Stochastic Three-Dimensional Rotating Navier-Stokes Equations: Averaging, Convergence, Regularity and 3D Nonlinear Dynamics Multiscale Analysis and High Performance Computation of Three-Dimensional Nonlinear Dynamics in Rotating Stratified Geophysical Flows ALEX MAHALOV Arizona State University F. Flandoli and A. Mahalov, Stochastic 3D Rotating Navier-Stokes equations: averaging, convergence and regularity, Archive for Rational Mechanics and Analysis, vol. 205, Issue 1, p. 195-237, 2012 .
Transcript
Stochastic Three-Dimensional Rotating Navier-Stokes Equations: Averaging, Convergence, Regularity and 3D Nonlinear Dynamics
Multiscale Analysis and High Performance Computation
of Three-Dimensional Nonlinear Dynamics in Rotating Stratified Geophysical Flows
ALEX MAHALOV
Arizona State University
F. Flandoli and A. Mahalov, Stochastic 3D Rotating Navier-Stokes equations: averaging, convergence and regularity, Archive for Rational Mechanics and Analysis, vol. 205, Issue 1, p. 195-237, 2012 .
Mathematics for Nonlinear Phenomena: Analysis and Computation Part 1. 3D Navier-Stokes Equations + Waves: Computation of Real Atmospheric Flows Part 2. Multi-Scale Analysis of the 3D Stochastic Rotating Navier-Stokes Equations: Averaging, Convergence and Regularity (joint work with Franco Flandoli, Pisa; Archive for Rational Mechanics and Analysis Vol 205 2012) Part 3. Mathematics of 3D Rotating Homogeneous Turbulence: Open Problems
Applications
● Mixing and Transport in Atmospheric Flows ● Wind and Solar Energy Research and Forecasting ● Remote Sensing ● Optical Communications ● Astronomy (Forecasting `Seeing’ Conditions) ● Aviation Industry ● Urban Atmospheres, Pollution ● Earth System Models at Decadal and Regional Scales:
Regional/Urban Climate and Energy
3D Navier-Stokes Equations + Waves: Computation of Real Atmospheric Flows
• Microscale nesting (space and time) and novel implicit relaxation computational techniques
• Boundary conditions from high resolution global/mesoscale datasets
• Targeted fine scale modeling and forecasts, nested simulations
• High Performance Computing (HPC)
3D Navier-Stokes Nonlinearity +
Mountain and Inertio-Gravity Waves
Fine Scale Modeling: Validation and Verification
Journal of Computational Physics 2008 Atmospheric Chemistry and Physics 2011
Physica Scripta 2015
Validation and Verification
Terrain-induced Rotor Experiment (T-REX) campaign of
measurements, Owens Valley, CA, March-April 2006 Targeted Simulations: IOPs of the T-REX campaign
NSF campaign of measurements
Global model
T-REX field campaign area and ground-based instrumentation layout (T-REX=Terrain-Induced Rotor Experiment Campaign of Measurements)
Global data and targeted microscale domain showing National Center for Atmospheric Research (NSF NCAR,
Boulder) research aircraft and balloon trajectories
v Nested simulations are initialized with analysis from NCEP GFS, ECMWF.
v four vertically&horizontally nested domains, microscale nest NCAR Research Aircraft and Two Balloons
Global model (NCEP GFS, ECMWF) Domain 1: Nested to the global model Domain 2: Nested to domain 1
Domain 3: Nested to domain 2 Domain 4: Nested to domain 3
Distributions of vector wind speed fields on 320 K isentrope on April 1, 2006. The dot indicates the location of balloon launching site.
3D Nonhydrostatic Navier-Stokes Eqs for Atmospheric Dynamics
( )( )
( )( )
( )
mQmmt
dt
dt
t
Wddt
Vydydt
Uxdxdt
FqVQ
gWV
V
FV
FpgwVW
FppvVV
FppuVU
=⋅∇+∂
=−∇⋅+∂
=⋅∇+∂
=⋅∇+Θ∂
=−∂−⋅∇+∂
=∂∂+∂+⋅∇+∂
=∂∂+∂+⋅∇+∂
Θ
010
φµ
φ
µ
θ
µαα
φαααµ
φαααµ
η
η
η
( )d
dhtdh ppµη −= dhtdhsd pp −=µ
The moist equations are formulated using a terrain-following pressure coordinate:
where
3D Fully Compressible Nonhydrostatic Equations for Atmospheric Dynamics
ddµαφη −=∂
;0
0
γ
αθ
⎟⎠⎞⎜
⎝⎛=
d
mdp
Rpp
gz=φd
d ρα 1=
( )WVUvV d ,,==
µ θµd=Θ
along with the diagnostic relation for the inverse density
and the equation of state
is the geopotential, is the inverse density of the dry air
and are the coupled velocity vector and potential temperature.
