Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions

Abstract

We analyze nonlinear Schrödinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.

Paracontrolled distributions and function spaces

We recall the definitions of Bony paraproducts, Besov and Sobolev spaces and collect some results about products of distributions. We work on the d-dimensional torus $\mathbb {T}^d:= \mathbb {R}^d/\mathbb {Z}^d$ for $d=2,3$. For any f in the space $\mathscr {S}' ( \mathbb T^d,\mathbb R)$ of tempered distributions on $\mathbb {T}^d$, the Fourier transform of f will be denoted by $\hat{f} : \mathbb {Z}^d\rightarrow \mathbb {C}$ (or sometimes $\mathscr {F}f$) and is defined for $k\in \mathbb {Z}^d$ by

$$\begin{aligned} \hat{f}(k) := \langle f,\exp (2\pi i \langle k, \cdot \rangle ) = \int _{\mathbb {T}^d} f(x) \exp (- 2\pi i\langle k,x\rangle ) dx. \end{aligned}$$

Recall that for any $f\in L^2(\mathbb {T}^d,\mathbb {R})$ and a.e. $x\in \mathbb {T}^d$, we have

$$\begin{aligned} f(x)=\sum _{k \in \mathbb {Z}^d} \hat{f}(k) \exp ( 2\pi i\langle k,x\rangle ). \end{aligned}$$

(66)

The Sobolev space $\mathscr {H}^\alpha (\mathbb T^d)$ with index $\alpha \in \mathbb {R}$ is defined as

$$\begin{aligned} \mathscr {H}^\alpha (\mathbb {T}^d) := \{f \in \mathscr {S}' ( \mathbb {T}^d;\mathbb R): \sum _{k\in \mathbb {Z}^d} (1+|k|^2)^\alpha \, |\hat{f}(k)|^2 < +\infty \}\,. \end{aligned}$$

Before introducing the Besov spaces, we recall the definition of Littlewood-Paley blocks. We denote by $\chi$ and $\rho$ two nonnegative smooth and compactly supported radial functions $\mathbb {R}^d\rightarrow \mathbb {R}$ such that

• 1. The support of $\chi$ is contained in a ball $\{x\in \mathbb {R}^d: |x| \le R\}$ and the support of $\rho$ is contained in an annulus $\{x\in \mathbb {R}^d: a\le |x| \le b\}$;

• 2. For all $\xi \in \mathbb {R}^d$, $\chi (\xi )+\sum _{j\ge 0}\rho (2^{-j}\xi )=1$;

• 3. For $j\ge 1$, $\chi \rho (2^{-j}\cdot )\equiv 0$ and $\rho (2^{-i}\cdot ) \rho (2^{-j}\cdot )\equiv 0$ for $|i-j|\ge 1$.

The Littlewood-Paley blocks $(\Delta _j)_{j\ge -1}$ acting on $f\in \mathscr {S}'(\mathbb {T}^d)$ are defined by

$$\begin{aligned} \mathscr {F}(\Delta _{-1} f) = \chi \hat{f}\ \text{ and } \text{ for } \ j \ge 0, \quad \mathscr {F}(\Delta _j f) = \rho (2^{-j}.) \hat{f}. \end{aligned}$$

Note that, for $f\in \mathscr {S}'(\mathbb {T}^d)$, the Littlewood-Paley blocks $(\Delta _j f)_{j\ge -1}$ define smooth functions, as their Fourier transforms have compact supports. We also set, for $f\in \mathscr {S}'$ and $j\ge 0$,

$$\begin{aligned} S_j f := \sum _{i=-1}^{j-1} \Delta _i f \end{aligned}$$

and note that $S_j f$ converges in the sense of distributions to f as $j\rightarrow \infty$.

