Affine Variant of Fractional Sobolev Space with Application to Navier-Stokes System

Abstract

It is proved that for $\alpha\in (0,1)$, $Q_\alpha(\rn)$, not only as an intermediate space of $W^{1,n}(\rn)$ and $BMO(\rn)$ but also as an affine variant of Sobolev space $\dot{L}^{2}_\alpha(\rn)$ which is sharply imbedded in $L^{\frac{2n}{n-2\alpha}}(\rn)$, is isomorphic to a quadratic Morrey space under fractional differentiation. At the same time, the dot product $\nabla\cdot\big(Q_\alpha(\rn)\big)^n$ is applied to derive the well-posedness of the scaling invariant mild solutions of the incompressible Navier-Stokes system in $\bn=(0,\infty)\times\rn$.

Full text

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Dynamics of PDE, Vol.4, No.3, 227-245, 2007
Homothetic Variant of Fractional Sobolev Space with
Application to Navier-Stokes System
Jie Xiao
Communicated by Y. Charles Li, received October 28, 2006.
Abstract. It is proved that for α ∈ (0,1), Qα(Rn), not only as an interme-
diate space of W1,n(Rn) and BMO(Rn) but also as a homothetic variant of
Sobolev space˙L2
phic to a quadratic Morrey space under fractional differentiation. At the same
time, the dot product ∇·(Qα(Rn))nis applied to derive the well-posedness of
the scaling invariant mild solutions of the incompressible Navier-Stokes system
in R1+n
+
= (0,∞) × Rn.
α(Rn) which is sharply imbedded in L
2n
n−2α(Rn), is isomor-
Contents
1.
2.
3.
References
Introduction and Summary
Proof of Theorem 1.2
Proof of Theorem 1.4
227
233
239
244
1. Introduction and Summary
We begin by the square form of John-Nirenberg’s BMO space (cf. [13]) which
plays an important role in harmonic analysis and applications to partial differen-
tial equations. For a locally integrable complex-valued function f defined on the
Euclidean space Rn, n ≥ 2, with respect to the Lebesgue measure dx, we say that
f is of BMO class, denoted f ∈ BMO = BMO(Rn), provided
?
?f?BMO=sup
I
??(I)?−n?
I
??f(x) − fI
??2dx
?1
2
< ∞.
1991 Mathematics Subject Classification. Primary 35Q30, 42B35; Secondary 46E30.
Key words and phrases. Homothety, W1,n(Rn), Qα(Rn), BMO(Rn), quadratic Morrey
space, sharp Sobolev imbedding, incompressible Navier-Stokes system.
The author was supported in part by both NSERC (Canada) and Dean of Science Startup
Fund (MUN, Canada).
c ?2007 International Press
227

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228JIE XIAO
Here and elsewhere supImeans that the supremum ranges over all cubes I ⊂ Rn
with edges parallel to the coordinate axes; ?(I) is the sidelength of I; and fI =
??(I)?−n?
been introduced in [11], as defined below.
If(x)dx stands for the mean value of f over I.
On the basis of the semi-norm ? · ?BMO, a large scale of function spaces has
Definition 1.1. For α ∈ (−∞,∞), let Qα be the space of all measurable
complex-valued functions f on Rnobeying
???(I)?2α−n?
This Qα is a natural extension of BMO according to the following result
(proved in [11] and [30]):


Here W1,n= W1,n(Rn) is the homothetic energy space of all C1complex-valued
functions f on Rnwith
??
More importantly, Qα, α ∈ (0,1), can be regarded as the homothetically invariant
counterpart of the homogeneous Sobolev space˙L2
complex-valued functions f on Rnwith the α-energy
??
The reason for saying this is at least that ? · ?Qαand ? · ?˙L2
property:
?f?Qα= sup
I
I
?
I
|f(x) − f(y)|2
|x − y|n+2αdxdy
?1
2
< ∞.
Qα=



BMO,
New space between W1,nand BMO,
C,
α ∈ (−∞,0),
α ∈ [0,1),
α ∈ [1,∞).
?f?W1,n =
Rn|∇f(x)|ndx
?1
n
< ∞.
α=˙L2
α(Rn) which consists of all
?f?˙L2
α=
Rn
?
Rn
|f(x) − f(y)|2
|x − y|n+2αdxdy
?1
2
< ∞.
αenjoy the following
?f ◦ φ?Qα= ?f?Qα and ?f ◦ φ?˙L2
for any homothetic map x ?→ φ(x) = λx + x0; λ > 0,x0∈ Rn.
In the six year period since the paper [11] appeared, it has been found that
Qα is a useful and interesting concept; see also [1], [2], [24], [8], [9], [16], [23],
[7] and [10].This means that the study of this new space has not yet ended
up – in fact, there are many unexplored problems related to Qα. In this paper,
although not attacking one of those open problems in Section 8 in [11], we go well
beyond the previous results by studying the relation between this space and the
quadratic Morrey space, but also giving an application of the induced facts to the
incompressible Navier-Stokes system.
To deal with the former, it is necessary to consider the following variant of [8,
Theorem 3.3] that expands Fefferman-Stein’s basic result for BMO in [12]: Given
a C∞real-valued function ψ on Rnwith
?
α= λα−n
2?f?˙L2
α
ψ ∈ L1,|ψ(x)| ? (1 + |x|)−(n+1),
Rnψ(x)dx = 0andψt(x) = t−nψ(x
t).

