Self Similar Solutions for Navier Stokes Equations in R

Description

Niveau: Supérieur, Licence, Bac+2
Self-Similar Solutions for Navier-Stokes Equations in R 3 M.Cannone ? CEREMADE, Universite Paris IX Dauphine, F-75775 Paris Cedex 16 F.Planchon Centre de Mathematiques, U.R.A. 169 du C.N.R.S., Ecole Polytechnique, F-91 128 Palaiseau Cedex Abstract We construct self-similar solutions for three-dimensional incom- pressible Navier-Stokes equations, providing some examples of func- tional spaces where this can be done. We apply our results to a par- ticular case of L2 initial data. Introduction We are interested in the Navier-Stokes equations for an incompressible vis- cous fluid filling the whole space. We denote the unknown velocity field by u(x, t) = (u1, u2, u3)(x, t), x ? R3 , ? · u = ∂∂x1u1 + ∂ ∂x2u2 + ∂ ∂x3u3; u · ? is the differential operator u1 ∂∂x1 +u2 ∂ ∂x2 +u3 ∂ ∂x3 . Then the Navier-Stokes equations are for t > 0 { ∂u ∂t + (u · ?) u = ?∆u??p ? · u = 0, (1) where the initial data is u(x, 0) = u0(x). For the sake of simplicity, we restrict ourselves to ? = 1, as a rescaling allows us to obtain any value.

• similar solution


• divergence free


• solutions exist


• let u0 ?


• navier stokes equations


• unique self


• self

Sujets

• École polytechnique
• Washington University in St. Louis
• Navier?Stokes equations

