Global solutions of a class of infinite energy three
dimensional Navier-Stokes and Euler equations
Namkwon Kim and Bataa Lkhagvasuren
Mathematics department
Chosun University
South Korea
October 20, 2016
Outline
Introduction
Reformulation
Existence result for Navier-Stokes equations
Blow-up solutions for Navier-Stokes equations
Existence result for Euler equations
Namkwon Kim and Bataa Lkhagvasuren
2 / 35
Global solutions of a class NS and Euler equation
Incompressible Navier-Stokes (Euler) equations
Incompressible Navier-Stokes (Euler) equations in R3 write:
∂t u + u · ∇u − ν∆u + ∇p = 0,
(1)
∇·u =0
(2)
where
u−velocity field
p−pressure of the fluid
Assumption:
u1 = u1 (x, y , t), u2 = u2 (x, y , t), u3 = zγ(x, y , t)+ϕ(x, y , t). (3)
for some scalar fields γ, ϕ : R2 × R+ → R.
Namkwon Kim and Bataa Lkhagvasuren
3 / 35
Global solutions of a class NS and Euler equation
Solutions of the form (3)
have infinite energy.
May be placed in a different category than those have finite
energy.
still satisfy the 3D fluid equations.
May enhance our understanding on the local behavior of the
flow.
Namkwon Kim and Bataa Lkhagvasuren
4 / 35
Global solutions of a class NS and Euler equation
History of the problem
J.D. Gibbon, A.S. Fokas, and C.R. Doering (1999) showed the
relation to 2D flow when γ depends only on time γ = γ(t).
A.J. Majda and A.L. Bertozzi (2002) studied when γ = 0.
For axisymmetric case, the exact blow-up solutions of Euler
equation were constructed by J.D. Gibbon and etc (2003).
K. Ohkitani (2007), a blow up problem was studied for
Navier-Stokes flow (largeness on initial data).
Peter Constantin (2000), the exact blow-up solutions for a
periodic domain.
K. Ohkitani and J.D. Gibbon (2000), Numerical study of
singularity formation.
Namkwon Kim and Bataa Lkhagvasuren
5 / 35
Global solutions of a class NS and Euler equation
With ue ≡ (u1 , u2 ), 2D part is:
u − ν∆e
u + ∇p = 0,
∂t ue + ue · ∇e
∇ · ue = − γ.
This is not 2D flow, because of divergence condition.
Third velocity field u3 = zγ(x, y , t) + ϕ(x, y , t) gives
−
∂p
∂γ
= z(
+ ue · ∇γ − ν∆γ + γ 2 )+
∂z
∂t
∂ϕ
+ ue · ∇ϕ − ν∆ϕ + γϕ).
(
∂t
Integrating
∂γ
1
−p(x, y , z) = z 2 (
+ ue · ∇γ − ν∆γ + γ 2 )+
2
∂t
∂ϕ
z(
+ ue · ∇ϕ − ν∆ϕ + γϕ) + f (x, y , t).
∂t
Namkwon Kim and Bataa Lkhagvasuren
6 / 35
Global solutions of a class NS and Euler equation
∂p
∂x
and
∂p
∂y
are independent of z. Thus
∂γ
+ ue · ∇γ − ν∆γ + γ 2 = c(t),
∂t
∂ϕ
+ ue · ∇ϕ − ν∆ϕ + γϕ = d(t).
∂t
In general, we have
∂ ue
+ ue · ∇e
u − ν∆e
u + ∇p = 0,
∂t
∂γ
+ ue · ∇γ − ν∆γ + γ 2 = c(t),
∂t
∂ϕ
+ ue · ∇ϕ − ν∆ϕ + ϕγ = d(t),
∂t
∇ · ue = −γ,
Namkwon Kim and Bataa Lkhagvasuren
7 / 35
(4)
Global solutions of a class NS and Euler equation
We first show that c(t) = d(t) = 0 in (4) if ∇u is smooth and ue
does not grow too fast near infinity.
Lemma
Let ue, γ, ϕ is a smooth solution of (4) for t ∈ [0, T ]. If ue grows
sublinearly near infinity and γ, ϕ ∈ W 1,p for all t ∈ [0, T ] for some
1 < p < ∞, then
c(t), d(t) = 0
in (4).
