On well-posedness of the Cauchy problem for MHD system in Besov spaces

0.28 MB·English

by  Changxing Miao

On well-posedness of the Cauchy problem for MHD
system in Besov spaces
Changxing Miao1, Baoquan Yuan2
8
0
0 1 Institute of Applied Physics and Computational Mathematics,
2
P.O. Box 8009, Beijing 100088, P.R. China.

n
(miao [email protected])
a
J

2 2 College of Mathematics and Informatics, Henan Polytechnic University,
1
Jiaozuo City, Henan Province, 454000, P.R. China.

] ([email protected])
P
A
.
h Abstract
t
a ThispaperisdevotedtothestudyoftheCauchyproblemofincompressiblemagneto-
m
hydrodynamics system in framework of Besov spaces. In the case of spatial dimension
[ n 3 we establish the global well-posedness of the Cauchy problem of incompressible
≥
magneto-hydrodynamics system for small data and the local one for large data in Besov
3
n−1
v space B˙ p (Rn), 1 p< and 1 r . Meanwhile, we also prove the weak-strong
p,r
8 uniquenessofsolutio≤nswit∞hdatain≤B˙np−≤1(∞Rn) L2(Rn)for n +2 >1. Incaseofn=2,
5 p,r ∩ 2p r
4 we establish the global well-posedness of solutions for large initial data in homogeneous
7 Besov space B˙p2−1(R2) for 2<p< and 1 r < .
p,r
0 ∞ ≤ ∞
6 AMS Subject Classification 2000: 76W05, 74H20, 74H25.
0
/
h
t Keywords: Incompressiblemagneto-hydrodynamicssystem,homogeneousBesovspace,
a
m well-posedness, weak-strong uniqueness.
:
v
i 1 Introduction
X
r
a In this paper we consider the n-dimensional incompressible magneto-hydrodynamics (MHD)
system
u u+(u )u (b )b p = 0 (1.1)
t
−△ ·∇ − ·∇ −∇
b b+(u )b (b )u= 0 (1.2)
t
−△ ·∇ − ·∇
divu = 0, divb = 0 (1.3)
with initial data
u(0,x) = u (x), (1.4)
0
b(0,x) =b (x). (1.5)
0
1
wherex Rn,t > 0. Hereu= u(t,x) = (u (t,x), ,u (t,x)),b = b(t,x) = (b (t,x), ,b (t,x))
1 n 1 n
∈ ··· ···
and p = p(t,x) are non-dimensional quantities corresponding to the flow velocity, the mag-
netic field and the pressure at the point (t,x), and u (x) and b (x) are the initial velocity
0 0
and initial magnetic field satisfying divu =0, divb =0, respectively. For simplicity, we have
0 0
included the quantity 1 b(t,x)2 into p(t,x) and we set the Reynolds number, the magnetic
2| |
Reynolds number, and the corresponding coefficients to be equal to 1.
Itiswellknownthatforanyinitialdata(u ,b ) L2(Rn)withn 2, theMHDequations
0 0
∈ ≥
(1.1)-(1.5) have been shown to possess at least one global L2 weak solution (u(t,x),b(t,x))
C ([0,T];L2(R2)) L2((0,T]);H˙1(R2)) for any T > 0 such that ∈
b
∩
t
(u,b) 2 +2 ( u(s), b(s)) 2 ds (u ,b ) 2 , (1.6)
k kL2(R2) k ∇ ∇ kL2(R2) ≤ k 0 0 kL2(R2)
Z0
but the uniqueness and regularity remain open besides the case of n = 2, [6, 13]. Usually,
we define a Leray weak solution by any L2 weak solution (u,b) to the MHD (1.1)-(1.5), i.e.
which satisfies the MHD equations in distribution sense, and satisfying the energy estimate
(1.6).
When n = 2, for initial data (u (x),b (x)) L2(R2) there exists a unique global solution
0 0
∈
to MHD system (1.1)-(1.3) with (u(t,x),b(t,x)) C ([0, );L2(R2)) L2((0, );H˙1(R2))
b
∈ ∞ ∩ ∞ ∩
C ((0, ) R2), where C (I) denotes the space of bounded and continuous functions on
∞ b
∞ ×
I [6, 13]. Note that the coupled relation between equations (1.1) and (1.2) as well as the
relation
((b )b,u)+((b )u,b) = 0, for any 0 t < ,
·∇ ·∇ ≤ ∞
where (, ) stands for the inner product in L2 with respect to the spatial variables. It follows
· ·
that the solution (u,b) satisfies the energy equality:
t
(u,b) 2 +2 ( u(s), b(s)) 2 ds = (u ,b ) 2 , (1.7)
k kL2(R2) k ∇ ∇ kL2(R2) k 0 0 kL2(R2)
Z0
for any 0 t < .
