Remarks on unsolved basic problems of the Navier-Stokes equations

7 Pages·0.11 MB·English

by  Alexander Rauh

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Remarks on unsolved basic problems of the Navier–Stokes
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equations

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a
J
Alexander Rauh ∗
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Fachbereich Physik, Carl von Ossietzky Universit¨at, D-26111 Oldenburg, Germany

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n
y
d Abstract
fundamentalsoftheNSE,seeforinstancethemono-
-
u graph of Doering and Gibbon [6], or a series of pa-
l
f Thereisrenewedinterestinthequestionofwhether pers by Lions [7] and references therein.
.
s the Navier–Stokes equations (NSE), one of the fun- This contribution focuses on the question of self-
c
i damental models of classical physics and widely consistency which arises, when one of the assump-
s
y used in engineering applications, are actually self- tions inherent to the NSE, namely the continuum
h consistent. After recalling the essential physical as- hypothesis, is confronted with the length scales
p
sumptions inherent in the NSE, the notion of weak emerging from solutions of the deterministic NSE.
[
solutions, possible implications for the energy con- First, the NSE will be briefly derived from physical
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servation law, as well as existence and uniqueness principleswithdueattention paidtothecontinuum
v
7 in the incompressible case are discussed. Emphasis hypothesis. After recalling the notion of weak solu-
2
will be placed on the possibility of finite time sin- tions, the state of theart of mathematical existence
0
1 gularities and their consequences for length scales and uniqueness proofs will be indicated. Theimpli-
0 which should be consistent with the continuum hy- cation of weak solutions upon energy conservation
1
pothesis. will be discussed. The possibility of finite time sin-
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/ gularities will be related to length scales and thus
s
c to the problem of self-consistency.
i
s 1 Introduction
y
2 Derivation of the NSE
h
p
Ascomputationalfluiddynamicsmakesprogressto-
:
v wards the simulation of realistic three-dimensional TheNSEarebasedontheconservation ofmassand
i on Newton’s second law. In addition, the more spe-
X flows, the validity of the Navier–Stokes equations
cial assumption of a so–called Newtonian fluid is
r (NSE) can be tested in a more and more refined
a adopted, which is justified in a great many cases
way. To put it from an applied point of view: Be-
of hydrodynamic flows. To formulate the conserva-
fore experiments in windtunnels are substituted by
tion laws it is customary to pick out a connected
computer simulations, one should make sure that
cluster of molecules contained in volume V which
the underlying theory is at least self-consistent. As t
is deformed in time and translated according to the
a matter of fact, after the classical mathematical
local velocity v(x,t) of the flow. Time derivatives
work by Leray [1], Hopf [2], Ladyzhenskaya [3], Ser-
of corresponding magnitudes are conveniently eval-
rin [4], Temam [5], to refer to important contribu-
uated by means of the Reynolds transport theorem
tions in the field, there is renewed interest in the
d
∗ dV f(x,t) ≡
Lecture given at 3rd Inter-
dt
national Summer School/Conference, LET’S FACE CHAOS ZVt
df ∂f
through NONLINEAR DYNAMICS, Maribor, Slovenia, 24
dV := dV +div(fv) (1)
June-5July 1996. dt ∂t
ZVt (cid:18) (cid:19) ZVt (cid:20) (cid:21)
1
where f is a scalar function. If ρ(x,t) denotes the where µ and ν are macroscopic viscosity parame-
mass density, then conservation of mass, namely ters. In the incompressible case, ν drops out, and
after making use of mass conservation we can write
d
dV ρ =0 (2) down the momentum balance as follows
dt
ZVt
∂v
dV [ρ +ρ (v·∇)v+
gives rise to the continuity equation 0 0
∂t
ZVt
dρ ∂ρ grad(p)−µ∆v−ρ0f] = 0. (10)
≡ +div(ρv) = 0, (3)
dt ∂t
Here V is an arbitrary local space volume. To be
t
whichifρ=ρ0 isconstant, leads to theincompress- sure of the existence of the above integral, one may
ibility condition adopt the sufficient conditions that the following
fields are locally square integrable
div(v) = 0. (4)
∂ ∂ ∂2 ∂
Newton’ssecondlawimpliesthatanychangeinmo- v, v, v, v, p, f. (11)
∂t ∂x ∂x ∂x ∂x
i i k i
mentum is caused by external forces which in con-
tinuum physics are described by the volume force This can be easily seen with the aid of the Schwarz
density f (e.g. gravity) and by a tensorial force Π. inequality. For instance, if xˆ is a cartesian unit
i
This tensor reflects the influence of the adjacent vector, then we can write
fluid on a given fluid particle. The momentum bal-
∂v ∂v·xˆ
ance reads | dV i|2 ≡ | dV i|2 ≤
∂t ∂t
ZVt ZVt
d ∂v ∂v
dV ρv = dV ρf + dSΠ◦nˆ (5)
V dV · . (12)
dt t
ZVt ZVt I∂Vt ZVt ∂t ∂t
where nˆ dS is the oriented surface element of the
As will be discussed later on, there may arise dif-
volume V . It is convenient to separate in Π an
t ficulties with the conservation laws when certain
isotropic part, the pressure p, which is present also
weak conditions on the velocity field v are adopted
in the hydrostatic case, from the so–called stress
as is customary in the frame of functional analysis.