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 12
• Langue: English

Extrait

Self-Similar Solutions for
3Navier-Stokes Equations in
M.Cannone F.Planchon
CEREMADE, Centre de Math´ematiques,
Universit´e Paris IX Dauphine, U.R.A. 169 du C.N.R.S.,
F-75775 Paris Cedex 16 Ecole Polytechnique,
F-91 128 Palaiseau Cedex
Abstract
We construct self-similar solutions for three-dimensional incom-
pressible Navier-Stokes equations, providing some examples of func-
tional spaces where this can be done. We apply our results to a par-
2ticular case of L initial data.
Introduction
We are interested in the Navier-Stokes equations for an incompressible vis-
cous fluid filling the whole space. We denote the unknown velocity field by
3 ∂ ∂ ∂u(x;t)= (u ;u ;u )(x;t), x2 ,ru = u + u + u ;ur is the1 2 3 1 2 3∂x ∂x ∂x1 2 3
∂ ∂ ∂differentialoperatoru +u +u . ThentheNavier-Stokes equations1 2 3∂x ∂x ∂x1 2 3
are for t> 0 (
@u
+(ur) u = Δu r p
(1)@t
ru = 0;
wheretheinitialdataisu(x;0)=u (x). Forthesakeofsimplicity, werestrict0
ourselves to = 1, as a rescaling allows us to obtain any value. p=p(x;t) is
the pressure field, which is determined from u up to an inessential function
∗Currently Departement of Mathematics, Washington University, St-Louis MO 63130,
USA
1
R
Rof time. We are interested in solutions which are (weakly) continuous from
[0;+1) to a Banach space of (vector-valued) distributions in x. We define
∂ ∂the non-linear term to be (u u)++ (u u), which is defined as a1 3∂x ∂x1 3
∂ ∂tempered distribution in more general situations thanu u++u u.1 3∂x ∂x1 3
2Note that ifu(x;t) is a solution of (1), thenu(x; t) is also a solution for
all> 0. Therefore it is natural to ask whether solutionsu(x;t) of (1) exist
and satisfy, for > 0,
2u(x;t)=u(x; t); (2)
or, equivalently
1 x
p pu(x;t)= U ; (3)
t t
whereU isa divergence-free vector field. Such solutionsu(x;t)arecalled, for
obvious reasons, self-similar solutions of the Navier-Stokes equations. Before
going into the details of the construction of self-similar solutions, let us ex-
plain, by means of anexample, where the difficulty arises from. Let us forget
for a while the divergence-free condition, and consider U(x) = 1=(1+jxj):p
then u(x;t) defined by (3) is well-defined and is 1=(jxj+ t), which tends
to 1=jxj for t ! 0. In other words, u (x) is homogeneous with degree 1,0
and every initial data that gives a self-similar solution must verify this prop-
erty. Unfortunately, those functions do not belong to the usual spaces where
3 1/2strong solutions exist, such as L or H . We shall therefore replace them
by other functional spaces that allow, among their elements, homogeneous
functions of degree 1. Then, we will prove the existence and uniqueness of
such solutions, up to a smallness condition on the initialdata. To ourknowl-
edge, [6] is the only recent work in connection to self-similar solutions. In
this paper, the authors construct self-similar solutions to the Navier-Stokes
equations in terms of vorticity, using measure spaces.
Definitions and Theorems.
Let us define the Besov spaces we will use in the following:
nDefinition 1 Let 2S( ) be a radial function such that
ˆ ˆSupp fjj< 1+"g and ()= 1 for jj< 1:
nj jDefine (x) = 2 (2 x), S the convolution operator with , andj j j
n0 nΔ =S S . Let f 2S ( ), s< , 1<a;b +1. The homogeneousj j+1 j a
2
R
Rs˙Besov spaces are defined as follows: f 2B if and only ifa,b
1" #
+1 bX
js b
a(2 kΔ fk ) < +1:j L
1
αThe non-homogeneous counterpart is defined by: f 2B if and only ifp,q
" #1
+1 bX
js b
a akS fk + (2 kΔ fk ) < +1:0 L j L
0
The reader should refer to [1] or [13] for details. We also have a simple
characterization of the homogeneous functions of degree 1 by considering
their restriction to the unit sphere (see [2]).
Theorem 1 Let 1 q 1 and = 1 3=q, then the following three
properties are equivalent for all functionsf which are homogeneous of degree
1.
α 3˙ f belongs to the (homogeneous) Besov space B ( ).q,1
2 The restriction of f to a neighborhood Ω of the unit sphere S belongs
αto the (non-homogeneous) Besov space B (Ω).q,q
2 The restriction of f to S belongs to the (non-homogeneous) Besov
α 2space B (S ).q,q
α˙The homogeneous Besov spaces B are not separable, so the convergenceq,1
α α 0˙ ˙willbetaken intheweak topology(B ;B ), withq theconjugateexpo-0q,1 q ,1
nent of q. Observe that an equivalent convergence condition in these spaces
is given by by the following definition:
Definition 2 Let B be a Banach functionnal space. Then a sequence f ofj
vectors in B converges weakly to f 2 B, if the sequence kf k is boundedj B
and f *f in the sense of distributions.j
n 0 nWe recall that a functionnal space is such that S( )B S ( ) where
both injections are continuous. We are now in the position to state the main
theorem of the paper, where S(t) denotes the heat operator:
3
R
R
R0˙Theorem 2 Let u 2 B be a divergence free vector field that is homo-0 3,1
geneous of degree 1. There exists an universal constant > 0 such that,
if
ku k 0 <; (4)0 ˙B3,1
then there exists a unique global solution u(x;t) of (1) such that