p is the pressure,
F represents forcing terms arising from model physics, turbulent mixing, rotation, …
...,,,, icvmmdm qqqqqQ == µ are the coupled mixing ratios of water vapor, cloud, ice, etc.
( )( ) ( )vvdvm qqRR 61.111 +≈+= θθθ
( ) 1....1 −+++++= ircvd qqqqαα
Topography and wind vector fields at 12 km altitude. The black curve shows the trajectory of balloon launched at (36.49 N, 118.84 W) on April 1, 2006 at 7:50 UTC
Potential temperature (black), eastward wind (blue), and northward wind (red) from balloon measurements during T-REX. The balloon was launched at (36.49 N, 118.84 W) on April 1, 2006 at 7:50 UTC. High levels of Optical Turbulence OT correspond to peaks of N2 (right panel).
Terrain-induced Rotor Experiment (T-REX) campaign of measurements, Owens Valley, CA , March-April 2006
TREX campaign Owens Valley, CA. Longitude (118.56 W, 117.42 W)-altitude cross-section at latitude 36.82 N for potential temperature (contour) and vertical velocity (color) for the microscale domain. Strong slanted optical layers in UTLS.
v Vertical cross-sections of horizontal wind component transverse to the valley (color, m/s) and potential temperature (K) on April 1, 2006 at 8 UTC: (a) across and (b) along the valley. (c) and (d) are the same as (a) and (b) respectively but at 6 UTC. v The horizontal axes X and Y indicate the distance with respect to the location (36.70 N, 118.50 W) and (36.29 N, 118.01 W) respectively.
Longitude-altitude cross-sections. Left panel: potential temperature (contour) and eastward wind (color); right panel: potential temperature (contour) and vertical velocity from the microscale domain.
Time series of (a) potential temperature, (b) vertical velocity , and (c) eastward (solid) and northward (dashed) winds from aircraft measurements. (d)-(f) are the same as (a)- (c) but from simulations.
simulations observations
Zooms of time series for (a) potential temperature and (b) vertical velocity from NCAR research aircraft HIAPER observations/measurements (dashed) and from simulations (solid).
Part 2 Rigorous Multi-Scale Analysis of the 3D Stochastic Rotating Navier-Stokes Equations: F. Flandoli and A. Mahalov, Stochastic 3D Rotating Navier-Stokes equations: averaging, convergence and regularity, Archive for Rational Mechanics and Analysis, vol. 205, Issue 1, p. 195-237, 2012 . Rigorous Multi-Scale Analysis of the 3D Navier-Stokes Nonlinearity + Waves (Deterministic: BMN 1995-2001) .
References
[1] W. Arendt, CJK Batty, M. Hieber and F. Neubrander (2001), Vector-Valued Laplace Transforms and Cauchy Problems, Birkhauser.
[2] V.I. Arnold and B.A. Khesin (1997), Topological Methods in Hydro-dynamics, Applied Mathematical Sciences, 125, Springer.
[3] A. Babin, A. Mahalov, and B. Nicolaenko (2001), 3D Navier-Stokesand Euler Equations with initial data characterized by uniformly largevorticity, Indiana University Mathematics Journal, 50, p. 1-35.
[4] A. Babin, A. Mahalov, and B. Nicolaenko (1999), Global regularityof the 3D Rotating Navier-Stokes Equations for resonant domains,Indiana University Mathematics Journal, 48, No. 3, p. 1133-1176.
[5] A. Babin, A. Mahalov, and B. Nicolaenko (1997), Global regularityand integrability of the 3D Euler and Navier-Stokes equations foruniformly rotating fluids, Asymptotic Analysis, 15, 103–150.
[6] A. Babin, A. Mahalov and B. Nicolaenko (1995), Long–time averagedEuler and Navier–Stokes equations for rotating fluids, In “Structureand Dynamics of Nonlinear Waves in Fluids”, K. Kirchgassnerand A. Mielke (eds), World Scientific, p. 145–157.
