TURBULENCE FOR THE GENERALISED BURGERS EQUATION Abbreviations Introduction

44 Pages·2017·0.41 MB·English

TURBULENCE FOR THE GENERALISED BURGERS
EQUATION
ALEXANDRE BORITCHEV
Abstract. In this survey, we review the rigorous results on turbu-
lence for the generalised space-periodic Burgers equation:
u +f(cid:48)(u)u = νu +η, x ∈ S1 = R/Z,
t x xx
studiedbyA.Biryukandtheauthorin[8,10,12,13]. Here, f issmooth
and strongly convex, whereas the constant 0 < ν (cid:28) 1 corresponds to
a viscosity coe(cid:30)cient.
We will consider both the unforced case (η = 0) and the randomly
forced case, when η is smooth in x and irregular (kick or white noise)
in t. In both cases, sharp bounds for Sobolev norms of u averaged in
time and in ensemble of the type Cν−δ, δ ≥ 0, with the same value of
δ for upper and lower bounds, are obtained. These results yield sharp
bounds for small-scale quantities characterising turbulence, con(cid:28)rming
the physical predictions [7].
Abbreviations
• 1d, 3d, multi-d: 1, 3, multi-dimensional
• a.e.: almost everywhere
• a.s.: almost surely
• (GN): the Gagliardo(cid:21)Nirenberg inequality (Lemma 1.1)
• i.i.d.: independent identically distributed
• r.v.: random variable
Introduction
The generalised 1d space-periodic Burgers equation
∂u ∂u ∂2u
+f(cid:48)(u) −ν = 0, ν > 0, x ∈ S1 = R/Z (1)
∂t ∂x ∂x2
(the classical Burgers equation [15] corresponds to f(u) = u2/2) is a
popular model for the Navier(cid:21)Stokes equation. Indeed, both of them
have similar nonlinearities and dissipative terms. Therefore, the physi-
cal arguments justifying various theories of hydrodynamical turbulence
Date: August 24, 2017.
1
2 ALEXANDRE BORITCHEV
can usually be applied to describe the behaviour of solutions to the
Burgers equation. Thus, this equation is often used as a benchmark
for turbulence theories. It is also used as a benchmark for the numeri-
cal methods for turbulent (cid:29)ows. For more information, see [7].
For ν (cid:28) 1 and f strongly convex, i.e. satisfying:
f(cid:48)(cid:48)(x) ≥ σ > 0, x ∈ R, (2)
solutions of (1) exhibit turbulent-like behaviour, called Burgers turbu-
lence or (cid:16)Burgulence(cid:17) [6, 7]. To simplify the presentation, we restrict
ourselves to solutions with zero mean value in space:
(cid:90)
u(t,x)dx = 0, ∀t ≥ 0. (3)
S1
The space mean value does not change in time. Indeed, since u is
1-periodic in space, we have:
(cid:90) (cid:90) (cid:90)
d
u(t,x)dx = − f(cid:48)(u(t,x))u (t,x)dx+ν u (t,x)dx = 0.
x xx
dt
S1 S1 S1
If the mean value of the initial value u on S1 is equal to b (cid:54)= 0, we
0
may consider the zero mean value function
v(t,x) = u(t,x+bt)−b,
which is a solution of (1) with f(y) replaced by g(y) = f(y +b)−by.
So the assumption (3) does not lead to a loss of generality.
In this survey, we consider both the unforced equation (1) and the
generalised Burgers equation with an additive forcing term, smooth in
space and irregular in time (see Subsection 1.2). We summarise the
estimates obtained by Biryuk and the author [8, 10, 12, 13] for the
Sobolev norms as well as for the dissipation length scale and the small-
scale quantities relevant for the theory of hydrodynamical turbulence:
the structure functions and the energy spectrum. This survey is par-
tiallybasedonthePh.D.thesisoftheauthor[11], wheresometechnical
points are covered in more detail.
