Incompressible Euler equation in 2d and 3d
Discuss problems open even in 2d
C.Bardos
Laboratoire Jacques-Louis Lions
Mathematical Hydrodynamics Moscow June 2006.
Good reasons to focus on the incompressible Euler Equation :
Applications often correspond to very large Reynolds number.
A theorem valid for any finite reynolds number should be compatible with
results concerning infinite Reynolds number. In fact it is the case
Reynolds= ∞ which drive other results I will give several examples of this
fact.
The parabolic structure does not carry enough information to deal with the
3d Navier Stokes equation. A basic exemple of Mongtomery-Smith (2001)
√
∂tu − ∆u − −∆(u2) = 0 No energy conservation
Mathematical Hydrodynamics Moscow June 2006.
Basic Fact in 3d
Existence of a smooth solution for finite time with initial data in
Hs , s ≥ n
2 + 1. Back to Lichtenstein based on the comparison with the
Riccati equation
3
0
y ≤ Cy 2 ; y = ||u||2
Hs
Existence uniqueness and propagation of the regularity of the intial data
(including analyticity B. , Benachour) persists as long as
Z T
0
||∇ ∧ u(., t)||L∞ dt < ∞ or Kozono
Z T
0
||∇ ∧ u(., t)||BMOdt < ∞
Instability in W 1,p for all 1 < p < ∞ Exemple on the Torus with pressure
less fluid :
(u1(x2), 0, u3(x1 − tu1(x2), x2))
Mathematical Hydrodynamics Moscow June 2006.
Blow up for some solutions of infinite energy in (R2/L)2 × R Constantin
u = (u1(x1, x2, t), u2(x1, x2, t), x3γ(x1, x2, t)) = (ũ, x3γ)
∂t(∇ ∧ ũ) + ũ∇(∇ ∧ ũ) = γũ
∇ · ũ = −γ ⇒ ∇ · u = 0
Z
2
I(t) = − 2
γ(x1, x2, t)dx1dx2 to enforce periodicity
2
2
L (R /L)
∂tγ + ũ∇γ = −γ 2 + I(t)
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Dissipative solutions
Let u(x, t) be a smooth solution of the Euler equation in Ω and v(x, t) ; v
are divergence free and tangent to the boundary ∂Ω. test functions.
E(v) = ∂tv + P (v∇v) , P Leray projectorv sol ⇒ E(v) = 0.
∂tu + ∇x(u ⊗ u) + ∇xp = 0
∂tw + ∇x(w ⊗ w) + ∇xq = E(w)
d|u − w|2
+ 2(D(w)(u − w), (u − w)) = 2(E(w), u − w)
dt
By integration :
Rt
2
|u(t) − w(t)| ≤ e 0 2||D(w)||∞(s)ds|u(0) − w(0)|2
Z t Rt
+2
e s 2||D(w)||∞(τ )dτ (E(w), u − w)(s)ds
0
Mathematical Hydrodynamics Moscow June 2006.
(1)
A divergence free, tangent to the boundary function
u ∈ C(Rt; (L2(Ω))d − w)
is called a dissipative solution if for any divergent free, tangent to the
boundary smooth test function it satisfies (1).
• Any classical solution is a dissipative solution.
• With w = 0 one obtains for the dissipative solution the relation
|u(t)|2 ≤ |u(0)|2 this is not in contradiction with an some non conservation
of energy coming from lack of regularity and justifies the name dissipative.
• Assume that w is a classical solution and u a dissipative solution then one
has
Rt
2
|u(t) − w(t)| ≤ e 0 2||D(w)||∞(s)ds|u(0) − w(0)|2
in particular if there exists a classical solution any dissipative solution with
the same initial data coincide with it.
Mathematical Hydrodynamics Moscow June 2006.
A weak-strong uniqueness result in the sense of Dafermos. At variance
with the definition of weak solution (cf Shnirelman).
Interior dissipative solution is a dissipative solution which satisfies (1) for
all smooth w with compact support.
Any Leray solution of Navier Stokes weakly converges (modulo
subsequence) to an interior dissipative solution of the Euler equation
With no boundary the existence of a smooth solution of the Euler equation
implies the convergence of Leray solution of Navier Stokes.
Mathematical Hydrodynamics Moscow June 2006.