[7] L. Ca↵arelli, R. Kohn, L. Nirenberg, Partial regularity of suitableweak solutions to the NavierStokes equations, Comm. Pure. Appl.Math. 35 (1982), 771831.
[8] Y. Giga, A. Mahalov and B. Nicolaenko (2007), The Cauchy problemfor the Navier-Stokes equations with spatially almost periodic initialdata, Annals of Mathematics Studies, 163, p. 213-223, PrincetonUniversity Press.
[9] Y. Giga, K. Inui, A. Mahalov, S. Matsui and J. Saal (2007), Rotat-ing Navier-Stokes equations in R3
+ with initial data nondecreasing atinfinity: the Ekman boundary layer problem, Archive for Rational
Mechanics and Analysis, 186, No. 2, p. 177-224.
[10] Y. Giga, K. Inui, A. Mahalov and S. Matsui (2006), Navier-Stokesequations in a rotating frame with initial data nondecreasing at infin-ity, Hokkaido Mathematical Journal, 35, No. 2, p. 321-364.
[11] T. Kato (1972), Nonstationary flows of viscous and ideal fluids in R3,J. Func. Anal., 9, p. 296-305.
[12] O.A. Ladyzhenskaya (1969), Mathematical Theory of Viscous Incom-pressible Flow, 2nd ed., Gordon and Breach, New York.
[13] J. Leray (1934), Sur le mouvement d’un liquide visqueux emplissantl’espace, Acta Math., 63, p. 193-248.
[14] H. Poincare (1910), Sur la precession des corps deformables, Bull.Astronomique, 27, p. 321-356.
[15] S. L. Sobolev (1954), Ob odnoi novoi zadache matematicheskoi fiziki,Izvestiia Akademii Nauk SSSR, Ser. Matematicheskaia, 18, No.1, p. 3–50.
[16] T. Yoneda (2010), Long-time solvability of the Navier-Stokes equa-tions in a rotating frame with spatially almost periodic large data,Archive for Rational Mechanics and Analysis, 10.1007/s00205-010-0360-4, p. 1-13.
3D Navier-Stokes Equations + Waves
Stochastic 3D Rotating Navier-Stokes Equations:Averaging, Convergence and Regularity
The stochastic 3D incompressible Navier-Stokes Equations
@tU � ⌫�U + (U ·r)U = �r⇡ +pQ@W@t
,
divU = 0,
U |t=0 = U 0
where U(t, x) = (U1, U2, U3), x = (x1, x2, x3) is the velocity field (arandom 3D vector field), ⇡(t, x) is the pressure (a random scalar field),⌫ > 0 is the kinematic viscosity, W(t) is a cylindrical Wiener process,defined on a filtered probability space (⌦, Ft,P), in the Hilbert spaceH = L2
s of square integrable solenoidal vector fields and the operator Qis a non-negative, symmetric, of trace class in H .
Cylindrical Wiener Processes
Consider a sequence {Wn (t)}n2N of independent 1-dimensional Wienerprocesses, together with a complete orthonormal system {bn}n2N of aHilbert space H . Formally speaking, the cylindrical Wiener process is
W (t) :=1X
n=1
Wn (t) bn
but this series does not converge in H in any natural probabilistic sense
(the formal idea is that EhkW (t)k2H
i=
P1n=1E
⇥W2
n (t)⇤=
P1n=1 t =
1). The series above converges in suitable “negative order” space Y � H ,but this fact is not used here. What is used here is that the series
pQW (t) :=
1X
n=1
�nWn (t) bn
converges in L2 (⌦;H) when Q is a trace class non-negative, self-adjointoperator in H such that Qbn = �2
nbn
(indeed Eh��pQW (t)
��2i
=P1
n=1 �2nE
⇥W2
n (t)⇤= t
P1n=1 �
2n < 1);
and, similarly, stochastic integrals of the formZ t
0Ts
pQdW (s) =
1X
n=1
�n
Z t
0TsbndWn (t)
converge in L2 (⌦;H) and define continuous martingales in H , when Ts
are bounded operators, strongly measurable and bounded in s (or moregeneral operator valued processes). A relevant example in fluid dynamics,especially for the investigation of turbulence and transport of energy be-tween scales, is the case when Q is finite range, namely when �n 6= 0 onlyfor a finite number of n’s. In this case
pQW (t) is a finite dimensional
noise, a random activation of a finite number of (low) modes bn.
