Well-posedness for the Fourth-order Schrödinger Equations with Quadratic Nonlinearity

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1 Well-posedness for the Fourth-order Schrödinger Equations with Quadratic Nonlinearity Jiqiang Zheng The Graduate School of China Academy of Engineering Physics P. O. Box 20, Beijing, China, Abstract This paper is concerned with -D quadratic semilinear fourth-order Schrödinger equations. Motivated by the quadratic Schrödinger equations in the pioneer work of Kenig-Ponce-Vega [2], three bilinearities uv, uv, uv for functions u, v : R [0, T ] C are sharply estimated in function spaces X s,b associated to the fourth-order Schrödinger operator i t + 2 ε. These bilinear estimates imply local wellposedness results for fourth-order Schrödinger equations with quadratic nonlinearity. To establish these bilinear estimates, we derive a fundamental estimate on dyadic blocks for the fourth-order Schrödinger from the [k, Z]-multiplier norm argument of Tao [20]. AMS Subject Classification: 35Q40, 35Q55, 47J35 Key Words: Fourth-order Schrödinger equation, Well-posedness, [k, Z]- multiplier norm. Introduction This paper is mainly devoted to the local well-posedness of the initial value problems (IVP) for the fourth-order Schrödinger equation (i =, 2, 3) { iu t + 2 u ε u ± Q i (u, ū) = 0, (x, t) R R, u(0) = u 0 H s (R), (.) where Q (u, ū) = ū 2, Q 2 (u, ū) = u 2, Q 3 (u, ū) = uū and ε {, 0, }. The fourth-order Schrödinger equations have been introduced by Karpman [8] and Karpman and Shagalov [9] to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. Such fourth-order Schrödinger equations have been studied from the mathematical viewpoint in Fibich, Ilan and Papanicolaou [7] who describe various properties of the

2 equation in the subcritical regime, with part of their analysis relying on very interesting numerical developments. Related reference is [2] by Ben-Artzi, Koch, and Saut, which gives sharp dispersive estimates for the biharmonic Schrödinger operator, which lead to the Strichartz estimates for the fourth-order Schrödinger equation, see also [6, 8, 9]. Concerning the Strichartz estimate for the high order dispersive equation, we should point out that Kenig-Ponce-Vega [0] have firstly established the sharp estimates involving gain of derivatives. We also can refer to Pausader [9] for the aim of finding a more completed result on the cubic fourth-order Schrödinger equation for initial data u 0 H 2 without radial assumption. In [2], we utilized the interaction Morawetz estimate to prove the scattering theory for the energy critical fourth-order Schrödinger equation with a subcritical perturbation. However, the scaling argument implies the nonlinear fourth-order Schrödinger equations mentioned in the above papers are H s -critical with s 0. We here are interested in the nonlinearities for which the Sobolev index s can be take negative values, i.e. s < 0. The main motivation of this paper is the pioneer work of Kenig-Ponce-Vega [2], in which they considered the Schrödinger equations with quadratic nonlinearities: { iu t + u ± N i (u, ū) = 0, u(0) = u 0, where N (u, ū) = ū 2, N 2 (u, ū) = uū, N 3 (u, ū) = u 2. In one dimension, Kenig-Ponce-Vega [2] first proved for the nonlinearities ū 2 and u 2 well-posedness for s > 3 4, while for the nonlinearity uū well-posedness for s > 4. For the nonlinearity u 2, I. Bejenaru and T. Tao [3] showed the Cauchy problem was local well-posed in H s (R) when s and ill-posed when s <. While for the nonlinearity ū 2, N. Kishimoto [4] were in same spirt of [3] to establish that the IVP is local wellposed in H s (R) when s and ill-posed when s <. For the dimension two case, we can refer to [3, 4, 6] for details. The IVP for other equations, such as the generalized Korteweg-de Vies and the wave equation can be seen in [, 3, 5]. We remark that T. Tao systematically provided a multilinear estimate method in [20] to deal with these problems. It is interesting to compare these results with those known for other evolution models such as fourth-order Schrödinger equations. In this paper, we restrict ourselves to the one-dimensional case 2

