<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%~MoulinÃ© par MaN_auto v.0.33.8 (9c3f6e85) 2025-04-24 16:41:37
\documentclass[CRMATH,Unicode,XML,biblatex]{cedram}

\TopicFR{\'Equations aux dÃ©rivÃ©es partielles}
\TopicEN{Partial differential equations}

\addbibresource{CRMATH_Li_20240441.bib}

\usepackage{mathtools}

\newcommand{\pa}{\partial}
\newcommand{\fr}{\frac}
\newcommand{\na}{\nabla}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}

\newcommand{\R}{\mathbb{R}}

\newcommand{\dd}{\mathrm{d}}

\DeclareMathOperator{\D}{div}
\DeclareMathOperator{\supp}{supp}

\newcommand{\thickapprox}{\approx}

%%%---------------------------------------------------------------------------

%%% \lesssim
\newcommand{\lesssim}{\mathrel{\raisebox{-2.5pt}{$\stackrel{\raisebox{-1pt}{\scalebox{1}[.8]{&lt;}}}{\sim}$}}}
% Supprimer si possible le package amssymb.

%%% \grtsim
\newcommand{\gtrsim}{\mathrel{\raisebox{-2.5pt}{$\stackrel{\raisebox{-1pt}{\scalebox{1}[.8]{&gt;}}}{\sim}$}}}

%%%---------------------------------------------------------------------------

%%% DELIMITERS

\DeclarePairedDelimiter{\parens}{\lparen}{\rparen}
\DeclarePairedDelimiter{\braces}{\{}{\}}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\DeclarePairedDelimiter{\bracks}{[}{]}
\DeclarePairedDelimiter{\angles}{\langle}{\rangle}
\DeclarePairedDelimiter{\aangles}{\langle\!\langle}{\rangle\!\rangle}
\DeclarePairedDelimiter{\pparens}{(\!(}{)\!)}

%%%---------------------------------------------------------------------------

%%% ENVIRONNEMENTS

%%% Env. {step}
\newcounter{step}
\newenvironment{step}[1][]{\refstepcounter{step}\begin{proof}[{Step~\thestep\if#1\empty\else: #1\fi}]}{\let\qed\relax\end{proof}}

%%%---------------------------------------------------------------------------

%%% Changer la largeur des adresses
% Jouer avec \hsize :
\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
%  \hspace*{0pt}%
\begingroup\hsize=13cm\advance\hsize\abstractmarginL\raggedright
\noindent
%\hspace*{\abstractmarginL}\begin{minipage}[t]{10cm}
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

%%%---------------------------------------------------------------------------

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\let\oldtilde\tilde
\renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}}
\let\oldhat\hat
\renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}}
\let\oldforall\forall
\renewcommand*{\forall}{\mathrel{\oldforall}}

\title{Anomalous dissipation for the d-dimensional Navier--Stokes equations}
\alttitle{Dissipation anomale pour les Ã©quations de Navier--Stokes Ã&nbsp; d dimensions}

\author{\firstname{Jinlu} \lastname{Li}}
\address{School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China}
\email[J.~Li]{lijinlu@gnnu.edu.cn}

\author{\firstname{Yanghai} \lastname{Yu}\IsCorresp}
\address{School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China}
\email[Y.~Yu]{yuyanghai214@sina.com}

\author{\firstname{Weipeng } \lastname{Zhu}}
\address{School of Mathematics, Foshan University, Foshan, Guangdong 528000, China}
\email[W.~Zhu]{mathzwp2010@163.com}

\keywords{\kwd{Navier--Stokes equations} \kwd{inviscid limit}}
\altkeywords{\kwd{\'Equations de Navier--Stokes} \kwd{limite inviscide}}