Assume U 0 2 H and D be a periodic domain or R3. Existence of weak(also in the probabilistic sense) solutions of the above stochastic 3DNSEwith the energy inequality property is known, E denoting expectation:
E
Z
D
|U(t, x)|2 dx�+ 2⌫E
Z t
0
Z
D
|rU(s, x)|2 dxds�
E
Z
D
��U 0(x)��2 dx
�
+ Trace (Q) t,
E
"sup
t2[0,T ]
Z
D
|U(t, x)|2 dx#< 1
If U 0 is su�ciently regular, existence and uniqueness of a regular solutionis also known on a local random time interval [0, ⌧U0), but a priori ⌧U0
may be very small.
Key Norm:
Z
D
|rU(t, x)|2 dx
Stochastic 3D Rotating Navier-Stokes Equations
The Stochastic 3D Rotating Navier-Stokes Equations (RNSE):
@tU � ⌫�U + (U ·r)U +1
✏e3 ⇥ U = �r⇡ +
pQ@W@t
, (1)
divU = 0, (2)
U |t=0 = U 0(x1, x2, x3). (3)
We consider these equations on the torus D = [0, 2⇡]3 with periodicboundary conditions and zero average. For stress-free boundary conditionsin x3 we only need to restrict Fourier series to be even in x3 for U1, U2 andodd in x3 for U3. In (1), e3 is the vertical axis, e3 ⇥ U = (�U2, U1, 0) =
JU . Here J =
0
@0 �1 01 0 00 0 0
1
A is the rotation matrix corresponding to the
Coriolis term. Eqs. (1)-(3) are fundamental in meteorology and geophys-ical fluid dynamics where the Coriolis force an Poincare waves play anessential role.
SAME ENERGY INEQUALITY AS THE 3D NAVIER-STOKESEQUATIONS WITHOUT ROTATION
Main Regularity Theorem
For the stochastic 3D RNSE (1)-(3) with fast rotation, we prove
global well posedness up to exceptional events. Precisely, assume the
initial condition U 0 2 H3. Locally in time there is a unique regular
solution. Denote the (potential) explosion time in the H1 norm by
⌧ "U0, a random variable. A priori, ⌧ "
U0 can be very small with very
large probability. We prove that, given any arbitrary large final time
horizon T ,
P�⌧ "U0 > T
�= 1, (4)
namely the probability of non explosion up to T is one if the intensity
of rotation is su�ciently high, 0 < ✏ < ✏0.
In general, one can prove only P (⌧U0 > 0) = 1 for the stochastic 3DNSE. However, it is not as strong as the corresponding deterministic result,where ⌧U0 = 1 when rotation is large enough ([3], [4], [5]). Given theincremental covariance and the final time horizon T , the noise producesarbitrarily large excursions of the solutions on [0, T ] with small but non-zero probability, and blow-up may occur in principle. Moreover, over[0,1), large excursions appear with probability one, hence we cannot hopeto throw away a small probability event and have global well posednesson [0,1).
The energy injected in the system by the noise may be large, the initialcondition may have large energy, and the observational time horizon canbe long; regularization is the consequence of a precise mechanism, not justabsence of relevant 3D nonlinear dynamics. To understand reality, thisresult is much more interesting that the simple case of small initial condi-tions and small noise; it deals with stochastic fluids having intense activityand energy, and proves that energy and vorticity remain bounded due tofast rotation. Since rotation (global or local) is one of the most commonfeatures of real fluids, the principle that fast rotation has a smoothinge↵ect is important for applications.
Besides regularity results, we prove also averaging theorems. We provethat, as " ! 0, the solution of equations (1)-(3) converges to the solutionof a 3D stochastic equation of Navier-Stokes type, the so called stochasticresonant averaged equations, which have better regularity properties thatthe original stochastic 3D NSE. Here the regularity comes out as a mainresult along with the averaging, due to the regularity of the stochasticaveraged equations.
The mathematical methods that we develop are applicable to generalstochastic PDEs (SPDEs) written in the operator form (6). We note thatour main focus is on the 3D Navier-Stokes equations + Waves for whichglobal regularity results are not known.
The case of flows in R3 with decaying initial condition and random forceacting in a compact region of space can also be considered in our frame-work. The periodic case studied here is more di�cult due to resonancesin the three-dimensional nonlinear stochastic dynamics andthe lack of dispersion.
