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 changxing@iapcm.ac.cn) a J 2 2 College of Mathematics and Informatics, Henan Polytechnic University, 1 Jiaozuo City, Henan Province, 454000, P.R. China. ] (bqyuan@hpu.edu.cn) 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 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 magnetic 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 ∈ ∞ Caldrón’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 bilinear 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 Caldrón’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 Hölder’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 Hölder 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 Hölder 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

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