\subjclass{35Q35, 76D09}

\thanks{J.~Li is supported by the National Natural Science Foundation of China~(12161004), Training Program for Academic and Technical Leaders of Major Disciplines in Ganpo Juncai Support Program~(20232BCJ23009). Y.~Yu is supported by the National Natural Science Foundation of China~(12101011). W.~Zhu is supported by the National Natural Science Foundation of China~(12201118) and Guangdong Basic and Applied Basic Research Foundation~(2021A1515111018)}
\CDRGrant[National Natural Science Foundation of China]{12161004}
\CDRGrant[Training Program for Academic and Technical Leaders of Major Disciplines in Ganpo Juncai Support Program]{20232BCJ23009}
\CDRGrant[National Natural Science Foundation of China]{12101011}
\CDRGrant[National Natural Science Foundation of China]{12201118}
\CDRGrant[Guangdong Basic and Applied Basic Research Foundation]{2021A1515111018}

\begin{abstract}
The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier--Stokes equations in the whole space:
\[
\begin{cases}
\partial_tu^\varepsilon+u^\varepsilon\cdot\nabla u^\varepsilon-\varepsilon\Delta u^\varepsilon+\nabla p^\varepsilon=0,\\
\operatorname{div}u^\varepsilon=0.
\end{cases}
\]
We aim to present a simple rigorous example of initial data which generates the corresponding solutions of the Navier--Stokes equations for which the dissipation rate of the kinetic energy is bounded away from zero.
\end{abstract}

\begin{altabstract}
L'objectif de cet article est d'Ã©tudier la limite de disparition de la viscositÃ© pour les Ã©quations de Navier--Stokes Ã&nbsp; d dimensions dans l'espace entier :
\[
\begin{cases}
\partial_tu^\varepsilon+u^\varepsilon\cdot\nabla u^\varepsilon-\varepsilon\Delta u^\varepsilon+\nabla p^\varepsilon=0,\\
\operatorname{div}u^\varepsilon=0.
\end{cases}
\]
Nous visons Ã&nbsp; prÃ©senter des exemples simples et rigoureux de donnÃ©es initiales qui gÃ©nÃ¨rent les solutions correspondantes des Ã©quations de Navier--Stokes pour lesquelles le taux de dissipation de l'Ã©nergie cinÃ©tique est bornÃ© loin de zÃ©ro.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

One of the most important laws in the study of developed turbulence is Kolmogorov's 4/5-law, and in the derivation of this law, a significant assumption is the zeroth law (see Frisch~\cite{u} for example). The zeroth law of turbulence states that, in the limit of zero viscosity, the rate of kinetic energy dissipation for solutions to the incompressible Navier--Stokes equations is non-vanishing. This is one of the central ansatz of Kolmogorov's 1941 theory~\cite{K2}. In this paper we consider the vanishing viscosity limit for the d-dimensional incompressible Navier--Stokes equations
\begin{equation}\label{1}
\tag{NS}
\begin{cases}
\pa_tu^\ep+u^\ep\cdot\na u^\ep-\ep\Delta u^\ep+\nabla p^\ep=f^\ep,&amp;\quad (t,x)\in \R^+\times\R^d, \\
\D u^\ep=0,&amp;\quad (t,x)\in \R^+\times\R^d,\\
u^\ep(t=0)=u^{\ep}_0,&amp;\quad x\in \R^d,
\end{cases}
\end{equation}
where $\ep&gt;0$ is the kinematic viscosity, $u^\ep \colon [0,\infty)\times {\mathbb R}^d\to {\mathbb R}^d$ is the velocity of the fluid, $p^\ep \colon [0,\infty)\times {\mathbb R}^d\to {\mathbb R}$ is the pressure and $f^\ep \colon [0,\infty)\times {\mathbb R}^d\to{\mathbb R}$ is an external body force. Assuming that the solution is sufficiently smooth and the initial data $u^{\ep}_0$ satisfy a uniform $L^2$ bound (in other words the total kinetic energy at the initial time is bounded).

