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Schrodinger方程随机问题的是定性,随机初值,这篇文章不错
WP AND SCATTERING FOR ANLS Corollary 1.3. Let r>0 be as in Theorem [. I For all uo E Br(H-72),there exists a solution∈C(0,∞);Hs)of位om(0,∞) and the solution scatters in H-1/2. More precisely, there exists u+E H-12 such that l(t)-c△+→0in-12ast→+a Moreover, we obtain the large data local in time well-posedness in the scalin critical Sobolev space. To state the result, we put BR(H°):={0∈H0=+0,og-12<6.‖lbo2<B} for s< Theorem 1. 4. Let d= l, m= 4 and P(u,U=I. Then the equation(II)is locally in time well-posed in H-12. More precisely, there erists>0 such that for nlR≥6 and uo∈B.r(H-12) there erists a solution ∈z 1/2 0,)CC([0,T);H foT=6R-80f① Furthermore, the same statement remains valid if we replace H-12 by H-12 as well as z-1/2(0,到)by2-1/2(0,]) Remark 1.5. For s>-1/ 2, the local in time well-posedness in H follows from the usual Fourier restriction norm method, which covers for all initial data in H It however is not of very much interest. On the other hand, since we focus on the scaling critical cases, which is the negative regularity, we have to impose that the part of initial data is small. But. Th loren is a large data result because the l- part is not restricted The main tools of the proof are the UP space and vp space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadad, Herr and Koch(阻,阿) We also consider the one dimensional cubic case and the high dimensional cases he second result in this paper is as follows Theorem 1.6.(i) Let d= l and m=3. Then the equation(I.Ib is locally well p0 sed in h°fors>0. (ii)Let d22 and(m- 1)d> 4. Then the equation(II) is globally well-posed for small data in hsc ( or hs for s>sc) and the solution scatters in hSc (or H for H HIRAYAMA AND M OKAMOTO The smoothing effect of the linear part recovers derivative in higher dimensional case. Therefore, we do not use the UP and vp type spaces. More precisely, to establish Theorem 1.6] we only use the Strichartz estimates and get the solution in C(0, T); Hse)n LPm([0, T); Wgm, 5c+i/(m-1)with Pmn= 2(m-1), qn=2(m 1)d/(m-1)d-2. Accordingly, the scattering follows from a standard argument Since the condition(m-1)d2 4 is equivalent to sc +1/(m-1)20, the solution space [Pm([0, T); Wgm, Sc+1/( m-D))has nonnegative regularity even if the data belongs to hSc with-1/(m-1)≤s<0. Our proof of Thorem正可(i) cannot applied for d= l since the Schrodingier admissible(a, b)in(5. 3) does not exist Remark 1.7. For the case d=l, m= 4 and P(u, u)u, we can obtain the local in time well-posedness of( 1.I in IIs fors20 by the same way of the proof of Theorem [7.6 Actually, we can get the solution in C(0, Tl; HS)nL(0, T): Ws+1/2,oc fors>0 by using the iteration argument since the fractional Leibnitz rule (see 1) and the Holder inequality imply ≤T/4‖V 18+214rou2|L4L∞3LL214I2 1 Lt°(0.,T);JL4) We give a remark on our problem, which shows that the standard iteration argu ment does not work Theorem 1.8.