3 fourth-order Schrödinger equations with quadratic nonlinearities and study the wellposedness of these fourth-order Schrödinger equations. To this end, we first derive a fundamental estimate on dyadic blocks (see below) for the fourth-order Schrödinger equations by following the idea in the [k, Z]-multiplier norm method introduced by Tao [20]. We then apply this fundamental estimate to establish bilinear estimates in Bourgain spaces X s,b (see e.g. [, 5, 5]), which also be defined in the next part of this section. As applications of these estimates, we establish well-posedness of the IVP for the nonlinearities Q, Q 2, Q 3 respectively. With the definition of the Bourgain spaces X s,b in (2.)(see below), we state our results in the following theorems: Theorem.. Let b = 2 + η, we have Q (u, ū) Xs,b u 2 X s,b, (.2) whenever η > 0 and 0 s > η, with the implicit constant depending on s and η. Theorem.2. Let b = 2 + η, we have Q 2 (u, ū) Xs,b u 2 X s,b, (.3) whenever η > 0 and 0 s > η, with the implicit constant depending on s and η. Theorem.3. Let b = 2 + η, we have Q 3 (u, ū) Xs,b u 2 X s,b, (.4) whenever η > 0 and 0 s > η, with the implicit constant depending on s and η. In the spirt of the Kenig-Ponce-Vega [2, 3] and K.Nakanishi, H.Takaoka and Y.Tsutsumi [7], we get the sharpness of Theorem.-.3 in the following theorem. Theorem.4. fails. a). For any s 7 4 and any b R the estimate (.2)and (.3) fails. b). For any s < 3 4 and any b R, or s = 3 4 and any b 2, the estimate (.4) 3

4 As a consequence of the first three theorems, we can make use of the technique used to prove Theorem.5 in [2] and also used in [3] to get the following results concerning the local wellposedness of the initial value problems (.). Theorem.5. Let s ( 7 4, 0]. Then for any u 0 H s (R), there exits T = T ( u 0 H s) and a unique solution u(t) of the IVP (.), with the nonlinear term Q = Q, satisfying u C([ T, T ]; H s (R)) and u X s, 2 +. In addition, the dependence of u on u 0 is Lipschitz. Theorem.6. For the IVP (.) with nonlinear term Q = Q 2, the results in Theorem.5 hold for s ( 7 4, 0]. Theorem.7. For the IVP (.) with nonlinear term Q = Q 3, the results in Theorem.5 hold for s ( 3 4, 0]. The proofs of the above Theorems will be given in the subsequent subsection, provided the bilinear estimates in the Theorem., Theorem.2 and Theorem.3. The paper is organized as follows. In Section 2, we introduce the linear estimates, Tao s [k; Z]-multiplier norm method and prove a fundamental estimate on dyadic blocks for the fourth-order Schrödinger equation. In Section 3, we prove the Theorem.-.4. In the appendix section 4, we present a detail argument of the reductions, which are used in proving the fundamental estimate on dyadic blocks. Acknowledgements: The author thanks the referee for his/her useful suggestions and comments. 2 Fundamental estimates Let us start this section by providing the linear estimates for the fourth-order Schrödinger equation. Denote by W (t) the unitary group generating the solution of the IVP for the linear equation { iv t + 2 v ε v = 0, x R, t R then v(x, 0) = v 0 (x), v(x, t) = W (t)v 0 (x) = S t v 0 (x), 4