We write stochastic equations (1)-(3) in a form convenient for averaging.Let A be the Stokes operator
A = Pcurl2P = �P�P, (5)
there P denotes Leray projection onto divergence free vector fields. Clearly,A = curl2 = �� on divergence free vector fields. Applying to theStochastic 3D RNSE (1)-(3) the Leray projection P , we obtain for U =PU
@tU + ⌫AU +1
✏SU + B(U,U) =
pQ@W
@t , (6)
U |t=0 = U 0, (7)
where
B(U,U) = P (U ·rU) = P (curlU ⇥ U), S = PJP. (8)
The Poincare-Coriolis operator S = PJP in (6) is skew-symmetric.
We consider (6) and define the new process u(t)
U(t) = ⌥(�t/✏)u(t), u(t) = ⌥(+t/✏)U(t) (9)
where ⌥(�t/✏) = e�St/✏ is a unitary group in the Hilbert space H .The Poincare-Coriolis operator S is the generator of that unitary group.Eqs (6) written for the process u(t) have the form:
@tu + ⌫Au + B(t/✏, u, u) = ⌥(t/✏)pQ@W@t
, (10)
B(t/✏, u, u) = ⌥(t/✏)B(⌥(�t/✏)u,⌥(�t/✏)u), (11)
u|t=0 = U 0, (12)
where B is given by Eqs. (8). The term B(t/✏, u, u) defined in (11) hashighly oscillatory operator coe�cients. In general, the operators Q and S
do not commute and, hence, the operators Q and ⌥(t/✏) do not commute.We note that all regularity results for u(t) immediately imply regularityresults for U(t) since the operators ⌥(±t/✏) are isometries in spaces H↵
for every ↵ � 0.
Averaged Operators and 3D Resonances
There are two essential ingredients in the definition of the stochasticresonant averaged 3D RNSE. First, the key ingredient is to introduce theaveraged covariance operator Q. Let Q be a positive trace class operatoron H . Let t1 > 0. Define
Q := lim✏!0
1t1
Z t1
0erS/✏Qe�rS/✏ dr = lim
T1!11T1
Z T1
0e⌘SQe�⌘S d⌘. (13)
These are Cesaro type averages. The above limit exists in operator norm,and Q is a positive trace class operator. Another key remark, which isalso done in the deterministic theory of 3D RNSE, is that the operatorB(t/", u, u) splits into two parts
B
✓t
", u, u
◆= eB (u, u) + Bosc
✓t
", u, u
◆(14)
where eB (u, u) is independent of t. It is called the resonant operator; theother operator, Bosc
�t", u, u
�, the non-resonant one, is highly oscillatory
for small " and will be averaged out in the limit ✏ ! 0. The averagedcovariance operator Q and nonlinear operator B form the 3D averagedstochastic equation (31), for which global regularity is proven and completetheory is developed.
For the fully three-dimensional stochastic problem (1)-(3), we prove: (i)averaging theorems for the corresponding stochastic problem in the case ofstrong rotation (0 < ✏ ✏0); (ii) regularity results for solutions of (1)-(3)by bootstrapping from global regularity of the limit stochastic equationand convergence theorems.
Convergence of Cesaro Averages
We refer to [1] for definitions of Cesaro averages for operator valuedfunctions. Let ⌥ = (⌥t)t2R be a strongly continuous unitary group on aHilbert space H .
Theorem.Let Q be a positive trace class operator on H. Then
CQ := limT!1
1T
Z T
0⌥tQ⌥�t dt
exists in operator norm, CQ is a positive trace class operator, and
Tr(CQ) = Tr(PQ).
This Theorem is used to define the averaged covariance operator Q (Q ⌘CQ). Here unitary group ⌥(t) = etS has the Poincare-Coriolis operator Sas its generator. We have Tr(Q) = Tr(Q).
Averaged operators and 3D resonances
For the stochastic 3D Rotating Navier-Stokes Equations (1)-(3), theoperator B
�t", u, u
�was defined above in (11). Now we introduce the
corresponding bilinear operator B�t", u, v
�:
B
✓t
", u, v
◆= �e
t"SP
⇣e�
t"Su⇥ e�
t"Scurlv
⌘.
The operator B�t", u, v
�can be decomposed into a sum of the resonant
and non-resonant (oscillatory) parts:
B
✓t
", u, v
◆= eB (u, v) + Bosc
✓t
", u, v
◆. (15)
The operator Bosc�t", u, v
�is highly oscillatory for small ".