1 x
u(x;t) = p U p (5)
t t
0˙with U 2B and3,1
U =S(1)u +W (6)0
3where W 2L satisfies
kWk 3 <C(): (7)L
The continuity at t= 0 of u(x;t) is defined by the definition 2.
The Besov spaces we are using here arise quite naturally in the study of the
existence of global solutions to the Navier-Stokes equations (see [2]). But, in
the present case, we can use other spaces which are far more simple. They
were introduced in [3], and their definition is the following:
q,mDefinition 3 Let q;m2 . The space E consists of all the functions in
3mC ( ) such
α qj αjj@ f(x)jC(1+jxj) (8)
holds for all jjm. We define the homogeneous space in the same way:
α qj αjj@ f(x)jCjxj if jjm ; (9)
q,m˙which we denote by E .
Thecorrespondingexistence theoremforself-similarsolutionsinthesespaces
reads as follows:
Theorem 3 For all m 0, there exists > 0 and >0 such that, if u (x)0
is divergence free, homogeneous of degree 1 and
ku k <; (10)˙1,m0 E
4
R
Nthen there exists a unique self-similar solution
1 x
u(x;t) = p U p (11)
t t
of (1) such that kUk <. Moreover,1,m˙E
U(x) =S(1)u +W (12)0
2,m 1,m˙where W 2E , and u(x;t) converges to u (x) weakly in E in the sense0
of definition 2.
We remark that, as in the previous theorem, the solution is expressed as a
sum, with two different kinds of terms. The heat term S(1)u is a trend,0
whereas the bilinear term is a fluctuation, which has more regularity (or
spatial decay) than the tendency. This is very clear in the second theorem
where the index m measures the decay. We can also give a local version
of Theorems 2 and 3, starting with u (x) with finite energy, which shall be0
homogeneous of degree 1 near 0.
Such a function u (x) is defined in the following way. We take U (x) a0 0
divergence free vector field, which is homogeneous of degree 1. We restrict
1 3ourselves to m 1. Then we define u˜ (x) =(x)U (x) where 2C ( )0 0 0
has value 1 in a neighborhood of 0 . Finally, we set u = (ue ) .0 0
mTheorem 4 There exists > 0 and T > 0 such that if the C norm of U0
on the unit sphere is not larger than , then the solution u(x;t) of (1) with
initial data u(x;0)=u (x) can be expressed as0

1 x
u(x;t) = p U p +r(x;t) (13)
t t

1 xp pwhere U is the self-similar solution with initial data U (x) and0t t
1 3r(x;t)2L ([0;T) ): (14)
5
P
R
RTheorem 5 There exists >0 and T >0 such that if
ku k <; (15)˙00 B
3,1
the solution u(x;t) of (1) with initial data u(x;0) = u (x) can be expressed0
as
1 x
u(x;t) = p U p +r(x;t) (16)
t t

1 xp pwhere U is the self-similar solution with initial data U (x) and0t t
1 3r(x;t)2L ([0;T) ): (17)
Proofs.
We first reformulate the problem in order to obtain an integral equation for
u. This is standard practice, and was first employed by Kato and Fujita (see
[8][9]),andusedveryoftensince(see[2][5][4][12]). Alloftheseauthorsused
semi-group theory, but in the present case, we do not need this formalism,
3for the exact expression of the heat kernel in allows us to obtain directly
the estimations we need (see [2] [7]). Let be the projection operator from
32 3 2(L ( )) onto the subspace of divergence-free vectors, denoted by L , and
ξjR the Riesz transform having symbol . We easily see thatj jξj
0 1 0 1 0 1
u u R 1 1 1
@ A @ A @ Au = u R (18)2 2 2
u u R 3 3 3
P
where = R u . It is well-known that can be extended to a boundedj jj
3 3p p s soperator from (L ) onto L , 1 < p < +1, and from (H ) onto H ,
tΔs 0. Note that commutes with S(t) = e , whereas on an open set
Ω, we would need to introduce the Stokes operator Δ and the associate
semi-group. Note that
Ker =fuj9 such that u =rg:
Using , (1) becomes the evolution equation
8
@u>< = Δu r(u
u);
@t (19)ru = 0;>:
u(x;0) = u (x):0
6
P
P
R
P
P
P
P
R
P
P
R
P
P
PWe replaced (ur)u byr(u
u) to avoid problems of definition (see [2]),
and this can be made only because ru = 0. It is then standard to study
(19) via the co