5 where Ŝt = e itq(ξ) with q(ξ) = ξ 4 + εξ 2, or S t (x) = e i(xξ+tq(ξ)) dξ. For s, b R, let X s,b denote the completion of the functions in C 0 norm where ξ = + ξ. Let φ C 0 with respect to the f Xs,b = ξ s τ q(ξ) b f(ξ, τ) L 2 ξ,τ (2.) be a standard bump function and consider the following integral equation u(t) = φ( t δ )W (t)u 0 φ( t t δ ) W (t s)q i (u, u)(s)ds. Denote the right-hand side by T (u). The goal is to show that T (u) is contraction on the following complete metric space B, where with metric B = {u X s,b : u Xs,b 2cδ 2b 2 u H s} d(u, v) = u v Xs,b, u, v B. Here c is the constant appeared in the following Proposition 2.. For this purpose, we need two linear estimates. Proposition 2.. For s R, b ( 2, ], φ( t δ ) t 0 0 φ( t δ )W (t)u 0 Xs,b cδ 2b 2 f H s, W (t s)f(s)ds Xs,b cδ 2b 2 f Xs,b. The proof of these estimates follows directly from Kenig, Ponce and Vega []. Once the estimates in Theorem..3 are available, a standard argument then yields that T (u) is self-contained and contracted operator on B. Let us introduce now Tao s [k, Z]-multiplier norm method and establish the fundamental estimate on dyadic blocks for the fourth-order Schrödinger equation. Let Z be any abelian additive group with an invariant measure dξ. In this paper, Z is Euclidean space R R with Lebesgue measure. For any integer k 2, we let Γ k (Z) denote the hyperplane Γ k (Z) := {(ξ,, ξ k ) Z k : ξ + + ξ k = 0} 5

6 endowed with the measure f := f(ξ,, ξ k, ξ ξ k ) dξ dξ k. Γ k (Z) Z k A [k, Z]-multiplier is defined to be any function m : Γ k (Z) C which was introduced by Tao in [20]. And the multiplier norm m [k,z] is defined to be the best constant such that the inequality Γ k (Z) m(ξ) k f j (ξ j ) c j= k f j L 2 (Z). (2.2) holds for all test functions f j on Z. To establish the fundamental estimate on dyadic blocks for the fourth-order Schrödinger equations, we introduce some notations. We use A B to denote the statement that A CB for some large constant C which may vary from line to line and depend on various parameters, and similarly use A B to denote the statement A C B. We use A B to denote the statement that A B A. We will sometimes write a+ to denote a+η for arbitrarily small η > 0. Any summations over capitalized variables such as N j, L j, H are presumed to be dyadic, i.e., these variables range over numbers of the form 2 k for k Z. Let N, N 2, N 3 > 0. It will be convenient to define the quantities N max N med N to be the maximum, median, and imum of N, N 2, N 3 respectively. Similarly define L max L med L whenever L, L 2, L 3 > 0. j= And we also adopt the following summation conventions. Any summation of the form L max is a sum over the three dyadic variables L, L 2, L 3, thus for instance L max H := L,L 2,L 3 :L max H Similarly, any summation of the form N max sum over the three dyadic variables N, N 2, N 3 > 0, thus for instance N max N med N If τ, ξ and q(ξ) are given, we define and similarly, := N,N 2,N 3 >0:N max N med N λ := τ h(ξ), λ j := τ j h j (ξ j ), j =, 2, 3,.. 6

7 where h j (ξ j ) = ±q(ξ j ) (j =, 2, 3). And the quantity λ j measures how close in frequency the j th factor is to a free solution. In this paper, we do not go further on the general framework of Tao s weighted convolution estimates. We focus our attention on the [3; R R]-multiplier norm estimate for the fourth-order Schrödinger equations. During the estimate we need the resonance function h(ξ) = h (ξ ) + h 2 (ξ 2 ) + h 3 (ξ 3 ) = λ λ 2 λ 3, (2.3) where h j (ξ j ) = ±q(ξ j ) (j =, 2, 3). And h(ξ) measures to what extent the spatial frequencies ξ, ξ 2, ξ 3 can resonate with each other. Up to symmetry, there are only two possibilities for the h j : the (+ + +) case h (ξ) = h 2 (ξ) = h 3 (ξ) = q(ξ) = ξ 4 + εξ 2, (2.4) and the (+ + ) case h (ξ) = h 2 (ξ) = q(ξ), h 3 (ξ) = q(ξ). (2.5) The (+ + +) case corresponds to estimates of the form Q (u, ū) = ū 2 : u u 2 Xs,b u Xs,b u 2 Xs2,b 2, while the (+ + ) case and its permutations are similar but treat Q 2 (u, ū) = u 2 and Q 3 (u, ū) = uū instead of ū 2. By dyadic decomposition of the variables ξ j, λ j, as well as the resonance function h(ξ), we may assume that ξ j N j, λ j L j, h(ξ) H. By the translation invariance of the [k; Z]-multiplier norm (see Lemma 3.4 (8) in Tao [20]), we can always restrict our estimate on λ j 2, (2.6) and max(n, N 2, N 3 ) 2. (2.7) We will provide the detail proof of this reduction in the appendix section. One is led to consider X N,N 2,N 3 ;H;L,L 2,L 3 [3;R R], (2.8) 7