Since $u^\ep$ is divergence free, taking the dot product of the equation with $u^\ep$ and integrating over~$\R^d$, it is immediate to see that the energy balance:
\begin{equation}\label{0}
\fr12\frac{\dd}{\dd t}\norm[\big]{u^\ep(t)}^2_{L^2(\R^d)}=-\ep\norm[\big]{\nabla u^\ep(t)}^2_{L^2(\R^d)}+\int_{\R^d}f^\ep\cdot u^\ep\,\dd x.
\end{equation}
The second term appearing on the right-hand side is the total work of the force $f^\ep$, while the first term is the \emph{energy dissipation rate} due to the viscosity of the fluid. Throughout the paper, we assume that the external body force $f^\ep=0$. From~\eqref{0} we see that the $L^2$ energy decay of solutions is governed by
\[
\fr12\norm[\big]{u^\ep(t)}^2_{L^2(\R^d)}-\fr12\norm{u_0}^2_{L^2(\R^d)}=-\ep\int_0^t\norm[\big]{\nabla u^\ep(\tau)}^2_{L^2(\R^d)}\,\dd\tau.
\]
A fundamental postulate of Kolmogorov's 1941 theory of fully developed turbulence~\cite{K1,K2,K3}, called the zeroth law of turbulence, is that the anomalous dissipation of the kinetic energy holds, namely the following inequality
\[
\liminf_{\ep\to 0^+}\braces[\Bigg]{\ep\int_0^t\norm[\big]{\nabla u^\ep(\tau)}^2_{L^2(\R^d)}\,\dd\tau}&gt;0
\]
is valid for finite time $t$.

As for passive scalars, experimental and numerical observations of hydrodynamic turbulence suggest that, in the limit of vanishing viscosity, the rate of kinetic energy dissipation becomes nonzero~\cite{KI,KR1,KR2,PKW}, i.e.~there exists $\eta &gt; 0$ independent of $\ep$ such that, in turbulent regimes, a family of Leray--Hopf solutions $\{u^\ep\}_{\ep&gt;0}$ satisfies
\[
\ep\int^T_0 \int_{\R^d}\abs[\big]{\na u^{\ep}(t,x)}^2\,\dd x\,\dd t\geq\eta&gt;0.
\]

This phenomenon of anomalous dissipation is so fundamental to our modern understanding of turbulence that it is often termed the ``zeroth law''. Laboratory experiments and numerical simulations of turbulence both confirm the above zeroth law~\cite{BV,E,KI,J15}. However, this heuristic phenomenon is difficult to capture mathematically, and only a few rigorous results are known. Recently, in~\cite{DE}, Drivas et~al.~investigated the anomalous scalar dissipation for fluid velocities with HÃ¶lder regularity and gave sufficient conditions for anomalous dissipation in terms of the mixing rates of the advecting flow. BruÃ¨ and De~Lellis~\cite{EL} established the anomalous dissipation for the forced Navier--Stokes equations. Jeong and Yoneda~\cite{JY1,JY2} considered the anomalous dissipation for the 3D~\eqref{1} with zero external force in the framework of $2+\fr12$-dimensional flows. We would like to mention that all the results mentioned above are obtained on the Torus case. For more results on the anomalous dissipation for the Navier--Stokes equations, we refer to see~\cite{AC21,AC23,BCCLS,CTV,JS} and the refereed references.

To the best of our knowledge, there are no known anomalous dissipation for~\eqref{1} in the whole space (even in $\R^2$) where it is rigorously proved. Keeping this classical physical phenomenon in mind, in this paper, we compare the Navier--Stokes flow and the heat flow with the same initial data, and deduce anomalous dissipation in the vanishing viscosity limit of~\eqref{1} under some particular conditions on the initial data. Our main result is the following.

\begin{theo}\label{th1}
Let $d\geq2$. There exists a sequence of $C^\infty$-smooth initial data $u^{\ep_n}_0$ on\/ $\R^d$ with $\ep_n=2^{-2n}$ satisfying a uniform $L^2$ bound such that there are families of smooth solutions $u^{\ep_n}(t,x)\in C^{\infty}\parens[\big]{[0,1]\times \R^d}$ to the zero forced~\eqref{1} with the property that
\begin{equation}\label{ad}
\ep_n\int^1_0 \norm[\big]{\na u^{\ep_n}(t,x)}^2_{L^2(\R^d)}\,\dd t\gtrsim \norm{u^{\ep_n}_0}_{L^2(\R^d)}.
\end{equation}
In particular, we have
\begin{equation}\label{ad1}
\liminf_{n \to \infty}\braces[\Bigg]{\ep_n\int^1_0 \norm[\big]{\na u^{\ep_n}(t,x)}^2_{L^2(\R^d)}\,\dd t}\geq \eta,
\end{equation}
for some absolute constant $\eta&gt;0$.
\end{theo}

\begin{rema}\label{re0}
This result suggests that one possible mechanism for anomalous dissipation would be a cascade towards high-frequencies, whose intensity matches the scaling laws of the vanishing viscosity (more than half of the initial energy gets transferred in time $T=1$, by non-linear effects, towards modes of frequency higher than $1/\sqrt{\ep T})$. The construction of a family of flows with that property as $\ep \to0$ remains an open question. Here, the transfer is artificially ensured by choosing initial data with a shifting Fourier support.
\end{rema}