The major di(cid:27)erence between the unforced and the white-forced gen-
eralised Burgers equation is the energy picture. In the (cid:28)rst case, we
have a dissipative system: the L norm is decreasing in time. Conse-
2
(cid:82)
quently, the regime where the energy u2/2 dissipates fast enough
S1
(which yields a time-averaged lower bound on the Sobolev norms) is
transient and depends on the initial condition through a certain quan-
tity D (see (15) for its de(cid:28)nition). On the contrary, in the second case,
after a time needed either to dissipate energy if u is large or to supply
0
energy if u is small, we are in a quasi-stationary regime, in the sense
0
that in average on large enough time intervals, we have an approximate
3
balance between the dissipation rate −νE(cid:107)u(cid:107)2 and the constant energy
1
supply rate I .
0
For the unforced Burgers equation, some upper estimates for small-
scale quantities are well-known. For example, Lemma 4.1 of our work
is an analogue in the periodic setting of the one-sided Lipschitz esti-
mate due to Oleinik, and the upper estimate for the structure function
S ((cid:96)) follows from an estimate for the solution in the class of bounded
1
variation functions BV. For references on these classical aspects of the
theory of scalar conservation laws, see [20, 45, 49]. For some upper es-
timates for small-scale quantities, see [38, 53]. To our best knowledge,
rigorous lower estimates were not known before Biryuk’s and our work.
Theresearchonthesmall-scalebehaviourofsolutionsforthe(forced)
generalisedBurgersequationismotivatedbytheproblemofturbulence.
It has been inspired by the pioneering works of Kuksin, who obtained
lower and upper estimates for Sobolev norms by negative powers of the
viscosity for a large class of equations (see [41, 42] and references in
[42]). For more recent results obtained by Kuksin, Shirikyan and oth-
ers for the 2D Navier(cid:21)Stokes equation, see the book [43] and references
therein.
The estimates for Sobolev norms and for small-scale quantities pre-
sented in our work are asymptotically sharp in the sense that viscosity
enters lower and upper bounds at the same negative power. Such esti-
mates are not available for the more complicated equations considered
in [41, 42, 43].
This survey is also concerned with the problem of the invariant mea-
sure for the stochastic generalised Burgers equation. This problem has
been treated prevously for the nonlinearity uu by Sinai [51, 52] and in
x
the inviscid limit ν → 0 by E, Khanin, Mazel and Sinai [22]; see also
[28, 31]. Here we present a simple approach to this problem, described
in [13], which consists in using L -contractivity of the (cid:29)ow correspond-
1
ing to the equation, and a coupling argument.
We do not consider other aspects of Burgulence, such as the inviscid
limit, the behaviour of solutions for spatially rough forcing and the
noncompact setting. We refer the reader to the survey by Bec and
Khanin [7], which is concerned with physical aspects of the theory of
Burgulence, and to the survey by Bakhtin [3], which discusses related
probabilistic and ergodic results.
Organisation of the paper: We begin by introducing the notation
and setup in Section 1. In Section 2, we present the K41 theory as well
as the physical predictions for Burgulence. In Section 3, we formulate
the main results.
4 ALEXANDRE BORITCHEV
InSection4, weconsiderthesolutionu(t,x)oftheunforcedequation
(1). In Subsection 4.1, we begin by recalling the upper estimate for the
quantity
max u (s,x), t ≥ 1.
x
s∈[t,t+1], x∈S1
Using this bound, we get upper and lower estimates for the Sobolev
norms of u. In Subsection 4.2 we study the implications of our re-
sults for Burgulence theory. Namely, we give sharp upper and lower
bounds for the dissipation length scale, the increments and the spec-
tral asymptotics for the (cid:29)ow u(t,x), which hold uniformly for ν ≤ ν .
0
The quantity ν > 0 depends only on f and on the initial condition.
0
These results justify rigorously the physical predictions for small-scale
quantities which characterise Burgulence.
Inthetwolastsectionsofthepaper, weconsidertherandomlyforced
generalised Burgers equation. In Section 5, we obtain analogues of the
results in Section 4, which also con(cid:28)rm the corresponding physical pre-
dictions [7]. Section 6 is concerned with the stationary measure.