tensor T
From eq.(10), the following standard NSE in the
form of partial differential equations are inferred
Π = −pδ +T , i,k = 1,2,3. (6)
ik ik ik
∂v
TheNewtonian fluidassumptionnowamounts to ρ0 +(v·∇)v = −grad(p)+µ∆v+ρ0f (13)
∂t
the following linear relations between T and the (cid:20) (cid:21)
strain (rate) tensor S: wherepisdeterminedthroughtheincompressibility
condition div(v) = 0.
3
T = C S (7)
ik ikmn mn
m,n=1 3 Continuum assumptions and
X
length scales
with
1 ∂v ∂v
m n
S = + . (8)
mn
2 ∂x ∂x The NSE describe macroscopic physical quantities
(cid:18) n m(cid:19)
The4thranktensorC isconstantanddescribesthe which constitute mean values with respect to the
effect of viscosity. In the isotropic case, C is of the underlying atomic degrees of freedom. The den-
sity ρ(x,t) at the space point x, for instance, has
form
to be understood as an average over some volume
C = νδ δ +µ(δ δ +δ δ ) (9) ∆V centered at x. If ∆V is chosen too small, a
ikmn ik mn im kn in km
2
single measurement of ρ may largely deviate from test vector fields Φ ∈ S are introduced with the
its mean value due to molecular fluctuations. An following properties
estimate for a physically reasonable lower bound of
∆V can be deduced from the mean thermal den- D := {Φ| Φ ∈ D(Ω); div(Φ) =0 } (18)
sity fluctuation ∆ρ as given in standard textbooks
∞
of thermodynamics [8] where D(Ω) is the Schwartz space (C and com-
pact support in Ω). Now v(x,t) is called a weak
∆ρ kTκ solution if it is locally square integrable and if the
= (14)
ρ s∆V following projections of the NSE and the continu-
ity equation hold for every Φ ∈ D and for every
where k is the Boltzmann constant, T the absolute C1 scalar function φ with compact support in Ω,
temperatureandκthecompressibility. Ifwerequire
respectively [4]
the relative fluctuation ∆ρ/ρ to be smaller than,
say 10−3, at T = 300◦ Kelvin, then we find that τ ∂v ∂Φ
k k
dt dV[Φ −v v −ν v ∆Φ −
the diameter d of the volume ∆V should be d ≥ k ∂t i k ∂x 0 k k
3·10−7m for air, or d≥ 10−8m for water. Zo ZΩ i Φ f ] = 0, (19)
k k
Asanimplication,ifsolutionsofthedeterministic
dV v·grad(φ) = 0 (20)
NSEturn outto vary on a space scale much smaller
ZΩ
than the above lower bounds, then we are outside
of the validity domain of these equations. Here is where ν0 = µ/ρ0 denotes the kinematic viscosity
the point where the self-consistency problem arises. and summation convention is adopted. The pres-
In the turbulent regime, length scales decrease sure term dropped out in (19) due to the solenoidal
with increasing Reynolds number R. As is listed property of Φ. A typical theorem reads [3]:
e.g. in [6], the Kolmogorov length δ below which
K
Theorem: A unique weak solution exists, at least
eddies are destroyed by dissipation, is given by
in the time interval t ∈ [0,τ ] with τ ≤ τ, pro-
δ = L/R3/4 where L is a typical external length, 1 1
K vided the initial velocity field α(x) ∈ W 2 and the
e.g. the diameter of the containment. As another 2
external force density f obeys the condition
example the thickness δ of a turbulent boundary
B
layer scales as δ ∼ L/(RlogR). If L = 1 cm, then
δK and δB reachBthe continuum limit at R ≈ 106. τ dt dV f2+ ∂f 2 1/2 < ∞ (21)
Z0 ZΩ " (cid:18)∂t(cid:19) #
4 Weak solutions and energy bal-
where W 2 denotes the Sobolev space with the sec-
2
ance
ond space derivatives being square integrable.
