Asymptotic behavior of global

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Asymptotic behavior of global solutions to the Navier-Stokes Equations in R 3 Fabrice Planchon ? Centre de Mathematiques, U.R.A. 169 du C.N.R.S., Ecole Polytechnique, F-91 128 Palaiseau Cedex Abstract We construct global solutions to the Navier-Stokes equations with initial data small in a Besov space. Under additional assumptions, we show that they behave asymptotically like self-similar solutions. Introduction When studying global solutions to an evolution problem, it is natural to study their asymptotic behavior, as it is usually a simpler way to describe the long term behavior than the solution itself. Global solution of the non-linear heat equation have been showed to be asymptotically close to self-similar solutions ?Currently Program in Applied and Computational Mathematics, Princeton University, Princeton NJ 08544-1000 , USA 1

  • following heuristic

  • differential operator

  • similar solution

  • navier stokes equations

  • let ? ?

  • when studying global


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Asymptotic behavior of global
solutions to the Navier-Stokes
3Equations in
∗Fabrice Planchon
Centre de Math´ematiques,
U.R.A. 169 du C.N.R.S.,
Ecole Polytechnique,
F-91 128 Palaiseau Cedex
Abstract
We construct global solutions to the Navier-Stokes equations with
initial data small in a Besov space. Under additional assumptions, we
show that they behave asymptotically like self-similar solutions.
Introduction
Whenstudyingglobalsolutionstoanevolutionproblem,itisnaturaltostudy
their asymptotic behavior, as it is usually a simpler way to describe the long
term behavior than the solution itself. Global solution of the non-linear heat
equationhavebeenshowedtobeasymptoticallyclosetoself-similarsolutions
∗CurrentlyPrograminAppliedandComputationalMathematics,PrincetonUniversity,
Princeton NJ 08544-1000 , USA
1
R[7]. Under certain conditions, we will show how to obtain similar results for
the incompressible Navier-Stokes system,.
We recall the equations

∂u = Δu−∇·(u⊗u)−∇p,
∂t
(1) ∇·u = 0, 3u(x,0) = u (x), x∈ , t≥ 0.0
As we are in the whole space, ifu(x,t) is a solution of (1), then for allλ> 0
2u (x,t) =λu(λx,λ t) is also a solution.λ
We now note that studying the asymptotic behavior ofu(x,t) for large time
is equivalent to studying the asymptotic behavior ofu (x,t) for largeλ withλ
fixed time. Actually, we shall show that, as t goes to ∞, the natural space

x√scale is t as in the heat equation. If we replacex by and lett→∞, we
t
obtainthesameresultasifweletλ→∞inu (x,t). Thisnewpointofviewisλ
interesting for the following heuristic reason: we expect that the limitv(x,t)
ofu (x,t) will also be a solution of (1). Furthermore, one might assume thatλ
v(x,t) is the solution with initial data v (x) = lim λu(λx,0). Of course,0 λ→∞
1 x√ √thelimiting solutionis invariant under thescaling, sov(x,t) = V( ), and
t t
v (x) is an homogeneous function of degree −1.0
Such self-similar solutions have been studied previously (see [4],[2]), and we
shall see in the present work how to make rigorous the previous heuristic
approach.
Let us define the projection operator onto the divergence free vector fields:
     