≤ ∞
The purpose of this paper can be divided into two aspects. At first, we prove that for
initial data (u0,b0) B˙pn,/rp−1(Rn), 1 r , 1 p < , the Cauchy problem (1.1)-(1.5)
∈ ≤ ≤ ∞ ≤ ∞
hastheuniquelocalstrongsolutionorglobalstrongsmallsolutioninBesovspaceB˙pn,/rp−1(Rn).
If wefurtherassumethatthe data(u ,b )is in L2(Rn), theabove solution coincides with any
0 0
Leray weak solution associated with (u ,b ). In fact, we shall establish the stability result of
0 0
the Leray weak solution and strong solution in Section 3 which implies the weak and strong
uniqueness.
Theorem 1.1. Let (u0,b0) B˙pn,/rp−1(Rn), 1 p < , 1 r , 2 < q and
∈ ≤ ∞ ≤ ≤ ∞ ≤ ∞
divu = divb = 0.
0 0
(i) For 1 r , there exists ε > 0 such that if (u ,b ) < ε , then (1.1)-(1.5)
≤ ≤ ∞ 0 k 0 0 kB˙pn,/rp−1 0
has a unique solution (u,b) satisfying
(u,b) C (R+;B˙n/p 1) Lq(R+;B˙sp+2/q(Rn)), r < , (1.8)
∈ b p,r − ∩ p,r ∞
or
e
(u,b) C (R+;B˙n/p 1) Lq(R+;B˙sp+2/q(Rn)), r = , (1.9)
∈ ∗ p,∞− ∩ p,∞ ∞
e
2
where s = n 1 > 1 4 is a real number.
p p − − q
(ii) For 1 r < , there exists a time T and a unique local solution (u(t,x),b(t,x)) to
≤ ∞
the system (1.1)-(1.5) such that
n+2 1
(u,b) C ([0,T];B˙n/p 1) Lq([0,T];B˙ p q− (Rn)), r < , (1.10)
∈ b p,r − ∩ p,r ∞
or
e
(u,b) C ([0,T];B˙n/p 1) Lq([0,T];B˙sp+2/q(Rn)), r = , (1.11)
∈ ∗ p,∞− ∩ p,∞ ∞
where p, q satisfying n + 2 > 1, C denote the continuity in t = 0 with respect to time
2p q e
∗
n+2 1
t in weak star sense, Lq([0,T];B˙ p q− (Rn)) denotes the mixed space-time space defined by
p,r
Littlewood-Paley theory, please refer to Section 2 for details.
e
Theorem 1.2. Let (u0,b0) B˙pn,/rp−1(Rn) L2(Rn) be a divergence free datum. Assume
∈ ∩
1 ≤ p < ∞ and 2 < r < ∞ such that 2np + 2r > 1. Let (u,b) ∈ C([0,T];B˙pn,/rp−1(Rn)) ∩
L (R+;L2(Rn)) L2(R+;H˙1(Rn)) be the unique solution associated with (u ,b ). Then all
∞ 0 0
∩
Leray solutions associated with (u ,b ) coincide with (u,b) on the interval [0,T].
0 0
Secondly, weshallestablish theglobalwell-posednessfortheCauchyproblemoftheMHD
system (1.1)-(1.5) for data in larger space than L2(R2) space, i.e. the homogeneous Besov
space B˙p2,/rp−1(R2) for 2 < p < and 1 r < . Let us give some rough analysis. If
∞ ≤ ∞
1 p < 2 and 1 r or p = 2 and 1 r 2, the global well-posedness is trivial because
≤ ≤ ≤ ∞ ≤ ≤
of the embedding relation B˙p2,/rp−1(R2) ֒ L2(R2); The case 2 p < and 1 r 2
→ ≤ ∞ ≤ ≤
can be deduced into the case 2 p < and 2 < r < because of Sobolev embedding
≤ ∞ ∞
B˙p2,/rp1−1(R2) ֒→ B˙p2,/rp2−1(R2) with r1 ≤ r2. An interesting question is whether the MHD
system (1.1)-(1.5) is global well-posedness for arbitrary data in the Besov space B˙p2,/rp−1(R2)
for 2 p < , r = .