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HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM229
Then for a measurable complex-valued function f on Rn,
(1.1)f ∈ Qα⇐⇒ sup
x∈Rn,r∈(0,∞)r2α−n
?r
0
?
|y−x|<r
|f ∗ ψt(y)|2t−(1+2α)dydt < ∞.
Here and henceforth, Lp= Lp(Rn) represents the complex Lebesgue space equipped
with p-norm ? · ?Lp; ∗ stands for the convolution operating on the space variable;
and U ? V means that there exists a constant c > 0 such that U ≤ cV .
Two particular choices of ψ in (1.1) yield two characterizations of Qαinvolving
the Poisson and heat semi-groups. As for this aspect, denote by e−t√−∆(·,·) and
et∆(·,·) are the Poisson and heat kernels respectively; that is,
e−t√−∆(x,y) = Γ?n + 1
and
et∆(x,y) = (4πt)−n
2
?π−n+1
2 t(|x − y|2+ t2)−n+1
2
2exp
?
−|x − y|2
4t
?
.
Of course, for β ∈ R, the notation (−∆)
operator
β
2 is the β/2-th power of the Laplacian
−∆ = −
n
?
j=1
∂2
j= −
n
?
j=1
∂2
∂x2
j
determined by the partial derivatives {∂j= ∂/∂xj}n
?
On the one hand, if
ψ0(x) =1 + |x|2− (n + 1)Γ?n+1
then
j=1and the Fourier transform
(·)
?
(−∆)
β
2f(x) = (2π|x|)βˆf(x).
2
?π−n+1
2
(1 + |x|2)
n+3
2
,
(ψ0)t(x) = t∂te−t√−∆(x,0)
and hence for α ∈ (0,1) and a measurable complex-valued function f on Rn,
(1.2)
?r
On the other hand, if
?xj
then
(ψj)t(x) = t∂jet2∆(x,0)
and so, for α ∈ (0,1) and a measurable complex-valued function f on Rn,
f ∈ Qα⇐⇒sup
x∈Rn,r∈(0,∞)r2α−n
0
?
|y−x|<r
???∂te−t√−∆f(y)
−|x|2
4
???
2
t1−2αdydt < ∞.
ψj(x) = −(4π)−n
2
2
?
exp
??
forj = 1,...,n,
(1.3)f ∈ Qα⇐⇒sup
x∈Rn,r∈(0,∞)r2α−n
?r
0
?
|y−x|<r
|∇et2∆f(y)|2t1−2αdydt < ∞.
With the help of the above-mentioned facts, we can establish the following
result.
Theorem 1.2. Let α ∈ (0,1). Then

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230JIE XIAO
(i) Qα= (−∆)−α
2L2,n−2α?→ BMO is proper with
sup
?f?Qα>0
?f?BMO
?f?Qα
≤
?
n
n+2α
2
2
,
where a measurable complex-valued function f on Rnbelongs to L2,n−2αif and only
if
???(I)?2α−n?
(ii)˙L2
?
2
?f?L2,n−2α= sup
I
I
|f(x) − fI|2dx
?1
2
< ∞.
α= (−∆)−α
?f?L
?f?˙L2
(iii) Q−1
and only if
2L2?→ L
2n
n−2αis sharp with
Γ?n−2α
sup
?f?˙L2
α>0
2n
n−2α
α
=
2
?
Γ?n+2α
?
?1
2?
Γ(n)
Γ?n
2
?
?α
n??
Rn
|e−2πy·(1,0,...,0)− 1|2
|y|n+2α
dy
?−1
2
.
α;∞= ∇ · (Qα)n, where a tempered distribution f on Rnbelongs to Q−1
α;∞if
?f?Q−1
α;∞=sup
x∈Rn,r∈(0,∞)
?
r2α−n
?r2
0
?
|y−x|<r
|et∆f(y)|2t−αdydt
?1
2
< ∞.
Note that L2,n−2α is the so-called Morrey space of square form (cf. [3] and
[22]) and L2,n = BMO. So Theorem 1.2 (i) keeps true for α = 0 in the sense
of (−∆)0BMO = BMO.
Strichartz’s (−∆)−α
???(I)?−n?
The imbedding without best constant in Theorem 1.2 (ii) is well-known (see for
example [21, Theorem 1] and the related references therein) and very useful in the
study of the semi-linear wave equations (cf. [20]). A close look at both (i) and (ii)
reveals that Qαbehaves like a homothetic Sobolev space. In addition, Theorem 1.2
(iii) extends [15, Theorem 1]: BMO−1= ∇·(BMO)nthat just says: f ∈ BMO−1
if and only if there are fj∈ BMO such that f =?n
Navier-Stokes system on the half-space R1+n