u u R σ1 1 1
     
(2) u = u − R σ     2 2 2
u u R σ3 3 3
where R is the Riesz transform of the symbolj
ξi
(3) σ (ξ) = ,Rj |ξ|
and where
(4) σ =R u +R u +R u .1 1 2 2 3 3
2
P
R
PTherefore is a pseudo-differential operator of order 0.
tΔWe transform the system (1) into an integral equation, where S(t) = e
denotes the heat kernel,
Z t
(5) u(x,t) =S(t)u (x)− S(t−s)∇·(u⊗u)(x,s)ds.0
0
This equation can be solved by a classical fixed point method (see [1],[5],[6]).
Following themethodof[1], we remarkthatthe bilinear termintheprevious
equation can be reduced to a scalar operator
Z t 1 ·
√(6) B(f,g) = G ∗(fg)ds,
2(t−s) t−s0
where G is analytic, such that
C
(7) |G(x)| ≤
41+|x|
C
(8) |∇G(x)| ≤ .
41+|x|
This comes easily from the study of the symbol of B, as we have an exact
expressionundertheintegral. Thematrixofthispseudo-differential operator
has components like
ξ ξ ξ 2j k l −t|ξ|(9) − e
2|ξ|
2−t|ξ|off the diagonal, with an additional term ξ e on it. The function G isj
then the inverse Fourier transform of any of these functions at t = 1. The
1 ∞only thing we will need is that G∈L ∩L .
Thispaperisorganizedasfollows. Inafirstpart,wewilldefinethefunctional
setting which is well-suited for our study, then study global existence in this
setting, and lastly the behavior of attracting solutions for large time, if they
exist. Then in a second part, we will try to state a partial converse to
the theorem 3, that is a condition on the initial data in order to obtain
a convergence to a self-similar solution for large time. The third part will
be devoted to a better understanding of this condition, and will include
reformulations of the condition and examples.
3
P
P1 Global existence in Besov spaces
3A well suited functional space to study (1) is L ([5]), as ku k 3 = kuk 3.λ L L
3But homogeneous functions of degree −1 are not in L , and we easily see
that the weak limit of u is 0. We therefore have to enlarge functional0,λ
space to include homogeneous functions of degree −1. We have chosen the
3−(1− ),∞
p˙homogeneous Besov spaces B . We will see later they arise naturallyp
in our problem. Let us recall their definition ([9],[10])
Definition 1
n cb bLet φ ∈ S( ) such that φ ≡ 1 in B(0,1) and φ ≡ 0 in B(0,2) ,
nj j ′ nφ (x) = 2 φ(2 x), S =φ ∗·, Δ =S −S . Let f be in S ( ).j j j j j+1 j
n n s,q˙• If s < , or if s = and q = 1, f belongs to B if and only of thepp p
following two conditions are satisfied
Pm ′– Thepartialsum Δ (f)convergetof forthetopologyσ(S,S)j−m
js q
p– The sequence ǫ = 2 kΔ (f)k belongs to l .j j L
n n n s,q˙• If s> , or s = and q > 1, let us denote m =E(s− ). Then Bpp p p
is the space of distributions f, modulo polynomials of degree less than
m+1, such that
P∞
– We have f = Δ (f) for the quotient topology.j−∞
js q
p– The sequence ǫ = 2 kΔ (f)k belongs to l .j j L
We remark that nothing in this definition restrictss from being negative. In
fact, wewilluses =−(1−3/p)whichisindeednegativeasp> 3. Inthepar-
ticular case wheres< 0, it is worth noting that we can replace the condition
js q js q
p pǫ = 2 kΔ (f)k ∈ l by the equivalent condition ǫ˜ = 2 kS (f)k ∈ l .j j L j j L
This second condition implies easily the first one, and conversely, we remark
( 1thatǫ˜ can be seen as a convolution betweenǫ andη = 2sj)∈l . We shallj j j
obtain the following theorem which extends the results of [1].
4
R
RTheorem 1
There exists a positive function η(q), q > 3 such that
3−(1− ),∞
p
if u ∈B , ∇·u = 0, p≥ 3, satisfies0 p 0
(10) ku k 3 <η(q)0 −(1− ),∞q
Bq
for a fixed q >p, then there exists a unique solution of (1) such that
3
−(1− ),∞
p˙(11) u∈C ([0+∞),B ),pw
where C denotes the weakly continuous functions, and, ifw
p≤ 6 and u =S(t)u +w(x,t), then0
∞ 3 3(12) w∈L ([0+∞),L ( ))
and
3(13) kwk <γ(q),L
where γ(q) depends only of η(q).
We remark that the restriction p ≤ 6 in order to obtain (12) is merely due
to the linear part: the equivalent of (12) actually holds for p > 6 if one
considers higher order terms, ifu is written as an infinite sum of multi-linear
operatorsofu . Forsakeofsimplicity, werestrict ourselves totthefirst term,0
which yields thisrestriction. Wewill prove thetheorem1, using afixedpoint
argument via the following abstract lemma (Picard’s theorem in a Banach
space).
Lemma 1
Let E be a Banach space, B a continuous bilinear application, x,y∈E
(14) kB(x,y)k ≤γkxk kyk .E E E
5
RThen, if 4γkx k < 1, the sequence defined by0 E
x =x +B(x ,x )n+1 0 n n
converges to x∈E such that
1
(15) x =x +B(x,x) and kxk < .0 E