The Besov space with parameters $p,q \in [1,\infty ),\alpha \in \mathbb {R}$ can now be defined as

$$\begin{aligned} B_{p,q}^{\alpha }(\mathbb {T}^d, \mathbb {R}):=\left\{ u\in \mathscr {S}'(\mathbb {T}^d); \quad \Vert u\Vert _{ B_{p,q}^{\alpha }}= \left( \sum _{j\ge -1}2^{jq\alpha }\Vert \Delta _ju\Vert ^q_{L^p} \right) ^{1/q} <+\infty \right\} .\nonumber \\ \end{aligned}$$

(67)

We also define the Besov-Hölder spaces

$$\begin{aligned} \mathscr {C}^{\alpha }:= B_{\infty ,\infty }^{\alpha } \end{aligned}$$

which are naturally equipped with the norm $\Vert f\Vert _{\mathscr {C}^\alpha }:=\Vert f\Vert _{B_{\infty ,\infty }^{\alpha }}=\sup _{j\ge -1} 2^{j \alpha } \Vert \Delta _j f\Vert _{L^\infty }$. For $\alpha \in (0,1)$ these spaces coincide with the classical Hölder spaces.

We can formally decompose the product fg of two distributions f and g as

$$\begin{aligned} fg=f\prec g+f\circ g+f\succ g \end{aligned}$$

where

$$\begin{aligned} f\prec g :=\sum _{j\ge -1} \sum _{i=-1} ^{j-2} \Delta _i f \Delta _j g&\text {and}&f\succ g:= \sum _{j \ge -1} \sum _{i=-1}^{j-2} \Delta _i g \Delta _j f \end{aligned}$$

are usually referred to as the paraproducts whereas

$$\begin{aligned} f\circ g:=\sum _{j\ge -1}\sum _{|i-j|\le 1}\Delta _i f \Delta _j g \end{aligned}$$

(68)

is called the resonant product.

Moreover, we define the notations $f\preccurlyeq g:=f\prec g+f\circ g$ and $f\succcurlyeq g:=f\succ g+f\circ g$.

The paraproduct terms are always well defined irrespective of regularities. The resonant product is a priori only well defined if the sum of regularities is strictly greater than zero. This is reminiscent of the well known fact that one can not multiply distributions in general. The following result makes those comments precise and gives simple but extremely vital estimates for paraproducts.

Proposition A.1

(Bony estimates,[2]) Let $\alpha , \beta \in \mathbb {R}$. We have the following bounds:

• 1. If $f \in L^2$ and $g \in \mathscr {C}^{\beta }$, then

  $$\begin{aligned} \Vert f \prec g \Vert _{\mathscr {H}^{\beta - \delta }} \le C_{\delta , \beta } \Vert f \Vert _{L^2} \Vert g \Vert _{\mathscr {C}^{\beta }} \end{aligned}$$

  for all $\delta > 0$.

• 2. if $f \in \mathscr {H}^{\alpha }$ and $g \in L^{\infty }$ then

  $$\begin{aligned} \Vert f \succ g \Vert _{\mathscr {H}^{\alpha }} \le C_{\alpha , \beta } \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }}. \end{aligned}$$

• 3. If $\alpha < 0$, $f \in \mathscr {H}^{\alpha }$ and $g \in \mathscr {C}^{\beta }$, then

  $$\begin{aligned} \Vert f \prec g \Vert _{\mathscr {H}^{\alpha + \beta }} \le C_{\alpha , \beta } \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }}. \end{aligned}$$

• 4. If $g \in \mathscr {C}^{\beta }$ and $f \in \mathscr {H}^{\alpha }$ for $\beta < 0$ then

  $$\begin{aligned} \Vert f \succ g\Vert _{\mathscr {H}^{\alpha + \beta }} \le C_{\alpha , \beta } \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }} \end{aligned}$$

• 5. If $\alpha + \beta > 0$ and $f \in \mathscr {H}^{\alpha }$ and $g \in \mathscr {C}^{\beta }$, then

  $$\begin{aligned} | | f \circ g \Vert _{\mathscr {H}^{\alpha + \beta }} \le C_{\alpha , \beta } \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }}. \end{aligned}$$

where $C_{\alpha , \beta }$ is a finite positive constant.