is to establish the existence of a solution (velocity)
u = u(t,x) =?u1(t,x),...,un(t,x)?
with a pressure p = p(t,x) of the fluid at time t ∈ [0,∞) and position x ∈ Rn
that assumes the given data (initial velocity) a = a(x) = (a1(x),...,an(x)). If the
solution exists, is unique, and depends continuously on the initial data (with respect
to a given topology), then we say that the Cauchy problem is well-posed in that
topology.
Quite surprisingly, this part corresponds nicely to
2BMO-equivalence [26, Theorem 3.3]:
?
f ∈ (−∆)−α
2BMO ⇐⇒ sup
I
II
|f(x) − f(y)|2
|x − y|n+2αdxdy
?1
2
< ∞.
j=1∂jfj.
As with the latter, we recall that the Cauchy problem for the incompressible
+
= (0,∞) × Rn:
in
(1.4)



∂tu − ∆u + (u · ∇)u − ∇p = 0,
∇ · u = 0,
u|t=0= a,
R1+n
+
;
in
in
R1+n
+
Rn
;

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HOMOTHETIC FRACTIONAL SOBOLEV SPACE WITH APPLICATION TO NS SYSTEM231
Of particularly significant is the invariance of (1.4) under the scaling changes:


So if (u(t,x),p(t,x),a(x)) satisfies (1.4) then (uλ(t,x),pλ(t,x),aλ(x)) is a solution
of (1.4) for any λ > 0. This leads to a consideration of the well-posedness for (1.4)
with a Cauchy data being of the scaling invariance. Through the scale invariance
?
Kato proved in [14] that (1.4) has mild solutions locally in time if a ∈ (Ln)nand
globally if ?a?(Ln)n is small enough (for some generalizations of Kato’s result, see
e.g. [28] and [31]). Furthermore, in [15], Koch-Tataru found, among other results,
that (1.4) still has mild solutions locally in time if a ∈ (V MO−1)nand globally
once
?



u(t,x) ?→ uλ(t,x) = λu(λ2t,λx);
p(t,x) ?→ pλ(t,x) = λ2p(λ2t,λx);
a(x) ?→ aλ(x) = λa(λx).
?aλ?(Ln)n =
n
j=1
?(aλ)j?Ln =
n
?
j=1
?(aj)λ?Ln =
n
?
j=1
?aj?Ln = ?a?(Ln)n,
n
?
j=1
?aj?BMO−1 =
n
?
j=1
sup
x∈Rn,r∈(0,∞)
r−n
?r2
0
?
|y−x|<r
|et∆aj(y)|2dydt
?1
2
is sufficiently small. Here and henceforward, by a mild solution u(t,x) of (1.4) we
mean that u(t,x) solves the integral equation
?t
where et∆a(x) = (et∆a1(x),...,et∆an(x)) and P is the Helmboltz-Weyl projection:
u(t,x) = et∆a(x) −
0
e(t−s)∆P∇ · (u ⊗ u)ds,
P = {Pjk}j,k=1,...,n= {δjk+ RjRk}j,k=1,...,n
with δjk being the Kronecker symbol and Rj= ∂j(−∆)−1
form.
Observe that ?·?BMO−1 and ?·?Q−1
a(x) ?→ λa(λx). So it is a natural thing to extend the results of Kato and Koch-
Tataru to the Qα-setting. To do this, we introduce the following concept.
2 being the Riesz trans-
α;∞are also invariant under the scale transform
Definition 1.3. Let α ∈ (0,1) and T ∈ (0,∞]. Then we say:
(i) A tempered distribution f on Rnbelongs to the space Q−1
?
α;Tprovided
?f?Q−1
α;T=sup
x∈Rn,r∈(0,T)
r2α−n
?r2
0
?
|y−x|<r
|et∆f(y)|2t−αdydt
?1
2
< ∞;
(ii) A tempered distribution f on Rnbelongs to V Q−1
0;
(iii) A function g on R1+n
+
belongs to the space Xα;T provided
√t?g(t,·)?L∞
α provided limT→0?f?Q−1
α;T=
?g?Xα;T
= sup
t∈(0,T)
+sup
x∈Rn,r2∈(0,T)
?
r2α−n
?r2
0
?
|y−x|<r
|g(t,y)|2t−αdydt
?1
2
< ∞.