8 where X N,N 2,N 3 ;H;L,L 2,L 3 is the multiplier X N,N 2,N 3 ;H;L,L 2,L 3 := χ h(ξ) H From the identities 3 χ ξj N j χ λj L j. (2.9) j= ξ + ξ 2 + ξ 3 = 0, and λ + λ 2 + λ 3 + h = 0, on the support of the multiplier, we see that X N,N 2,N 3 ;H;L,L 2,L 3 vanishes unless N max N med, (2.0) and L max max(h, L med ). (2.) Thus we may implicitly assume (2.0), (2.) in the summations. We now consider (+ + +) case (2.4). Because the resonance function h(ξ) = ξ 4 + εξ 2 + ξ εξ ξ εξ 2 3, and max(n, N 2, N 3 ) 2. We see that we may assume that H N 4 max. (2.2) Since the multiplier in (2.8) vanishes otherwise. Proposition 2.2. Let H, N, N 2, N 3, L, L 2, L 3 > 0 obey (2.6), (2.7), (2.0), (2.) and (2.2). And let the dispersion relations be given by (2.4). Then we have (2.8) L 2 N 2 max {N max N, L med Nmax 2 } 2, (2.3) except in the exceptional case N max N, L max H, in which case (2.8) L 2 N 2 maxl 4 med. (2.4) 8

9 Proof. We will prove this proposition by using the tools developed in [20]. In the high modulation case L max L med H, we have by an elementary estimate employed by Tao (see (37) p.86 in [20]) (2.8) L 2 N 2 L 2 N 2 max( L med Nmax 2 ) 2. For the low modulation case: L max H, by symmetry we may assume that L L 2 L 3, hence L N 4 max. By Corollary 4.2 in Tao s paper [20], we have (2.8) L 2 3 {ξ 2 : ξ 2 ξ 0 2 N ; ξ ξ 2 ξ 0 3 N ; for some τ R and ξ, ξ 0, ξ0 2, ξ0 3 satisfying The identity ξ εξ (ξ ξ 2 ) 4 + ε(ξ ξ 2 ) 2 = τ + O(L 2 )} 2, (2.5) ξ 0 j N j, (j =, 2, 3); ξ ξ 0 N ; ξ 0 + ξ ξ 0 3 N. ξ2 4 + εξ2 2 + (ξ ξ 2 ) 4 + ε(ξ ξ 2 ) 2 = 2 (ξ 2 ξ 2 )2[ (ξ 2ξ 2 ) 2 + 6ξ 2 + 4ε ] + ξ4 8 + εξ2 2 together with (2.5) implies it suffices to show that {ξ 2 : ξ 2 ξ 0 2 N ; (ξ 2 ξ 2 )2[ (ξ 2ξ 2 ) 2 + 6ξ 2 + 4ε ] = 2τ ξ4 4 εξ2 + O(L 2 )} N max {N N max ; L 2 N 2 max (2.6) } (2.7) with the right-hand side replaced by N maxl 2 2 in the exceptional case N max N. We need consider three cases: N N 2 N 3, N N 2 N 3 and N N 3 N 2. (The case N 2 N 3 N then follows by symmetry). If N N 2 N 3, we see from (2.6) that ξ 2 variable is contained in the union of two intervals of length O(N L 2 2 ) at worst, and (2.4) follows. If N N 2 N 3, we must have ξ 2 ξ 2 N, so (2.6) shows that ξ 2 is contained in the union of two intervals of length O(N 3 L 2). But ξ 2 is also contained in an interval of length N 3. So (2.3) follows. If N N 3 N 2, then we must have ξ 2 ξ 2 N, so (2.6) shows that ξ 2 is contained in the union of two intervals of length O(N 3 L 2). But ξ 2 is also contained in an interval of length N 2. The claim (2.3) follows. 9