Namkwon Kim and Bataa Lkhagvasuren
8 / 35
Global solutions of a class NS and Euler equation
Reformulation
From now on, we set c(t) = d(t) = 0 and present a more favorable
formulation of (4). The vorticity of this type of flow is given by
(zγy + ϕy , −zγx − ϕx , u2,x − u1,y ) and if we denote the third
component of vorticity by ω, then ω is decoupled:
∂ω
+ ue · ∇ω − ν∆ω = γω.
∂t
(5)
Using Helmholtz-Hodge decomposition and the relations
∇ · ue = −γ,
∇ × ue = ω,
ue can be expressed by
ue = −∇(∆−1 γ) + ∇⊥ (∆−1 ω).
Namkwon Kim and Bataa Lkhagvasuren
9 / 35
(6)
Global solutions of a class NS and Euler equation
Reformulation
More precisely, we can express ue with singular integral operators as
follows.
Z
u1 = −
Z
K1 (x, y)γ(y)dy −
2
ZR
u2 = −
K2 (x, y)ω(y)dy + U1 ,
2
ZR
K2 (x, y)γ(y)dy +
R2
(7)
K1 (x, y)ω(y)dy + U2 ,
R2
In order to make the integrals in (7) meaningful , this Biot-Savart
type formula has to be modified depending on the integrability of
argument functions.
Namkwon Kim and Bataa Lkhagvasuren
10 / 35
Global solutions of a class NS and Euler equation
Reformulation
1
If ω, γ ∈ Lp (R2 ) with 1 < p < 2, then
K1 (x, y) =
1 x1 − y1
1 x2 − y2
, K2 (x, y) =
2π |x − y|2
2π |x − y|2
2p
and ue ∈ L 2−p (R2 ) by Hardy-Littlewood-Sobolev inequality. In
this case, Ui = 0 for i = 1, 2.
2
If ω, γ ∈ Lp (R2 ) with 2 ≤ p < ∞, then
1 x1 − y1
y1
1 x2 − y2
y2
K1 (x, y) =
+
, K2 (x, y) =
+
2π |x − y|2 |y|2
2π |x − y|2 |y|2
2
1−
and |e
u | ≤ C |x| p k(ω, γ)kLp (R2 ) , ue ∈ Lploc (R2 ) and uniformly
Hölder continuous with exponent 1 − p2 if p > 2, and
ue ∈ BMO if p = 2. We take Ui = ui (0), i = 1, 2.
Namkwon Kim and Bataa Lkhagvasuren
11 / 35
Global solutions of a class NS and Euler equation
Reformulation
3
s (R2 ), Besov spaces, then
If ω, γ ∈ Bp,q
K1 (x, y) =
1 x1 − y1
1 x2 − y2
, K2 (x, y) =
2
2π |x − y|
2π |x − y|2
where the kernels Kj , j = 1, 2 are defined by the Fourier
multipliers
−iξj
Kbj (ξ) =
, j = 1, 2.
|ξ|2
Note that the kernels Kj are homogeneous of order −1.
s+1 (R2 ) by Besov embedding. In this case,
Therefore, ue ∈ Bp,q
Ui = 0 for i = 1, 2.
Namkwon Kim and Bataa Lkhagvasuren
12 / 35
Global solutions of a class NS and Euler equation
Reformulation
Given U ∈ R2 and γ0 , ω0 , ϕ0 , find (γ, ω, ϕ) satisfying
∂ω
+ ue · ∇ω − ν∆ω − γω = 0,
∂t
∂γ
+ ue · ∇γ − ν∆γ + γ 2 = 0,
∂t
∂ϕ
+ ue · ∇ϕ − ν∆ϕ + γϕ = 0,
∂t
(8)
where ue is given in (7).
Namkwon Kim and Bataa Lkhagvasuren
13 / 35
Global solutions of a class NS and Euler equation
Weak solution for Navier-Stokes equation (ν = 1)
First, we give the definition of the weak solution for the system (8).
