An invariant in shock clustering and Burgers turbulence

0.13 MB·English

by  Ravi Srinivasan

An invariant in shock clustering and Burgers
turbulence
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1 Ravi Srinivasan1
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2 January 9, 2012

n
a
J
Abstract

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1-DscalarconservationlawswithconvexfluxandMarkovinitialdata

] are now known to yield a completely integrable Hamiltonian system. In
I
S this article, we rederive the analogue of Loitsiansky’s invariant in hy-
drodynamicturbulencefromtheperspectiveofintegrablesystems. Other
.
n relevantphysicalnotionssuchasenergydissipationandspectrumarealso
i
l discussed.
n
[ MSC classification: 60J35, 35R60,37K10, 35L67,82C99

2 Keywords: Shock clustering, stochastic coalescence, kinetic theory, Burgers
v
turbulence, integrable systems, Loitsiansky invariant.
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. 1 Introduction
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0
1 The inviscid Burgers equation
1
: u2
v ∂ u+∂ =0, x∈R,t>0, u(0,x)=u (x) (1)
i t x(cid:18) 2 (cid:19) 0
X
r with random initial data u has been extensively studied over the course of
a 0
several decades. As a caricature of hydrodynamic turbulence, the earliest fun-
damentalresultsareduetoBurgershimself,whostudied(1)foru awhitenoise
0
process in space [7]. When considered with an added random forcing term, the
equation becomes a model of random growth with deposition. Taken together,
these form the basis of Burgers-KPZ turbulence. A broad overview of these
areas can be found in [25] and the review article [1].
Many essential features of Burgers turbulence are preserved in the more
general context of 1-D scalar conservation laws with strictly convex flux and
random initial data. In recent work [21], we have shown the class of Markov
1DepartmentofMathematics,TheUniversityofTexasatAustin,Austin,TX78712. Email:
[email protected]
1
processes in space with only downwardjumps is preservedby the deterministic
dynamics. Althoughitisoneofthesimplestexamplesofanontrivialflowonthe
space of probability measures, this model has surprisingly rich behavior. The
evolution is given in terms of a Lax equation, and it has deep connections to
kinetic theory,randommatrices,andstatistics. Moreover,it hasbeen shownto
be a completely integrable Hamiltonian system [19]. As an integrable system,
its evolution is constrained by conserved quantities. In this article we examine
how one particular conserved quantity of physical interest, well-known in the
turbulence literature, arises in our setting.
1.1 Entropy solutions to scalar conservation laws
To begin, let us review some basic facts about the 1-D scalar conservation law
∂ u+∂ f(u)=0, x∈R,t>0 (2)
t x
with strictly convex flux f ∈ C1 and initial data u(x,0) = u (x). Classical
0
solutionsfoundusing themethodofcharacteristicstypicallyceasetoexistafter
a finite time due to the formation of shocks. Instead, a suitable global-in-time
solutionto(2)is the entropy solution, derivedviaavariationalprinciple forthe
action functional
x−s
I(s;x,t)=U (s)+tf∗ . (3)
0
(cid:18) t (cid:19)
Here,U0(s)= 0su0(r)dr isthe initialpotentialandf∗(s)=supu∈R{us−f(u)}
the Legendre tRransform of the flux. We assume that U0 has no upward jumps
and that lim I(s;x,t) = +∞ always holds, ensuring that the infimum of
|x|→∞
I is achieved. The inverse Lagrangian function a(x,t) is then defined by the
Hopf-Lax formula:
a(x,t)=arg+minI(s;x,t). (4)
s∈R
The + denotes that we choose a(x,t) to be the largest value where I achieves
its minimum. In terms of infinitesimal particles, a gives the (rightmost) initial
location of the particle at location x at time t. For fixed t > 0, a(x,t) is
increasing in x. At points of continuity of a(·,t) the entropy solution is
x−a(x,t)
u(x,t)=(f′)−1 . (5)
(cid:18) t (cid:19)
Downwardjumpsinucorrespondtoshocks,andthevelocityofashockatxwith
left- and right-limits u =u(x ,t) satisfies the Rankine-Hugoniot condition
± ±
f(u )−f(u )
− +
u(x,t)= =:[f] . (6)
u −u u−,u+
− +
2
1.2 Loitsiansky’s invariant
Now suppose that u (x) is a mean-zero velocity field which is statistically sta-
0
tionary and ergodic in x. Then for each t > 0, the entropy solution u(x,t)
remains stationary ergodic and ensemble averages are equivalent to spatial av-
erages(whichwedenotebybrackets). Ithaslongbeenknownthat(1)admitsa
conserved quantity analogous to Loitsiansky’s invariant in hydrodynamic tur-
bulence [17]:
∞
J (t)= hu(x,t)u(x+h,t)idh. (7)
0
Z
0
J is simply the integralofthe two-pointcorrelationfunction, andis essentially
0
thevalue ofthe energyspectruminthe limitoflowwavenumbers. Inaseriesof
articles[3,4,5, 6], Burgersprovidedthree distinct derivationsof the invariance
of J . Upon doing so he investigated its implications on the evolution of a
0
variety of shock statistics. Intriguingly, many of Burgers’ calculations involve
(unclosed) kinetic equations describing shock coalescence as in [11], and are
quite prescient in that respect.
