\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 104, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/104\hfil Distinction of turbulence from chaos]
{Distinction of turbulence from chaos -- rough dependence on initial data}


\author[Y. C. Li\hfil EJDE-2014/104\hfilneg]
{Y. Charles Li}  % in alphabetical order


\address{Yanguang Charles Li \newline
Department of Mathematics, University of Missouri,
Columbia, MO 65211, USA}
\email{liyan@missouri.edu}
\urladdr{http://www.math.missouri.edu/~cli}

\thanks{Submitted December 9, 2013. Published April 15, 2014.}
\subjclass[2000]{76F20, 35Q30, 37D45}
\keywords{Rough dependence on initial data; chaos; turbulence;
\hfill\break\indent dependence on initial data}

\begin{abstract}
 This article presents a new theory on the nature of turbulence:
 when the Reynolds number is large, violent fully developed turbulence is due
 to ``rough dependence on initial data'' rather than chaos which is caused by
 ``sensitive dependence on initial data''; when the Reynolds number is moderate,
 (often transient) turbulence is due to chaos. The key in the validation
 of the theory is estimating the temporal growth of the initial perturbations
 with the Reynolds number as a parameter. Analytically, this amounts
 to estimating the temporal growth of the norm of the derivative of the solution
 map of the Navier-Stokes equations, for which here I obtain an upper bound
 $e^{C \sqrt{t Re} + C_1 t}$. This bound clearly indicates that when the Reynolds
 number is large, the temporal growth rate can potentially be large in short time,
 i.e. rough dependence on initial data.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

For a long time, fluid dynamists have suspected that turbulence is ``more than''
chaos. Many chaoticians including the present author have believed that
turbulence is ``no more than'' chaos in Navier-Stokes equations.
A recent result \cite{Inc12} on Euler equations forced
the present author to have to change mind.

The signature of chaos is ``sensitive dependence on initial data''; here I want to
address ``rough dependence on initial data'' which is very different from sensitive
dependence on initial data. For solutions (of some system) that exhibit sensitive
dependence on initial data, their initial small deviations usually amplify
exponentially
(with an exponent named Liapunov exponent), and it takes time for the deviations
to accumulate to substantial amount (say order $O(1)$ relative to the small initial
deviation). If $\epsilon$ is the initial small deviation, and $\sigma$ is the
Liapunov exponent, then the time for the deviation to reach $1$ is about
 $\frac{1}{\sigma} \ln \frac{1}{\epsilon}$. On the other hand, for solutions that exhibit
rough dependence on initial data, their initial
small deviations can reach substantial amount instantly.
Take the 3D or 2D Euler equations of fluids as the example, for any $t \neq 0$
(and small for local existence), the solution
map that maps the initial condition to the solution value at time $t$ is nowhere
locally uniformly continuous and nowhere differentiable \cite{Inc12}.
In such a case, any small deviation of the initial condition can potentially
reach substantial amount instantly. My theory is that the high Reynolds number
violent turbulence is due to such rough dependence on initial data, rather than
sensitive dependence on initial data of chaos. When the Reynolds number is
sufficiently large (the viscosity is sufficiently small), even though the
solution map of the Navier-Stokes equations
is still differentiable, but the derivative of the solution map should be
 potentially extremely large everywhere (of order $e^{C \sqrt{t Re}}$
as shown below) since the solution map of
the Navier-Stokes equations approaches
the solution map of the Euler equations when the viscosity approaches zero
 (the Reynolds number approaches infinity). Such everywhere large derivative
of the solution map of the Navier-Stokes equations manifests itself as the
development of violent turbulence in a short time. In summary, moderate
Reynolds number turbulence is due to sensitive dependence on initial data of chaos,
while large enough Reynolds number turbulence is due to rough dependence on initial
data. This is an important new understanding on the nature of turbulence \cite{Li13}.
One may call this the new complexity of turbulence \cite{Mit09} \cite{LLW13}.

