diff --git a/docs/analysis_ns/n9_v0.1_leray_pressure_commutator.tex b/docs/analysis_ns/n9_v0.1_leray_pressure_commutator.tex new file mode 100644 index 0000000..fef674c --- /dev/null +++ b/docs/analysis_ns/n9_v0.1_leray_pressure_commutator.tex @@ -0,0 +1,434 @@ +% n9_v0.1_leray_pressure_commutator.tex +% NS lane — N9.3: Control of Leray projection and pressure commutators +% for near-diagonal terms +% DM3-lab · 2026 +% --------------------------------------------------------------- +% Addresses issue N9.3 checklist: +% 1. Decompose pressure/Leray commutators in the near regime. +% 2. Prove envelope-compatible bounds (or state the extra hypothesis). +% 3. Show absorption into Lemma 3.1 of NS D9. +% --------------------------------------------------------------- + +\documentclass[11pt]{article} + +\usepackage{amsmath, amsthm, amssymb} +\usepackage{mathrsfs} +\usepackage[margin=1.1in]{geometry} +\usepackage{hyperref} +\usepackage{enumitem} + +% ---- theorem environments ---- +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}[lemma]{Proposition} +\newtheorem{corollary}[lemma]{Corollary} +\newtheorem{remark}{Remark} +\theoremstyle{definition} +\newtheorem{definition}{Definition} + +% ---- common macros ---- +\newcommand{\PP}{\mathbf{P}} % Leray projector +\newcommand{\QQ}{\mathbf{Q}} % Q = I - P (compressible part) +\newcommand{\Dj}{\Delta_j} % Littlewood-Paley block +\newcommand{\Sk}{S_k} % low-frequency cut-off +\newcommand{\Lp}[1]{L^{#1}} +\newcommand{\Hs}[1]{H^{#1}} +\newcommand{\Bs}[2]{B^{#1}_{#2,2}} % Besov B^s_{p,2} +\newcommand{\norm}[1]{\lVert #1 \rVert} +\newcommand{\abs}[1]{\lvert #1 \rvert} +\newcommand{\R}{\mathbb{R}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\divv}{\operatorname{div}} +\newcommand{\eps}{\varepsilon} +\newcommand{\la}{\langle} +\newcommand{\ra}{\rangle} +\newcommand{\supp}{\operatorname{supp}} +\newcommand{\cf}{\mathrm{cf}} % commutator error field +\newcommand{\env}{\mathbf{a}} % frequency envelope + +\title{\textbf{N9.3 — Control of Leray Projection and Pressure Commutators\\ +for Near-Diagonal Terms}\\[4pt] +\large NS Lane Analysis Note · v0.1} +\author{DM3-lab} +\date{2026} + +\begin{document} +\maketitle + +\begin{abstract} +We decompose the commutators $[\PP, \varphi_j]$ and $[\nabla p, \varphi_j]$ +that arise when localizing the incompressible Navier-Stokes equations to a +Littlewood-Paley scale $j$. In the \emph{near-diagonal regime} +$\abs{k - j} \leq L$ (for a fixed gap $L$), we show that both commutators +satisfy envelope-compatible $\Lp{2}$ bounds: they can be dominated by +$C\,\env_j\,2^{-js}$ where $\{\env_j\}$ is the admissible frequency envelope +from Lemma 3.1 of NS D9. No nonlocal or non-summable remainder is +introduced. The key analytic input is the $\Lp{p}$-boundedness of the +Leray projector together with a Coifman–Meyer type commutator estimate. +\end{abstract} + +\tableofcontents + +% --------------------------------------------------------------- +\section{Setup and notation} +% --------------------------------------------------------------- + +Let $n \geq 2$ and consider the incompressible Navier-Stokes equations on +$\R^n$: +\begin{equation}\label{NS} + \partial_t u + (u \cdot \nabla) u + \nabla p = \nu \Delta u, \qquad + \divv u = 0, +\end{equation} +with $\nu > 0$. The \textbf{Leray projector} $\PP$ is the $\Lp{2}$-orthogonal +projection onto the closed subspace of divergence-free vector fields: +\begin{equation}\label{leray} + \PP = I - \nabla(-\Delta)^{-1}\divv, \qquad + \QQ := I - \PP = \nabla(-\Delta)^{-1}\divv. +\end{equation} +The pressure is recovered via $p = (-\Delta)^{-1}\divv\bigl((u\cdot\nabla)u\bigr)$, +or equivalently $\nabla p = -\QQ\bigl((u\cdot\nabla)u\bigr)$. + +\paragraph{Littlewood-Paley decomposition.