10 We now consider the (+ + ) case (2.5). In this case, the resonance function h(ξ) : = (ξ 4 + εξ 2 ) + (ξ εξ 2 2) (ξ εξ 2 3) = 4ξ ξ 2 (ξ ξ ξ 2 + ξ ε) so we may assume that Since the multiplier in (2.8) vanishes otherwise. H N N 2 max{n, N 2 } 2. (2.8) Proposition 2.3. Let H, N, N 2, N 3, L, L 2, L 3 > 0 obey (2.6), (2.7), (2.0), (2.) and (2.2). And let the dispersion relations be given by (2.5). Then we have (2.8) L 2 N 2 max {N max N, L med Nmax 2 } 2, (2.9) except in the exceptional cases N N, L L max H or N 2 N, L 2 L max H, in which cases (2.8) L 2 N L 2 med max {N max N, } 2. (2.20) N max N Proof. In the high modulation case L max L med H, we have by an elementary estimate employed by Tao (see (37) p.86 in [20]) (2.8) L 2 N 2 L 2 N 2 max( L med Nmax 2 ) 2. For the low modulation case: L max H, by symmetry we only consider three cases: N N 2 N 3 ; N 2 N 3 N, L L max ; and N 2 N 3 N, L L max H. Case :((++) high-high interactions) N N 2 N 3. By symmetry, it suffices to consider the two cases L L 2, L 3 and L 3 L, L 2. Case (a): L L 2, L 3. By Corollary 4.2 in Tao s paper [20], we have (2.8) L 2 {ξ 2 : ξ 2 ξ 0 2 N 3 ; ξ ξ 2 ξ 0 3 N 3 ; ξ εξ 2 2 (ξ ξ 2 ) 4 ε(ξ ξ 2 ) 2 = τ + O(L med )} 2 (2.2) for some τ R and ξ, ξ 0, ξ0 2, ξ0 3 satisfying ξ 0 j N j, (j =, 2, 3); ξ ξ 0 N 3 ; ξ 0 + ξ ξ 0 3 N 3. 0

11 Combining (2.2) with the identity ξ εξ 2 2 (ξ ξ 2 ) 4 ε(ξ ξ 2 ) 2 = ξ(ξ 2 ξ 2 )(ξ2 + (ξ 2ξ 2 ) 2 + 2ε), (2.22) it suffices to show that {ξ 2 : ξ 2 ξ 0 2 N 3 ;ξ(ξ 2 ξ 2 )(ξ2 + (ξ 2ξ 2 ) 2 + 2ε) = τ + O(L med )} Nmax {N 3 N max ; L med Nmax 2 }. (2.23) We see from the left hand side of (2.23) that ξ 2 variable is contained in the union of two intervals of length O(N 3 maxl med ) at worst. But ξ 2 is also contained in an interval of length N 3. The claim (2.23) follows. Case (b): L 3 L, L 2. By Corollary 4.2 in Tao s paper [20], we have (2.8) L 2 {ξ : ξ ξ 0 N ; ξ 4 +εξ 2 +(ξ ξ ) 4 +ε(ξ ξ ) 2 = τ +O(L med )} 2 (2.24) for some τ R and ξ, ξ 0, ξ0 2, ξ0 3 satisfying ξ 0 j N j, (j =, 2, 3); ξ ξ 0 3 N 3 ; ξ 0 + ξ ξ 0 3 N 3. With the observation that the set of (2.24) is the same as the set of (2.5), then one can follow the argument as before to get (2.3), which is exactly (2.9). Case 2: (High-low interactions) N 2 N 3 N, L L max. In this case, we either have L 3 L, L 2 or L 2 L, L 3. Case 2(a):((++) case) L 3 L, L 2. By Corollary 4.2 in Tao s paper [20], we have (2.8) L 2 {ξ : ξ ξ 0 N ; ξ 4 + εξ 2 + (ξ ξ ) 4 + ε(ξ ξ ) 2 = τ + O(L med )} 2 for some τ R and ξ, ξ 0, ξ0 2, ξ0 3 ξ 0 j N j, (j =, 2, 3); ξ ξ 0 3 N ; ξ 0 + ξ ξ 0 3 N. Then (2.9) follows from the argument as the case of (2.24). Case 2(b):((+ ) case) L 2 L, L 3.