Proposition A.2

Given $\alpha \in (0, 1)$, $\beta , \gamma \in \mathbb {R}$ such that $\beta + \gamma < 0$ and $\alpha + \beta + \gamma > 0$, there exists a trilinear operator C with the following bound

$$\begin{aligned} \Vert C (f, g, h) \Vert _{\mathscr {H}^{\alpha + \beta + \gamma }} \lesssim \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }} \Vert h \Vert _{\mathscr {C}^{\gamma }} \end{aligned}$$

for all $f \in \mathscr {H}^{\alpha }$, $g \in \mathscr {C}^{\beta }$ and $h \in \mathscr {C}^{\gamma }$.

The restriction of C to the smooth functions satisfies

$$\begin{aligned} C (f, g, h) = (f \prec g) \circ h - f (g \circ h). \end{aligned}$$

Proof

This is a restatement of the result (commutator Lemma) in [2], and the proof follows the same lines, with slight modifications. $\square$

We also prove the following modified version of the above Proposition, which suits our framework.

Proposition A.3

Let $\alpha \in (0, 1)$, $\beta , \gamma \in \mathbb {R}$ such that $\beta + \gamma < 0$ and $\alpha + \beta + \gamma > 0$. Then, there exists a trilinear operator $C_N$ with the following bound

$$\begin{aligned} \Vert C_N (f, g, h) \Vert _{\mathscr {H}^{\alpha + \beta + \gamma }} \lesssim \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }} \Vert h \Vert _{\mathscr {C}^{\gamma }} \end{aligned}$$

for all $f \in \mathscr {H}^{\alpha }$, $g \in \mathscr {C}^{\beta }$ and $h \in \mathscr {C}^{\gamma }$.

The restriction of $C_N$ to the smooth functions satisfies

$$\begin{aligned} C_N (f, g, h)&:= \left( \Delta _{> N} (f\prec g)\right) )\circ h- f(g \circ h) \end{aligned}$$

Proof

Observe that we have

$$\begin{aligned} C (f, g, h) - C_N (f, g, h)&= \left( \Delta _{\le N} (f\prec g)\right) )\circ h. \end{aligned}$$

So, we only need to show

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) )\circ h \Vert _{\mathscr {H}^{\alpha + \beta + \gamma }} \lesssim \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }} \Vert h \Vert _{\mathscr {C}^{\gamma }}. \end{aligned}$$

By product estimates, we obtain right away

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) )\circ h \Vert _{\mathscr {H}^{\alpha + \beta + \gamma }} \lesssim \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }} \Vert h \Vert _{\mathscr {C}^{\gamma }}. \end{aligned}$$

We need to show

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }} \lesssim \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }}. \end{aligned}$$

We can write

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }}^2 = \sum _{k=-1}^{\infty } 2^{2k(\alpha +\beta )} \Vert \Delta _k \left( \Delta _{\le N} (f\prec g)\right) )\Vert _{L^2}^2. \end{aligned}$$

By the support of Fourier transforms we have that $\Delta _k \left( \Delta _{\le N} (f\prec g)\right) ) = 0$ for $k> N+1$ so we obtain

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }}^2 = \sum _{k=-1}^{N+1} 2^{2k(\alpha +\beta )} \Vert \Delta _k \left( \Delta _{\le N} (f\prec g)\right) )\Vert _{L^2}^2. \end{aligned}$$

By using the convention $\Delta _{<k} f:= \sum _{i=-1}^{k-2} \Delta _k f$ we rewrite

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }}^2 = \sum _{k=-1}^{N+1} 2^{2k(\alpha +\beta )} \Vert \Delta _k \left( \Delta _{\le N} (\sum _{i=-1}^\infty \Delta _{<i} f \Delta _i g)\right) )\Vert _{L^2}^2. \end{aligned}$$

Again by support arguments this boils down to

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }}^2 = \sum _{k=-1}^{N+1} 2^{2k(\alpha +\beta )} \Vert \Delta _k \left( \Delta _{\le N} \left( \sum _{i=-1}^{N+1} \Delta _{<i} f \Delta _i g)\right) \right) \Vert _{L^2}^2. \end{aligned}$$