Let us define the space
q(16) F ={f(x,t)| supkf(x,t)k < +∞}.q L
t>0
The following characterization will be very useful.
Proposition 1
nTake α> 0, γ ≥ 1, f ∈S( ), then
α
2(17) kfk = supt kS(t)fk γL
t>0
−α,∞˙is a norm in B equivalent to the usual dyadic one.γ
Therefore, using the Sobolev inclusion
3 3−1,∞ −1,∞
p q˙ ˙B ֒→B ,p q
3−1,∞
q˙for p≤q, we see that u ∈B , so thatq0
√ √
t[S(t)u ]( tx)∈F .0 q
Then, in order to apply lemma 1 to F , we are left to prove that ifq
√ √
Df = tf( tx,t),t
6
R−1 −1 ˜then DB(D ·,D ·) is bicontinuous on F . Take f = Df and g˜ = Dgt q t tt t
˜in F . We denote M = fg˜ ∈ F . We observe that the bilinear operatorq q/2
(renormalized with D ) can be written as followst
Z 1 1 x x dλe ˜B(f,g˜) = G(√ )∗M(√ ,λt) .
2(1−λ) λ1−λ λ0
Then, by H¨older and Young inequalities, we obtain
Z 1 Cdλe ˜ ˜(18) kB(f,g˜)k ≤ kfk kg˜kF F Fq 1 3 3 q q+ 1−
2 2q q0 (1−λ) λ
which gives us η(q). Proceeding the same way, if p≤ 6, gives
Z
1 Cdλ˜ ˜e(19) kB(f,g˜)k ≤ kfk kg˜k .F F F3 3 3 q q1−
q q0 (1−λ) λ
This proves (12) and (13). We remark that the restriction over p arises in
(19), and that ifp> 6, we obtainF instead ofF . Therefore if we considerp/2 3
e e ˜B(B(f,g˜),g˜), it will be in F . From a simple iteration, we will eventuallyp/3
reach F . We have now to prove the weak convergence when t→ 0. Clearly3
∞ 3S(t)u ⇀ u by a duality argument. As for the bilinear term, ifφ∈C ( )0 0 0
t→0
1 .√and if we denote by Q(θ) the convolution operator with G( ),2θ θ
ehQ(t−s)fg(s),φi=hfg(s),S(t−s)Qφi,
c ξ ξ ξj k ℓ 1e e b ewhereQ is defined byQφ(ξ) = φ(ξ), so thatQφ∈L , like the function2|ξ|
γeG defined previously. ThereforeS(t−s)Qφ is(uniformly int−s) inL , with
1 2+ = 1. Thus
γ q
Z Zt t
(20) |h Q(t−s)fg(s)ds,φi| ≤ C kfg(s)k dsq/2L
0 0
Z t ds ˜(21) ≤ C kfk kg˜kF F3 q q1−
q0 s
3
q(22) ≤ Ct → 0 .
7
RThe uniqueness part of the theorem follows from the construction part, so
we have proved the theorem 1, in the case where p =q, with q for which 10
is verified. We next remark that the solution u actually satisfies
√ √
′(23) tu( tx,t)∈F ′, for all q ≥p ,q> 3,q
and that moreover the bilinear term w satisfies
√ √ p ′
′(24) tw( tx,t)∈F , for ≤q ≤p .q
2
(23) is of course true for the linear part. Then, the bilinear term is in F3
and in F for the particular q we have fixed. And by interpolation betweenq

