Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces

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by  D. Q. Khai

Well-posedness for the Navier-Stokes equations with data
in homogeneous Sobolev-Lorentz spaces
D. Q. Khai, N. M. Tri
Institute of Mathematics, VAST
18 Hoang Quoc Viet, 10307 Cau Giay, Hanoi, Vietnam
6
1
0
2
Abstract: In this paper, we study local well-posedness for the Navier-Stokes

n
equations (NSE) with the arbitrary initial value in homogeneous Sobolev-
a
J Lorentz spaces H˙ s (Rd) := ( ∆)−s/2Lq,r for d 2,q > 1,s 0, 1
Lq,r − ≥ ≥ ≤
8 r , and d 1 s < d, this result improves the known results for
≤ ∞ q − ≤ q
] q > d,r = q,s = 0 (see [4, 7]) and for q = r = 2, d 1 < s < d (see [4, 9]).
P 2 − 2
In the case of critical indexes (s = d 1), we prove global well-posedness for
A q −
NSE provided the norm of the initial value is small enough. The result that
.
h
is a generalization of the result in [5] for q = r = d,s = 0.
t
a
m
[
1. Introduction

1 §
v
2 We consider the Navier-Stokes equations in Rd:
4
7
∂ u = ∆u .(u u) p,
1 t −∇ ⊗ −∇
0  .u = 0,
.  ∇
1 u(0,x) = u ,
0
0

6
which is a condensed writing for
1
:
v 1 k d, ∂ u = ∆u d ∂ (u u ) ∂ p,
i ≤ ≤ t k k − l=1 l l k − k
X  d ∂ u = 0, P
r  l=1 l l
a P1 k d, uk(0,x) = u0k.
≤ ≤

The unknown quantities are the velocity u(t,x) = (u (t,x),...,u (t,x)) of
1 d
the fluid element at time t and position x and the pressure p(t,x).
In the 1960s, mild solutions were first constructed by Kato and Fujita ([18],
[19]) that are continuous in time and take values in the Sobolev spaces
12010 Mathematics Subject Classification: Primary 35Q30; Secondary 76D05, 76N10.
2Keywords: Navier-Stokesequations,existenceanduniqueness oflocalandglobalmild
solutions, Sobolev-Lorentz.
3e-mail address: [email protected]@math.ac.vn
1
Hs(Rd),(s d 1), say u C([0,T];Hs(Rd)). In 1992, a modern treat-
≥ 2 − ∈
ment for mild solutions in Hs(Rd),(s d 1) was given by Chemin [9]. In
≥ 2 −
1995, using the simplified version of the bilinear operator, Cannone proved
the existence of mild solutions in H˙ s(Rd),(s d 1), see [4]. Results on the
≥ 2 −
existence of mild solutions with value in Lq(Rd),(q > d) were established in
the papers of Fabes, Jones and Rivi`ere [11] and of Giga [14]. Concerning the
initial data in the space L∞, the existence of a mild solution was obtained by
Cannone and Meyer in ([4], [7]). In 1994, Kato and Ponce [23] showed that
the NSE are well-posed when the initial data belong to the homogeneous
Sobolev spaces H˙ dq−1(Rd),(d q < ). Recently, the authors of this article
q
≤ ∞
have considered NSE in the mixed-norm Sobolev-Lorentz spaces, see [17].
In this paper, for d 2,q > 1,s 0, 1 r , and d 1 s < d, we
≥ ≥ ≤ ≤ ∞ q − ≤ q
investigate mild solutions to NSE in the spaces L∞ [0,T];H˙ s (Rd) when
Lq,r
the initial data belong to the Sobolev-Lorentz spac(cid:0)es H˙ s (Rd), wh(cid:1)ich are
Lq,r
more general than the spaces H˙ s(Rd), (H˙ s(Rd) = H˙ s (Rd)). We obtain
q q Lq,q
the existence of mild solutions with arbitrary initial value when T is small
enough, and existence of mild solutions for any T > 0 when the norm of
the initial value in the Besov spaces B˙s−d(1q−q1˜),∞(Rd), 1(1 + s) < 1 <
q˜ 2 q d q˜
min 1 + s , 1 is small enough. (cid:0)
2 2d q
In th(cid:8)e particu(cid:9)la(cid:1)r case (q > d,r = q,s = 0), we get the result which is more
general than that of Cannone and Meyer ([4], [7]). Here we obtained a state-
ment that is stronger than that of Cannone and Meyer but under a much
weaker condition on the initial data.