In terms of phase space dynamics of dynamical systems, when the Reynolds number
is very high, fully developed turbulence is not the result of a strange attractor,
rather a result of super fast deviation amplifications (facilitated
by the large derivative of turbulent solutions in their initial data). Strange
attractor is a long time object, while the development of such violent turbulence
is of short time. Such fully developed turbulence is maintained by constantly super
fast deviation amplifications. When the Reynolds number is set to
infinity, deviation amplification rate is infinity. So the dynamics of Euler
equations is very close to a random process.
In contrast, chaos in finite dimensional conservative systems often manifests
itself as the so-called stochastic layers. Dynamics inside the stochastic layers
has the long term sensitive dependence on initial data.
When the Reynolds number is moderate, viscous diffusive term in Navier-Stokes
equations is stronger, deviation
amplification rate is moderate. At this stage, turbulence is basically chaos
in Navier-Stokes equations \cite{VK11,KUV12,KE12,KK01}. In some cases,
strange attractor, homoclinic orbits, and bifurcation routes to chaos can
be observed \cite{VK11}. When the Reynolds number is lowered to its critical value,
the initiator for the transition from the basic laminar flow is the linear
instability of the laminar flow near the basic laminar flow
(i.e. the basic laminar flow plus high spatial frequency deviations).
This the resolution of the Sommerfeld (turbulence) paradox \cite{LL11}.

The type of rough dependence on initial data shared by the solution map of
the Euler equations is difficult to find in finite dimensional systems.
The solution map of the Euler equations is still continuous in initial data.
Such a solution map (continuous, but nowhere locally uniformly continuous)
does not exist in finite dimensions. This may be the reason that one usually
finds chaos (sensitive dependence on initial data) rather than rough dependence
on initial data in finite dimensions. If the
solution map of some special finite dimensional system is nowhere continuous, then
the dependence on initial data is rough, but may be too rough to have any realistic
application. In infinite dimensions, irregularities of solution maps are quite
common, e.g. in water wave equations \cite{CL13,CMSW14}.

Even though the relation between Liapunov exponent and chaos (and instability)
can be complicated \cite{LK07}, generically a positive Liapunov exponent is a
 good indicator of chaotic dynamics. In connection with turbulence,
 Liapunov exponent and its extensions have been studied \cite{PV94,ABCPV97}.
To distinguish that turbulence is exhibiting rough or sensitive dependence
on initial data, one needs to study the derivative of the solution map.

\section{Derivative of the solution map}

Let $S^t$ be the solution map which maps the initial value $u(0)$ to the
solution's value $u(t)$ at time $t$. So for any fixed time $t$, $S^t$ is a
map defined on the phase space. The temporal growth of the norm of the
derivative $DS^t$ of the solution map $S^t$ describes the amplification
of the initial perturbation. The well-known Liapunov exponent is defined
by $DS^t$:
\[
\sigma = \lim_{t \to +\infty} \frac{1}{t} \ln \| DS^t \| .
\]
A positive Liapunov exponent implies that nearby orbits deviate exponentially
in time, i.e. sensitive dependence on initial data. The Liapunov exponent
is a measure of long term temporal growth of the norm
of the derivative $DS^t$. The temporal property of the norm of $DS^t$ can
of course be much more complicated than simple long term exponential growth.
In particular, the norm of $DS^t$ can be large in short time
(i.e. super fast temporal growth). In such a case, the dynamics
(described by $S^t$) exhibits short term unpredictability
(i.e. rough dependence on initial data). One can define the following exponent
\[
\eta = \lim_{t \to 0^+} \frac{1}{t^\alpha} \ln \| DS^t \|, \quad \text{where } \alpha >0 .
\]
When $\eta$ is large (e.g. approaching infinity as a parameter approaches a limit),
one has short term unpredictability. In the case of Navier-Stokes equations
to be studied later, $\eta$ can potentially be as large as $C \sqrt{Re}$ with
 $\alpha = 1/2$.