By a weak solution of (8) for initial data U ∈ R2 ,
ω0 , γ0 , ϕ0 ∈ Lp , 1 < p < ∞, we mean a triple
(ω, γ, ϕ) ∈ L∞ ([0, T ); Lp ) satisfying
−2,p
),
∂t (ω, γ, ϕ) ∈ L1 ([0, T ); Wloc
and
Z tZ
(ω, γ, ϕ)(x, 0) = (ω0 , γ0 , ϕ0 )
Z tZ
Z tZ
ω ue · ∇φdx,
Z tZ
Z tZ
Z tZ
Z tZ
2
∂t γφdx =
γ∆φdx + 2
γ φdx +
γ ue · ∇φdx,
0
R2
0
R2
0
R2
0
R2
Z tZ
Z tZ
Z tZ
Z tZ
∂t ϕφdx =
ϕ∆φdx + 2
γϕφdx +
ϕe
u · ∇φdx
ω∆φdx +
∂t ωφdx =
0
R2
0
R2
0
R2
0
R2
R2
0
0
R2
0
R2
(9)
for all φ ∈ C0∞ ([0, T ] × R2 ) with ue given in (7).
Namkwon Kim and Bataa Lkhagvasuren
14 / 35
Global solutions of a class NS and Euler equation
Local existence for Navier-Stokes equation
We first give the local in time existence for the system (8) for initial
q
data (ω0 , γ0 , ϕ0 ) ∈ Lq ∩ L q−1 , 1 < q < 2.
Theorem
q
q−1 . Given
0
Lq (R2 ), there
Let 1 < q < 2 and q 0 =
U ∈ R2 ,
ω0 , γ0 , ϕ0 ∈ Lq (R2 ) ∩
exists a time T > 0 such that
the system (8) has a weak solution
0
ω, γ, ϕ ∈ L∞ (0, T ; Lq ∩ Lq (R2 )) ∩ L2 (0, T ; W 1,2 ). Furthermore,
ω, γ, ϕ are smooth for 0 < t < T and ue ∈ L∞ (0, T ; L∞ ).
Outline of Proof:
Iteration by linear equations
Energy type estimates
Strong convergence
Namkwon Kim and Bataa Lkhagvasuren
15 / 35
Global solutions of a class NS and Euler equation
Local existence for Navier-Stokes equation
Next, we deal with the existence for weaker initial data using the
theorem 1.
Theorem
Given U ∈ R2 and ω0 , γ0 , ϕ0 ∈ Lp for some 1 < p < ∞, there
exists a time T > 0 such that the system (8) has a weak solution
ω, γ, ϕ ∈ L∞ ([0, T ); Lp ) satisfying
p
|(ω, γ, ϕ)| 2 −1 (ω, γ, ϕ) ∈ L2 (0, T ; W 1,2 ).
(10)
Outline of Proof:
Mollify the initial data by ω,0 , γ,0 , ϕ,0 ∈ L6/5 ∩ L6 ∩ Lp ,
which approximates ω0 , γ0 , ϕ0 in Lp .
Energy type estimates
Aubin-Nitsche type compactness lemma
Namkwon Kim and Bataa Lkhagvasuren
16 / 35
Global solutions of a class NS and Euler equation
Global existence for Navier-Stokes equation
We show that a local in time solution we find in the previous section
becomes global if γ0 ∈ Lp , 1 < p < 2 and γ0 ≥ 0 further.
Theorem
Suppose that ω0 , γ0 , ϕ0 ∈ Lp (R2 ), p ∈ (1, 2). If γ0 ≥ 0 further,
then there exists a global solution of (8) corresponding to
ω0 , γ0 , ϕ0 .
Namkwon Kim and Bataa Lkhagvasuren
17 / 35
Global solutions of a class NS and Euler equation
Blow-up solutions for NS equations
We give nontrivial exact blow-up solutions to (4) and give some
subsolutions which blow up in a finite time. Let us assume the
velocity field is of the form
u = (ur (r , t), uθ (r , t), zγ(r , t)),
(11)
with cylindrical coordinates (r , θ, z) and with the assumption
(ω, ϕ) = (0, 0).