1. Notation and setup
All functions which we consider in this paper are real-valued, except
in Section 2, where vectors in R3 are written in bold script.
1.1. Functional spaces and Sobolev norms. Consider a zero mean
valueintegrablefunctionv onS1. Forp ∈ [1,∞], wedenoteitsL norm
p
by|v| . TheL normisdenotedby|v|, and(cid:104)·,·(cid:105)standsfortheL scalar
p 2 2
product. From now on L , p ∈ [1,∞], denotes the space of zero mean
p
value functions in L (S1). Similarly, C∞ is the space of C∞-smooth
p
zero mean value functions on S1.
For a nonnegative integer m and p ∈ [1,∞], Wm,p stands for the
Sobolev space of zero mean value functions v on S1 with (cid:28)nite homo-
geneous norm
(cid:12) (cid:12)
(cid:12)dmv(cid:12)
|v| = (cid:12) (cid:12) .
m,p (cid:12)dxm(cid:12)
p
In particular, W0,p = L for p ∈ [1,∞]. For p = 2, we denote Wm,2 by
p
Hm and abbreviate the corresponding norm as (cid:107)v(cid:107) .
m
Since the length of S1 is 1, we have:
|v| ≤ |v| ≤ |v| ≤ |v| ≤ ··· ≤ |v| ≤ |v| ≤ ...
1 ∞ 1,1 1,∞ m,1 m,∞
We recall a version of the classical Gagliardo(cid:21)Nirenberg inequality (see
[21, Appendix]):
5
Lemma 1.1. For a smooth zero mean value function v on S1,
|v| ≤ C|v|θ |v|1−θ,
β,r m,p q
where m > β ≥ 0, and r is de(cid:28)ned by
1 (cid:16) 1(cid:17) 1
= β −θ m− +(1−θ) ,
r p q
under the assumption θ = β/m if p = 1 or p = ∞, and β/m ≤ θ < 1
otherwise. The constant C depends on m,p,q,β,θ.
From now on, we will refer to this inequality as (GN).
For any s ≥ 0, Hs stands for the Sobolev space of zero mean value
functions v on S1 with (cid:28)nite norm
(cid:16)(cid:88) (cid:17)1/2
(cid:107)v(cid:107) = (2π)s |k|2s|vˆ(k)|2 , (4)
s
k∈Z
where vˆ(k) are the complex Fourier coe(cid:30)cients of v(x). For an integer
s = m, this norm coincides with the previously de(cid:28)ned Hm norm. For
s ∈ (0,1), (cid:107)v(cid:107) is equivalent to the norm
s
(cid:32) (cid:33)1/2
(cid:90) (cid:16)(cid:90) 1 |v(x+(cid:96))−v(x)|2 (cid:17)
(cid:48)
(cid:107)v(cid:107) = d(cid:96) dx (5)
s (cid:96)2s+1
S1 0
(see [1, 54]).
Subindices t and x, which can be repeated, denote partial di(cid:27)erenti-
ation with respect to the corresponding variables. We denote by v(m)
the m-th derivative of v in the variable x. For brevity, the function
v(t,·) is denoted by v(t).
1.2. Well-posedness and di(cid:27)erent types of forcing. In Section 4,
we consider the unforced equation (1) with a C∞-smooth initial con-
dition u . This equation has a unique solution in C∞: see for instance
0
[39, Chapter 5].
In Section 5, we consider the generalised Burgers equation with two
di(cid:27)erent types of additive forcing in the right-hand side, taking again
a C∞-smooth initial condition u . Since the forcing always has zero
0
mean value in space and the initial condition satis(cid:28)es (3), its solutions
satisfy (3) for all time.
First, we consider the kick force. We begin by providing the space L
2
with the Borel σ-algebra (Ω,F). Then we consider an L -valued r.v.
2
ζ = ζω on a probability space (Ω,F,P). We suppose that ζ satis(cid:28)es
the following three properties.
6 ALEXANDRE BORITCHEV
(i) (Non-triviality)
P(ζ ≡ 0) < 1.