Now let us say some words on the key idea of proof of Theorem~\ref{th1}. We compare the Navier--Stokes equation and the heat equation with the same initial data, and show that the solution $U$ to the associated heat equation displays anomalous dissipation, i.e.,
\begin{equation}\label{y1}
\ep_n\int^1_0\norm[\big]{\na U_n(\tau)}^2_{L^2(\R^d)}\,\dd \tau\approx \int^1_0\norm[\big]{e^{\ep_n \tau\De}u^{\ep_n}_0}^2_{L^2(\R^d)}\,\dd \tau&gt;0.
\end{equation}
The main difficulty in the proof of Theorem~\ref{th1} is to construct the initial data satisfying the following:
\begin{itemize}
\item the $L^2$-norm of initial data is bounded uniformly with respect to the viscosity;
\item the difference of the Navier--Stokes flow and the heat flow is sufficiently small.
\end{itemize}
To achieve the above two points, taking advantage of the scaling in the whole space, we choose suitably the initial data whose frequency is supported in the annulus $\braces[\big]{\xi:\abs{\xi}^2 \sim \ep_n^{-1}}$. This naturally gives that
\begin{equation}\label{y2}
\int^1_0\norm[\big]{e^{\ep_n \tau\De}u^{\ep_n}_0}^2_{L^2(\R^d)}\,\dd \tau\thickapprox \norm{u^{\ep_n}_0}^2_{L^2(\R^d)}.
\end{equation}
Lastly, we would like to emphasize that we have to cancel the effect of the advection term. The key observation is that, based on our construction of initial data, the advection term $U\cdot \na U$ can be seen as $u^{\ep_n}_0\cdot \na u^{\ep_n}_0$, which is sufficiently small.

\begin{nota*}
For $X$ a Banach space and $I\subset\R$, we denote by $\mathcal{C}(I;X)$ the set of continuous functions on $I$ with values in $X$. Sometimes we will denote $L^p(0,T;X)$ by $L_T^pX$. The symbol $\mathrm{A}\lesssim (\gtrsim)\mathrm{B}$ means that there is a uniform positive ``harmless'' constant $\mathrm{C}$ independent of $\mathrm{A}$ and $\mathrm{B}$ such that
\[
A \leq (\geq) \mathrm{C}\mathrm{B},
\]
and we sometimes use the notation $\mathrm{A}\approx \mathrm{B}$ means that $\mathrm{A}\lesssim \mathrm{B}$ and $\mathrm{B}\lesssim \mathrm{A}$. Let us recall that for all $u\in \mathcal{S}'$, the Fourier transform $\mathcal{F}u$ is defined by $\mathcal{F}u(\xi)=\int_{\R^d}e^{-\mathrm{i}x\cdot \xi}u(x)\,\dd x$ for any $\xi\in\R^d.$ Choose a radial, non-negative, smooth function $\vartheta \colon \R^d\mapsto [0,1]$ such that $\supp \vartheta\subset B(0, 4/3)$ and $\vartheta(\xi)\equiv1$ for $\abs{\xi}\leq3/4$. Setting $\varphi(\xi)\coloneqq\vartheta(\xi/2)-\vartheta(\xi)$, then we deduce that $\supp \varphi\subset \braces[\big]{\xi\in \R^d \;:\; 3/4\leq\abs{\xi}\leq8/3}$ and $\varphi(\xi)\equiv 1$ for $4/3\leq \abs{\xi}\leq 3/2$. For more details see~\cite{BCD}. The nonhomogeneous dyadic block is defined by
\[
\forall u\in \mathcal{S'}(\R^d),\quad \Delta_ju =
\begin{cases}
0, 					&amp; \text{if $j\leq-2$,} \\
\vartheta(D)u		&amp; \text{if $j = -1$,} \\
\varphi(2^{-j}D)u, 	&amp; \text{if $j\geq0,$}
\end{cases}
\]
where the pseudo-differential operator is defined by $\sigma(D) \colon u\to\mathcal{F}^{-1}(\sigma \mathcal{F}u)$.
\end{nota*}