\section{Derivative estimate for Navier-Stokes equations}

To verify the rough dependence on initial data for the solution map of the
 Navier-Stokes equations, we need to estimate the temporal growth of the norm
of the derivative of the solution map of the Navier-Stokes equations.
The Navier-Stokes equations are given by
\begin{gather}
 u_t + \frac{1}{Re} \Delta u  = - \nabla p - u\cdot \nabla u , \label{ONS1} \\
 \nabla \cdot u = 0 , \label{ONS2}
\end{gather}
where $u$ is the $d$-dimensional fluid velocity ($d=2,3$), $p$ is the
fluid pressure, and
$Re$ is the Reynolds number. Applying the Leray projection, one gets
\begin{equation}
u_t + \frac{1}{Re} \Delta u  = - \mathbb{P} \left ( u\cdot \nabla u \right ) . \label{NS}
\end{equation}
The Leray projection is an orthogonal projection in $L^2(\mathbb{R}^d)$, given by
\[
\mathbb{P} g = g - \nabla \Delta^{-1} \nabla \cdot g .
\]
Setting the Reynolds number to infinity $Re = 0$, the Navier-Stokes
equation \eqref{NS} reduces to
the Euler equation
\begin{equation}
u_t = - \mathbb{P} \left ( u\cdot \nabla u \right ) . \label{E}
\end{equation}
Let $H^n(\mathbb{R}^d)$ be the Sobolev space of divergence free fields.
By the local wellposedness
result of Kato \cite{Kat72,Kat75}, when $n > \frac{d}{2} +1$ ($d=2,3$), for any
$u \in H^n(\mathbb{R}^d)$, there
is a neighborhood $B$ and a short time $T>0$, such that for any $v \in B$
there exists a unique
solution to the Navier-Stokes equation \eqref{NS} in
$C^0([0,T]; H^n(\mathbb{R}^d))$; as $Re
\to \infty$, this solution converges to that of the Euler equation \eqref{E}
in the same space. For any
$t \in [0, T]$, let $S^t$ be the solution map:
\begin{equation}
S^t :   B  \mapsto H^n(\mathbb{R}^d), \quad  S^t (u(0)) = u(t),
\end{equation}
which maps the initial condition to the solution's value at time $t$.
The solution map is continuous for both Navier-Stokes equation \eqref{NS}
and Euler equation \eqref{E} \cite{Kat72,Kat75}. A recent result of
Inci \cite{Inc12} shows that for Euler equation \eqref{E} the solution map is
nowhere differentiable. Then it is natural to theorize that the norm of
the derivative of the solution map approaches infinity (at most places)
as the Reynolds number approaches infinity. Estimating the temporal growth
of the norm of the derivative of the solution map is a daunting task.
The entire subject of hydrodynamic stability is a special case where the
 base solution (where the derivative of the solution map is taken) is steady.
Below I obtain an upper bound on the temporal growth of the norm of the
derivative of the solution map. I believe the upper bound is sharp,
i. e. there is no smaller upper bound.

\begin{theorem} \label{thm3.1}
The norm of the derivative of the solution map of Navier-Stokes equation
 \eqref{NS}  has the upper bound
\begin{equation}
\| DS^t(u(0)) \| \leq e^{C \sqrt{t Re} + C_1 t}, \label{UB}
\end{equation}
where
\[
C = \frac{8}{\sqrt{2e}} \max_{\tau \in [0,T]} \| u(\tau )\|_n, \quad
C_1 = 4\max_{\tau \in [0,T]} \| u(\tau )\|_n = \frac{\sqrt{2e}}{2} C.
\]
\label{UBT}
\end{theorem}