≤ ∞ ∞
Theorem 1.3. Let (u0(x),b0(x)) B˙p2,/rp−1(R2) be divergence free vector field. Assume that
∈
2 p < and 1 r < . Then there exists a unique solution to the MHD system (1.1)-
≤ ∞ ≤ ∞
(1.5) such that (u,b) ∈ C([0,∞);B˙p2,/rr−1(R2)). Moreover, if p, r satisfy also 2p + 2r > 1 and
1 r < , the following estimate holds:
≤ ∞
1+β
(u,b) C (u ,b ) (1.12)
k kB˙p2,/rp−1 ≤ k 0 0 kB˙p2,/rp−1
for any t 0, where β > p.
≥ 2
From the above discussion, it is sufficient to prove the case 2 p < and 2 < r <
≤ ∞ ∞
in Theorem 1.3. Since (u0,b0) C([0, );B˙p2,/rr−1(R2)) has infinite energy, so we have to use
∈ ∞
Caldr´on’s argument [4, 8] and perform a interpolation between the L2-strong solution and
the solution in C([0,∞);B˙p2¯,/r¯r¯−1(R2)) with p < p¯ < ∞ and r < r¯ < ∞. In detail, let us
decompose data
(u (x),b (x)) = (v (x),g (x))+(w (x),h (x)), (1.13)
0 0 0 0 0 0
with (v0,g0) L2(Rn) and (w0,h0) B˙p2¯,/r¯p¯−1(R2) for some p < p¯< and r < r¯< with
∈ ∈ ∞ ∞
smallnorm. Thecorrespondingsolutionsaredenotedby(v(t,x),g(t,x)) and(w(t,x),h(t,x)),
3
where the solutions (w,h) satisfies the MHD system and (v,g) satisfies MHD-like equations.
The global existence of solution (w,h) in the Besov space Lq((0, );B˙pn,/rp+2/q−1(Rn)), 1
∞ ≤
p < , 2 < q and 1 r for n 2 can be generally proved. The MHD-like
∞ ≤ ∞ ≤ ≤ ∞ ≥
system is locally solved, then by the energy inequality we prove (v,g) is global solvable for
n = 2. TheideacomesfromI.Gallagher andF.Planchon[8]whodealwiththeNavier-Stokes
equations, however we have give a different proof for the strong solutions to the MHD system
n+2 1
(1.1)-(1.5) on the mixed time-space Besov spaces Lq([0,T];B˙ p q− (Rn)).
p,r
The remaining parts of the present paper are organized as follows. Section 2 gives some
definitions and preliminary tools. In Section 3 wee establish some linear estimates and bi-
linear estimates of the solution in framework of mixed space-time Besov space by Fourier
localization and Bony’s para-product decomposition, and by which we complete the proof of
Theorem1.1 and Theorem1.2. Theorem1.3 willbeproved in Section 4 by Caldr´on’s argument
in conjunction with the real interpolation method.
We conclude this section by introducing some notations. Denote by (Rn) and (Rn)
′
S S
the Schwartz space and the Schwartz distribution space, respectively. For any interval I R
⊂
and any Banach space X we denote by C(I ;X) the space of strongly continuous functions
from I to X, and by (I;B) the time-weighted space-time Banach space as follows
σ
C
1
σ(I;X) = f C(I;B) : f; σ(I;X) = suptσ f X < .
C ∈ k C k k k ∞
t I
n ∈ o
we denote by Lq(I;X) and Lq1,q2(I;X) the space of strongly measurable functions from I to
X with u();X Lq(I) and u();X Lq1,q2, respectively. Lq1,q2 denotes usual Lorentz
k · k ∈ k · k ∈
space, please refer to [1, 9, 14] for details .
