The distinction of turbulence from chaos — rough dependence on initial data
I propose 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 . 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.
Key words and phrases:Rough dependence on initial data, sensitive dependence on initial data, chaos, turbulence
1991 Mathematics Subject Classification:Primary 76, 35; Secondary 34
For a long time, fluid dynamists have suspected that turbulence is “more than” chaos. Many chaoticians including the current author have believed that turbulence is “no more than” chaos in Navier-Stokes equations. A recent result  on Euler equations forced the current 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 relative to the small initial deviation). If is the initial small deviation, and is the Liapunov exponent, then the time for the deviation to reach is about . 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 (and small for local existence), the solution map that maps the initial condition to the solution value at time is nowhere locally uniformly continuous and nowhere differentiable . 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 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 . One may call this the new complexity of turbulence  .
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  .
Even though the relation between Liapunov exponent and chaos (and instability) can be complicated , generically a positive Liapunov expoent is a good indicator of chaotic dynamics. In connection with turbulence, Liapunov exponent and its extensions have been studied  . When the Reynolds number is moderate, homoclinic orbits, strange attractors and bifurcation routes to chaos of the Navier-Stokes equations are all dug out   . To distinguish that turbulence is exhibiting rough or sensitive dependence on initial data, one needs to study the derivative of the solution map.
2. Derivative of the solution map
Let be the solution map which maps the initial value to the solution’s value at time . So for any fixed time , is a map defined on the phase space. The temporal growth of the norm of the derivative of the solution map describes the amplification of the initial perturbation. The well-known Liapunov exponent is defined by :
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 . The temporal property of the norm of can of course be much more complicated than simple long term exponential growth. In particular, the norm of can be large in short time (i.e. super fast temporal growth). In such a case, the dynamics (described by ) exhibits short term unpredictability (i.e. rough dependence on initial data). One can define the following exponent
When 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, can potentially be as large as with .
3. The derivative estimate for Navier-Stokes equations
To verify the rough dependence on initial data for the solution map of the Navier-Stokes equations, I 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
where is the -dimensional fluid velocity (), is the fluid pressure, and is the Reynolds number. Applying the Leray projection, one gets
The Leray projection is an orthogonal projection in , given by
Setting the Reynolds number to infinity , the Navier-Stokes equation (3.3) reduces to the Euler equation
Let be the Sobolev space of divergence free fields. By the local wellposedness result of Kato  , when (), for any , there is a neighborhood and a short time , such that for any there exists a unique solution to the Navier-Stokes equation (3.3) in ; as , this solution converges to that of the Euler equation (3.4) in the same space. For any , let be the solution map:
which maps the initial condition to the solution’s value at time . The solution map is continuous for both Navier-Stokes equation (3.3) and Euler equation (3.4)  . A recent result of Inci  shows that for Euler equation (3.4) 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.
The norm of the derivative of the solution map of Navier-Stokes equation (3.3) has the upper bound,
Applying the method of variation of parameters, one converts the Navier-Stokes equation (3.3) into the integral equation
Taking the differential in , one gets the differential form
The norm of the derivative is given by
Applying the inequality
Applying the Gronwall’s inequality, one gets the estimate (3.6). ∎
By Theorem 3.1, for any initial perturbation , the deviation of the corresponding solutions can potentially amplifies according to
When the Reynolds number is large, the amplification can potentially reach substantial amount in short time.
The beauty of the upper bound (3.6) 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 . In the zero-viscosity limit, some of the eigenvalues may persist to be the eigenvalues of the corresponding linear Euler equations ; 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 exponent in (3.6) covers the growth induced by persistent unstable eigenvalues, while the exponent in (3.6) covers the growth induced by the rest eigenvalues. When is large, the 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  of the linear Euler equations leads to the nonexistence of the derivatives of the solution map of Euler equations.
The upper bound (3.6) is sharp when the base solution (where the derivative of the solution map is taken) is the zero solution . In this case,
and the upper bound (3.6) is also . In general, estimating the lower bound of 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.
-  E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, A. Vulpiani, Predictability in the large: an extension of the concept of Lyapunov exponent, J. Phys. A 30 (1997), 1-26.
-  R. Chen, Y. Liu, On the ill-posedness of a weakly dispersive one-dimensional Boussinesq system, Journal d’Analyse Mathématique 121, no.1 (2013), 299-316.
-  R. Chen, J. Marzuola, D. Spirn, J. Wright, On the regularity of the flow map for the gravity-capillary equations, arXiv: 1111.5361, to appear in J. Funct. Anal. (2014).
-  H. Inci, On the well-posedness of the incompressible Euler equation, Dissertation, University of Zurich (2012), arXiv: 1301.5997.
-  T. Kato, Nonstationary flows of viscous and ideal fluids in , J. Funct. Anal. 9 (1972), 296-305.
-  T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Lect. Notes in Math., Springer 448 (1975), 25-70.
-  G. Kawahara, M. Uhlmann, L. van Veen, The significance of simple invariant solutions in turbulent flows, Ann. Rev. Fluid Mech. 44 (2012), 203-225.
-  T. Kreilos, B. Eckhardt, Periodic orbits near onset of chaos in plane Couette flow, Chaos 22 (2012), 047505.
-  J. Ladyman, J. Lambert, K. Wiesner, What is a complex system? European J. for Philosophy of Sci. 3 (2013), 33-67.
-  Y. Lan, Y. Li, On the dynamics of Navier-Stokes and Euler equations, J. Stat. Phys. 132 (2008), 35-76.
-  G. Leonov, N. Kuznetsov, Time-varying linearization and the Perron effects, Intl. J. Bifurcation and Chaos 17, no.4 (2007), 1079-1107.
-  Y. Li, Invariant manifolds and their zero-viscosity limits for Navier-Stokes equations, Dynamics of PDE 2, no.2 (2005), 159-186.
-  Y. Li, Major open problems in chaos theory and nonlinear dynamics, Dynamics of PDE 10, no.4 (2013), 379-392. arXiv: 1305.2864
-  M. Mitchell, Complexity, Oxford U. Press, 2009.
-  G. Paladin, A. Vulpiani, Predictability in spatially extended systems, J. Phys. A 27 (1994), 4911-4917.
-  L. van Veen, G. Kawahara, Homoclinic tangle on the edge of shear turbulence, Phys. Rev. Lett. 107 (2011), 114501.