The nonlinear interactions in the resonant operator eB are restricted tothe resonant set
D (k,m, �1, �2) = ��1k3|k| � �2
m3
|m| + �k,m,�1,�2
(k +m)3|k +m| = 0, (16)
where 3D wavevectors k = (k1, k2, k3), |k| =pk21 + k22 + k23 and similar
for wavevectors m and n; n = k +m is Fourier convolution. The bilinearresonant operator eB satisfies the important inequality
����⇣eB (u, u) , Au
⌘
L2
���� C kuk2H1kukH2
, u 2 H2, (17)
where u(x1, x2, x3) are three-dimensional vector fields. Here A = curl2 =�� is the Stokes operator. The above inequality follows from Lemmason restricted convolutions in [3] and [4]. The significance of this esti-mate is that the operator eB acting on 3D vector fields has fully three-dimensional nonlinear interactions restricted on a resonant manifold inFourier space. This restriction leads to an improvement in estimates forthe three-dimensional operator eB in comparison with the operator B.
Moreover, both eB and Bosc satisfy
(B (u, u) , u)L2= 0 (18)
and Kato inequalities [11], true for s � 2,
kB (u, v)kHs C kukHs
kvkHs+1(19)
| (B (u, v) , v)Hs| C kukHs
kvk2Hs(20)
| (B (u, v) , v)H2| C kukH3
kvk2H2(21)
| (B (u, v) , u)Hs| C kuk2Hs
kvkHs+1. (22)
We note that the operator B�t", u, v
�also satisfies inequalities (19)-
(22). For example, using the fact that the operators e±t"S are isometries
in Hs and (19), we obtain����B✓t
", u, v
◆����Hs
=����e
t"SB
⇣e�
t"Su, e�
t"Sv
⌘���Hs
=���B
⇣e�
t"Su, e�
t"Sv
⌘���Hs
C���e�
t"Su
���Hs
���e�t"Sv
���Hs+1
= C kukHskvkHs+1
,(23)
where C is a constant independent of ✏.Now we show how resonances appear in the averaged covariance op-
erator Q defined by (13). The operator eQ is a trace class non-negative,self-adjoint operator in H , and we have
DeQg, h
E
H:= lim
"!0
1
t1
Z t1
0
DQe�
r"Sg, e�
r"Sh
E
Hdr (24)
for all g, h 2 H , independent of t1 > 0. Clearly, eQ = Q if the operatorsQ and S commute (equivalently, Q and curl commute). The averagedoperator eQ includes new resonant terms in the non-commutative caseQcurl 6= curlQ.
The averaged covariance operator Q and nonlinear opera-tor B form the stochastic resonant averaged 3D RNSE thatis globally well-posed.
Averaging of Holder Continuous Random Processes
Let (⌦, F,P) be a probability space with expectation E. Let f =f (t,!), t 2 [0, T ], ! 2 ⌦, be a stochastic process. Assume that supt2[0,T ]|f (t)|is measurable.
LemmaLet p � 1, � 6= 0 be given. Assume that
E
"sup
t2[0,T ]|f (t)|p
# C0
E [|f (t)� f (s)|p] C0 |t� s|↵ for all t, s 2 [0, T ] .
Then (for a new constant C1 = C1 (C0, T, p))
E
"sup
t2[0,T ]
����Z t
0ei
s"�f (s) ds
����p# C1
⇣���"
�
���↵
+���"
�
���⌘.
Stochastic Limit Resonant Equations
Let�⌦, F, (Ft)t�0 ,P
�be a standard filtered probability space and
(W (t))t�0 be a cylindrical Wiener process in H .
Let
✓e⌦, eF ,
⇣eFt
⌘
t�0, eP ,
⇣fW (t)
⌘
t�0
◆be another copy of it. Let Q be
a trace class non-negative, self-adjoint operator in H . Let M" (t) be theGaussian martingale defined as
M" (t) =
Z t
0er"Sp
QdW (r) . (25)
The martingaleM" (t) is associated with the stochastic term⌥(t/✏)pQ@W
@t
in (10). The laws of the Gaussian martingales M" (t) are tight in propertopologies so we can prove that they converge weakly (as " ! 0) to thelaw of a Gaussian martingale M (t). We may identify the limit processM (t) as a Wiener process in H with covariance eQ defined by (13).