Theorem Laure Saint -Raymond. Let F be a family of renormalized
solutions of the rescaled Boltzmann equation.
1
∂tF + v∇xF = q B(F, F) ,
q>1
Assume the existence of a divergence free vector field uin ∈ L2(IT 3) for
which
1
in|M
H(F
) → 0 as → 0 ,
in
2
1,u
,1
M
1,uin,1
=
1
3
|v−uin |2
2
e−
(2π) 2
Then the family
1
(
Z
vFdv)
is relatively compact in w − L∞(R+; L1(IT 3)) and each of its limit points is
an interior dissipative solution of the 3d Euler equation. With no boundary
(periodic or the whole space) the existence on an intervall [0, T ] of a
smooth solution implies both the convergence of the Navier Stokes
solution and of the moment of the solution of the rescaled Boltzmann
equation on the same intervall
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Spectra and Wigner Transform
∂tuν − ν∆uν + ∇(uν ⊗ uν ) = −∇pν ,
∇ · uν = 0 , uν (x, 0) = u0(x) given
! t!
1
1
|uν (., t)|2 + ν
|∇uν (x, t)|2dx = |u(., 0)|2
(2)
2
2
0 Ω
With the energy equality (inequality) (2) the solution of the NS converges
weakly to a function u . The obstruction of to u being a weak solution of
the Euler equation lies in the expression :
uν ⊗ uν may be $= uν ⊗ uν
Defect of convergence related to a deterministic version of the Reynold
stress tensor and the Kolmogorov spectra :
Mathematical Hydrodynamics Moscow June 2006.
O. Reynolds
A.N. Kolmogorov
E.P. Wigner
uν ⊗ uν = uν ⊗ uν + (uν − uν ) ⊗ (uν − uν ))
RT (u) = (uν − uν ) ⊗ (uν − uν ));
∂tu + ∇u ⊗ u + ∇RT (u) + ∇p = 0 RT (u) sym. pos. meas. val. matrix
e ν = uν − u by zero outside Ω and introduce the deterministic
Extend u
√
correlation spectra or Wigner transform at the scale ν :
√
√
Z
1
ν
ν
iky u
e ν (x +
e ν (x −
y)
⊗
u
y)dy
RT ˆ
(uν )(x, t, k) =
e
d
d
2π IRy
2
2
uν − u =
Z
IRdk
RT ˆ
(uν )(x, t, k)dk
For the weak limit RT (u)(x, t, k) one has
Z
Rkd
RT (u)(x, t, k)dk = RT (u)(x, t)
√
√
Z
1
ν
ν
2
iky
e
e
φ(x) RT (u)(x, t) =
e (φuν )(x −
y) ⊗ (φuν )(x +
y)dy
2π d Ryd
2
2
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RT is a local object with local isotropy : Galilean invariance one has
E(x, t, |k|)
k⊗k
ˆ
RT (u)(x, t, k) =
(Id −
)
d−1
2
|k|
|k|
(3)
Furthermore under the assumption (traditional in turbulence) that RT is
invariant under rotation and depends only on the tensor (∇u + ∇⊥u) one
has in 3d the formulas :
RTˆ(u)(x, t, k) = a(|(∇u + ∇⊥u)|)Id + µ(|(∇u + ∇⊥u)|)(∇u + ∇⊥u)
+ λ(|(∇u + ∇⊥u)|)(∇u + ∇⊥u)2
with in 2d λ ≡ 0 .
Mathematical Hydrodynamics Moscow June 2006.
Remark and open problems
• Observe that the term a(|(∇u + ∇⊥u)|)Id can be absorbed in the pressure
gradient.
• When ever there is a smooth solution for the Euler equation (with no
boundary) RT (u) ≡ 0
• Taking in account all the hypothesis made giving a formula that would
relate µ and λ and the spectra E(|k|, t, x) should be a first step in a
mathematical treatment of turbulent models say like − k
• A second step would be to prove that in any case and for smooth initial
data the defect measure is invariant under rotation and therefore the
formula (3) is valid.