′F and F it is in all F with 3<q <q. We are left to prove (23) for theq/2 q q
′bilinear term when q > q. An easy modification of (18) takes care of this
situation :
Z
1 Cdλ ˜e ˜(25) kB(f,g˜)k ≤ kfk kg˜kF F F′ 1 3 3 q q3q + − 1−′2 q 2q q0 (1−λ) λ
′and if q > 6 we get all the q >q. Otherwise, we have to proceed in several
′steps to reach a value q > 6. Note that the great amount of flexibility
provided by inequalities of type (18), (25) allows us to obtain this result
in many different ways. In particular, we could establish the bicontinuity
′ ′of the renormalized operator from F ×F to F and carry along the fixedq q q
point iterations all the properties we want, provided the different continuity
constantsverifyinequalities inthecorrect way, whichhappenstobethecase.
3,∞By the way, we remark that initial data in the the space L are included.
In fact, we have the following embedding,
Theorem 2
3−(1− ),∞3,∞ 3 p˙(26) L ( )֒→Bp
for all p> 3.
8
RIn order to prove this, we will make use of the following characterization of
weak Lebesgue spaces:
Z
23,∞
3(27) f ∈L ⇔ |f(x)|dx≤C|E|
E
∞for all Borel setsE. In particular, ifϕ∈S thenϕ∗f ∈L , and therefore is
p 3,∞ 3,∞in L , for all p> 3. In fact ϕ∗f ∈L , and all bounded functions in L
pare also in L , as the following estimate shows
X X
−jp −j −j+1 j(3−p)2 |2 ≤|g|≤ 2 | ≤C 2 < +∞.
j≥0 j≥0
Thus,
Z
3j j jS (f) = 2 ϕ(2 x−2 y)f(y)dyj
Z
j −j= ϕ(2 x−y)f(2 y)dy
Z
j j −j −j= 2 ϕ(2 x−y)2 f(2 y)dy
j j= 2 h(2 x).
3,∞Also, as h and f have the same norm in L , we obtain
3(1− )
pkS (f)k p ≤ 2 kfk 3,∞,j L L
which achieves the proof.
Now that we have solutions in the proper functional setting, we can study
the asymptotic behavior of these solutions. We begin with a definition:
Definition 2
pWe say thatu(x,t) “converges inL norm” to a functionV(x) if and only of
one of the two equivalent conditions is satisfied:
1. For all compact intervals [a,b]⊂ (0+∞)
pL (dx) 1 x
√ √u (x,t) −→ V( ) , λ→∞λ
t t
uniformly for t∈ [a,b]
9√ √ pL (dx)
2. tu( tx,t) −→ V(x), when t→∞.
Then we will show the following
Theorem 3
Let us take 3<p< +∞. Let u(x,t) be a solution of (1) such that
√ √
(28) supk tu( tx,t)k p < +∞L
t>0
(29) u(x,t) converges weakly to u (x) when t→ 0.0
If
p(30) u “ converges in L norm ” to V ,
3−(1− ),∞
p 1 x√ √then the initial data u (x) belongs to B , V( ) is a self-similar0 p t t
solution of (1), and
p(31) S(t)u “ converges in L norm ” to v (x) ,0 1
where v (x) =S(1)v , and v is the initial data of the self-similar solution.1 0 0
Note that we did not make any smallness assumption on the initial data. In
3−(1− ),∞
pother respects, when u ∈B , the condition (31) implies that0 p
(32) λu (λx) converges weakly to v when λ→ 0 ,0 0
but this is not equivalent, and we postpone the discussion on that matter to
part 3. We recall that the integral equation is
Z t√ √ √ √
tu( tx,t) = t[S(t)u ]( tx)− D{S(t−s)∇· u⊗u(s)}ds.0 t
0
√ √
Let us denote U(t) = tu( tx,t). Then we have
√ √
e(33) U(t) = t[S(t)u ]( tx)−B(U,U)(t),0
10
P