As pointed out in K. Ohkitani, (2007), we have a considerable
simplification in our system. With the notation γ
e = −γ, the
equations become
∂e
γ
1 ∂
∂e
γ
∂e
γ
2
+ ur
=γ
e +
r
,
∂t
∂r
r ∂r
∂r
(12)
Z
1 r
τγ
e(τ )dτ.
ur (r ) =
r 0
Namkwon Kim and Bataa Lkhagvasuren
18 / 35
Global solutions of a class NS and Euler equation
Blow-up solutions for NS equations
First, we look for an example of blow-up solution in the form
γ(t, r ) = ξ(t)e −r
2 h(t)
(13)
,
where ξ and h are functions of time only. Under this assumption,
our system becomes
ξ 0 − ξ 2 − r 2 ξh0 = −4hξ + 4r 2 h2 ξ,
ξ(t)(1 − e −r
ur (r ) =
2rh(t)
2 h(t)
)
(14)
,
and it implies that
h0 = −4h2 ,
ξ 0 + 4hξ = ξ 2 .
Namkwon Kim and Bataa Lkhagvasuren
19 / 35
(15)
Global solutions of a class NS and Euler equation
Blow-up solutions for NS equations
The solutions of the system (15) can be computed explicitly.
h(t) =
h(0)
,
4h(0)t + 1
h(0)
ξ(t) =
.
(4h(0)t + 1)(k − ln(4h(0)t + 1))
(16)
with k is an arbitrarily constant.
If we take h(0) > 0 and k > 0, then the corresponding solution
blows up and the blow-up time will be
t∗ =
ek − 1
.
4h(0)
(17)
We remark that the solutions we have found blow up everywhere and
Z ∞
1
p
C ξ(t)
C
−r 2 ph
(18)
kγkp = C ξ(t)
e
rdr
=
,
1 = ∗
t −t
0
(2ph) p
as t → t ∗ . Therefore, the blow-up is of type I.
Namkwon Kim and Bataa Lkhagvasuren
20 / 35
Global solutions of a class NS and Euler equation
Blow-up solutions for NS equations
Though the above blow-up solution is exact, it is nonnegative on
whole plane R2 for all t ≥ 0. We now show that there are blow-up
solutions developed from sign-changing initial data. It is shown in K.
Ohkitani, (2007) that the system (12) enjoys the comparison
principle. Hence, we will look for sign-changing subsolutions which
blow up in a finite time.
Theorem
Any smooth solution γ
e(t, r ) of the equation (12) blows up in a
finite time if
γ
e(0, r ) ≥ c1 exp(−
c0 2
r ) − c0
4
(19)
for some constants c1 > 3c0 > 0.
Namkwon Kim and Bataa Lkhagvasuren
21 / 35
Global solutions of a class NS and Euler equation
Blow-up solutions for NS equations
Proof:
Let us consider subsolutions of the form
γ(t, r ) = ξ(t)e −r
2 h(t)
− c0 ,
(20)
where c0 > 0 and ξ(0) > c0 . Plugging γ(t, r ) in (12), the equation
becomes
−r 2 ξh0 + ξ 0 − ξ 2 + r 2 hξc0 = −2ξc0 +
c0
−r
e 2h
+ (−4ξh)(1 − r 2 h).
(21)
c0
> 0, γ(t, r ) = ξ(t)e −r
Since −r
e 2h
(12) if
2 h(t)
− c0 will be a subsolution of
h0 − c0 h = −4h2 ,
ξ 0 + (2c0 + 4h)ξ = ξ 2 .
Namkwon Kim and Bataa Lkhagvasuren
22 / 35
(22)
Global solutions of a class NS and Euler equation
Blow-up solutions for NS equations
We find that
h(t) =
c0
,
4
ξ(t) =
3c0 e −3c0 t
0
e −3c0 t + 3c
c1 − 1
is a solution of (22) if c1 > 0. Clearly, ξ(t) blows up at
0
t = − 3c10 ln(1 − 3c
c1 ), provided that c1 > 3c0 .