• Denote by
(t) = (T, ν)(t) =
Z
Ω
|∇uν (x, t)|2dx ,
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Z ∞
1
(t)dt ≤ |u(., 0)|2
2
0
then the determinist counter part of the statistical Kolmogorov law would
be the relation
2
k⊗k )
(Id
−
5
|k|2
RT ˆ
(uν )(x, t, k) ' 1.5(t) 3 |k|− 3
|k|2
for ν → 0
(5)
The existence of estimate (5) implies the convergence to a weak solution
(even if no regular solution do exist). The existence of the smooth solution
will give
lim ν
ν→0
Z TZ
0
Ω
|∇uν (x, t)|2dxdt = 0
Mathematical Hydrodynamics Moscow June 2006.
Boundary value problem for Navier Stokes
Describe the very few results that do exist for the limit ν → 0 for solution
of the Navier Stokes equation in a domain Ω ⊂ IRd , d = 2, 3 with
homogenous boundary condition. The Dirichlet boundary condition uν = 0
is not the easiest one (for instance in 2d u · n = 0 and ∇ ∧ u = 0 on the
boundary are much easier for our purpose). However it is the one to
consider.
• It can be deduced in the smooth regime from the Boltzmann equation
when the interaction with the boundary is described by a scattering kernel.
• It generates the pathology which observed in physical experiments.
Problem come from the boundary layer generate by the jump in tangential
component of the vorticity. The non linearity may imply the propagation of
this jump inside the media.
Proposition In any case (modulo extraction of subsequence) uν converges
weakly to an interior dissipative solution u.
Mathematical Hydrodynamics Moscow June 2006.
This is not enough because for an interior dissipative solution w the test
functions are of compact support or at least should belong to H01(Ω) and
and the solution u is not zero on the boundary.
Mathematical Hydrodynamics Moscow June 2006.
Theorem Kato For divergence free flows (∇ · u = 0) with the same initial
data u0(x)
∂tuν − ν∆uν + ∇(uν ⊗ uν ) + ∇pν = 0 , , uν (x, t) = 0 on ∂Ω
(6)
∂tu + ∇(u ⊗ u) + ∇pν = 0 , u(x, 0) = u0(x) , u(x, t) · n = 0 on ∂Ω (7)
Assume that the equation (7) has a smooth solution for 0 < t < T then
the following fact are equivalent :
RT R
(iii) ν 0 Ω |∇uν (x, t)|2dxdt → 0
RT R
(iv) ν 0 Ω∩{d(x,∂Ω)<ν} |∇uν (x, t)|2dxdt → 0
tang
ν
(v) ν ∂u∂n
→ 0 in D0(∂Ω × [0, T ])
(vi) uν (t) → u(t) weakly in L2(Ω) for each t ∈ [0, T ] u(t) is smooth and
RT (u)(x, t, k) ≡ 0
For proofs Use energy estimate and a for (v) a boundary layer correction.
Mathematical Hydrodynamics Moscow June 2006.
It does not seems that anything more can be said in full generality.
However in 2d it is a good example of a non trivial situation in spite of the
fact that the limit equation has a smooth solution.
The Reynold Tensor and its spectra RT (u)(x, t) and RTˆ(u)(x, t, k) should
be non zero in the “Von Karman vortex street” and since it is propagated
by a turbulent flow local isotropy could be assumed leading to formulas :
RTˆ(u)(x, t, k) =
E(|k|)
k⊗k
⊥ u)|)(∇u + ∇⊥ u)
(Id
−
)dk
'
µ(|(∇u
+
∇
|k|2
IRdk 2π|k|
Z
Eventually the pathology of the problem is related to the instability of the
Prandlt equations.
Mathematical Hydrodynamics Moscow June 2006.
Von Karman Vortex street behind a cylinder 2d
Numerical simulation done by S.Popinet Re= 160
Von Karman Vortex street over
Guadalupe Island
Prandlt and Von Karman
The Prandlt Boundary Layer =
√
ν
∂tu1 − 2∆u1 + u1∂x1 u1 + u2∂x2 u1 + ∂x1 p = 0
∂tu2 − 2∆u2 + u1∂x1 u2 + u2∂x2 u2 + ∂x2 p = 0
∂x1 u1 + ∂x2 u2 = 0 u1(x1, 0) = u2(x1, 0) = 0 on x1 ∈ R
x2
,
X1 = x1, X2 =
e 1(x1, X2) = u1(x1, X2), u
e 2(x1, X2) = u2(x1, X2) .