(i)Let d= 1, m=3,s<0 and P3(u, t)=a2u. Then the flow map of(II)from HS Lo C(R; Hs) is nol snooth i)Letm≥2,s<sand= V or ao for some1≤k≤d. Then the flow map of① omhs to O(R;H°)ism0 t smooth More precisely, we prove that the Hlow map is not C3 if d=1, m=3, s<0and P(u,u)=apu or Cm if d 21, m 22, and s< sc. It leads that the standard iteration argument fails, because the flow map is smooth if it works. Of course, there is a gap between ill-posedness and absence of a smooth flow map Since the resonance appears in the case d=l, m=3 and P3(u, l=alu, there exists an irregular flow map even for the subcritical Sobolev regularity Notation. We denote the spatial Fourier transform by or Fa, the Fourier transform in time by Fi and the Fourier transform in all variables by. or Fix. The free evolution S(t) t△ given as a fourier multiplier F[S(t)月()=e-if(5) WP AND SCATTERING FOR ANLS We will use a<b to denote an estimate of the form A< CB for some constant C and write A n b to mean A< B and< A. We will use the convention that capital letters denote dyadic numbers, e.g. N=2n for n E Z and for a dyadic summation we write∑yaN:=∑nea2and∑N>naN:=∑ mE, 2n>M a2r for brevity. Let X E Co((2, 2)) be an even, non-negative function such that x(t)=1 for t<1 We define v(t): x(t)-x(2t) and N(t):v(N-t). Then, ENN(t) whenever tf0. We define frequency and modulation projections u()(),Qa( s|4)a(x,5) Furthermore, we define QSM: -CNSM QN and Q&M: -1d-Qs M The rest of this paper is planned as follows. In Section 2, we will give the definition and properties of the UP space and Vp space. In Section 3, we will give the multilinear estimates which are main estimates to prove Theorems [I and L.4 In Section 4, we will give the proof of the well-posedness and the scattering(Theorem 1.1 Corollary 1.3 and Theorem 1. 41). In Section 5, we will give the proof of Theorem 1. 6 In Section 6, we will give the proof of Theorem 1.8 2. THE UP. VP SPACES AND THEIR PROPERTIES In this section, we define the UP space and the vp space, and introduce the properties of these spaces which are proved by Hadac, Herr and Koch(4,5) We define the set of finite partitions z as 2:={tA=0K∈N,-∞<t<t<…<tk≤∞} and if tK=oo, we put v(tK): =0 for all functions v: R-7L2 Definition21.let1≤n<∞.For{h3k=0∈2amd{qk}k=c2mth ck-o llor li2= l we call the function a:R+L2 given by a(t ∑14-14()=1 k=1 a "Up-atom". Furthermore, we define the atomic space ∑A1:Un-aomn,y∈ C such that∑M< writh the norm ∑A=∑入 U-atom,入;∈C H HIRAYAMA AND M OKAMOTO Definition 2.2. Let 1<p<o. We define the space of the bounded p-varialion VP:={:R→L2‖l with the norm Sup ∑o(t)-(t-)z2 {tk}A0∈z Likewise, let Vrc denote the closed subspace of all right-continuous functions v E Vp ith lim;→-∞0(t)=0, endowed with the same norn‖l·‖v Proposition 2.3(4 Proposition 2.2, 2.4, Corollary 2.6). Let 1<p<q<oo (i)Up,Vp and V rc are Banach spaces i) For every”∈VP,lim→-∞o(t) and lim→x(t) east (iii) The embeddings UP e,Verc 7U9c7 L(R; L2(R ))are continuous Theorem 2.4(4 Proposition 2, 10, Remark 2.12). Let 1<p<oo and 1/p+1/p 1.l ue v re be absolutely continuous on every compact intervals, then 1 Up Ip ((t).0()1 ∈p,|ll Definition2.5.Let1≤p<∞. we define 2:={u:R→>L2s(-)∈Un} wIth, the norm callus S(-)|lp, 3:={v:R→L2|S(-)∈Vm with the norm vve: =S(-ullvp Remark 2.6. The embeddings UP<)Vp< U9>L(R; I2)hold for 1<p<q< ∞ by Proposition23 Proposition 2.7(4 Corollary 2.18). Let 1<p<oo. We haie SUlz≤M-l/2lvg Smally save,‖ RsMullvp≤lv, (22 Proposition 2.8(4 Proposition 2.19). Let 0:L2(R4)×……×L2(4)→L( WP AND SCATTERING FOR ANLS be a m-linear operator:. Assume thal Jor some 1 <p, g<o e(S()q,…,S()m):)≤II L2(Rd Then, there erists t:U3×…xU3→?