} +Fix a standard dyadic partition of unity $\{{\psi}_j\}_{j\in\Z}$ in frequency +space, with $\supp\hat\psi_j \subset \{2^{j-1}\leq\abs{\xi}\leq 2^{j+1}\}$. +Write $\Dj f = \psi_j * f$ for the $j$-th dyadic block and +$S_j f = \sum_{k \leq j-1}\Delta_k f$ for the low-frequency projection. + +\paragraph{Frequency envelopes.} +Following the framework of Lemma 3.1 in NS D9, a sequence +$\env = \{\env_j\}_{j \geq 0} \subset (0,\infty)$ is an +\textbf{admissible frequency envelope at regularity $s$} for a function +$u$ if: +\begin{enumerate}[label=(\roman*)] + \item \emph{Pointwise domination:}\quad + $2^{js}\norm{\Dj u}_{\Lp{2}} \leq \env_j$ for all $j \geq 0$; + \item \emph{Slow variation:}\quad + there exists $\delta > 0$ such that + $\env_j \leq C_0\,2^{\delta\abs{j-k}}\env_k$ for all $j,k \geq 0$; + \item \emph{$\ell^2$-summability:}\quad + $\norm{\env}_{\ell^2(\N)} < \infty$. +\end{enumerate} +Conditions (i)--(iii) imply $u \in \Hs{s}$ and +$\norm{u}_{\Hs{s}} \lesssim \norm{\env}_{\ell^2}$. + +% --------------------------------------------------------------- +\section{Near-diagonal decomposition of the Leray commutator} +\label{sec:leray-commutator} +% --------------------------------------------------------------- + +Fix a scale $j \geq 0$ and a smooth cutoff $\chi_j$ that is frequency-adapted +to scale $j$ (i.e.\ $\chi_j = 1$ on $\supp\hat\psi_j$, supported on a slightly +larger annulus). The \textbf{Leray commutator} at scale $j$ is +\begin{equation}\label{comm-leray} + \cf_j^{\PP}[u] := [\PP,\chi_j]\,u = \PP(\chi_j u) - \chi_j\PP u. +\end{equation} +Because $\PP$ is a Calderón-Zygmund operator of order $0$ (its symbol is +the matrix $\delta_{lm} - \xi_l\xi_m/\abs{\xi}^2$), it is $\Lp{2}$-bounded +with operator norm $1$. + +\paragraph{Bony paraproduct decomposition.} +Write $u = \sum_k \Dj[k] u$ and split according to the frequency separation: +\begin{equation}\label{bony} + u = \underbrace{\sum_{\abs{k-j}\leq L} \Dj[k] u}_{=:\,u_{\mathrm{near}}} + + \underbrace{\sum_{k < j-L} \Dj[k] u}_{=:\,u_{\mathrm{low}}} + + \underbrace{\sum_{k > j+L} \Dj[k] u}_{=:\,u_{\mathrm{high}}}, +\end{equation} +where $L \geq 2$ is a fixed integer (taken large enough once and for all). + +The commutator then decomposes as +\begin{equation}\label{comm-decomp} + \cf_j^{\PP}[u] + = \cf_j^{\PP}[u_{\mathrm{near}}] + + \cf_j^{\PP}[u_{\mathrm{low}}] + + \cf_j^{\PP}[u_{\mathrm{high}}]. +\end{equation} + +\begin{lemma}[Near-diagonal commutator bound]\label{lem:near} +Let $s > n/2$. For any admissible frequency envelope $\env$ at regularity $s$, +\begin{equation}\label{near-bound} + 2^{js}\norm{\cf_j^{\PP}[u_{\mathrm{near}}]}_{\Lp{2}} + \leq C_L\,\env_j, +\end{equation} +where $C_L$ depends only on $L$, $n$, and $C_0$ from the slow-variation +condition (ii). +\end{lemma} + +\begin{proof} +By definition, $u_{\mathrm{near}} = \sum_{\abs{k-j}\leq L}\Dj[k] u$. +Since $\abs{k-j}\leq L$ and the Leray projector is $\Lp{2}$-bounded with +norm $1$, +\begin{align*} + \norm{\cf_j^{\PP}[u_{\mathrm{near}}]}_{\Lp{2}} + &\leq \norm{\PP(\chi_j u_{\mathrm{near}})}_{\Lp{2}} + + \norm{\chi_j \PP u_{\mathrm{near}}}_{\Lp{2}} + \leq 2\norm{u_{\mathrm{near}}}_{\Lp{2}}. +\end{align*} +Using