Since Leray’s pioneering work [1], one has been As should be noticed, even if the condition (21)
looking for generalized solutions v(x,t) of the in- on the external field f holds for arbitrarily large τ,
compressible NSE in the space time domain Ω := uniquenesscanbeguaranteedbytheabovetheorem
τ
Ω×[0,τ] with the following properties: only for the smaller time interval t ∈ [0,τ1]. While
this is typical in space dimension three, one has
v(x,t = 0) = α(x), (15) τ = τ in the case of two-dimensional flows.
1
v| = 0, (16) Whichpricedowehavetopayforacceptingweak
∂Ω
div(v) = 0. (17) solutions? To discuss a possible implication for en-
ergy conservation, we recall the notion of weak and
The above equations correspond to the initial con- strong convergence of a sequence of real functions
dition, no-slip boundary condition and incompress- a(1),a(2),..a(N),... This sequence is called to con-
∗
ibility, respectively. To establish weak solutions, verge weakly against the function a , if for any
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square integrable function g for the weak solution v∗ together with the test field
Φ = v(N) ∈ D and obtain
dV a(N)a(N) < ∞ and τ ∂v(N) ∂v(N)
ZΩ E(N)(τ)−E(N)(0)+ν dt dV k k
0
lim dV a(N)g = dV a∗g. (22) Z0 ZΩ ∂xi ∂xi
N→∞ZΩ ZΩ − τ dt dV v(N)f = R(N) (29)
k k
It converges strongly, if Z0 ZΩ
with
lim dV a(N)a(N) = dV a∗a∗. (23) τ ∂r(N) ∂v(N)
N→∞ZΩ ZΩ R(N)= dt dV −vk(N) ∂kt +vi∗rk(N) ∂kx
In the case of weak convergence, we have the iden- Z0 ZΩ h i
(N) (N) (N)
+ν r ∆v +r f . (30)
tity [1] 0 k k k k
i
Apart from partial integrations, we made useof the
lim { dV (a(N) −a∗)2− dV a(N)a(N) + incompressibility condition (20) which implies the
N→∞ ZΩ ZΩ relation
∗ ∗
dV a a }= 0 (24)
(N)
∂v 1
ZΩ dV v(N)v(N) k = dS(v(N))2nˆ ·v(N).
i k ∂x 2
which is true because dV a(N)a∗ converges ZΩ i Z∂Ω (31)
∗ ∗
(weakly) to dV a a and the two non-converging The above surface integral vanishes because vN ∈
R
terms ΩdV Ra(N)a(N) cancel each other identically. D. It should be noticed that eq.(29) holds true for
As a consequence one has in particular [1] any cutoff N; it follows strictly from the definition
R
(19) of a weak solution; in particular, no approxi-
liminf dV a(N)a(N) ≥ dV a∗a∗. (25) mate projection scheme was adopted as is common
N→∞ ZΩ ZΩ in Galerkin representations.
Here, the equality sign is guaranteed only in the In the case of strong solutions with v∗ ∈ W2
2
case of strong convergence where simultaneously in the space time domain Ω , one can show that
τ
liminf = limsup. R(N) → 0 in the limit N → ∞ so that we would
To derive the energy balance for a sequence of have the physically plausible energy balance
approximationsv(N) whichconvergeweaklyagainst τ ∂v∗ ∂v∗
a solution v∗ of (19), we use basis functions Φ(ν) ∈ E∗(τ)+ν0 dt dV k k
∂x ∂x
D with the properties (18) as Z0 ZΩ i i
τ
∗ ∗
= E (0)+ dt dV v f , (32)
k k
N Z0 ZΩ
v(N)(x,t) := c (t)Φ(ν)(x), c ∈ R. (26)
(ν) (ν) or in words: the kinetic energy at time τ plus the
νX=1 energy dissipated up to τ equals the initial kinetic
It is convenient to introduce the following abbrevi- energy plus the work done by the volume force f up
ation for the kinetic energy at time t to time τ.