It is easily seen that J is preserved for any 1-D scalar conservation law. A
0
formal derivation is as follows. Denoting u(x) = u(x,t), one has that for any
h>0,
∂ hu(x)u(x+h)i=hu(x)∂ u(x+h)+∂ u(x)u(x+h)i
t t t
=−hu(x)∂ f(u(x+h))+∂ f(u(x))u(x+h)i.
x x
Therefore,thetwo-pointcorrelationfunctionθ(h)=hu(x)u(x+h)iitselfsatisfies
a conservation law with flux τ(h)=hu(x)(f(u(x+h))−f(u(x−h)))i:
∂ θ(h)+∂ τ(h)=0. (8)
t h
For Burgers’ flux f(u) = u2/2, this is the analogue of the K´arm´an-Howarth
equation for three-dimensional isotropic turbulence in the vanishing viscosity
limit. Assuming that the correlation length of the initial velocity field decays
sufficiently fast, one has that τ(h)→0 as h→∞ and
∞
∂ J (t)=∂ θ(h)dh=0.
t 0 t
Z
0
If, in addition, u(x) is statistically invariantunder the transformationx7→−x,
u7→−u, the correlation flux takes the form
τ(h)=−2hf(u(x))u(x+h)i. (9)
Whenf(u)=u2/2,thisimpliesthatτ =−1S withthird-orderstructurefunc-
6 3
tion S (h)= (u(x+h)−u(x))3 . This is particular to Burgersequation since
3
τ typically ca(cid:10)nnot be expressed i(cid:11)n a simple manner using structure functions.
Finally, we briefly remark on the physical relevance of the conserved quan-
tity (7) for isotropic Navier-Stokes turbulence in the infinite domain. In this
3
context, it is widely believed that certain universal scaling laws, such as a law
of energy decay, should hold for the statistics of the velocity field. Dimensional
analysis dictates that the mean energy dissipation should be proportional to
U3/L, where U(t) is the root-mean square velocity and L(t) is the integral
length scale. A self-similarity ansatz for the velocity correlation function [24]
relates the scaling of U(t) and L(t)—however, an additional relation is needed
to obtain the scaling exponents themselves. The invariance of J , commonly
0
referred to as the “permanence of large eddies,” was utilized by Kolmogorov
[16] to obtain a power law for the energy decay. While this was later shown
to hold only under particular assumptions on the initial spectrum, it remains a
fundamental result in the theory. In the simpler setting of Burgers turbulence,
many of the same issues have been thoroughly investigated and settled. We
direct the interested reader to [1, 14, 15] for a comprehensive discussion.
1.3 Lax equation and integrability of shock clustering
To start, we take u (x) to be a stationary Markovprocess with only downward
0
jumps. In [21], it was proven that the entropy solution to (2) then satisfies a
closure property: for each fixed t>0, u(x,t) is a stationary Markov process in
x with only downward jumps. This was also shown to hold if the initial data
is the derivative of a stationary L´evy process in x with only downward jumps
(e.g., white noise). A formalderivationofthis closurefor Burgersequationwas
obtained in [10].