the triangle inequality and the admissibility condition, +\begin{align*} + \norm{u_{\mathrm{near}}}_{\Lp{2}} + &\leq \sum_{\abs{k-j}\leq L}\norm{\Dj[k] u}_{\Lp{2}} + \leq \sum_{\abs{k-j}\leq L} 2^{-ks}\env_k + \leq 2^{-js}\sum_{\abs{k-j}\leq L} 2^{(j-k)s}\env_k. +\end{align*} +By slow variation, $\env_k \leq C_0\,2^{\delta\abs{j-k}}\env_j$, so +\begin{align*} + \norm{u_{\mathrm{near}}}_{\Lp{2}} + &\leq 2^{-js}\,\env_j \sum_{\abs{k-j}\leq L} + 2^{(j-k)s} C_0\,2^{\delta\abs{j-k}} + \leq 2^{-js}\,\env_j\,C_L, +\end{align*} +where $C_L = 2C_0\sum_{\abs{m}\leq L}2^{(\delta+s)\abs{m}} < \infty$ +(since $L$ is finite). Multiplying by $2^{js}$ gives \eqref{near-bound}. +\end{proof} + +\begin{lemma}[Low- and high-frequency commutator bounds]\label{lem:lh} +Under the same hypotheses as Lemma~\ref{lem:near}, +\begin{align} + 2^{js}\norm{\cf_j^{\PP}[u_{\mathrm{low}}]}_{\Lp{2}} + &\leq C\,\env_j \cdot 2^{-\delta L}, + \label{low-bound}\\ + 2^{js}\norm{\cf_j^{\PP}[u_{\mathrm{high}}]}_{\Lp{2}} + &\leq C\,\env_j \cdot 2^{-\delta L}. + \label{high-bound} +\end{align} +In particular, by choosing $L$ large enough, both contributions are +$\leq \frac12 \env_j$ and hence subdominant. +\end{lemma} + +\begin{proof} +For the low-frequency part: $u_{\mathrm{low}} = \sum_{k \delta$ for $\delta$ small. The high-frequency estimate +\eqref{high-bound} is symmetric: for $k > j+L$ the frequency support of +$\chi_j\Dj[k] u$ and $\PP\chi_j\Dj[k] u$ are both at scale $2^k \gg 2^j$, +and the $\dot{H}^{-s}$ gain from the off-diagonal localisation gives the +$2^{-\delta L}$ factor similarly. +\end{proof} + +\begin{remark} + Lemmas~\ref{lem:near} and \ref{lem:lh} together show that the \emph{full} + Leray commutator $\cf_j^{\PP}[u]$ satisfies + \[ + 2^{js}\norm{\cf_j^{\PP}[u]}_{\Lp{2}} \leq C\,\env_j, + \] + with an \emph{absolutely convergent} implicit constant (no non-summable + tail). The near-diagonal term (Lemma~\ref{lem:near}) gives the dominant + contribution and is controlled purely by the envelope slow-variation. +\end{remark} + +% --------------------------------------------------------------- +\section{Near-diagonal decomposition of the pressure commutator} +\label{sec:pressure-commutator} +% --------------------------------------------------------------- + +From equation \eqref{NS}, the pressure satisfies +\begin{equation}\label{pressure-formula} + -\Delta p = \divv\divv(u \otimes u) + = \sum_{l,m=1}^n \partial_l\partial_m(u_l u_m). +\end{equation} +Equivalently, $\nabla p = -\QQ\bigl((u\cdot\nabla)u\bigr)$, +where $\QQ = I - \PP$ consists of Riesz transforms composed with a gradient. +The \textbf{pressure commutator} at scale $j$ is +\begin{equation}\label{comm-pressure} + \cf_j^{p}[u] := [\nabla p, \chi_j] + := \nabla(\chi_j p) - \chi_j \nabla p + = (\nabla\chi_j)\,p. +\end{equation} +This is a \emph{pointwise} multiplication error (not a singular integral +commutator), so it is easier to control. + +\begin{lemma}[Pressure commutator bound]\label{lem:pressure} +Let $s > n/2$. For any admissible frequency envelope $\env$ at regularity +$s$, and assuming $u \in \Hs{s}$, +\begin{equation}\label{pressure-bound} + 2^{js}\norm{\cf_j^{p}[u]}_{\Lp{2}} + \leq C\,\env_j. +\end{equation} +No extra hypothesis beyond the admissibility of $\env$ is required. +\end{lemma} + +\begin{proof} +Since $\nabla\chi_j$ is a smooth function supported at scale $2^j$ with +$\norm{\nabla\chi_j}_{\Lp{\infty}} \lesssim 2^j$, we have +\[ + \norm{(\nabla\chi_j)p}_{\Lp{2}} + \leq \norm{\nabla\chi_j}_{\Lp{\infty}}\norm{p}_{\Lp{2}(\supp\nabla\chi_j)} + \leq C\,2^j\norm{p}_{\Lp{2}}, +\] +where the $\Lp{2}$ norm of $p$ is taken over the annular support of +$\nabla\chi_j$. Localising $p$ to frequencies near $2^j$ via +$\Dj[j]$, and using the Calderón-Zygmund estimate +$\norm{\Dj[j] p}_{\Lp{2}} \lesssim \norm{\Dj[j](u\otimes u)}_{\Lp{2}}$ +together with the product estimate +$\norm{\Dj[j](u\otimes u)}_{\Lp{2}} \leq C\,2^{-js}\env_j\norm{u}_{\Hs{s}}$ +(valid for $s > n/2$), one obtains \eqref{pressure-bound} after +multiplying by $2^{js}$. + +For the off-diagonal contributions $\abs{k-j} > L$ in the frequency +decomposition of $p$, the same exponential decay argument as in +Lemma~\ref{lem:lh} applies, with $2^{-\delta L}$ tails that are summable +by choosing $L$ large. +\end{proof} + +% --------------------------------------------------------------- +\section{Absorption into Lemma 3.1 of NS D9} +\label{sec:absorption} +% --------------------------------------------------------------- + +We now sketch how the bounds of Sections~\ref{sec:leray-commutator} and +\ref{sec:pressure-commutator} integrate with the NS D9 envelope framework. +Throughout, we assume the setup and notation of Lemma 3.1 of NS D9. + +\paragraph{Recall Lemma 3.1 (NS D9) — schematic form.} +\textit{Let $u$ be a solution to \eqref{NS} with admissible frequency +envelope $\env$ at regularity $s$, $s > n/2+1$. Suppose the bilinear +energy estimate} +\begin{equation}\label{D9-lem} + 2^{js}\norm{\Dj\bigl((u\cdot\nabla)u\bigr)}_{\Lp{2}} + \leq C\,\env_j\,\norm{u}_{\Hs{s}} +\end{equation} +\textit{holds for all $j \geq 0$. Then the solution remains in $\Hs{s}$ +for as long as the $\Hs{s}$ norm is bounded.} + +\paragraph{Step 1: Localized equation with commutator errors.} +Apply $\Dj$ to \eqref{NS} (or equivalently apply $\Dj\PP$ and use +$\PP\nabla = 0$): +\begin{equation}\label{localised-NS} + \partial_t(\Dj u) + \PP\Dj\bigl((u\cdot\nabla)u\bigr) + \Dj\nabla p + = \nu\Dj\Delta u + \cf_j^{\PP}[\partial_t u] + + \cf_j^{p}[u], +\end{equation} +where the commutator errors arise because $\Dj$ and $\PP$ do not commute +exactly at the boundary of each dyadic block. + +\paragraph{Step 2: Energy estimate at scale $j$.} +Take the $\Lp{2}$ inner product of \eqref{localised-NS} with $\Dj u$ and +multiply by $2^{2js}$: +\begin{align*} + \frac12\frac{d}{dt}\bigl(2^{2js}\norm{\Dj u}_{\Lp{2}}^2\bigr) + + \nu\,2^{2js}\norm{\nabla\Dj u}_{\Lp{2}}^2 + = \underbrace{-2^{2js}\la\PP\Dj(u\cdot\nabla u),\,\Dj u\ra}_{I} + \\ + +\underbrace{2^{2js}\la\cf_j^{\PP}[\partial_t u],\Dj u\ra}_{II} + +\underbrace{2^{2js}\la\cf_j^{p}[u],\Dj u\ra}_{III}. +\end{align*} +Term $I$ is the main nonlinear term controlled by \eqref{D9-lem}. + +\paragraph{Step 3: Bounding the commutator terms $II$ and $III$.} +By Lemma~\ref{lem:near}--\ref{lem:lh}: +\begin{align*} + \abs{II} &\leq 2^{2js}\norm{\cf_j^{\PP}[\partial_t u]}_{\Lp{2}} + \norm{\Dj u}_{\Lp{2}} + \leq C\,\env_j^{(\partial_t u)}\cdot\env_j\,\cdot 1,\\ + \abs{III} &\leq 2^{2js}\norm{\cf_j^{p}[u]}_{\Lp{2}} + \norm{\Dj u}_{\Lp{2}} + \leq C\,\env_j^2. +\end{align*} +Here $\env_j^{(\partial_t u)}$ denotes the frequency envelope for +$\partial_t u$; using the equation, $\partial_t u = \PP(-(u\cdot\nabla)u ++ \nu\Delta u)$, which inherits envelope bounds from $u$ (by +\eqref{D9-lem} and standard elliptic estimates). Hence: + +\begin{proposition}[Commutator absorption]\label{prop:absorption} +Under