\begin{proof}
Applying the method of variation of parameters, one converts the Navier-Stokes
equation \eqref{NS} into the integral equation
\begin{equation}
u(t) = e^{\frac{t}{Re}\Delta } u(0) - \int_0^t e^{\frac{t-\tau }{Re}\Delta }
\mathbb{P} \left ( u\cdot \nabla u \right ) d\tau .                   \label{INS}
\end{equation}
Taking the differential in $u(0)$, one gets the differential form
\begin{equation}
du(t) = e^{\frac{t}{Re}\Delta } du(0) - \int_0^t e^{\frac{t-\tau }{Re}\Delta } \mathbb{P} \left ( du\cdot \nabla u
+u \cdot \nabla du \right ) d\tau .   \label{DNS}
\end{equation}
The norm of the derivative $DS^t(u(0)) = \partial u(t) / \partial u(0)$ is given by
\begin{equation}
\| DS^t(u(0)) \| = \sup_{du(0)} \frac{\| du(t)\|_n}{\| du(0)\|_n} . \label{ND}
\end{equation}
Applying the inequality
\[
\| e^{\frac{t}{Re}\Delta } u \|_n
\leq \Big(\frac{1}{\sqrt{2e}} \sqrt{\frac{Re}{t}} + 1 \Big)\| u \|_{n-1} ,
\]
one gets
\begin{align*}
&\| du(t) \|_n\\
& \leq  \| du(0) \|_n +  4 \max_{\tau \in [0,T]} \| u(\tau )\|_n  \int_0^t
\Big( \frac{\sqrt{Re}}{\sqrt{2e}} \frac{1}{\sqrt{t-\tau }} +1 \Big)
\| du(\tau ) \|_n  d\tau .
\end{align*}
Applying the Gronwall's inequality, one gets the estimate
\[
\| du(t) \|_n \leq  e^{C \sqrt{t Re} + C_1 t} \| du(0) \|_n ,
\]
where
\[
C = \frac{8}{\sqrt{2e}} \max_{\tau \in [0,T]} \| u(\tau )\|_n, \quad
C_1 = 4\max_{\tau \in [0,T]} \| u(\tau )\|_n
= \frac{\sqrt{2e}}{2} C.
\]
By \eqref{ND},
\[
\| DS^t(u(0)) \| \leq e^{C \sqrt{t Re} + C_1 t}.
\]
\end{proof}

\begin{remark} \rm
By Theorem \ref{UBT}, for any initial perturbation $\delta u(0)$,
the deviation of the corresponding solutions can potentially amplifies
according to
\[
\| \delta u(t) \|_n \leq  e^{C \sqrt{t Re}\ + \ C_1 t} \| \delta u(0) \|_n .
\]
When the Reynolds number is large, the amplification can
potentially reach substantial amount in short time.
\end{remark}

\begin{remark} \rm
The same upper bound \eqref{UB} also holds for the periodic boundary condition;
i.e. when the Navier-Stokes equations \eqref{ONS1}-\eqref{ONS2} are defined
on $d$-dimensional torus $\mathbb{T}^d$ instead of the whole space $\mathbb{R}^d$.
\end{remark}

\begin{remark} \rm
The beauty of the upper bound \eqref{UB} can be revealed when the base
solution (where the derivative of the solution map is taken) is steady.
In such a case, one is dealing with hydrodynamic stability theory.
The zero-viscosity limit of the eigenvalues of the linear Navier-Stokes
equations at the steady state can be complicated \cite{LL08}.
In the zero-viscosity limit, some of the eigenvalues may persist to be
the eigenvalues of the corresponding linear Euler equations \cite{Li05};
some eigenvalues may condense into continuous spectra; and other eigenvalues
 may approach a set that is not in the spectra of the corresponding linear
Euler equations. The $C_1t$ exponent in \eqref{UB} covers the growth induced
by persistent unstable eigenvalues, while the $C\sqrt{tRe}$ exponent
in \eqref{UB} covers the growth induced by the rest eigenvalues.
When $Re$ is large, the $C\sqrt{tRe}$ can be large in short time.
During such short time, stable eigenvalues do not imply ``decay".
Even though its derivative does not exist, directional derivatives of
the solution map of Euler equations can exist as shown by the existence
of solutions to the well-known Rayleigh equations. The unbounded continuous
 spectrum \cite{Li05} of the linear Euler equations leads to the nonexistence
of the derivatives of the solution map of Euler equations. Figure \ref{ge11-12}
is the spectra of the 2D linear NS (Euler) operator at the shear
$\omega = 2 \cos x$ where $\omega$ is the vorticity and the spatial periodic domain
is [$2\pi, \frac{2\pi}{0.7}$]. Figure \ref{kat5-7m1} is the spectra of the
2D linear NS (Euler) operator at the cat's eye
\[
\omega = 2\cos x +  \cos y \ ,
\]
where the spatial periodic domain is [$2\pi, 2\pi$].
\end{remark}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\textwidth]{fig1a}\quad % fige11.eps
\includegraphics[width=0.4\textwidth]{fig1b}   % fige12.eps
\end{center}
\caption{The spectra of the 2D linear NS operator (at a linear shear) where
$\epsilon = 1/Re$, the isolated dots are the eigenvalues, and the bold face lines
are unbounded continuous spectra.}
\label{ge11-12}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig2a}\\   % cat5m1.eps
(a)\\
\includegraphics[width=0.45\textwidth]{fig2b}\quad % cat6m1.eps
\includegraphics[width=0.45\textwidth]{fig2c} \\ % cat7m1.eps
(b)\hfil (c)
\end{center}
\caption{The spectra of the linear NS operator (at the cat's eye) where
$\epsilon = 1/Re$, all the dots in (a) are eigenvalues, (b) is a half
plane continuous spectrum, and (c) is the whole plane continuous spectrum.}
\label{kat5-7m1}
\end{figure}