In the particular case (q = r = 2, d 1 < s < d), we get the result which is
2 − 2
more general than those of Chemin in [9] and Cannone in [4]. Here we ob-
tained a statement that is stronger than those of Chemin in [9] and Cannone
in [4] but under a much weaker condition on the initial data.
In the case of critical indexes (1 < q d,r 1,s = d 1), we get a
≤ ≥ q −
result that is a generalization of a result of Cannone [5]. In particular, when
q = r = d,s = 0, we get back the Cannone theorem (Theorem 1.1 in [5]).
The paper is organized as follows. In Section 2 we prove some inequalities
for pointwise products in the Sobolev spaces and some auxiliary lemmas. In
Section 3 we present the main results of the paper. In the sequence, for a
space of functions defined on Rd, say E(Rd), we will abbreviate it as E.
2. Some auxiliary results
§
In this section, we recall the following results and notations.
Definition 1. (Lorentz spaces). (See [1].)
2
For 1 p,r , the Lorentz space Lp,r(Rd) is defined as follows:
≤ ≤ ∞
A measurable function f Lp,r(Rd) if and only if
∈
f (Rd) := ∞(tp1f∗(t))rdt 1r < when 1 r < ,
Lp,r 0 t ∞ ≤ ∞
(cid:13)(cid:13)f(cid:13)(cid:13) (Rd) :=(cid:0)sRup tp1f∗(t) < (cid:1) when r = ,
Lp,∞ ∞ ∞
(cid:13) (cid:13) t>0
(cid:13)wh(cid:13)ere f∗(t) = inf τ : d( x : f(x) > τ ) t , with d being the
M { | | } ≤ M
Lebesgue measure i(cid:8)n Rd. (cid:9)
Before proceeding to the definition of Sobolev-Lorentz spaces, let us in-
troduce several necessary notations. For real number s, the operator Λ˙s is
defined through Fourier translation by
Λ˙sf ∧(ξ) = ξ sfˆ(ξ).
| |
(cid:0) (cid:1)
For 0 < s < d, the operator Λ˙s can be viewed as the inverse of the Riesz
potential I up to a positive constant
s
f(y)
I (f)(x) = dy for x Rd.
s Z x y d−s ∈
Rd
| − |
For q > 1,r 1, and 0 s < d, the operator I is continuous from Lq,r to
≥ ≤ q s
Lq˜,r, where 1 = 1 s, see ([26], Theorem 2.4 iii), p. 20).
q˜ q − d
Definition 2. (Sobolev-Lorentz spaces). (See [12].)
For q > 1,r 1, and 0 s < d, the Sobolev-Lorentz space H˙ s (Rd) is
≥ ≤ q Lq,r
defined as the space I (Lq,r(Rd)), equipped with the norm
s
f := Λ˙sf .
(cid:13) (cid:13)H˙Lsq,r (cid:13) (cid:13)Lq,r
(cid:13) (cid:13) (cid:13) (cid:13)
Lemma 1. Let q > 1,1 r r˜ , and 0 s < d. Then we have the
≤ ≤ ≤ ∞ ≤ q
following imbedding maps
(a)
H˙ s ֒ H˙ s ֒ H˙ s ֒ H˙ s .
Lq,1 → Lq,r → Lq,r˜ → Lq,∞
(b) H˙ s = H˙ s (equality of the norm).
q Lq,q
Proof. It is easily deduced from the properties of the standard Lorentz
spaces.
In the following lemmas, we estimate the pointwise product of two functions
in H˙ s(Rd),(d 2) which is a generalization of the Holder inequality. In
q ≥
the case when s = 0 we get back the usual Holder inequality. Pointwise
multiplication results for Sobolev spaces are also obtained in literature, see
for example [10], [26], [22] and the references therein.
3
Lemma 2. Assume that
1 1 1
1 < p,q < d, and + < 1+ .
p q d
Then the following inequality holds
uv . u v , u H˙ 1,v H˙ 1,
(cid:13) (cid:13)H˙r1 (cid:13) (cid:13)H˙p1(cid:13) (cid:13)H˙q1 ∀ ∈ p ∈ q
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
where 1 = 1 + 1 1.
r p q − d
Proof. By applying the Leibniz formula for the derivatives of a product
of two functions, we have
uv ∂α(uv) (∂αu)v + u(∂αv) .