Now, let ρR be the characteristic function of the interval [0, R]. For
any c0 , c1 > 0 satisfying c1 > 3c0 , we consider an initial data
c0 2
γR = c1 e − 4 r − c0 ρR (r ).
c0 2
c0 2
Since c1 e − 4 r > 0 and c1 e − 4 r ∈ L1 (R2 ),
Z
γR (r )rdr = 0
R2
for some R = R0 . Since γR0 (r ) ∈ L1 (R2 ) ∩ L∞ (R2 ), it implies that
γR0 belongs to the Hardy space H1 . Then, the initial velocity ue0
corresponding to γR0 belongs to L2 (R2 ). This means that there is
an initial data ue0 belonging to L2 which blows up in a finite time.
Namkwon Kim and Bataa Lkhagvasuren
23 / 35
Global solutions of a class NS and Euler equation
Result for Euler equation (ν = 0)
Given U ∈ R2 and γ0 , ω0 , ϕ0 , find (γ, ω, ϕ) satisfying
∂ω
+ ue · ∇ω − γω = 0,
∂t
∂γ
+ ue · ∇γ + γ 2 = 0,
∂t
∂ϕ
+ ue · ∇ϕ + γϕ = 0,
∂t
(23)
where ue is given in (7).
The previous computations are not applicable to Euler equations.
Namkwon Kim and Bataa Lkhagvasuren
24 / 35
Global solutions of a class NS and Euler equation
Result for Euler equation (ν = 0)
s (R2 ), where
We obtain the existence results in the Besov spaces Bp,q
s, p and q satisfy the following conditions with d = 2:
d
, if q < ∞,
p
d
s ≥ 1 + , if q = 1,
p
p ∈ (1; ∞) and q ∈ [1; ∞].
s >1+
(24)
s−1 ,→ L∞
The conditions (24) enable us to use the embedding Bp,q
and the key estimates.
Moreover, the conditions (24) give us the needed regularity for ue (at
least Lipschitzian) so that the initial regularity is preserved during
the evolution.
Namkwon Kim and Bataa Lkhagvasuren
25 / 35
Global solutions of a class NS and Euler equation
Notations
In the sequel, for the sake of convenience, we use the notations
kf kp = kf kLp (R2 ) , θ = (ω, γ, ϕ) and define
kθkp = kωkp + kγkp + kϕkp ,
s
s
s
s .
kθkBp,q
= kωkBp,q
+ kγkBp,q
+ kϕkBp,q
for p, q ∈ [1, ∞] and s ∈ R. All the constants will be denoted by C ,
which if necessary can be made larger from one line to the other.
Namkwon Kim and Bataa Lkhagvasuren
26 / 35
Global solutions of a class NS and Euler equation
A priori estimates
First, we give Lp estimate for the solution of the following general
scalar transport equation:
∂f
+ ue · ∇f = g ,
∂t
f |t=0 = f0 .
(25)
Proposition (Chapter 3, Danchin’s Note)
Let ∇ · ue = −γ and ∇e
u ∈ L∞ ([0, T ] × R2 ). For r ∈ [1; ∞], the
solution of (25) satisfies:
Z t
R
R
1 t
1 τ
0
kγk∞ dτ
0
r
(kf0 kr +
e − r 0 kγk∞ dτ kg (τ )kr dτ ).
kf (t)kr ≤ e
0
(26)
Namkwon Kim and Bataa Lkhagvasuren
27 / 35
Global solutions of a class NS and Euler equation
A priori estimates
Proposition
s be a solution of (23) with p, q and s satisfy (24). If
Let γ ∈ Bp,q
γ0 ≥ 0, then we have
0 ≤ γ(t) ≤ max γ0 (x)
x∈R2
(27)
for t ∈ [0, T ] and T is the maximal existence time of the solution.
Morever, if the initial data of the system (23) satisfies θ0 ∈ Lr (R2 )
for r ∈ [1; ∞], then
2
kθ(t)kr ≤ kθ0 kr e r tγ
∗
for t ∈ [0, T ],
(28)
with γ ∗ = max γ0 (x).
x∈R2
Namkwon Kim and Bataa Lkhagvasuren
28 / 35
Global solutions of a class NS and Euler equation
A priori estimates: Commutator estimate in Besov space
Proposition (Lemma 2.100 in Bahouri and Danchin)
Let s > 0 and p, q ∈ [1, ∞]. For the commutator
Rj = [e
u · ∇, ∆j ]f = ue · ∇∆j f − ∆j (e
u · ∇f ),
(29)
the estimate
k 2js kRj kp
q
s
s
k
≤
C
k∇e
u
k
kf
k
+
kf
k
k∇e
u
k
∞
∞
Bp,q
Bp,q
j l
(30)
is valid.