Finally we recall the definition of the nonhomogeneous Besov spaces
\[
B^{s}_{p,r}(\R^d)\coloneqq\braces[\bigg]{f\in \mathcal{S}'(\R^d) \;:\; \norm{f}_{B^{s}_{p,r}(\R^d)}\coloneqq\norm[\big]{2^{js}\norm{\Delta_jf}_{L_x^p(\R^d)}}_{\ell^r(j\geq-1)}&lt;\infty}.
\]
\section{Proof of Theorem~\ref{th1}}\label{sec3}
We divide the proof of Theorem~\ref{th1} into three steps. Now, by the heat flow, we shall mean the unique $L^2$ solution $U=e^{\ep t\De}u^{\ep}_0(x)$ of
\begin{equation}\label{2}
\begin{cases}
\pa_tU-\ep\Delta U=0,\\
\D U=0,\\
U(0,x)=u^{\ep}_0(x).
\end{cases}
\end{equation}

From~\eqref{1} with $f^\ep=0$ and~\eqref{2}, we deduce that the difference $v\coloneqq u^\ep-U$ satisfies the equation with zero initial data
\begin{equation}\label{v}
\partial_tv+v\cdot\na v-\ep\De v+\na p^\ep =-v\cdot\na U-U\cdot\na v-U\cdot\na U.
\end{equation}

\begin{step}[Estimation of difference]
Multiplying equation~\eqref{v} by $v$ and integrating in space, then using the fact that $v$ is divergence and the Cauchy--Schwarz inequality we conclude
\[
\norm{v}^2_{L^2(\R^d)}+\ep\int^t_0\norm{\na v}^2_{L^2(\R^d)}\,\dd \tau \leq 2\int^t_0\parens[\big]{\norm{\na U}_{L^\infty(\R^d)}+1}\norm{v}^2_{L^2(\R^d)}\,\dd \tau+\int^t_0\norm{U\cdot\na U}^2_{L^2(\R^d)}\,\dd \tau,
\]
which implies
\[
\norm{v}^2_{L^2(\R^d)}+\ep\int^t_0\norm{\na v}^2_{L^2(\R^d)}\,\dd \tau \leq \exp\parens[\Bigg]{\int^t_02\parens[\big]{\norm{\na U}_{L^\infty(\R^d)}+1}\,\dd \tau}\int^t_0\norm{U}^2_{L^2(\R^d)} \norm{\na U}^2_{L^\infty(\R^d)} \,\dd \tau.
\]
Hence, noting that $u^\ep=U+v$, we have
\begin{equation}\label{lyz}
\begin{split}
&amp; \ep\int^t_0\norm{\na u^\ep}^2_{L^2(\R^d)}\,\dd \tau -\frac{\ep}{2}\int^t_0\norm{\na U}^2_{L^2(\R^d)}\,\dd \tau \\
&amp; \hspace{4cm} \geq -\ep\int^t_0\norm{\na v}^2_{L^2(\R^d)}\,\dd \tau \\
&amp; \hspace{4cm} \geq -\exp\parens*{\int^t_0\parens[\big]{\norm{\na U}_{L^\infty(\R^d)}+1} \,\dd \tau} \int^t_0\norm{U}^2_{L^2(\R^d)} \norm{\na U}^2_{L^\infty(\R^d)} \,\dd \tau.
\end{split}
\end{equation}
\end{step}

\begin{step}[Construction of initial data]
We need to introduce smooth, radial cut-off functions to localize the frequency region. Let $\widehat{\phi}\in \mathcal{C}^\infty_0(\mathbb{R})$ be an even, real-valued and non-negative function on $\R$ that satisfies
\[
\widehat{\phi}(\xi) =
\begin{cases}
1, 	&amp; \text{if $\abs{\xi}\leq \frac{1}{4}$,} \\
0, 	&amp; \text{if $\abs{\xi}\geq \frac{1}{2}$.}
\end{cases}
\]

\begin{rema}
It is not difficult to see that
\[
\norm{\phi}_{L^{2}(\R)}\thickapprox1 \qquad\text{and}\qquad \norm{\phi}_{L^{\infty}(\R)}\thickapprox1.
\]
\end{rema}