Applying two successive Young’s we obtain

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }}^2 \le \sum _{k=-1}^{N+1} 2^{2k(\alpha +\beta )} \Vert \phi _k\Vert _{L^1} \Vert \phi _{\le N}\Vert _{L^1} \sum _{i=-1}^{N+1} \Vert \Delta _{<i} f \Delta _i g\Vert _{L^2}^2 \end{aligned}$$

where on the right hand side we can write

$$\begin{aligned}&\Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }}^2 \le \sum _{k=-1}^{N+1} 2^{2k(\alpha +\beta )} \Vert \phi _k\Vert _{L^1}^2 \Vert \phi _{\le N}\Vert _{L^1}^2 \sum _{i=-1}^{N+1} \Vert \Delta _{<i} f \Delta _i g\Vert _{L^2}^2\\&\quad \le \sum _{k=-1}^{N+1} 2^{(2k-2i)\beta } 2^{(2k-2i)\alpha }\Vert \phi _k\Vert _{L^1}^2 \Vert \phi _{\le N}\Vert _{L^1}^2 \left( \sup _{1\le i \le N+1}2^{2 i \beta }\Vert \Delta _i g\Vert _{L^\infty }^2 \right) \\&\quad \quad \sum _{i=-1}^{N+1} 2^{2i\alpha } \Vert \Delta _{<i} f\Vert _{L^2}^2. \end{aligned}$$

At this point, it is clear that for a constant depending on N we readily have

$$\begin{aligned} \Vert \left( \Delta _{\le N} (f\prec g)\right) ) \Vert _{\mathscr {H}^{\alpha + \beta }}^2 \lesssim \Vert f \Vert _{\mathscr {H}^{\alpha }}^2 \Vert g \Vert _{\mathscr {C}^{\beta }}^2 \end{aligned}$$

and the result follows.

Lemma A.4

(Bernstein’s inequality, [14]) Let $\mathscr {A}$ be an annulus and $\mathscr {B}$ be a ball. For any $k \in \mathbbm {N}, \lambda > 0,$and $1 \le p \le q \le \infty$ we have

• 1. if $u \in L^p (\mathbbm {R}^d)$ is such that ${\text {supp}} (\mathscr {F}u) \subset \lambda \mathscr {B}$ then

  $$\begin{aligned} \underset{\mu \in \mathbbm {N}^d : | \mu | = k}{\max } \Vert \partial ^{\mu } u \Vert _{L^q} \lesssim _k \lambda ^{k + d \left( \frac{1}{p} - \frac{1}{q} \right) } \Vert u \Vert _{L^p} \end{aligned}$$

• 2. if $u \in L^p (\mathbbm {R}^d)$is such that ${\text {supp}} (\mathscr {F}u) \subset \lambda \mathscr {A}$ then

  $$\begin{aligned} \lambda ^k \Vert u \Vert _{L^p} \lesssim _k \underset{\mu \in \mathbbm {N}^d : | \mu | = k}{\max } \Vert \partial ^{\mu } u \Vert _{L^p}. \end{aligned}$$

Proposition A.5

(Paralinearisation, [15]) Let $\alpha \in (0, 1)$ and $F \in C^2 .$ Then there exists a locally bounded map $R_F : \mathscr {C}^{\alpha } \rightarrow \mathscr {C}^{2 \alpha }$ such that

$$\begin{aligned} F (f) = F' (f) \prec f + R_F (f) \ {\text {for}} {\text {all}} \ f \in \mathscr {C}^{\alpha }. \end{aligned}$$

Lemma A.6

Let $\alpha , \beta , \gamma \in \mathbbm {R}$ with $\alpha + \beta <0$, $\alpha + \beta + \gamma \ge 0$, and $f \in \mathscr {H}^{\alpha }, g \in \mathscr {C}^{\beta }, h \in \mathscr {H}^{\gamma },$ then there exists a map D(f, g, h) with the following bound

$$\begin{aligned} | D (f, g, h) | \lesssim \Vert g \Vert _{\mathscr {C}^{\beta }} \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert h \Vert _{\mathscr {H}^{\gamma }}. \end{aligned}$$

(69)

Moreover the restriction of D(f, g, h) to the smooth functions f, g, h is as follows:

$$\begin{aligned} D (f, g, h) = \langle f, h \circ g \rangle - \langle f \prec g, h \rangle . \end{aligned}$$