Denote by p(dy,t) and {q (y,dz,t)} the stationary and transition mea-
h h>0
suresin x of the processu(x,t). The evolutionof the statistics canbe stated in
terms of the generator of the solution process. First, define the 1- and 2-point
operators through their action on appropriate test functions ϕ:
P(t)ϕ(y)= p(dy,t)ϕ(y), Q (t)ϕ(y)= q (y,dz,t)ϕ(z). (10)
h h
ZR ZR
ConsiderthegeneratorA(t)ϕ=lim 1 (Q (t)ϕ−ϕ)oftheMarkovsemigroup
h↓0 h h
in x (assuming the Feller property holds). For each fixed t>0, it has the form
A(t)ϕ(y)=b(y,t)ϕ′(y)+ n(y,dz,t)(ϕ(z)−ϕ(y)), (11)
ZR
where b(y,t) is the drift coefficient of the process and n(y,dz,t) is the jump
measure. Now we define an operator B(t) which, for each fixed x, serves as a
‘generator’of the solution in t:
B(t)ϕ(y)=−b(y,t)f′(y)ϕ′(y)− n(y,dz,t)[f] (ϕ(z)−ϕ(u)). (12)
y,z
ZR
B(t) truly is the generator of a Markov semigroup if f is decreasing. As shown
in [21], the evolution of the 1- and 2-point operators then takes the form
∂ P =PB, ∂ Q =[Q ,B] (13)
t t h h
4
where square brackets denote the commutator. In terms of generators, this is
the Lax equation
∂ A=[A,B]. (14)
t
Wehaveshownthat(14)isequivalenttoakineticequationfortheshockstatis-
tics. Theseequations(remarkably!) admitexplicitsolutionswhenu isaBrow-
0
nianmotion[2,8,9,22]orawhitenoise[7,12,13]. Inaddition,therearemany
intriguing links to random matrices and integrable systems. This is discussed
at length in [21].
2 The Manakov relation and an invariant of the
Lax equation
WhiletheLaxequationyieldssomeconservedquantities,thesearenotsufficient
to fully describe the evolution of the system. One can, in addition, verify that
the operators A and B satisfy the Manakov relation
[A,N]−[M,B]=0. (15)
with multiplication operators
Mϕ(y)=yϕ(y), Nϕ(y)=f(y)ϕ(y). (16)
Equation(15)isrigidinthatitonlyholdswithmultiplicationoperatorsMand
N: if we let M ϕ(y) = ψ(y)ϕ(y) for any ψ, then [A,M ]−[M ,B]= 0 if
ψ ψA ψB
and only if ψ (y)=f(y) and ψ (y)=y.
A B
The Manakov relation yields the necessary additional integrals by allowing
a spectral parameter to be introduced as in [18]. That is, by (14) and (15),
∂ (A−µM)=[A−µM,B+µN], µ∈C. (17)
t
IfoneconsidersdiscretizationsofAandB byN×N matrices,thisimplies that
the spectral curve
Γ= (λ,µ)∈C2|det(A−λId−µM)=0 . (18)
(cid:8) (cid:9)
remains unchangedin time. The conservedquantities are then givenby the co-
efficientsofthecharacteristicpolynomial. IthasrecentlybeenshownbyMenon
[19] that this discretized system, the Markov N-wave model, is a completely
integrableHamiltoniansystem. Whiletheargumentaboveshowsthatthespec-
trum is invariant in the discrete setting, this does not remain true as N → ∞
since the continuous spectrum may evolve. Therefore, it is an interesting and
open problem to find conserved quantities that survive in the continuum limit,
and to determine which of these are finite. Since the discrete model does not
allow for nontrivial statistically stationary solutions unless one takes N → ∞,
we note that the invariant J can only appear in this limit.
0
5
Letusnowstateourmainresult. Inwhatfollowswewillassumethatp(dy,t)
andq (y,dz,t)decay fastenoughfor the appropriateintegralsto converge. For
h
stationary Markov solutions to (2), we find that the invariance of
∞ ∞
J = E[u(x,t)u(x+h,t)]dh= p(dy,t)y q (y,dz,t)z dh (19)
0 h
Z0 Z0 (cid:18)ZR ZR (cid:19)
is equivalent to the identity
PA(f(y)Q y)=0. (20)
h
Sinceuisstationaryinx,thisistheweakformoftheforwardequation0=A†p
with test function f(y)Q y. More generally, fix n ∈ N, denote h = x −x
h i i i−1
and Q = Q for i = 1,...,n−1, and let h = (h ,...,h ). We demon-
i hi 1 n−1
strate that the Manakov relation (15) implies the n-point function θ(h,t) =
PyQ y···Q y satisfies the conservation law
1 n−1
∂ θ(h)+∇·T(h)=0 (21)
t
withfluxT (h,t)=PyQ y···Q (f(y)Q y−yQ f(y))Q y···Q y. Inte-
i 1 i−1 i i i+1 n−1
gration with respect to h shows that
J(n) = E[u(x ,t)u(x ,t)···u(x ,t)]dx ···dx (22)
0 Z 0 1 n−1 1 n−1
x0<x1<···<xn−1
is conserved by the evolution.