(ii) (Finiteness of moments for Sobolev norms) For every
m ≥ 0, we have:
I = E(cid:107)ζ(cid:107)2 < +∞.
m m
(iii) (Vanishing of the expected value)
Eζ ≡ 0.
It is not di(cid:30)cult to construct explicitly ζ satisfying (i)-(iii). One
possibilty is to suppose that the real Fourier coe(cid:30)cients of ζ, de(cid:28)ned
for k > 0 by
√ (cid:90) √ (cid:90)
a (ζ) = 2 cos(2πkx)u(x); b (ζ) = 2 sin(2πkx)u(x), (6)
k k
S1 S1
are independent r.v. with zero mean value and exponential moments
tending to 1 fast enough as k → +∞.
Now let ζ = ζω, i ∈ N be i.i.d. r.v.’s having the same distribution as
i i
ζ. The sequence (ζ ) is a r.v. de(cid:28)ned on a probability space which
i i≥1
is a countable direct product of copies of Ω. From now on, this space
will itself be called Ω. The meaning of F and P changes accordingly.
For ω ∈ Ω, the kick force ξω is by de(cid:28)nition the distribution de(cid:28)ned
by
+∞
(cid:88)
ξω(t,x) = δ ζω(x),
t=i i
i=1
where δ denotes the Dirac measure at the time moment i.
t=i
The kick-forced equation corresponds to the case where, in the right-
hand side of (1), 0 is replaced by the kick force:
∂u ∂u ∂2u
+f(cid:48)(u) −ν = ξω. (7)
∂t ∂x ∂x2
This means that for integers i ≥ 1, at the moments i the solution u(x)
instantly increases by the kick ζω(x), and that between these moments
i
u solves (1). We make the additional assumption that the solution is
a right-continuous function of time.
Existence and uniqueness of spatially smooth solutions to (7) follows
directly from the corresponding fact for the unforced equation.
The other type of forcing considered here is the white force. Heuris-
tically this force corresponds to a scaled limit of kick forces with more
7
and more frequent kicks.
We provide each space Wm,p with the Borel σ-algebra. Then we
consider an L -valued Wiener process
2
w(t) = wω(t), ω ∈ Ω, t ≥ 0,
de(cid:28)ned on a complete probability space (Ω, F, P), and an adapted
(cid:28)ltration {F , t ≥ 0} (i.e., for t ≥ 0, w(t) is F -measurable, and F
t t t
and the σ-algebra generated by the r.v.’s w(t + s) − w(t), s ≥ 0 are
independent).
We assume that for each m and each t ≥ 0, w(t) ∈ Hm, almost
surely. That is, for ζ,χ ∈ L ,
2
E((cid:104)w(s),ζ(cid:105)(cid:104)w(t),χ(cid:105)) = min(s,t)(cid:104)Qζ,χ(cid:105),
where Q is a symmetric operator which de(cid:28)nes a continuous mapping
Q : L → Hm for every m. Thus, w(t) ∈ C∞ for every t, almost surely.
2
From now on, we rede(cid:28)ne the Wiener process so that this property
holds for all ω ∈ Ω. We will denote w(t)(x) by w(t,x). For m ≥ 0, we
denote by I the quantity
m
I = Tr (Q) = E(cid:107)w(1)(cid:107)2 .
m Hm m
FormoredetailsonWienerprocessesinHilbertspaces, see[18, Chapter
4] and [44].
It is not di(cid:30)cult to construct w(t) explicitly. For instance, we could
consider the particular case of a (cid:16)diagonal(cid:17) noise:
√ √
(cid:88) (cid:88)
w(t) = 2 a w (t)cos(2πkx)+ 2 b w˜ (t)sin(2πkx),
k k k k
k≥1 k≥1
where w (t), w˜ (t), k > 0, are standard independent Wiener processes
k k
and
(cid:88)
I = (a2 +b2)(2πk)2m < ∞
m k k
k≥1
for each m. From now on, dw(s) denotes the stochastic di(cid:27)erential
corresponding to the Wiener process w(s) in the space L .
