3}\Vert _{L_{t,x}^{3}}, \end{aligned}$$

(B.14)

where without loss of generality assumed $\psi (t)=\phi (t)^{4}$ for some well localized $\phi (t)$.

Using Estimate (3.7) to control the $L^{3}$ norm in (B.14) and applying a Cauchy Schwarz in J, the desired estimate then follows.

Lemma 3.7 can be proved similarly as Lemma 3.6.

Appendix C: A Cauchy–Schwarz type inequality

We summarize a (deterministic) Cauchy–Schwarz type inequality, that is often used in random data type problems. For simplicity, let $a_{ij}, b_{j}$ be real numbers, assume that

$$\begin{aligned} \sum _{j}b_{j}^{2}\lesssim 1, \end{aligned}$$

(C.1)

which of course implies

$$\begin{aligned} \sum _{j,j'}b_{j}^{2}b_{j'}^{2}\lesssim 1. \end{aligned}$$

(C.2)

Then, we have

$$\begin{aligned} \begin{aligned}&\sum _{i}|\sum _{j}a_{ij}b_{j}|^{2} =\sum _{i}\sum _{j,j'}a_{ij}a_{ij'}b_{j}b_{j'} =\sum _{i}\sum _{j}a_{ij}a_{ij}b_{j}^{2} +\sum _{i}\sum _{j\ne j'}a_{ij}a_{ij'}b_{j}b_{j'} \end{aligned} \end{aligned}$$

(C.3)

Note that

$$\begin{aligned} \sum _{i}|\sum _{j}a_{ij}a_{ij}b_{j}|^{2}\lesssim \sup _{j}\sum _{i}a^{2}_{ij} \end{aligned}$$

(C.4)

and, by Cauchy inequality,

$$\begin{aligned} \sum _{i}\sum _{j\ne j'}a_{ij}a_{ij'}b_{j}b_{j'}=\sum _{j\ne j'} b_{j}b_{j'}\sum _{i}a_{ij}a_{ij'}\lesssim \left( \sum _{j,j'}b^{2}_{j}b_{j'}^{2}\right) ^{1/2}\left\{ \sum _{j\ne j'}|\sum _{i}a_{ij}a_{ij}|^{2}\right\} ^{1/2} \end{aligned}$$

(C.5)

To summarize, and by simple generalization to the complex case, one has

Lemma C.1

Assume $\sum _{j}|b_{j}|^{2}\lesssim 1$, then

$$\begin{aligned} \begin{aligned} \sum _{i}|\sum _{j}a_{ij}b_{j}|^{2}\lesssim \max _{j}\sum _{i} |a_{ij}|^{2}+\left( \sum _{j\ne j'}|\sum _{i}a_{ij}\bar{a}_{ij'}|^{2}\right) ^{1/2} \end{aligned} \end{aligned}$$

(C.6)

One can also easily write down, via the dual estimate,

Lemma C.2

Assume $\sum _{j}|b_{j}|^{2}\lesssim 1$, then

$$\begin{aligned} \begin{aligned} \sum _{i}|\sum _{j}a_{ij}b_{j}|^{2}\lesssim \max _{i}\sum _{j} |a_{ij}|^{2}+\left( \sum _{i\ne i'}|\sum _{j}a_{i'j}\bar{a}_{ij}|^{2}\right) ^{1/2} \end{aligned} \end{aligned}$$

(C.7)