We note thattheseinvariantscanalsobe arrivedatbythe formalargument
intheintroduction,astheessentialpropertiesarethestationarityofthevelocity
field and sufficient decay in the correlation length scale. The purpose of this
article,however,is not to provea new resultbut to provide a novelperspective
on the matter from the viewpoint of integrable systems.
2.1 Derivation of main result
Forsimplicitywewillsuppresstinthenotationwheneverpossible. Webeginby
showingtheequivalenceof(19)and(20). Todothis,weneedonlydemonstrate
that the conservation law for the one-point function
∂ (PyQ y)=−∂ (P{−f(y)Q y+yQ f(y)}) (23)
t h h h h
holds. Recall that ∂ Q =AQ =Q A. Then (23) is equivalent to
h h h h
P{f(y)AQ y−yQ Af(y)−ByQ y+y[Q ,B]y}=0. (24)
h h h h
A short computation gives that
f(y)AQ y−yQ Af(y)=b(y)f(y)(∂ Q y)−yQ (b(y)f′(y))
h h y h h
+f(y) n(y,dz)(Q z−Q y)
h h
ZR
−yQ n(y,dz)(f(z)−f(y)) .
h
(cid:18)ZR (cid:19)
6
Similarly,
ByQ y+y[Q ,B]y =−b(y)f′(y)Q y−yQ (b(y)f′(y))
h h h h
− n(y,dz)(f(z)−f(y))Q z
h
ZR
−yQ n(y,dz)(f(z)−f(y)) .
h
(cid:18)ZR (cid:19)
Substituting these into (24) and collecting terms yields (20).
Similar calculations hold for the n-point function and become clearer upon
explicit use of the Manakov relation (15). Using that constants are in the
nullspace of A and B, equations (14) and (16) imply that
∂ (PyQ y···Q y)
t 1 n−1
n−1
=PByQ y···Q y+ PyQ y···Q y[Q ,B]yQ y···Q y
1 n−1 1 i−1 i i+1 n−1
Xi=1
n−1
=P[B,M]Q y···Q y+ PyQ y···Q [B,M]Q y···Q y.
1 n−1 1 i i+1 n−1
Xi=1
Substituting (15), the previous expression becomes
n−1
P[N,A]Q y···Q y+ PyQ y···Q yQ [N,A]Q y···Q y
1 n−1 1 i−1 i i+1 n−1
Xi=1
n−1
= ∂ {PyQ y···Q (f(y)Q y−yQ f(y))Q y···Q y},
hi 1 i−1 i i i+1 n−1
Xi=1
which is equivalent to (21).
We remarkthat the aboveargumentclarifieswhy quantities notofthe form
(22), such as E[u2(x)u(x+h)]dh, are typically not conservedin time:
R
∞ ∞
∂ Py2Q ydh =∂ PyQ yQ y| dh
tZ 2 2 tZ 1 2 h1=0 2
0 0
∞
= P(f(y)Ay−yAf(y))Q ydh 6=0
2 2
Z
0
except when y and f(y) are in the nullspace of A.
3 Exact solutions, energy, and spectra
3.1 White-noise initial data
The invariant J is expressed in terms of the 1- and 2-point functions for solu-
0
tions which retain the Markov property in space. In certain special cases, its
7
value can be explicitly obtained. For example, it is straightforwardto compute
J when u (x) is a stationary Ornstein-Uhlenbeck (mean-reverting) process:
0 0
du (x)=−βu (x)dx+γdB , β,γ >0, (25)
0 0 x
where B is Brownian motion. Since u has zero mean,
x 0
γ2
E[u (x)u (x+h)]=Cov(u (x),u (x+h))= e−βh. (26)
0 0 0 0
2β
Therefore, J =γ2/(2β2).