2
Now (cid:28)x m ≥ 0. By Fernique’s Theorem [44, Theorem 3.3.1], there
exist λ ,C > 0 such that
m m
(cid:16) (cid:17)
Eexp λ (cid:107)w(T)(cid:107)2 /T ≤ C , T ≥ 0. (8)
m m m
Therefore by Doob’s maximal inequality for in(cid:28)nite-dimensional sub-
martingales [18, Theorem 3.8. (ii)] we have:
(cid:16) p (cid:17)p
E sup (cid:107)w(t)(cid:107)p ≤ E(cid:107)w(T)(cid:107)p < +∞, (9)
m p−1 m
t∈[0,T]
8 ALEXANDRE BORITCHEV
for any T > 0 and p ∈ (1,∞).
The white-forced equation is obtained by replacing 0 by the weak
derivativeηω = ∂wω/∂tintheright-handsideof(1). Here,wω(t), t ≥ 0
is the Wiener process de(cid:28)ned above.
Definition 1.2. For T ≥ 0, we say that an H1-valued process u(t,x) =
uω(t,x) is a solution of the equation
∂uω ∂uω ∂2uω
+f(cid:48)(uω) −ν = ηω (10)
∂t ∂x ∂x2
for t ≥ T if:
(i) For every t ≥ T, ω (cid:55)→ uω(t,·) is F -measurable.
t
(ii) For every ω, t (cid:55)→ uω(t,·) is continuous in H1 for t ≥ T and
satis(cid:28)es
(cid:90) t(cid:16) 1 (cid:17)
uω(t) =uω(T)− νLuω(s)+ B(uω)(s) ds
2
T
+wω(t)−wω(T), (11)
where
B(u) = 2f(cid:48)(u)u ; L = −∂ .
x xx
For brevity, solutions for t ≥ 0 will be referred to as solutions.
Existence and uniqueness of smooth solutions to (10) is proved by
the mild solution technique (cf. [19, Chapter 14]). Since the forcing
and the initial condition are smooth in space, the mapping t (cid:55)→ u(t) is
time-continuous in Hm for every m, and t (cid:55)→ u(t)−w(t) has a space
derivative in C∞ for all t, a.s.
Nowconsider,forasolutionu(t,x)of(10),thefunctionalG (u(t)) =
m
(cid:107)u(t)(cid:107)2 and apply It(cid:244)’s formula [18, Theorem 4.17]:
m
(cid:90) t
(cid:107)u(t)(cid:107)2 =(cid:107)u (cid:107)2 − (cid:0)2ν(cid:107)u(s)(cid:107)2 +(cid:104)Lmu(s), B(u)(s)(cid:105)(cid:1)ds+tI
m 0 m m+1 m
0
(cid:90) t
+2 (cid:104)Lmu(s), dw(s)(cid:105)
0
(we recall that I = Tr(Q ).) Consequently,
m m
d
E(cid:107)u(t)(cid:107)2 = −2νE(cid:107)u(t)(cid:107)2 −E (cid:104)Lmu(t), B(u)(t)(cid:105)+I .
dt m m+1 m
As (cid:104)u, B(u)(cid:105) = 0, for m = 0 this relation becomes
d
E|u(t)|2 = I −2νE(cid:107)u(t)(cid:107)2. (12)
dt 0 1
9
1.3. Notation and agreements. When considering a Sobolev norm
in Wm,p, the quantity γ = γ(m,p) denotes max(0, m−1/p).
In Subsection 2.1, v(t,x) denotes the velocity of a 3d (cid:29)ow with pe-
riod 1 in each spatial coordinate. In the whole paper, u(t,x) denotes a
solution of the generalised Burgers equation with a given initial condi-
tion u = u(0,·). In Section 4, we deal with the equation (1) under the
0
assumptions (2-3). In Section 5 we deal with the equation (10) under
the assumptions (2-3) and under the additional condition
∀m ≥ 0, ∃h ≥ 0, C > 0 : |f(m)(x)| ≤ C (1+|x|)h, x ∈ R, (13)
m m
where h = h(m) is a function such that
1 ≤ h(1) < 2 (14)
(the lower bound on h(1) follows from (2)). The results in that section
also hold for the kicked equation (7), under the same assumptions as
for (10), except (14), which is unnecessary.