(cid:13) (cid:13)H˙r1 ≃ X (cid:13) (cid:13)Lr ≤ X (cid:13) (cid:13)Lr X (cid:13) (cid:13)Lr
|α|=1 |α|=1 |α|=1
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
By applying the H¨older and Sobolev inequalities we obtain
(∂αu)v ∂αu v . u v ,
X (cid:13) (cid:13)Lr ≤ X (cid:13) (cid:13)Lp(cid:13) (cid:13)Lq1 (cid:13) (cid:13)H˙p1(cid:13) (cid:13)H˙q1
|α|=1 |α|=1
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
where
1 1 1
= .
q q − d
1
Similar to the above reasoning, we have
u(∂αv) . u v .
Lr H˙1 H˙1
X (cid:13) (cid:13) (cid:13) (cid:13) p(cid:13) (cid:13) q
|α|=1
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
This gives the desired result
uv . u v .
H˙1 H˙1 H˙1
(cid:13) (cid:13) r (cid:13) (cid:13) p(cid:13) (cid:13) q
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Lemma 3. Assume that
1 s 1 s 1 1 s
0 s 1, > , > , and + < 1+ . (1)
≤ ≤ p d q d p q d
Then the following inequality holds
uv . u v , u H˙ s,v H˙ s,
(cid:13) (cid:13)H˙rs (cid:13) (cid:13)H˙ps(cid:13) (cid:13)H˙qs ∀ ∈ p ∈ q
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
where 1 = 1 + 1 s.
r p q − d
4
Proof. It is not difficult to show that if p,q,and s satisfy (1) then there
exists numbers p ,p ,q ,q (1,+ ) (may be many of them) such that
1 2 1 2
∈ ∞
1 1 s s 1 1 s s 1 1
= − + , = − + , + < 1,
p p p q q q p q
1 2 1 2 1 1
1 1 1
p < d,q < d, and + < 1+ .
2 2
p q d
2 2
Setting
1 1 1 1 1 1 1
= + , = + ,
r p q r p q − d
1 1 1 2 2 2
we have
1 1 s s
= − + .
r r r
1 2
Therefore, applying Theorem 6.4.5 (page 152) of [1] (see also [25] for H˙ s),
p
we get
H˙ s = [Lp1,H˙ 1 ] ,H˙ s = [Lq1,H˙ 1 ] ,H˙ s = [Lr1,H˙ 1 ] .
p p2 s q q2 s r r2 s
Applying the Holder inequality and Lemma 2 in order to obtain
uv . u v , u Lp1,v Lq1,
Lr1 Lp1 Lq1 ∀ ∈ ∈
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13)uv (cid:13) . (cid:13)u(cid:13) (cid:13)v(cid:13) , u H˙ 1 ,v H˙ 1 .
(cid:13) (cid:13)H˙r12 (cid:13) (cid:13)H˙p12(cid:13) (cid:13)H˙q12 ∀ ∈ p2 ∈ q2
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
From Theorem 4.4.1 (page 96) of [1] we get
uv . u v .
H˙s H˙s H˙s
(cid:13) (cid:13) r (cid:13) (cid:13) p(cid:13) (cid:13) q
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Lemma 4. Assume that
s 1 1 1 1 s
q > 1,p > 1,0 < min , , and + < 1+ . (2)
≤ d np qo p q d
Then we have the inequality
uv . u v , u H˙ s,v H˙ s,
(cid:13) (cid:13)H˙rs (cid:13) (cid:13)H˙ps(cid:13) (cid:13)H˙qs ∀ ∈ p ∈ q
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
where 1 = 1 + 1 s.
r p q − d
5
Proof. Denote by [s] the integer part of s and by s the fraction part
{ }
of the argument s. Using the formula for the derivatives of a product of two
functions, we have
uv = Λ˙s(uv) = Λ˙{s}(uv)
(cid:13) (cid:13)H˙rs (cid:13) (cid:13)Lr (cid:13) (cid:13)H˙r[s] ≃
(cid:13) (cid:13)∂αΛ˙{s(cid:13)}(uv) (cid:13)= (cid:13) Λ˙{s}∂α(cid:13)(uv)
Lr Lr
X (cid:13) (cid:13) X (cid:13) (cid:13)
|α|=[s] |α|=[s]
(cid:13) (cid:13) (cid:13) (cid:13)
= ∂α(uv) . ∂γu∂βv .