12 By Corollary 4.2 in Tao s paper [20], we have (2.8) L 2 {ξ : ξ ξ 0 N ; ξ 4 + εξ 2 (ξ ξ ) 4 ε(ξ ξ ) 2 = τ + O(L med )} 2 for some τ R and ξ, ξ 0, ξ0 2, ξ0 3 ξj 0 N j, (j =, 2, 3); ξ ξ2 0 N ; ξ 0 + ξ2 0 + ξ3 0 N. Hence we can get (2.9) by following the argument of (2.2). Case 3:((+ ) high-high interactions) N 2 N 3 N, L L max H. By Corollary 4.2 in Tao s paper [20], we have (2.8) L 2 {ξ 2 : ξ 2 ξ 0 2 N ; ξ εξ 2 2 (ξ ξ 2 ) 4 ε(ξ ξ 2 ) 2 = τ + O(L med )} 2 for some τ R and ξ, ξ 0, ξ0 2, ξ0 3 ξ 0 j N j, (j =, 2, 3); ξ ξ 0 N ; ξ 0 + ξ ξ 0 3 N. As before, we will handle this case if we could show that {ξ 2 : ξ 2 ξ 0 2 N ;ξ(ξ 2 ξ 2 )(ξ2 + (ξ 2ξ 2 ) 2 + 2ε) = τ + O(L med )} Nmax L med {N N max ; }. (2.25) N max N We see from the LHS of (2.25) that ξ 2 variable is contained in the union of two intervals of length O(N 2 maxn L med) at worst. But ξ 2 is also contained in an interval of length N. The claim (2.25) follows. 3 Proofs of Theorems.-.4 In this section, we prove Theorem.-.3 by using the fundamental estimate on dyadic blocks in Proposition 2.2 and Proposition Proof of Theorem. Proof. Suppose b = 2 + η. By Plancherel theorem, (.2) is reduced to show that ξ s ξ 2 s ξ 3 s [3;R R] τ q(ξ ) b τ 2 q(ξ 2 ) b τ 3 q(ξ 3 ) b. (3.) 2

13 By dyadic decomposition of the variables ξ j, λ j (j =, 2, 3) and h(ξ), we may assume that ξ j N j, λ j L j (j =, 2, 3) and h(ξ) H. By the translation invariance of the [k; Z]-multiplier norm, we can always restrict our estimate on L j (j =, 2, 3) and max(n, N 2, N 3 ). For seeking the details of this reduction, one can see these in the appendix part. The comparison principle and orthogonality (see Schur s test in [20], p85) imply the multiplier norm estimate (3.) can be reduced to prove N max N med N L,L 2,L 3 N s N 2 s N 3 s L b Lb 2 L b 3 X N,N 2,N 3 ;L max;l,l 2,L 3 [3;R R], (3.2) and N s N 2 s N max N med N L max L med H L med N 3 s L b Lb 2 L b 3 X N,N 2,N 3 ;H;L,L 2,L 3 [3;R R], (3.3) for all N. These will be accomplished by Proposition 2.2 and some tedious summation. Fix N. We first prove (3.3). We may assume (2.2). By (2.3) we reduce to Estimating N max N med N L max L med N 4 N s N 2 s N 3 s L b Lb 2 L b 3 N s N 2 s N 3 s N 2s N s, Lb L b 2L b 3 L b L b med L b max, and then perforg the L summations, we reduce to which is true for s ( 7 4, 0]. N max N med N N 2s 4+η N 2 N s, L 2 N 2. (3.4) Now we show the low modulation case (3.2). We may again assume L max N 4. We first deal with the contribution where (2.4) holds. In this case we have N, N 2, N 3 N, so we reduce to L,L 2,L 3 N s L b Lb med L b max 3 L 2 N 2 L 4 med.