WhenweconsidertherandomlyforcedgeneralisedBurgersequation,
PetEdenote,respectively,theprobabilityandtheexpectedvaluewith
respect to the probability measure Ω (cf. Section 1.2).
All quantities denoted by C with sub- or superindices are positive
and nonrandom. Unless otherwise stated, they depend only on the
following parameters:
• When dealing with the K41 theory, the statistical properties of
the forcing.
• When studying the unforced generalised Burgers equation, the
function f determining the nonlinearity f(cid:48)(u)u , as well as the
x
parameter
D = max(|u |−1, |u | ) (15)
0 1 0 1,∞
which characterises how generic the initial condition is.
• When studying the randomly forced generalised Burgers equa-
tion, the function f determining the nonlinearity f(cid:48)(u)u , as
x
well as the statistical properties of the forcing. In the case of
a kick force, by statistical properties we mean the distribution
function of the i.i.d. r.v.’s ζ . In the case of a white force,
i
we mean the correlation operator Q for the Wiener process w
de(cid:28)ning the random forcing.
In particular, these quantities never depend on the viscosity coe(cid:30)cient
ν.
Constants which also depend on parameters a ,...,a are denoted
1 k
10 ALEXANDRE BORITCHEV
a1,...,ak
by C(a ,...,a ). By X (cid:46) Y we mean that X ≤ C(a ,...,a )Y.
1 k 1 k
The notation X a1,∼...,ak Y stands for
a1,...,ak a1,...,ak
Y (cid:46) X (cid:46) Y.
In particular, X (cid:46) Y and X ∼ Y mean that X ≤ CY and C−1Y ≤
X ≤ CY, respectively. Note that this notation is never used wit the
parameter ν: in other words, dependence on the viscosity is always
explicitly speci(cid:28)ed.
We use the notation g− = max(−g,0) and g+ = max(g,0).
InSubsection2.1, thebrackets(cid:104)·(cid:105)denotetheexpectedvalue. Forthe
meaningofthebrackets{·}, seeSubsection4.1inthedeterministiccase
(where they correspond to averaging in time) and Subsection 5.3 in the
random case (where they correspond to averaging in time and taking
the expected value). The de(cid:28)nitions of the relevant ranges and the
length scales, as well as of the small-scale quantities, i.e. the structure
functions S and S = S and the spectrum E(k) depend on the
p,α p,1 p
setting: see Subsections 2.1, 2.2, 4.2 and 5.3.
2. Turbulence and the Burgers equation
2.1. Turbulence, K41theory, intermittency. Itiswell-knownthat
giving a precise de(cid:28)nition of turbulence is problematic. However, some
features are generally recognised as characteristic of turbulence: many
degrees of freedom, unpredictability/chaos, (small-scale) irregularity...
For a more detailed discussion, see [27, 55]. Here, we will only present
(in a slightly modi(cid:28)ed form) the vocabulary of the theory of turbulence
which is relevant to the study of the Burgers model. In particular, we
will proceed as if the (cid:29)ow v(t,x) under consideration is periodic in
space, without concerning ourselves with the physical relevance of K41
in this setting. Without loss of generality, we may assume that v is 1-
periodic in each coordinate x ,x ,x . Let us denote by ν the viscosity
1 2 3
coe(cid:30)cient; we only consider the turbulent regime 0 < ν (cid:28) 1.
We de(cid:28)ne the space scale as the inverse of the frequency under con-
sideration. In particular, the Fourier coe(cid:30)cients vˆ(k) for large values
of k or, in the physical space, the increments v(x+r)−v(x) for small
values of r, are prototypical small-scale quantities.
The theory which may be considered as a starting point for the
modern study of turbulence is essentially contained in three articles by
Kolmogorov which have been published in 1941 [33, 34, 35]. Thus, it
is referred to as the K41 theory.
The philosophy behind K41 is that although large-scale characteris-
tics of a turbulent (cid:29)ow are clearly individual (depending on the forcing