Notation: Throughout the paper, C stands for a generic constant. We will use the notation
A. B to denotethe relation A CB and thenotation A B todenote therelations A. B
≤ ≈
andB . A. Further, denotes thenormoftheLebesguespaceLp and (f ,f , ,f ) a
k·kp k 1 2 ··· n kX
denotes f a + + f a . Thetimeinterval I may beeither [0,T) foranyT > 0or [0, ).
k 1kX ··· k nkX ∞
2 Preliminary
In this section we first introduce Littlewood-Paley decomposition and the definition of Besov
spaces. Given f(x) (Rn), define the Fourier transform as
∈S
fˆ(ξ)= f(ξ) = (2π) n/2 e ixξf(x)dx, (2.1)
− − ·
F Rn
Z
and its inverse Fourier transform:
fˇ(x) = 1f(x) = (2π) n/2 eixξf(ξ)dξ. (2.2)
− − ·
F Rn
Z
Choose two nonnegative radial functions χ, ϕ (Rn) supported respectively in = ξ
∈ S B { ∈
Rn, ξ 4 and = ξ Rn, 3 ξ 8 such that
| | ≤ 3} C { ∈ 4 ≤ | | ≤ 3}
χ(ξ)+ ϕ(2 jξ) = 1, ξ Rn, (2.3)
−
∈
j 0
X≥
ϕ(2 jξ)= 1, ξ Rn 0 . (2.4)
−
∈ \{ }
j Z
X∈
4
Set ϕ (ξ) = ϕ(2 jξ) and let h = 1ϕ and h˜ = 1χ. Define the frequency localization
j − − −
F F
operators:
∆ f = ϕ(2 jD)f = 2nj h(2jy)f(x y)dy, (2.5)
j −
Rn −
Z
S f = ∆ f = χ(2 jD)f = 2nj ˜h(2jy)f(x y)dy. (2.6)
j k −
Rn −
k j 1 Z
≤X−
Formally, ∆ = S S is a frequency projection into the annulus ξ 2j , and S is a
j j j 1 j
frequency projectio−n int−o the ball ξ . 2j . One easily verifies that{w|i|th≈the}above choice
{| | }
of ϕ
∆ ∆ f 0 if j k 2 and ∆ (S f∆ f) 0 if j k 5. (2.7)
j k j k 1 k
≡ | − | ≥ − ≡ | − |≥
We now introduce the following definition of Besov spaces.
Definition 2.1. Let s R,1 p,q . The homogenous Besov space B˙s is defined by
∈ ≤ ≤ ∞ p,q
B˙s = f (Rn) : f < ,
p,q { ∈ Z′ k kB˙ps,q ∞}
where
1
q
2jsq ∆ f q , for q < ,
kfkB˙ps,q =  (cid:18)suXjp∈2Zjs ∆kf j ,kp(cid:19)for q = , ∞
 j p
j Z k k ∞
∈
and Z′(Rn) can be identified bythe quotient space S′/P with the space P of polynomials.
Definition 2.2. Let s R,1 p,q . The inhomogeneous Besov space Bs is defined by
∈ ≤ ≤ ∞ p,q
Bs = f (Rn) : f < ,
p,q { ∈ S′ k kBps,q ∞}
where
1
q
2jsq ∆ f q + S (f) , for q < ,
k j kp k 0 kp ∞
f Bs = (cid:18)j 0 (cid:19)
k k p,q  suXp≥2js ∆ f + S (f) , for q = .
 j p 0 p
k k k k ∞
j 0
≥

If s > 0, then Bps,q = Lp∩B˙ps,q and kfkBps,q ≈ kfkp+kfkB˙ps,q. We refer to [1, 15] for details.
ThefollowingDefinition2.3givesthemixedtime-spaceBesovspacedependentonLittlewood-
Paley decomposition (cf. [5]).
Definition 2.3. Let u(t,x) (Rn+1), s R,1 p, q, ρ . We say that u(t,x)
′
∈ S ∈ ≤ ≤ ∞ ∈
Lρ I;B˙s (Rn) if and only if
p,q
(cid:16) (cid:17) 2js u lq,
j Lρ(I;Lp)
e k△ k ∈
and we define
1/q
kukLeρ(I;B˙ps,q) , 2jsqk△jukqLρ(I;Lp) . (2.8)
(cid:18)j Z (cid:19)
X∈
5
For the convenience we also recall the definition of Bony’s para-product formula which
gives the decomposition of the product of two functions f(x) and g(x) (cf. [2, 3]).
Definition 2.4. The para-product of two functions f and g is defined by
T f = g f = S g f. (2.9)
g i j j 1 j
△ △ − △
i j 2 j Z
≤X− X∈
The remainder of the para-product is defined by
R(f,g) = g f. (2.10)
i j
△ △
i j 1
|−X|≤
Then Bony’s para-product formula reads
f g = T f +T g+R(f,g). (2.11)
g f
·
Using Bony’s para-product formula and the definition of homogeneous Besov space, one
can prove the following trilinear estimates, for details, see [8].