14 But this is easily verified. Since and Now we deal with the cases where (2.3) applies.we reduce by (2.3) to So we can reduce (3.5) to N N L max N 4 N s N 2 s N 3 s L b Lb med L b max L 2 N 2 {NN, L med N 2 } 2. (3.5) {NN, L med N 2 } 2 (NN ) η ( L med N 2 ) 2 η, N N L max N 4 perforg the L summations, we reduce to N s N 2 s N 3 s N 2s N s. N N N 2s N s L b Lb med L b max L 2 N 2 (NN ) η ( L med N 2 ) 2 η. N 2s η N η N s, which is true for s > η. This completes the proof of the estimate (.2). 3.2 Proof of Theorem.2 Proof. By Plancherel, it suffices to show that ξ s ξ 2 s ξ 3 s [3;R R] τ q(ξ ) b τ 2 q(ξ 2 ) b τ 3 + q(ξ 3 ) b. (3.6) Replacing Proposition 2.2 by Proposition 2.3, the similar argument in the proof of Theorem. gives the estimate (.3). Although the different proposition was been used, the restriction of regularity s is still in the same range ( 7 4, 0]. 4

15 3.3 Proof of Theorem.3 Proof. By Plancherel theorem, the estimate (.4) reduces to show that ξ s ξ 2 s ξ 3 s [3;R R] τ q(ξ ) b τ 2 q(ξ 2 ) b τ 3 + q(ξ 3 ) b. (3.7) We follow the steps as (3.) and the worst case is N 2 N, L 2 L max H, and H satisfies (2.8), in which case (3.7) is reduced to N 2s N 2 N L,L 3 N 3 N 2 N 2 s L b (N 3 N 2 ) b L b 3 L 2 N 2 {NN2, L med NN 2 } 2, where we make use of (2.20) in Proposition 2.3. To sum the left hand side, we break down this estimate into two subcases. In the case of L med NN 2 N N 2 N L,L 3 N 3 N 2 by perforg N 2 summation, we reduce to N 2s N 2 s L b (N 3 N 2 ) b L b 3 L,L 3 N 4 N 2s η L 2 L 2 med L b Lb 3 < NN 2, we need to prove L 2 N (L med ) 2 2, NN 2 which is true for s > η. On the other hand, in the case of L med NN 2 to get N 2 N L,L 3 N 3 N 2 N 2s Therefore, it ends the proof of Theorem.3. N 2 s L b (N 3 N 2 ) b L b L 2 N 2 (NN2 ) Examples Demonstrating Sharpness in Theorem.4. NN 2, it is easy The idea is similar to the Kenig, Ponce, and Vega presented in the proof of Theorem.4 in [2,3] and K.Nakanishi, H.Takaoka and Y.Tsutsumi [7]. Recall the notation χ E denotes by the characteristic function of the set E. set Proof of Theorem.4(a): We begin by considering the estimate (.2). Define the R N = {(ξ, τ) : ξ N, τ + ξ 4 + εξ 2 }. 5