u
2 u
e 1 − ν∂x
e + u∂
e x1 u 1 + u
e 2∂x2 u
e 1 + ∂x1 pe = 0
∂tu
2 1
e 1, x2, t) = p(x
e 1, t)
∂x2 pe = 0, p(x
e 1 + ∂x2 u
e2 = 0
∂x1 u
e 1(x1, 0) = u
e 2(x1, 0) = 0 for x1 ∈ R
u
e 2(x1, x2) = 0, lim u
e 1(x1, x2) = U (x1, t)
lim u
x2 →∞
x2 →∞
|U (x1, t)|2
e 1, t) = Constant
+ p(x
2
Mathematical Hydrodynamics Moscow June 2006.
• The validity of the Prandlt approximation is consistant with the Kato
theorem
Z TZ
√
2
ν
|∇ ∧ uν (x, s)| dxds ≤ C ν
0 Ω∩d(x,∂Ω)≤cν
• As long as the solution of of the Prandlt equations is smooth the
solution of Navier Stokes converges to the solution of the Euler equation.
• The possibility of detachment and recirculation appear in the fact that
the Prandlt problem is highly unstable.
• If initially the velocity profile is monotone then (Oleinik) global solution
do exist for the Prandlt equation.
• The loss of regularity in finite time is proven for some class of non
monotone initial profile (E W. B. Enquist)
• The existence of solution which is not described by the Prandlt equation
has been constructed by Grenier using unstable solutions of the 2d Euler
equation.
Mathematical Hydrodynamics Moscow June 2006.
• With analytic initial data (in fact analytic with respect to the tangential
variable is enough) one can prove abstract version of the Cauchy
Kowalewsky theorem the existence of a smooth solution of the Prandlt
equation for a fintie time and the convergence to the solution of the Euler
equation during this same time (Asano, Caflisch-Sammartino and
Cannone-Lombardo-Sammartino)
Mathematical Hydrodynamics Moscow June 2006.
Relation between Prandlt and Kelvin Helmholtz
Comparison beetwen Prandlt and Kelvin Helmholtz problems in 2d
The Kelvin Helmholtz problem concerns the evolution of a solution of the
2d Euler equation
∂tu + ∇(u ⊗ u) + ∇p = 0 ,
∇·u=0
(8)
with initial vorticity Ω being a measure concentrated on a curve
• Existence of a weak solution when Ω(x, 0) is a Radon measure with
distinguished sign (Delort) or with controlled change of sign (Lopes Filho,
Nussenzveig Lopes and Xin). This is impaired by the non uniqueness result
of Shnirelman
• For density located on a smooth curve the problem is equivalent to the
Birkhoff Rott equation in the complex plane.
1
∂tz =
p.v.
2πi
Z
Mathematical Hydrodynamics Moscow June 2006.
dλ0
z(t, λ) − z(t, λ0)
(9)
• As for the Prandlt one has for (9) a local in time existence uniqueness
result in the class of analytic initial data (B. Frisch Sulem and Sulem) with
a version of the Cauchy Kowalewsky theorem.
• As for the Prandlt equation one can construct a solution with blows up in
finite. Use reversibility write (Caflish Orellana)
z(t, λ) = λ + s0 + r(λ, t)
(10)
with s0 given by :
− 2t −iλ 1+ν
− 2t +iλ 1+ν
s0(λ, t) = (1 − i){(1 − e
)
− (1 − e
)
}
(11)
> 0 small enough , r(λ, t) proven to be analytic for t > 0 and much
smaller (O(2)) in C 2(λ) Since s0(λ, 0) ∼ λ1+ν :
0
z(λ, 0) ∈ C 1+ν , z(λ, 0) ∈
/ C 1+ν for ν 0 > ν
Mathematical Hydrodynamics Moscow June 2006.
(12)
• Different with some regularity ( Ctα(Cλ1+ν ) the problem can be “locally ”
reduced to the non linear perturbation of an elliptic equation :
Ω(0, 0) = 1 , z(λ, t) = (αt + β(λ+f (t, λ)) , f (0, 0) = ∇f (0, 0) = 0 .