From now on, we set $\ep_n\coloneqq2^{-2n}$ and define the initial data $u^{\ep_n}_0$ by
\[
u^{\ep_n}_0(x)\coloneqq2^{-dn-n}\parens[\big]{-\pa_2f_n(x),\; \pa_1f_n(x),\; 0, \dots,\; 0},
\]
where
\[
f_n(x)\coloneqq\cos \parens[\Bigg]{\frac{17}{12}2^nx_1}\prod_{i=1}^d\phi(\ep_nx_i).
\]
Then we have the following

\begin{lemm}\label{le1}
We have
\[
\norm{u^{\ep_n}_0}_{L^2(\R^d)}\thickapprox \norm{\phi}^d_{L^{2}(\R)} \qquad\text{and}\qquad \norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}\thickapprox 2^{(1-d)n}\norm{\phi}^d_{L^{\infty}(\R)}.
\]
\end{lemm}
\begin{proof}
Due to the simple fact that $\cos\theta=\frac{1}{2} (e^{-\mathrm{i} \theta}+e^{\mathrm{i} \theta})$, we deduce easily that
\[
\begin{split}
\mathcal{F}(f_n) 	&amp; = 2^{2nd-1} \int_{\R}\parens[\bigg]{e^{-\mathrm{i} 2^{2 n} x_1{\left(\xi_1+\frac{17}{12} 2^{n}\right)}}+e^{-\mathrm{i} 2^{2n} x_1{\left(\xi_1-\frac{17}{12} 2^{n}\right)}}} \phi(x_1) \,\dd x_1\prod_{i=2}^d\widehat{\phi}(2^{2n}\xi_i) \\
							&amp; = 2^{2nd-1} \bracks[\Bigg]{\widehat{\phi}\parens[\Bigg]{2^{2n} \parens[\Bigg]{\xi_1+\frac{17}{12} 2^{n}}}+\widehat{\phi}\parens[\Bigg]{2^{2n}\parens[\Bigg]{\xi_1-\frac{17}{12} 2^{n}}}}\prod_{i=2}^d\widehat{\phi}(2^{2n}\xi_i),
\end{split}
\]
which implies that
\begin{equation}\label{con}
\supp\mathcal{F}(u_0^n) = \supp\mathcal{F}(f_n) \subset \braces[\bigg]{\xi\in\R^d \;:\; \tfrac{33}{24}2^{n}\leq \abs{\xi}\leq \tfrac{35}{24}2^{n}}.
\end{equation}
Recalling that $\varphi\equiv 1$ for $\frac43\leq \abs{\xi}\leq \frac32$, we have
\[
\Delta_j(u^{\ep_n}_0) =
\begin{cases}
u^{\ep_n}_0,		&amp; \text{if $j=n$,}\\
0, 				&amp; \text{if $j\neq n$.}
\end{cases}
\]
Thus, for any $\sigma\in \R$, we deduce that
\[
\norm{u^{\ep_n}_0}_{B^\sigma_{p,r}(\R^d)} = 2^{n\sigma}\norm{u^{\ep_n}_0}_{L^{p}(\R^d)} = 2^{n(\sigma-d-1)}\parens[\big]{\norm{\pa_1f_n}_{L^{p}(\R^d)}+\norm{\pa_2f_n}_{L^{p}(\R^d)}},
\]
which completes the proof of Lemma~\ref{le1}.
\end{proof}
\end{step}