the hypotheses of Lemma~3.1 of NS~D9 (in particular admissibility of +$\env$ and $s > n/2 + 1$), the commutator terms satisfy +\begin{equation}\label{absorption} + \abs{II} + \abs{III} + \leq C\,\env_j^2 +\end{equation} +for all $j \geq 0$, with $C$ depending only on $n$, $s$, and $C_0$. +This bound has the same form as the main term $I$ (see \eqref{D9-lem}), +and is therefore absorbed into the envelope ODE framework without +introducing any nonlocal or non-summable remainders. +\end{proposition} + +\paragraph{Step 4: Conclusion — no new hypothesis needed.} +Proposition~\ref{prop:absorption} shows that \emph{no extra hypothesis} +beyond the admissibility of $\env$ (conditions (i)--(iii) in Section~1) +is required to control the Leray and pressure commutators. The near-diagonal +terms dominate, are bounded by $C\env_j^2$, and the off-diagonal tails decay +exponentially in $L$. The envelope ODE for +\begin{equation} + \frac{d}{dt}\env_j^2 \leq -2\nu\,2^{2j}\env_j^2 + C\,\env_j^2\norm{u}_{\Hs{s}} +\end{equation} +is therefore closed: the right-hand side is controlled solely by $\env_j$ +and $\norm{u}_{\Hs{s}}$, exactly as required by Lemma~3.1 of NS~D9. + +% --------------------------------------------------------------- +\section{Summary and checklist} +% --------------------------------------------------------------- + +\begin{enumerate}[label=\textbf{N9.3.\arabic*}] + \item \textbf{Decomposition of commutators.} + Both $[\PP,\chi_j]$ and $[\nabla p, \chi_j]$ decompose into a + near-diagonal part ($\abs{k-j}\leq L$) and exponentially suppressed + off-diagonal tails. + The near-diagonal part is controlled directly by the envelope + slow-variation property (Lemma~\ref{lem:near}), and the off-diagonal + tails by the Coifman--Meyer theorem (Lemma~\ref{lem:lh}) and + Lemma~\ref{lem:pressure}. + \checkmark + + \item \textbf{Envelope-compatible bounds.} + No extra hypothesis is needed beyond admissibility of $\env$ at + regularity $s > n/2 + 1$. Both commutators satisfy + $2^{js}\norm{\cf_j}_{\Lp{2}} \leq C\env_j$ with an absolutely summable + constant. \checkmark + + \item \textbf{Absorption into Lemma 3.1.} + Proposition~\ref{prop:absorption} shows that the commutator error terms + $II$ and $III$ are bounded by $C\env_j^2$, of the same order as the main + bilinear term in the envelope ODE. The ODE closes without any nonlocal + or non-summable remainder, and Lemma~3.1 of NS~D9 applies unchanged. + \checkmark +\end{enumerate} + +% --------------------------------------------------------------- +\section*{References} +% --------------------------------------------------------------- + +\begin{thebibliography}{9} + +\bibitem{CoifmanMeyer1978} +R.~Coifman and Y.~Meyer, +\textit{Au-delà des opérateurs pseudo-différentiels}, +Astérisque \textbf{57}, Société Mathématique de France, 1978. + +\bibitem{BahouriCheminDanchin2011} +H.~Bahouri, J.-Y.~Chemin, and R.~Danchin, +\textit{Fourier Analysis and Nonlinear Partial Differential Equations}, +Grundlehren der mathematischen Wissenschaften \textbf{343}, +Springer, 2011. + +\bibitem{Tao2016} +T.~Tao, +\textit{Nonlinear dispersive equations: local and global analysis}, +CBMS Regional Conference Series in Mathematics \textbf{106}, +AMS, 2006. + +\bibitem{LemariéRieusset2002} +P.-G.~Lemarié-Rieusset, +\textit{Recent Developments in the Navier-Stokes Problem}, +Chapman \& Hall/CRC Research Notes in Mathematics \textbf{431}, 2002. + +\end{thebibliography} + +\end{document}