Proposition 2.1. Let n 2 be the spatial dimension and let r and σ be two real numbers
≥
such that 2 r < , 2 < σ < and n + 2 > 1. Define the trilinear form as
≤ ∞ ∞ r σ
t
T(a,b,c) = (a(s,x) b(s,x)) c(s,x)dxds, (2.12)
Z0 ZRn ·∇ ·
for a,b L∞([0, );L2(Rn)) L2([0, );H˙1(Rn)) and c Lσ([0,T];B˙rn,/σr+2/σ−1(Rn)), 0 <
∈ ∞ ∩ ∞ ∈
t T. Then T(a,b,c) is continuous and satisfies estimates as follows:
≤
1/σ 1 1/σ 1/σ 1 1/σ
|T(a,b,c)| . kakL∞(R+;L2)k∇akL−2(R+;L2)kbkL∞(R+;L2)k∇bkL−2(R+;L2)kckLσ([0,T];B˙rnr,σ+σ2−1)
2/σ 1 2/σ
+k∇akL2(R+;L2)kbkL∞(R+;L2)k∇bkL−2(R+;L2)kckLσ([0,T];B˙rnr,σ+σ2−1)
2/σ 1 2/σ
+kakL∞(R+;L2)k∇akL−2(R+;L2)k∇bkL2(R+;L2)kckLσ([0,T];B˙rnr,σ+σ2−1), (2.13)
and
T(a,b,c) C(ε)( a L2(R+;L2)+ b L2(R+;L2))
| | ≤ k∇ k k∇ k
t
+C(ε 1) ( a(s) 2 + b(s) 2 ) c(s) σ ds. (2.14)
− Z0 k kL2 k kL2 k kB˙rnr,σ+σ2−1
In particular,
t
|T(a,a,c)| ≤ C(ε)k∇akL2(R+;L2)+C(ε−1)Z0 ka(s)k2L2kc(s)kσB˙rnr,σ+σ2−1ds. (2.15)
Here C(ε) and C(ε 1) are constants that can be arranged by ε and 1, respectively, for ε > 0.
− ε
Remark 2.1. In reference [8] authors only proved the estimates (2.13) and (2.15). Actually
the proof also implies the estimate (2.14).
6
Nextwegive thetime-space estimateof theheatsemigroupu(t,x) = S(t)u , e t u (x),
0 − △ 0
which has been proved in [8]. But the proof has a misprint that is the inequality (4.3) in [8]
should be
u . u 1−p2∓ 1 εu p2∓ , (2.16)
k kLpt∓(I;B˙2s,±2) k kLe∞t (I;B˙2±,2ε)k|∇| ± kL2t,x(I×Rn)
where I [0, ) or I = [0, ).
⊂ ∞ ∞
Proposition 2.2. Let 2 < p < , u (x) L2(Rn). Denote u(t,x) = S(t)u (x), then we
0 0
∞ ∈
have
u C u , (2.17)
k kLpt,2(I;Lqx) ≤ k 0kL2
for 2 + n = n, Lp,2(I) denotes Lorentz space with respect to t I.
p q 2 t ∈
The following propositions describe the H¨older’s and Young’s inequalities in Lorentz
spaces, which will be used in this paper, for their proofs we refer to [12].
Proposition 2.3. (Generalized Ho¨lder’s inequality) Let 1 < p , p , r < , such that
1 2
∞
1 1 1
= + < 1,
r p p
1 2
and 1 q , q , s with
1 2
≤ ≤ ∞
1 1 1
+ .
q q ≥ s
1 2
If f Lp1,q1, g Lp2,q2, then h = fg Lr,s such that
∈ ∈ ∈
h r f g , (2.18)
k k(r,s) ≤ ′k k(p1,q1)k k(p2,q2)
where r stands for the dual to r, i.e. 1 + 1 = 1.
′ r r′
Proposition 2.4. (Generalized Young’s inequality)
Let 1 < p , p , r < such that
1 2
∞
1 1 1 1 1
+ > 1, = + 1,
p p r p p −
1 2 1 2
and 1 q , q , s with
1 2
≤ ≤ ∞
1 1 1
+ .
q q ≥ s
1 2
If f Lp1,q1, g Lp2,q2, then h = f g Lr,s with
∈ ∈ ∗ ∈
h 3r f g . (2.19)
k k(r,s) ≤ k k(p1,q1)k k(p2,q2)
In particular, we have the weak Young’s inequality
h C(p,q) f g , (2.20)
(r, ) (p, ) (q, )
k k ∞ ≤ k k ∞ k k ∞
where 1< p, q, r < and 1 = 1 + 1 1.