0
By takingappropriatelimits we obtainJ inthe caseofawhite noiseinitial
0
velocityfieldforanyfluxf. Thatis,supposetheinitialpotentialisU (x)=σB
0 x
withσ >0thestrengthofthenoiseandB atwo-sidedBrownianmotionpinned
x
atthe origin. This correspondsto letting γ =σβ and taking β →∞ in (25), so
that J =σ2/2. The solution to Burgers equation with white noise initial data
0
wasobtainedexplicitlyintermsofaPainlev´etranscendentbyGroeneboom[13],
andlaterrediscoveredbyFrachebourgandMartin[12]. Theabovecomputation
thus determines the value of an integral of the form (19) with stationary and
transitionmeasuresasin[12,13]. Foramorethoroughexaminationofphysically
relevant quantities we refer the reader to [23].
3.2 Energy dissipation and power spectrum
J is closely related to other quantities of physical interest, including the mean
0
energy dissipation per unit interval and power spectrum. We now discuss some
basic facts regarding these quantities for stationary Markov solutions to (2).
The mean dissipation in an interval I is computed as follows. By Fubini’s
theorem and equation (13),
1 1 1 1
∂ E u(x,t)2dx = ∂ Py2 = PBy2. (27)
t t
(cid:20)|I|Z 2 (cid:21) 2 2
I
Let h(y)= y wf′(w)dw be the entropy flux associated to y2/2. Then
y0
R
1 z2 y2
By2 =−b(y)yf′(y)− n(y,dz)[f] −
y,z
2 ZR (cid:18) 2 2 (cid:19)
z2 y2
=−Ah(y)− n(y,dz) [f] − −(h(z)−h(y)) . (28)
y,z
ZR (cid:26) (cid:18) 2 2 (cid:19) (cid:27)
Applying P and using that PAh=0, we obtain
1 1
∂ E u(x,t)2dx
t
(cid:20)|I|Z 2 (cid:21)
I
1 y
=− p(dy) n(y,dz) (z+y)(f(z)−f(y))+ wf′(w)dw . (29)
ZR ZR (cid:26)2 Zz (cid:27)
8
This can also be arrived at by considering traveling wave solutions to (2) as in
[20]. In particular, the energy dissipated at any instant by a shock connecting
the states u >u is
− +
1 u−
(u +u )(f(u )−f(u ))+ wf′(w)dw
+ − + −
2 Z
u+
1 u−
= (u −u )(f(u )−f(u ))− f(w)dw >0.
− + − +
2 Z
u+
Summingupovertheexpectednumberofshocksfromu tou perunitinterval
− +
implies (29).
Next, consider the power spectral density of the velocity field u(x,t):
1
E(k,t)= e−ikhE[u(x,t)u(x+h,t)]dh, k ∈R,t≥0. (30)
2π ZR
Again,wedroptfromthenotation. ForstationaryMarkovsolutions,thepower
spectrum is well-defined and can be given in terms of the λ-resolvent of the
transition semigroup (Q ) . With ϕ a test function, let
h h≥0
∞
R ϕ(y)= e−λhQ ϕ(y)dh, Re λ>0. (31)
λ h
Z
0
Extending R to λ=ik with 06=k ∈R, we have that
λ
1
E(k,t)= Re(P(t)yR (t)y). (32)
ik
π
TheLaplacetransformof(23)withrespecttohisthereforeseentobeequivalent
to an evolution equation for the power spectrum:
k
∂ E =− Re(P{f(y)R y−yR f(y)}). (33)
t ik ik
π
To obtain R in (32), one first needs to solve the Lax equation (14). Finally,
ik
we note that J =lim πE(k,t) is simply a constant multiple of the spectral
0 k→0
density in the low-wavenumber limit.
Let us contrast the situation considered here with that of Burgers equation
with one-sided L´evy initial data as in [20]. In the latter case, it was shown
that there exist solutions with finite energy and infinite dissipation per unit
intervalduetoaninfluxofenergythroughtheboundaries. Here,thestationarity
of the solution forbids such energy fluxes. It was also demonstrated in [20]
that the traditional notion of power spectrum is irrelevant when considering
solutions with independent increments, as processeswith the same energy have
indistinguishablespectra. Ausefulnotionofspectrumwasinsteadgivenby the
Fourier-Laplacetransformofprocesspathswithoutaveraging. Incontrast,here
we have that the power spectrum is nontrivial and can be computed via the
resolvent of the solution process.
9
4 Acknowledgements
I would like to thank GovindMenon for bringing this problemto my attention,
and for the pleasure of a continuing collaboration. This work was partially
funded by NSF grant DMS 06-36586.
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