X (cid:13) (cid:13)H˙r{s} X (cid:13) (cid:13)H˙r{s}
|α|=[s] |γ|+|β|=[s]
(cid:13) (cid:13) (cid:13) (cid:13)
Set
1 1 s γ s 1 1 s β s
= −| |−{ }, = −| |−{ }.
p˜ p − d q˜ q − d
Applying Lemma 3 and the Sobolev inequality in order to obtain
∂γu∂βv . ∂γu ∂βv . u v . u v .
(cid:13) (cid:13)H˙r{s} (cid:13) (cid:13)H˙p{˜s}(cid:13) (cid:13)H˙q˜{s} (cid:13) (cid:13)H˙p|˜γ|+{s}(cid:13) (cid:13)H˙q˜|β|+{s} (cid:13) (cid:13)H˙ps(cid:13) (cid:13)H˙qs
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
This gives the desired result
uv . u v .
H˙s H˙s H˙s
(cid:13) (cid:13) r (cid:13) (cid:13) p(cid:13) (cid:13) q
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Lemma 5. Let 1 p,q and s R.
≤ ≤ ∞ ∈
(a) If s < 1 then the two quantities
∞ dt 1/p
(cid:16)Z t−2s et∆t12Λ˙f q p t (cid:17) and f B˙qs,p are equivalent.
0 (cid:0) (cid:13) (cid:13) (cid:1) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(b) If s < 0 then the two quantities
∞ dt 1/p
(cid:16)Z t−2s et∆f q p t (cid:17) and f B˙qs,p are equivalent,
0 (cid:0) (cid:13) (cid:13) (cid:1) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
where B˙s,p is the homogeneous Besov space.
q
Proof. See ([13], Proposition 1, p. 181 and Proposition 3, p. 182), or
see ([26], Theorem 5.4, p. 45).
The following lemma is a generalization of the above lemma.
Lemma 6. Let 1 p,q , α 0, and s < α. Then the two quantities
≤ ≤ ∞ ≥
∞ dt 1
(cid:16)Z (t−2s et∆tα2Λ˙αf Lq)p t (cid:17)p and f B˙qs,p are equivalent,
0 (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
6
Proof. Note that Λ˙s0 is an isomorphism from B˙s,p to B˙s−s0,p, see [3],
q q
then we can easily prove the lemma.
Lemma 7. Assume that q > 1,1 r , and 0 s < d. The fol-
≤ ≤ ∞ ≤ q
lowing statement is true: If u H˙ s then et∆u L∞([0, );H˙ s ) and
0 ∈ Lq,r 0 ∈ ∞ Lq,r
et∆u u .
(cid:13) 0(cid:13)L∞([0,∞);H˙Lsq,r) ≤ (cid:13) 0(cid:13)H˙Lsq,r
(cid:13) (cid:13) (cid:13) (cid:13)
Proof. We have
(cid:13)et∆u0(cid:13)H˙Lsq,r = (cid:13)et∆Λ˙su0(cid:13)Lq,r = (4π1t)d/2(cid:13)ZRd e−|4ξt|2Λ˙su0( .−ξ)dξ(cid:13)Lq,r
(cid:13) (cid:13)
(cid:13) (cid:13) ≤(cid:13) (4π1t)d/2(cid:13)Z e−4|ξt|2 Λ˙su0(cid:13)( .−ξ) Lq,rdξ (cid:13)
Rd (cid:13) (cid:13)
1 −|ξ|2 (cid:13) (cid:13)
= (4πt)d/2 ZRd e 4t (cid:13)u0(cid:13)H˙Lsq,rdξ = (cid:13)u0(cid:13)H˙Lsq,r.
(cid:13) (cid:13) (cid:13) (cid:13)
Let us recall following result on solutions of a quadratic equation in Banach
spaces (Theorem 22.4 in [26], p. 227).
Theorem 1. Let E be a Banach space, and B : E E E be a continuous
× →
bilinear map such that there exists η > 0 so that
B(x,y) η x y ,
k k ≤ k kk k
for all x and y in E. Then for any fixed y E such that y 1 , the
∈ k k ≤ 4η
equation x = y B(x,x) has a unique solution x E satisfying x 1 .