∞ r p q −
Proposition 2.5. Let 1 q and 1 q satisfy 1 + 1 1, p and p be conjugate
≤ 1 ≤ ∞ ≤ 2 ≤ ∞ q1 q2 ≥ ′
indices, i.e. 1 + 1 = 1. If f(x) Lp,q1 and g(x) Lp′,q2, then h(x) = f g L such that
p p′ ∈ ∈ ∗ ∈ ∞
h f g . (2.21)
k k∞ ≤ k k(p,q1)k k(p′,q2)
7
3 Well-posedness in Besov spaces: Case n 2
≥
This section is devoted to the proof of Theorem 1.1. One easy sees that (1.1)-(1.5) can be
rewritten as
u u+P (u u) P (b b)= 0, (3.1)
t
−△ ∇· ⊗ − ∇· ⊗
b b+P (u b) P (b b)= 0, (3.2)
t
−△ ∇· ⊗ − ∇· ⊗
divu= divb = 0, (3.3)
u(0,x) = u (x), b(0,x) = b (x). (3.4)
0 0
or their integral form
t
u= et u e(t s) P (u u) P (b b) ds, (3.5)
△ 0 − △
− ∇· ⊗ − ∇· ⊗
Z0 h i
t
b = et b e(t s) P (u b) P (b u) ds. (3.6)
△ 0 − △
− ∇· ⊗ − ∇· ⊗
Z0 h i
Here P stands for the Leray projector onto divergence free vector field.
3.1 Linear and nonlinear estimates
To prove the results of global or local well-posedness of the Cauchy problem (3.1)-(3.4) or
(3.5)-(3.6) in Besov space B˙pn,/rp−1(Rn), we need to establish linear and nonlinear estimates in
framework of mixed space-time space by Fourier localization. First we consider the solution
to linear parabolic equation
u u= f(t,x),
t
−△ (3.7)
(u(0,x) = u0.
Applying frequency projection operator to both sides of (3.7), one arrives at
j
△
∂
( u)+ ( u)= f. (3.8)
j j j
∂t △ △ △ △
Multiplying up 2 u on both sides of (3.8), we obtain
j − j
|△ | △
∂
u up 2 u u up 2 u = f up 2 u. (3.9)
j j − j j − j j j − j
∂t△ |△ | △ −△△ |△ | △ △ |△ | △
We integrate both sides of (3.9) and apply the divergence theorem to obtain
1 d
u p + u ( up 2 u)dx f u p 1. (3.10)
pdtk△j kp Rn∇△j ·∇ |△j | − △j ≤ k△j kpk△j kp−
Z
Since
u ( up 2 u)dx = (p 1) up 2 u2dx
j j − j j − j
Rn∇△ ·∇ |△ | △ − Rn|△ | |∇△ |
Z Z
= 4(pp−2 1) Rn|∇(|△ju|p2)|2dx= k∇(|△ju|p2)k22
Z
c 22j u p. (3.11)
≥ p k△j kp
8
We have
d
u +22jc u f . (3.12)
j p p j p j p
dtk△ k k△ k ≤ k△ k
Integrating both sides of (3.12) with respect to t we arrive at
u e cp22jt u (x) +e 22jcpt ( f χ(τ)), (3.13)
j p − j 0 p − j p
k△ k ≤ k△ k ∗ k△ k
where χ(τ) is a character function
1, if 0 τ t,
χ(τ) = ≤ ≤ (3.14)
(0, if others.
Taking Lq norm with respect to t in interval I in both sides of (3.14), by Young inequality
−
one has
1 2j 2j
k△jukLq(I;Lp) ≤ c−p q2−q k△ju0(x)kp +C(p,q)2−q′k△jfkLq/2(I;Lp). (3.15)
Here 1+ 1 = 1. Multiplying 2js+2qj on both sides of (3.15) and taking lr norm with respect
q q′ −
to j yields
u C(p,q) u + f . (3.16)
k kLeq(I;B˙ps,+r2/q) ≤ k 0kB˙ps,r k kLeq/2(I;B˙ps,+r4/q−2)
(cid:18) (cid:19)
Thus we arrive at
Lemma 3.1. Let 1 p < , 2 q , 1 r and s R. Assume u(t,x) is a
≤ ∞ ≤ ≤ ∞ ≤ ≤ ∞ ∈
solution to the Cauchy problem (3.7). Then there exists a constant C depending on p, q, n
so that
u C u + f . (3.17)
k kLeq(I;B˙ps,+r2/q) ≤ k 0kB˙ps,r k kLeq/2(I;B˙ps,+r4/q−2)
(cid:18) (cid:19)
In particular, if q r q we have
2 ≤ ≤
u C u + f , (3.18)
k kLq(I;B˙ps,+r2/q) ≤ k 0kB˙ps,r k kLq/2(I;B˙ps,+r4/q−2)
(cid:18) (cid:19)
by Minkowski inequality.