Proof:
The proof is identical to the proof of Lemma 2.100 in Bahouri and
Danchin except that we shift a derivative to the other term using
Bernstein’s lemma at the necessary step.
Namkwon Kim and Bataa Lkhagvasuren
29 / 35
Global solutions of a class NS and Euler equation
A priori estimates
We recall the product estimates and logarithmic inequality in Besov
spaces.
Proposition (Lemma 2.2 in Chae, 2004)
Let s > 0 and p, q ∈ [1, ∞]. There exists a constant C such that
the following inequality is true.
s ).
s
s
+ kg k∞ kf kBp,q
≤ C (kf k∞ kg kBp,q
kfg kBp,q
(31)
Proposition (Proposition 1.1 in Chae, 2004)
Let s > p2 , p ∈ (1, ∞) and q ∈ [1, ∞]. There exists a constant C
such that the following inequality is true.
kf k∞ ≤ C (1 + kf kḂ s
∞,∞
Namkwon Kim and Bataa Lkhagvasuren
30 / 35
s
(log+ kf kBp,q
+ 1)).
(32)
Global solutions of a class NS and Euler equation
A priori estimates
We recall the definition of time-space Besov spaces and the
properties.
s and L
er B s are
For r ∈ [1; ∞], the time-space Besov spaces Lrt Bp,q
t p,q
defined by the following norms:
X
1/q s
=
2jsq k∆j f kqLp
kf kLrt Bp,q
r ,
L[0,t]
j≥−1
kf kLer B s =
t
X
p,q
2jsq k∆j f kqLr
[0,t)
j≥−1
1/q
(Lp )
.
Proposition
According to the Minkowski inequality, we have
s
kf kLrt Bp,q
≤ kf kLer B s , if q ≤ r ,
t
kf k
s
Lrt Bp,q
Namkwon Kim and Bataa Lkhagvasuren
p,q
≥ kf kLer B s , if q ≥ r .
t
31 / 35
(33)
p,q
Global solutions of a class NS and Euler equation
A priori estimates
Proposition (Remark 2.2.1 in Danchin’s Note)
Let s > 0 and p, q, r ∈ [1, ∞]. There exists a constant C such that
the following inequality is true:
kfg kLer B s ≤ C (kf kLr1 (L∞ ) kg kLer2 B s + kg kLr3 (L∞ ) kf kLer4 B s ), (34)
p,q
p,q
t p,q
for r1 , r2 , r3 , r4 ∈ [1, ∞] and
Namkwon Kim and Bataa Lkhagvasuren
1
r
=
32 / 35
1
r1
+
1
r2
=
1
r3
+
1
r4 .
Global solutions of a class NS and Euler equation
Local existence result for Euler equations
Theorem
s and s, p and q satisfy (23). There exists a
Let ω0 , γ0 , ϕ0 ∈ Bp,q
time T such that (23) has a unique solution (ω, γ, ϕ) in
s × B s × B s ).
C ([0, T ); Bp,q
p,q
p,q
Namkwon Kim and Bataa Lkhagvasuren
33 / 35
Global solutions of a class NS and Euler equation
Global existence result for Euler equations
We prove that the local solution becomes global solution by
obtaining a priori estimate in the corresponding space.
Theorem
s and s, p and q satisfy (23). There exists a
Let ω0 , γ0 , ϕ0 ∈ Bp,q
global unique solution (ω, γ, ϕ) for the system (23) and (ω, γ, ϕ) in
s × B s × B s ).
C ([0, ∞); Bp,q
p,q
p,q
Namkwon Kim and Bataa Lkhagvasuren
34 / 35
Global solutions of a class NS and Euler equation
Thank you for the attention!
Lkhagvasuren Bataa