Consider the following four stochastic equations.Equation 1:
duR" +
⌫AuR" + eB
�uR" , u
R"
�+ �R
⇣��uR"��2H3
⌘Bosc
✓t
", uR" , u
R"
◆�dt = dM"
(26)where �R : [0,1) ! [0, 1] is a smooth function, equal to 1 on [0, R] andto zero on [R + 1,1). Here R > 0 is a large parameter.
Equation 2:
du" +
⌫Au" + eB (u", u") + Bosc
✓t
", u", u"
◆�dt = dM" (27)
Equation 3:
dw" +h⌫Aw" + eB (w", w")
idt = dM" (28)
Equation 4:
dw +h⌫Aw + eB (w,w)
idt =
qeQdfW (t) (29)
all with the same smooth initial condition u0 = U 0. We note that equa-tions (26)-(28) have the same noise. Clearly, for solutions of (26) and (27)
we have uR" (t) = u"(t) up to time for which��uR" (t)
��2H3
reaches R.
Equation 2:
du" +
⌫Au" + eB (u", u") + Bosc
✓t
", u", u"
◆�dt = dM". (30)
This equation is precisely equation (10) with the operator B written asa sum of the resonant eB and non-resonant Bosc parts. Our choice of thespace H3 for initial data in the above definition is natural to considersolutions of the stochastic 3D NSE in the inviscid limit ⌫ ! 0 (under ap-propriate scalings). We recall that initial data for the 3D Euler equationsis required to be at least in H3 for local regularity ([11]).
Equation 4:
dw +h⌫Aw + eB (w,w)
idt =
qeQdfW (t) (31)
Both with the same smooth initial condition u0 = U 0.The bilinear resonant operator eB satisfies the important inequality
����⇣eB (w,w) , Aw
⌘
L2
���� C kwk2H1kwkH2
, w 2 H2, (32)
where w(x1, x2, x3) are three-dimensional vector fields. Here A = curl2 =�� is the Stokes operator.
Stochastic Averaging Convergence Theorems
When we compare equations (27) and (28) which have the same noise,we can prove convergence in probability. On the other hand, since thefamily of Wiener processes M✏(t) (Q (", t) = e
t"SQe�
t"S) converges only in
law as ✏ ! 0, we prove only convergence in law for the corresponding so-lutions of SPDEs (27) and (31). We recall that equation (27) is equivalentto the stochastic 3D RNSE (1)-(3).
Let u✏, w✏ and w be solutions of stochastic equations (27)-(31), respec-tively. We have the following type of convergence for solutions of SPDEsas " ! 0:
u✏ � w✏ ! 0 in probability, (33)
u✏ ! w in law, (34)
w✏ ! w in law. (35)
Here we assume U 0 2 H3+⌘ for some ⌘ > 0.The same result holds for equation (1)-(3) (equivalently, (6)-(7)) since
⌥(�t/✏) = e�St/✏ preserves all Sobolev norms H↵. Let T > 0 arbitrarylarge. In law, the solution U✏(t) of the original stochastic 3D RNSE (1)-(3) is close on [0, T ] to the transformation of the (moderately varying)solution w(t) to the SPDE (31) without parameter ✏, by the fast rotationunitary group e�tS/✏. Namely, as ✏ ! 0,
⇣U✏(t)� e�tS/✏w(t)
⌘! 0 (36)
on arbitrary large time intervals [0, T ] as ✏ ! 0. Here U✏(t) and w(t)are solutions of the corresponding equations (1)-(3) and (31) with thesame initial condition. The stochastic dynamics is fully three-dimensional with dependence on all three space variablesand not close to any 2D manifold.
Part 3 Mathematics of 3D Rotating Homogeneous Turbulence: Open Problems Y. Giga, A. Mahalov and B. Nicolaenko (2007), The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data, Annals of Mathematics Studies, vol. 163, p. 213-223, Princeton University Press. T. Yoneda (2010), Long-time solvability of the Navier-Stokes equations in a rotating frame with spatially almost periodic large data, Archive for Rational Mechanics and Analysis, 10.1007/s00205-010-0360-4, p. 1-13.
We are very grateful to the Organizing Committee and Hosts for this
Fantastic Conference
Mathematics for Nonlinear Phenomena: Analysis and Computation, Sapporo 2015
Thank you very much!
Dedicated to Professor Yoshikazu Giga on his 60th Birthday with Admiration