(R;Ia(R):0 misfiring T(1,mn)Z(政:(a) such that T(ul, ,,. um)(t)(r)=To(u1(t),.., am(t))()ae Now we refer the Strichartz estimate for the fourth order Schrodinger equation proved by pausader. We say that a pair(P,q) is admissible if2≤p,q≤∞o, (P,、q,d)≠(2,∞,2),and ×少 Proposition 2.9(16 Proposition 3. 1). Let(p, q)and(a, b)be admissible pairs Then. we have S()y2|L‖ (t t)F(t)dt's 2/p-2/ 0 LPL and b are conjugate exponents of a and b respectively Propositions 2. 8 and 2g imply the following Corollary 2. 10. Let(p, q be an admissible pair U∈U .3 Next, we define the function spaces which will be used to construct the solution We define the projections P1 and Pe <l as Definition 2.11. Let s<0 (i) We define 7s: =[u EC(R; H(Rd)nU3 Illio < o] with the norm al2:(∑N2li (i) we define Zs:={u∈C(R:;H(R4)‖l|z<∞9}0 ith the norm H HIRAYAMA AND M. OKAMOTO i) We define y:={u∈C(R:H(R4)nV3|‖ly。<} with the e norm ∑N2l (iv) We define ys: fuE C(R; H(Rd)) ullys <a with the norm 0+‖P1l 3. MULTILINEAR ESTIMATE FOR PA(u, T)=T IN ld In this section, we prove multilinear estimates for the nonlinearity a(t)in ld, which plays a crucial role in the proof of Theorem 1.1 Lemma31. We assume that(o,50),(71,51),…,(7,5)∈R×R4sat9∑:07 0 and i-o S;=0. Then, we have nax max 50<j<4 (3.1) 0≤j≤4 Proof. By the triangle inequality, we obtain(. I) 3. 1. The homogeneous case Proposition3.2.Letd=1amd0<T≤∞, For a dyadic number n1∈22,we define the set A1 (Ni)as A1(N1):={(N2N,N4)∈(2) If No N Ni, then we have dcdi A1(N1)0R (3,2) I PN Lll IPN ulva l uill y 1/2 =2 Proof. We define WiN,T: =1 j=1,…,4) and put M:=Na ve decompose Id=QsM+QSM. We divide the integrals on the left-hand side of(B 2) into 10 pieces of the form 3, N,,T' (3.3 0 ithQ3∈{QM,Q3M}( 0. By the Plancherel's theorem, we have 4 WP AND SCATTERING FOR ANLS where c is a constant. Therefore, Lemma B Implies that L/INOIIQSMUs, r) drdt So, let us now consider the case that Q,=Qsm for some 0<j<4 First, we consider the case Qs=QSM. By the Cauchy-Schwartz inequality, we have eSMuo, oTIS ≤MQ2MMx2|Q01ltI∑okx Furthermore by(2. 1) and M No, we have QSM0|z2.≤N0210xl and by( 2.3) and Vs >Us, we have SUl -1 /2 ILALo <NQSu1N Slug<N/Q While by the Sobolev inequality, (2.31, Vs<+ U52 and the Cauchy-Schwartz inequal ity for the dyadic sun, we have T ∑ ∑ 2 N,T 0,T)uj (34 GIo 4. Therefore, we obtain ∑//( NoSMO, o, L d cdt A1(N)R lvIiluill by②2 since‖1o.u| v2 <ulva for any T'∈(0,。∞]. For the case= QSM is proved in same way 10 H HIRAYAMA AND M. OKAMOTO Next, we consider the case Qi=Qsm for some 2< i < 4. By the Holder inequality, we have NoQSMui, N,T I QSu; N, T A1(N1) 0≤j≤4 K NoQ01o. No, TL1216Q5u1,N1.TIILALSo ∑Q Mai, N, T Ⅱ∑Q4xr j4N≤ By L orthogonality and 2.1), we have ∑ >Mai,N:./ MWi, Ni,/L2 (3.5) 1/2 since MN No. While, by the calculation way as the case Qo=QSm, we have 0, No,TIL Q 1,N1,T T 0T)24y Therefore. we obtain NOQSMUi II QSu, N, 7 dadl RJ IR 0≤j<4 by22snel1llvn≈ Iv2 for any T'∈(0,∞) Proposition3.3.Letd=1and0<T≤∞. For a dyadic number m2∈22,e define the set A2(N2)as A2(N2):={(N3,N4)∈(2)4N2≥N≥N4}

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