− ∈ k k ≤ 2η
3. Main results
§
Now, for T > 0, we say that u is a mild solution of NSE on [0,T] cor-
responding to a divergence-free initial datum u when u solves the integral
0
equation
t
u = et∆u e(t−τ)∆P . u(τ,.) u(τ,.) dτ.
0 −Z ∇ ⊗
0 (cid:0) (cid:1)
Above we have used the following notation: For a tensor F = (F ) we define
ij
thevector .F by( .F) = d ∂ F andfortwovectorsuandv,wedefine
∇ ∇ i j=1 j ij
their tensor product (u v) P= u v . The operator P is the Helmholtz-Leray
ij i j
⊗
projection onto the divergence-free fields
(Pf) = f + R R f , (3)
j j j k k
X
1≤k≤d
7
where R is the Riesz transforms defined as
j
∂ iξ
j j
R = , i. e. R g(ξ) = gˆ(ξ)
j j
√ ∆ ξ
− d | |
withˆdenoting the Fourier transform. The heat kernel et∆ is defined as
et∆u(x) = ((4πt)−d/2e−|.|2/4t u)(x).
∗
If X is a normed space and u = (u ,u ,...,u ),u X,1 i d, then we
1 2 d i
∈ ≤ ≤
write
d
1/2
u X, u = u 2 .
∈ k kX (cid:16)Xk ikX(cid:17)
i=1
We define the auxiliary space s,q˜ which is made up by the functions u(t,x)
Kq,r,T
such that
α
u Ks,q˜ := sup t2 u(t,.) H˙s < ∞,
(cid:13) (cid:13) q,r,T 0<t<T (cid:13) (cid:13) Lq˜,r
(cid:13) (cid:13) (cid:13) (cid:13)
and
α
lt→im0t2(cid:13)u(t,.)(cid:13)H˙Lsq˜,r = 0, (4)
(cid:13) (cid:13)
where r,q,q˜,s being fixed constants satisfying
s 1 1 s+1
q,q˜ (1,+ ),r 1,s 0, < ,
∈ ∞ ≥ ≥ d q˜ ≤ q ≤ d
and
1 1
α = α(q,q˜) = d .
(cid:16)q − q˜(cid:17)
In thecase q˜= q, it is also convenient to define the space s,q˜ as thenatural
Kq,r,T
space L∞([0,T];H˙ s (Rd)) with the additional condition that its elements
Lq,r
u(t,x) satisfy
lim u(t,.) = 0. (5)
t→0(cid:13) (cid:13)H˙Lsq,r
(cid:13) (cid:13)
Remark 1. The auxiliary space := 0,q˜ (q˜ d) was introduced by
Kq˜ Kd,q˜,T ≥
Weissler and systematically used by Kato [20] and Cannone [5].
Lemma 8. Let 1 r r˜ . Then we have the following imbedding maps
≤ ≤ ≤ ∞
s,q˜ ֒ s,q˜ ֒ s,q˜ ֒ s,q˜ .
Kq,1,T → Kq,r,T → Kq,r˜,T → Kq,∞,T
Proof. It is easily deduced from Lemma 1 (a) and the definition
of s,q˜ .
Kq,r,T
8
Lemma 9. If u H˙ s (Rd) with q > 1,r 1,s 0, and s < 1 s+1
0 ∈ Lq,r ≥ ≥ d q ≤ d
then for all q˜ satisfying
s 1 1
< < ,
d q˜ q
we have
et∆u s,q˜ ,
0 ∈ Kq,1,∞
and the following imbedding map
H˙ s (Rd) ֒ B˙s−(dq−qd˜),∞(Rd). (6)
Lq,r → q˜
Proof. Beforeproving thislemma, we need toprove thefollowing lemma.
Lemma 10. Suppose that u Lq,r(Rd) with 1 q and 1 r < .
0
∈ ≤ ≤ ∞ ≤ ∞
Then lim u = 0, where n N, (x) = 0 for x x : x <
n→∞ Xn 0 Lq,r ∈ Xn ∈ { | |
(cid:13) (cid:13)
n x : u(cid:13) (x) (cid:13)< n and (x) = 1 otherwise.