Remark 3.1. For s R and 1 p,r , (B˙s , ) is a normed space. It is easy to
∈ ≤ ≤ ∞ p,r k·kB˙ps,r
check that (B˙s , ) is a Banach space if and only if s< n or s = n, r = 1.
p,r k·kB˙ps,r p p
Using Bony’s para-product decomposition we study the bilinear estimates. Consider two
tempered distributions u(t,x) and v(t,x), then
uv = T v+T u+R(u,v). (3.19)
u v
First we deal with the para-product term T v or T u as following lemma.
u v
9
Lemma 3.2. (1) Let B˙s (Rn) be a Banach space, then
p,r
kTuvkLeq/2(I;B˙ps,r) ≤kukLq(I;L∞)kvkLeq(I;B˙ps,r). (3.20)
(2) Let s < 0 and 1 = 1 + 1, and B˙s2 (Rn) be a Banach space. Then
1 r r1 r2 p,r2
kTuvkLeq/2(I;B˙ps,1r+s2) ≤ CkukLeq(I;B˙∞s1,r1)kvkLeq(I;B˙ps,2r2). (3.21)
Proof. By the definition of Lq(I;B˙s ) and H¨older inequality, direct computation yields
p,r
1/r
e
kTuvkLeq/2(I;B˙ps,r) = 2jsrkSj−1u△jvkrLq/2(I;Lp)
(cid:18)j Z (cid:19)
X∈
1/r
u 2jsr v r
≤ k kLq(I;L∞) k△j kLq(I;Lp)
(cid:18)j Z (cid:19)
X∈
≤ kukLq(I;L∞)kvkLeq(I;B˙ps,r). (3.22)
Noting that the equivalent definition of negative index Besov space
1/r 1/r
2jsr S u r 2jsr u r (3.23)
k j kLq(I;Lp) ≃ k△j kLq(I;Lp)
(cid:18)j Z (cid:19) (cid:18)j Z (cid:19)
X∈ X∈
for s < 0 (cf. [3]), we can derive similarly by H¨older inequality
1/r
kTuvkLeq/2(I;B˙ps,1r+s2) ≤ 2j(s1+s2)rkSj−1ukrLq(I;L∞)k△jvkrLq(I;Lp)
(cid:18)j Z (cid:19)
X∈
≤ CkukLeq(I;B˙∞s1,r1)kvkLeq(I;B˙ps,2r2). (3.24)
Next we estimate the remainder of para-product decomposition.
Lemma 3.3. Let s , s R, 1 p , p , p, r , r , r and 2 q such that
1 2 1 2 1 2
∈ ≤ ≤ ∞ ≤ ≤ ∞
1 1 1 1 1 1
= + , = + (3.25)
p p p r r r
1 2 1 2
and Lq(I;B˙s1 ), Lq(I;B˙s2 ) and Lq/2(I;B˙s1+s2) are Banach spaces. Assume 0 < s +s <
p1,r1 p2,r2 p,r 1 2
n, then
p
e e e
kR(u,v)kLeq/2(I;B˙ps,1r+s2) ≤ CkukLeq(I;B˙ps11,r1)kvkLeq(I;B˙ps22,r2). (3.26)
Moreover, if s +s = 0 and 1 + 1 = 1, then one has
1 2 r1 r2
kR(u,v)kLeq/2(I;B˙p0,∞) ≤ CkukLeq(I;B˙ps11,r1)kvkLeq(I;B˙ps22,r2). (3.27)
If s +s = n and r = 1, then
1 2 p
kR(u,v)kLeq/2(I;B˙pn,/1p) ≤ CkukLeq(I;B˙ps11,r1)kvkLeq(I;B˙ps22,r2). (3.28)
10