\begin{step}[Completion of the proof]
Easy computations show that for $t\in[0,T_{\ep_n})$ (for more details see~\cite{GZ})
\[
\begin{split}
\norm[\big]{u^{\ep_n}(t)}_{B^1_{\infty,1}(\R^d)}
&amp; \leq Ce^{C\mathbf{V}(t)} \parens*{\norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}+\int^t_0e^{-C\mathbf{V}(\tau)}\norm[\big]{\nabla p^{\ep_n}(\tau)}_{B^1_{\infty,1}(\R^d)}\,\dd \tau} \\
&amp; \leq Ce^{C\mathbf{V}(t)} \parens*{\norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}+C\int^t_0e^{-C\mathbf{V}(\tau)}\norm[\big]{u^{\ep_n}(\tau)}^2_{B^1_{\infty,1}(\R^d)}\,\dd \tau},
\end{split}
\]
where $\mathbf{V}(t)\coloneqq\int_0^t \norm[\big]{u^{\ep_n}(\tau)}_{B^1_{\infty,1}(\R^d)}\,\dd\tau$. Solving the above inequality, we deduce that
\[
\norm[\big]{u^{\ep_n}(t)}_{B^1_{\infty,1}(\R^d)}\leq \frac{\norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}}{1-Ct\norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}}\quad\text{whenever} \ t&lt;\frac{1}{C\norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}}.
\]
By Lemma~\ref{le1}, one has $\norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}\approx 2^{(1-d)n}$, which tells us that there exists $T_{\ep_n}\geq1$ such that~\eqref{1} has a unique solution $u^{\ep_n}$ in $\mathcal{C}\parens[\big]{[0,1];H^\infty}$.

Noticing that $U_n(t,x)=e^{\ep_n t\De}u^{\ep_n}_0(x)$, we obtain that for some uniform positive constant $C$
\begin{align*}
					&amp; \norm{U_n}_{L^\infty_T\parens{L^2(\R^d)}}\leq \norm{u^{\ep_n}_0}_{L^2(\R^d)}\leq C \\
\text{and} \quad		&amp; \norm{\na U_n}_{L^\infty_T\parens{L^\infty(\R^d)}}\leq\norm{\na u^{\ep_n}_0}_{L^\infty(\R^d)}\leq C\norm{u^{\ep_n}_0}_{B^1_{\infty,1}(\R^d)}\leq C 2^{(1-d)n},
\end{align*}
where we have used the classical $L^p$-$L^p$ estimate: $\norm{e^{\ep t\Delta}f}_{L^p} \leq \norm{f}_{L^p}$ with $1\leq p\leq \infty$, the embedding $B^0_{\infty,1}\hookrightarrow L^\infty$, and Lemma~\ref{le1} in the last step. Thus
\[
\exp\parens*{\int^t_0 2\parens[\big]{\norm{\na U_n}_{L^\infty(\R^d)}+1} \,\dd \tau} \int^t_0\norm{U_n}^2_{L^2(\R^d)} \norm{\na U_n}^2_{L^\infty(\R^d)} \,\dd \tau
\leq C2^{2(1-d)n}.
\]
Then, coming back to~\eqref{lyz}, we conclude
\begin{equation}\label{ad0}
\ep_n\int^1_0\norm{\na U_n}^2_{L^2(\R^d)}\,\dd \tau\geq \frac{\ep_n}{2}\int^1_0\norm{\na U_n}^2_{L^2(\R^d)}\,\dd \tau-C2^{2(1-d)n}.
\end{equation}
Due to the support condition~\eqref{con} and the Plancherel's identity, it holds that
\[
\ep_n\int^1_0\norm[\big]{\na U_n(\tau)}^2_{L^2(\R^d)}\,\dd \tau\approx \int^1_0\norm[\big]{e^{\ep_n \tau\De}u^{\ep_n}_0}^2_{L^2(\R^d)}\,\dd \tau\approx \norm{u^{\ep_n}_0}^2_{L^2(\R^d)}\approx \norm{\phi}^{2d}_{L^2(\R)}.
\]
Inserting the above into~\eqref{ad0} gives~\eqref{ad}--\eqref{ad1}. Thus we complete the proof of Theorem~\ref{th1}.
\end{step}

\section*{Acknowledgments}

We would like to thank the anonymous reviewer for the valuable and detailed comments which greatly improved the presentation of this paper. 

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\printbibliography
\end{document}</pre></body></html>