However, if v∗ is a weak solution, then we have
1
E(N)(t) := dV v(N)(x,t)v(N)(x,t). (27) only the property of boundedness of the integrals
2 ZΩ k k in (29) and (30), except for the fk-integrals and the
∗ initial energy E(N)(0) which converges under the
E (t)denotestheenergy correspondingtotheweak
assumptions specified in Theorem A. Making useof
solution
v∗ ≡ v(N) +r(N) (28) the inequality (25) we can write
wherer(N) istheremaindertotheapproximatefield τ ∂v(N) ∂v(N)
liminf E(N)(τ)+ν dt dV k k =
v(N). We now insert into (19) the above expression N→∞ " 0Z0 ZΩ ∂xi ∂xi #
4
τ ∂v∗ ∂v∗ If I(τ) exists only up to some time τ∗, then
E∗(τ)+ν0 dt dV ∂xk ∂xk +L∗ (33) kDvk∞ is singular at t = τ∗ in a way, that there
Z0 ZΩ i i
is at least one space point x ∈ Ω, where one of
where L∗ ≥ 0. With R∗ denoting the limes inferior 0
the components ∂v /∂x diverges, for instance as
of R(N), the energy balance (29) reads in the same i k
follows
limit
τ ∂v∗ ∂v∗
E∗(τ)τ+ν0Z0 dtZΩdV ∂xki ∂xki (cid:12)∂v∂i(xxk,t)(cid:12)x=x0 −→ (τ∗α−2t)γ, t < τ∗, γ ≥ 1.
= E∗(0)+ dt dV vk∗fk +R∗−L∗. (34) (cid:12)(cid:12) (cid:12)(cid:12) (37)
Z0 ZΩ Sin(cid:12)ce for t(cid:12) near τ∗ the behaviour (37) implies
changes of the velocity field over arbitrarily small
Thus, in the case of weak solutions there may be
length scales, it is in conflict with the continuum
unphysical sources or sinks (depending on the sign
assumption. The length scales are then small com-
of R∗ − L∗) of kinetic energy due to the presence
pared to the diameter of the volume ∆V of a
of singularities. The latter are connected with the
space gradients of the velocity field, since E(N)(t), fluidparticlewiththeconsequencethatmicroscopic
molecular forces come into play and can no longer
t ∈ (0,τ) can be shown to converge under rather
be neglected. In other words we are then out of the
general assumptions, see also [9]. If R∗ −L∗ < 0,
∗ validity domain of the deterministic NSE and we
then the kinetic energy E (t) is smaller than phys-
would have to consider stochastic forces in addition
ically expected; this is known as Leray inequality,
to the deterministic external forces. It is therefore
see e.g. p. 104 of [6].
not yet settled, whether the phenomenon of hydro-
dynamic turbulence is a manifestation of determin-
5 Uniqueness and finite time sin-
istic chaos alone.
gularities As should be noted, the problem of finite time
singularities cannot beovercome by some averaging
As already mentioned, one gets square integrable recipe, becausetheexistence of I(τ)is connected to
solutionsvunderrathergeneralassumptionsonthe the uniquenessof solutions as a sufficient condition,
external data. The main basic problem of the NSE and it may turn out to be also necessary.
is related to uniqueness which so far is tied to the
Similarly, in the case of compressible flows finite
existence of the following time integral, for a recent
time singularities could not be excluded so far [7].
discussion see [6],
The proof or disproof of the existence of finite time
τ singularities constitute one of the basic unsolved
I(τ) := dt kDvk∞ (35) problems in the analysis of the NSE. In the inviscid
Z0 case of the Euler equation, there is a general ar-
with the supremum norm gument for possible finite time singularities, see for
instance Frisch [10]. From a direct numerical sim-
∂v
kDvk∞ := maxmax| k|. (36) ulation of the Euler equations, Grauer and Sideris
i,k x∈Ω ∂xi [11] recently reported on evidence for a singularity
of the type as given in (37) with γ = 1.