0 n
}∩{ } X
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. With δ > 0 being fixed, we have
x : u (x) > δ x : u (x) > δ , (7)
n 0 n+1 0
|X | ⊇ |X |
(cid:8) (cid:9) (cid:8) (cid:9)
and
∞
x : u (x) > δ = x : u (x) = + . (8)
n 0 0
n∩=0{ |X | } { | | ∞}
We prove that
d( x : u (x) = + ) = 0, (9)
0
M { | | ∞}
with d being the Lebesgue measure in Rd, assuming on the contrary
M
d( x : u (x) = + ) > 0.
0
M { | | ∞}
We have u∗(t) := inf τ : d x : u (x) > τ t = + for all t
0 M { | 0 | } ≤ ∞
such that 0 < t < (cid:8)d( x : u(cid:0)(x) = + ) and(cid:1)then (cid:9)u = + , a
M { | 0 | ∞} 0 Lq,r ∞
contradiction. (cid:13) (cid:13)
(cid:13) (cid:13)
Note that
d x : u (x) > δ = d x : u (x) > δ .
0 0 0
M { |X | } M { | | }
(cid:0) (cid:1) (cid:0) (cid:1)
We prove that
d x : u (x) > δ < , (10)
0
M { | | } ∞
(cid:0) (cid:1)
assuming on the contrary
d x : u (x) > δ = .
0
M { | | } ∞
(cid:0) (cid:1)
9
We have u∗(t) δ for all t > 0, from the definition of the Lorentz space, we
0 ≥
get
∞ dt 1 ∞ dt 1 ∞ 1
u = (t1qu∗(t))r r (t1qδ)r r = δ tqr−1dt r = ,
0 Lq,r (cid:16)Z 0 t (cid:17) ≥ (cid:16)Z t (cid:17) (cid:16)Z (cid:17) ∞
(cid:13) (cid:13) 0 0 0
(cid:13) (cid:13)
a contradiction.
From (7), (8), (9), and (10), we infer that
lim d x : u (x) > δ = d( x : u (x) = + ) = 0. (11)
n 0 0
n→∞M { |X | } M { | | ∞}
(cid:0) (cid:1)
Set
u∗(t) = inf τ : d x : u (x) > τ t .
n M { |Xn 0 | } ≤
(cid:8) (cid:0) (cid:1) (cid:9)
We have
u∗(t) u∗ (t). (12)
n ≥ n+1
Fixed t > 0. For any ǫ > 0, from (11) it follows that there exists a number
n = n (t,ǫ) large enough such that
0 0
d x : u (x) > ǫ t, n n .
n 0 0
M { |X | } ≤ ∀ ≥
(cid:0) (cid:1)
From this we deduce that
u∗(t) ǫ, n n ,
n ≤ ∀ ≥ 0
therefore
lim u∗(t) = 0. (13)
n
n→∞
From (12) and (13), we apply Lebesgue’s monotone convergence theorem to
get
∞ dt 1
nl→im∞ Xnu0 Lq,r = nl→im∞(cid:16)Z (tq1u∗n(t))r t (cid:17)r = 0.
(cid:13) (cid:13) 0
(cid:13) (cid:13)
Now we return to prove Lemma 9. We prove that
0<sut<p∞tα2(cid:13)et∆u0(cid:13)H˙Lsq˜,1 . (cid:13)u0(cid:13)H˙Lsq,r. (14)
(cid:13) (cid:13) (cid:13) (cid:13)
Set
1 1 1
= 1+ .
h q˜− q
Applying Proposition 2.4 (c) in ([26], pp. 20) for convolution in the Lorentz
spaces, we have
(cid:13)et∆u0(cid:13)H˙Lsq˜,1 = (cid:13)et∆Λ˙su0(cid:13)Lq˜,1 = (4π1t)d/2(cid:13)e−|.4|t2 ∗Λ˙su0(cid:13)Lq˜,1 .
(cid:13) (cid:13) (cid:13) (cid:13)
td1/2(cid:13)(cid:13)e−|4.|t2(cid:13)L(cid:13)h,1(cid:13)Λ˙su0(cid:13)(cid:13)Lq,∞ = t(cid:13)−α2(cid:13)e−|.4|2(cid:13)Lh,1(cid:13)u(cid:13)0(cid:13)H˙Lsq,∞ . t−(cid:13)α2(cid:13)u0(cid:13)H˙Lsq,r.
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
10