1
Ω(t, λ0)dλ0
2
p.v.
|β| ∂tf =
0 ) +E(t, r(tλ))
f
(t,λ)−f
(t,λ
0
2πi
z(t,λ )∈U (λ − λ0 )(1 − )
λ−λ0
Z
(13)
Next one use the expansion
1
pv
2π
Z
=
2π
Z
dλ0
(λ − λ0)(1 +
0
dλ
f (λ,t)−f (λ0 ,t)
)
λ−λ0
X n
f (λ, t) − f (λ0, t) 0
dλ +
(λ − λ0)2
n≥2 2π
Z
(f (λ, t) − f (λ0, t))n 0
dλ .
0
(n+1)
(λ − λ )
(14)
and the formulas (Hilbert transform)
1
2π
Z
f (λ, t) − f (λ0, t) 0
i
1
dλ
=
−
sign(D)f
,
vp
0
2
(λ − λ )
2
2π
Z
f (λ, t) − f (λ0, t) 0
dλ = |D|f
0
2
(λ − λ )
and deduce from (13) and (14) that the real and imaginary part of
Mathematical Hydrodynamics Moscow June 2006.
f (t, λ) = X(t, λ) + iY (t, λ) are locally solutions of the system :
Ω2
∂tX = 02 |Dλ|Y + R1(X, Y ) + E1(t, λ) †
2β
Ω2
∂tY = 02 |Dλ|X+R2(X, Y )+E2(t, λ)
2β
The consequence is that a non analytic solution as the one given by
Caflsich and Orellana cannot remain in Ctα(Cλ1+ν ) after the apparition of
an Holder singularity it has to be more singular. In this case the challenge
is threshold of regularity that will imply analyticity.
Mathematical Hydrodynamics Moscow June 2006.
Improving the regularity threshold
Experiments and numerical simulation show the existence and the stability
of vortex sheet after the singularity. Furthermore these vortex sheet roll up
and seem to lead to rectifiable curves but of infinite length. Therefore the
“threshold of regularity” should be above and may include spirals with
finite length. In fact the best (to the best of my knowledge) result is due to
α (IR ; C 1+β (IR ) is replaced by H 1 (IR × IR ) .
Sijue Wu. The hypothesis Cloc
t
t
λ
λ
loc
loc
Estimates are done explicitly using theorems of G. David saying that for all
chord arc curves Γ : s 7→ ξ(s), s the arc length the Cauchy integral operator
CΓ(f ) = pv
Z
f (s0)
0)
dξ(s
ξ(s) − ξ(s0)
is bounded in L2(ds) . It is interesting to notice that these results will apply
to logarithmic spirals r = eθ but not to infinite length algebraic spirals.
Mathematical Hydrodynamics Moscow June 2006.
Conclusion
• I tried to show many crucial the issues lie in the analysis of the Euler
equation.
• In 3d space variables with no boundary the solution of incompressible
Navier Stokes equation converges with ν → 0 to a dissipative solution of
the Euler equation u. Therefore to the classical solution whenever such
classical solution exist. It will not be in the non uniqueness With the
formula :
(uν (x, t) − u) ⊗ (uν (x, t) − u) =
Z
RT ˆ
(uν )(x, t, k)dk
one observes that if a Onzager Kolmogorof type hypothesis holds
2
E(|k|)
−5
ˆ
3
|RT (uν )(x, t, k)| =
≤ |k| 3
(15)
2
4π|k|
then uν converges to a weak solution of the Euler equation. As conclusion
one can observe that with no boundary (15) implies :
lim ν
ν→0
Z TZ
0
Ω
|∇uν |2dxdx 6= 0 ⇒ no regular solution of the Euler equation
(16)
Mathematical Hydrodynamics Moscow June 2006.
• As observed above the situation is completely different when boundary
effects are present and this is already true in 2d.
• In this setting turbulent effect may appear and some results (invariance
under rotation) of the Wigner -Measure -Rayleigh Tensor may be available.
On the other hand the full description of what happen is surely out of
reach even in 2d.
• Existence of Kelvin Helmholtz type solution appear in the wake of
turbulence generated by boundary effect. In agreement with the previous
remark the support of the vorticity density is a very singular curve. On the
other hand such phenomena is very stable leading to the need of stability
theorem for singular solution. Surely much more difficult but may be more
natural than result based on the Arnold Criteria.
Mathematical Hydrodynamics Moscow June 2006.
Kelvin
Helmholtz
type interfaces.
Kelvin and Helmholtz