The origin of this integral will be indicated in the
Appendix. Up to now, in three dimensions the ex-
istence of I(τ) has been corroborated only for finite
time intervals τ. If I(τ) exists for arbitrarily large Acknowledgement The author is indepted to
τ, then bothuniquenessandexistence of weak solu- M. Boudouridesfor makinghim aware of J.Serrin’s
tions can be established for arbitrarily large times contribution to the field. He is also thankful to A.
under quite general conditions. Spille for a critical reading of the manuscript.
5
Appendix
whichbyGronwall’slemmacanbeintegratedtothe
final inequality
In the following it is sketched how the integral
I(τ)showsupinuniquenessproofs,see[6]. At vari- ku(t)k2 ≤ ku(0)k2exp −4ν0t+18I(t) . (44)
ance with [6] we do not adopt periodic boundary l2
(cid:20) (cid:21)
conditions. Let us assume there are two different
This result tells that, since the two supposed solu-
solutions v and v′ of the NSE (13). Then we define
tions v,v′ possess the same initial conditions and
u := v−v′ and obtain after subtracting the NSE
therefore u(0) = 0, we have u(t) = 0 for times
for v and v′
t ∈ (0,τ) for which I(t) exists. This conclusion
∂uk ∂uk ∂uk ∂vk ∂(p−p′) holds true also in the inviscid limit ν0 → 0.
−u +v +u = − +ν ∆u .
i i i 0 k
∂t ∂x ∂x ∂x ∂x
i i i k
(38)
References
When this equation is scalarly multiplied by u and
integrated over the volume Ω, then, apart from the
[1] Leray J., Acta Math. 63 (1934), 193
pressureterm,thesecondandthirdtermsoftheleft
hand side vanish by the same argument used before [2] Hopf E., Math. Nachrichten 4 (1951), 213
in (31). With the abbreviation
[3] Ladyzhenskaya O.A., The Mathematical The-
ory of Viscous Incompressible Flow, second
kuk2 = dV u·u (39)
edition, Gordon and Breach (1963), New York
ZΩ
we can write [4] Serrin J., The initial value problem for the
Navier-Stokes equations, in: Nonlinear Prob-
1 d
kuk2 = A+B; lems,ed.R.E.Langer,UniversityofWisconsin
2dt
Press (1963), Madison USA
∂u ∂u ∂v
k k k [5] Temam R., The Navier-Stokes Equations,
A := −ν dV ; B := − dV u u .
0 k i
ZΩ ∂xi ∂xi ZΩ ∂xi North-Holland (1978), Amsterdam
(40)
Nowtheviscosity termisestimated bymeansofthe [6] DoeringCh.R.andGibbonJ.D.,Applied Anal-
Poincar´e inequality [12] ysis of the Navier–Stokes Equations, Cam-
bridge University Press (1995), Cambridge
2
−A ≡ |A| ≥ kuk2 (41) USA
l2
[7] Lions P.-L., C. R. Acad. Sci. Paris 316 (1993),
where l denotes the smallest distance between two
S´erie I, 1335
parallel planes which just contain Ω. The B term
is estimated by using the definition (36) and the [8] Landau L.D. and Lifshitz E.M., Statistical
Schwarz inequality as follows Physics, Academie- Verlag (1976), Berlin
3 [9] Rauh A., Global stability of systems related to
∂v
|B|= | dV uk kui| ≤ kDvk∞ dV |ukui| the Navier-Stokes equations, (this conference
∂x
ZΩ i k,i=1ZΩ proceedings)
X
≤ 9kDvk∞kuk2. (42) [10] Frisch U., Fully developed turbulence and sin-
gularities, in: Chaotic behaviour of deter-
One arrives at the ordinary differential inequality
ministic systems, Les Houches XXXVI, eds.
1 d 2ν G. Iooss, R.H.G. Helleman, R.Stora, North-
2dtkuk2 ≤ − l20 +9kDvk∞ kuk2, (43) Holland (1983), Amsterdam and N.Y., p.668
(cid:20) (cid:21)
6
[11] Grauer R. and Sideris T.C., Physica D 88
(1995), 116
[12] Joseph D.D., Stability of Fluid Motion I,
Springer-Verlag (1976), Berlin
7