Proof

We define

$$\begin{aligned} D (f, g, h) := \left( \sum _{i \ge k - 1, | j - k | \le L} - \sum _{i \sim k, 1 < | j - k | \le L} \right) \langle \Delta _i f, \Delta _j h \Delta _k g \rangle . \end{aligned}$$

So we get, for some $\delta >0$,

$$\begin{aligned} | D (f, g, h) |&\lesssim \sum _{i \gtrsim k, j \sim k} | \langle \Delta _i f, \Delta _j h \Delta _k g \rangle |\\&\le \sum _{i \gtrsim k, j \sim k} \Vert \Delta _i f \Vert _{L^2} \Vert \Delta _j h \Vert _{L^2} \Vert \Delta _k g \Vert _{L^{\infty }}\\&\le \Vert g \Vert _{\mathscr {C}^{- 1 - \delta }} \sum _{i \gtrsim k, j \sim k} 2^{k (1 + \delta )} \Vert \Delta _i f \Vert _{L^2} \Vert \Delta _j h \Vert _{L^2} \\&\le \Vert g \Vert _{\mathscr {C}^{- 1 - \delta }} \Vert f \Vert _{\mathscr {H}^{(1 + \delta ) / 2}} \Vert h \Vert _{\mathscr {H}^{(1 + \delta ) / 2}} \end{aligned}$$

and this argument can be adapted to show (69) by simply observing $1 \le 2^{k(\beta + \alpha + \gamma )} = 2^{k\beta }2^{k ( \alpha + \gamma )}$, since $\beta + \alpha + \gamma \ge 0$ . Moreover, for smooth functions f, g, h; we can compute

$$\begin{aligned} \langle f, h \circ g \rangle - \langle f \prec g, h \rangle&= \sum _{i, | j - k | \le 1} \langle \Delta _i f, \Delta _j h \Delta _k g \rangle - \sum _{i< k - 1, j} \langle \Delta _i f \Delta _k g, \Delta _j h \rangle \\&= \left( \sum _{i, | j - k | \le 1} - \sum _{i< k - 1, | j - k | \le L} \right) \langle \Delta _i f \Delta _k g, \Delta _j h \rangle \\&= \left( \sum _{i, | j - k | \le L} {-} \sum _{i< k - 1, | j - k | \le L} {-} \sum _{i, 1< | j - k | \le L} \right) \langle \Delta _i f, \Delta _j h \Delta _k g \rangle \\&= \left( \sum _{i \ge k - 1, | j - k | \le L} - \sum _{i, 1< | j - k | \le L} \right) \langle \Delta _i f, \Delta _j h \Delta _k g \rangle \\&= \left( \sum _{i \ge k - 1, | j - k | \le L} - \sum _{i \sim k, 1 < | j - k | \le L} \right) \\&\qquad \langle \Delta _i f, \Delta _j h \Delta _k g \rangle = D (f, g, h). \end{aligned}$$

Hence the result. $\square$

Remark A.7

Proposition A.6 says that the paraproduct is almost the adjoint of the resonant product, meaning up to a more regular remainder term as is often the case in paradifferential calculus.

Lemma A.8

Let $f \in \mathscr {H}^{\alpha }, g \in \mathscr {C}^{\beta },$ with $\alpha \in (0, 1), \beta \in \mathbbm {R},$ there exists a bilinear map R(f, g) that satisfies the following bound

$$\begin{aligned} \Vert R (f, g) \Vert _{\mathscr {H}^{\alpha + \beta + 2}} \lesssim \Vert f \Vert _{\mathscr {H}^{\alpha }} \Vert g \Vert _{\mathscr {C}^{\beta }}, \end{aligned}$$

and restricts to smooth functions as

$$\begin{aligned} R (f, g) = (1 - \Delta )^{- 1} (f \prec g) - f \prec (1 - \Delta )^{- 1} g. \end{aligned}$$

Proof

The proof is basically a straightforward modification of the proof of [2, Proposition A.2], which has almost the same statement. $\square$