\begin{remark} \rm
The upper bound \eqref{UB} is sharp when the base solution
(where the derivative of the solution map is taken) is the zero solution $u(t)=0$.
In this case,
\[
\| DS^t(0)\| = 1 ,
\]
and the upper bound \eqref{UB} is also $1$. In general, estimating the lower
bound of $\| DS^t(u(0))\|$ may only be done on a case by case base for the base
solutions. When the base solutions are steady, this is the theory of hydrodynamic
instability.
\end{remark}


\section{Classical hydrodynamic instability -- directional derivative}

Classical hydrodynamic instability theory mainly focuses on the so-called linear
instability of steady fluid flows.  We can think that the linear instability
theory is based on Taylor expansion of the solution map for
Navier-Stokes equations \eqref{ONS1}-\eqref{ONS2}. Let $u^*$ be the steady
flow (a fixed point in the phase space),
$v_0$ be its initial perturbation, and $u^* + v(t)$ be the solution to the
Navier-Stokes equations \eqref{ONS1}-\eqref{ONS2}
with the initial condition  $u^* + v_0$.  According to Taylor expansion,
\[
v(t) = dv(t) + d^2v(t) + \dots ,
\]
where $dv(t)$ is the first differential in $v_0$ of the solution map at the steady
flow $u^*$, similarly for $d^2v(t)$ etc..
Under the Euler dynamics, this expansion fails since the first differential
does not exist \cite{Inc12}. The
first differential satisfies the differential form
\begin{equation} \label{dNS}
\begin{gathered}
 dv_t + \frac{1}{Re} \Delta dv  = - \nabla dp - dv\cdot \nabla u^* - u^*\cdot \nabla dv ,  \\
 \nabla \cdot dv = 0 ,
\end{gathered}
\end{equation}
where $dp$ is the pressure differential. The linear instability refers to
the instability of the differential form
\eqref{dNS}. In most cases studied, the steady flow $u^*$ depends on only one
spatial variable $y$ (the so-called
channel flow). This permits the following type solutions to the differential form,
\begin{equation}
dv(t) = \exp \{ i (\sigma t + k_1 x + k_3 z) \} V(y) , \label{FM}
\end{equation}
where ($x,y,z$) are the spatial coordinates, $\sigma$ is a complex parameter, and
($k_1,k_3$) are real parameters.
One can view \eqref{FM} as a single Fourier mode out of the Fourier transform of
$dv(t)$. In the phase space of the
dynamics, \eqref{FM} is a directional differential with the specific direction
specified by the ($k_1,k_3$) Fourier mode.
$V(y)$ satisfies the well-known Orr-Sommerfeld equation (Rayleigh equation in
the inviscid case $Re = \infty$).
Even though the first differential $dv(t)$ does not exist in the inviscid case
(\eqref{dNS} with $Re = \infty$), the
directional differential \eqref{FM} can exist with $V(y)$ solving the Rayleigh
equation. The classical
hydrodynamic instability theory mainly focuses on the studies of the directional
differential \eqref{FM}.



\begin{thebibliography}{99}

\bibitem{ABCPV97} E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, A. Vulpiani;
Predictability in the large: an extension of the concept of Lyapunov exponent,
 \emph{J. Phys. A} \textbf{30} (1997), 1-26.

\bibitem{CL13} R. Chen, Y. Liu;
On the ill-posedness of a weakly dispersive one-dimensional Boussinesq system,
\emph{Journal d'Analyse Math\'ematique} \textbf{121, no.1} (2013), 299-316.

\bibitem{CMSW14} R. Chen, J. Marzuola, D. Spirn, J. Wright;
 On the regularity of the flow map for the gravity-capillary equations,
\emph{arXiv: 1111.5361, to appear in J. Funct. Anal.} (2014).

\bibitem{Inc12} H. Inci;
 On the well-posedness of the incompressible Euler equation,
\emph{Dissertation, University of Zurich} (2012), arXiv: 1301.5997.

\bibitem{Kat72} T. Kato;
Nonstationary flows of viscous and ideal fluids in $R^3$,
\emph{J. Funct. Anal.} \textbf{9} (1972), 296-305.

\bibitem{Kat75} T. Kato;
Quasi-linear equations of evolution, with applications to partial differential
equations, \emph{Lect. Notes in Math., Springer} \textbf{448} (1975), 25-70.

\bibitem{KK01} G. Kawahara, S. Kida;
Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst,
\emph{J. Fluid Mech.} \textbf{449} (2001), 291-300.

\bibitem{KUV12} G. Kawahara, M. Uhlmann, L. van Veen;
The significance of simple invariant solutions in turbulent flows,
\emph{Ann. Rev. Fluid Mech.} \textbf{44} (2012), 203-225.

\bibitem{KE12} T. Kreilos, B. Eckhardt;
Periodic orbits near onset of chaos in plane Couette flow,
\emph{Chaos} \textbf{22} (2012), 047505.

\bibitem{LLW13} J. Ladyman, J. Lambert, K. Wiesner;
 What is a complex system?  \emph{European J. for Philosophy of Sci.}
\textbf{3} (2013), 33-67.

\bibitem{LL08} Y. Lan, Y. Li;
On the dynamics of Navier-Stokes and Euler equations, \emph{J. Stat. Phys.}
\textbf{132} (2008), 35-76.

\bibitem{LK07} G. Leonov, N. Kuznetsov;
Time-varying linearization and the Perron effects,
\emph{Intl. J. Bifurcation and Chaos} \textbf{17, no.4} (2007), 1079-1107.

\bibitem{Li05} Y. Li;
Invariant manifolds and their zero-viscosity limits for Navier-Stokes equations,
\emph{Dynamics of PDE} \textbf{2, no.2} (2005), 159-186.

\bibitem{Li13} Y. Li;
 Major open problems in chaos theory and nonlinear dynamics,
\emph{Dynamics of PDE} \textbf{10, no.4} (2013), 379-392. arXiv: 1305.2864

\bibitem{LL11} Y. Li, Z. Lin;
 A resolution of the Sommerfeld paradox,
 \emph{SIAM J. Math. Anal.} \textbf{43, no.4} (2011), 1923-1954.

\bibitem{Mit09} M. Mitchell;
 \emph{Complexity}, Oxford U. Press, 2009.

\bibitem{PV94} G. Paladin;
 A. Vulpiani, Predictability in spatially extended systems,
\emph{J. Phys. A} \textbf{27} (1994), 4911-4917.

\bibitem{VK11} L. van Veen, G. Kawahara;
 Homoclinic tangle on the edge of shear turbulence, \emph{Phys. Rev. Lett.}
 \textbf{107} (2011), 114501.

\end{thebibliography}
\end{document}