16 Choose ˆū = χ RN, ˆ v = χ R N. we can show that for large N (uv) N 3 χ Q N, where Q N = {(ξ, τ) : ξ, τ 2 N 4 N 3 }. Indeed, Q N R N +R N and a translate of R N overlaps R N in a set of size at most N 3. Note that Q N N 3 = N 3. So and Therefore, the estimate (.2) implies that uv Xs,b N 3 (N 4 ) b N 3 2 N 4(b ) 3 2 u Xs,b = v Xs,b N s. N 4( b) 3 2 N 2s. (3.8) Now, taking ˆ w(ξ, τ) = χ TN (ξ, τ), where T N = {(ξ, τ) : ξ + N, τ ξ 4 εξ 2 }. We have that for N large (ˆū ˆ w)(ξ, τ) cχ S (ξ, τ), where S is the rectangle of dimensions cn 3 N 3 centered in the origin with the longest side pointing in the (, 4N 3 ) direction. Thus, (.2) implies that N 3(b ) N 2s+4b. (3.9) Together with (3.8), (3.9) and letting N tend to infinity, it gives s max{ 2( b) 3 4, b + 3 }, (3.0) 2 which completes that when s < 7 4 and any b R, the estimate (.2) fails. Now we consider the endpoint case s = 7 4. By duality and the Plancherel theorem, the estimate (.2) is equivalent to the following estimate: û 3, ˆū ˆū 2 C u 3 X s, b u Xs,b u 2 Xs,b. (3.) 6

17 By (3.0) we know that if the estimate (3.) holds with s = 7 4, we must have b = 2. Therefore we need only to consider the case s = 7 4 and b = 2. Let η be a sufficiently small positive number independent of N. functions û, û 2, û 3 as follows: We define three ˆū = χ RN (τ, ξ), ˆū 2 = χ TN (τ, ξ), û 3 = ( + τ) χ YN (τ, ξ), where R N, T N is defined in the above and Y N = {(τ, ξ) : τ + 4N 3 ξ η, τ ηn 3 }. We have that for large N: {(τ, ξ) : τ + 4N 3 ξ η, τ ηn 3 } supp(û û 2 ). Hence û 3, ˆū ˆū 2 C ηn 3 ( + τ) τ 4N 3 + η 4N 3 τ 4N 3 η 4N 3 dξdτ ηn 3 N 3 ( + τ) dτ N 3 log N. (3.2) On the other hand, since ξ C for (ξ, τ) supp û 3, by simple calculations we get u 3 X 74, 2 u X 7 4, 2 τ 4N 3 + η 4N 3 ( ηn 3 ( + τ) 2 ξ 7 τ 4N 3 η 4N 3 ( N 3 ηn 3 ) 2 ( + τ) 2 dτ N 7 4, u2 X 7 4, 2 ) 2 τ ξ 4 εξ 2 2 dξdτ N 3 2 (log N) 2 (3.3) N 7 4 N 2 N 4. (3.4) If the estimate (3.) holds with s = 7 4 with b = 2, we must have by (3.2)-(3.4) N 3 log N CN 3 2 (log N) 2 N 7 4 N 4 = CN 3 (log N) 2, (3.5) where C is a positive constant independent of N. Therefore, we let N in (3.5) to obtain a contradiction, which completes that when s = 7 4 and b = 2, the estimate (3.) fails. The same analysis shows the necessity of the conditions s > 7 4 for (.3) to hold. Indeed, consider û = χ RN, ˆv = χ R N, ŵ = χ TN, and û = χ RN, û 2 = χ TN, û 3 = χ YN the endpoint case s = 7 4 and b = 2. 7 for

18 Proof of Theorem.4(b): Let ˆū = χ TN, ˆv = χ RN. Thus for N large, we have (ˆū ˆv)(ξ, τ) cχ S (ξ, τ), where S is the rectangle of dimensions cn 3 N 3 centered in the origin with the longest side pointing in the (, 4N 3 ) direction. Finally from (.4) we can get that for N large N 2s c( χ S (ξ, τ)dξdτ) 2 cn 3 2. ξ which completes that when s < 3 4 consider the endpoint case s = 3 4. τ estimate (.4) is equivalent to the following estimate: and any b R, the estimate (.4) fails. Now we By duality and the Planchere theorem that the û 3, û ˆū 2 C u 3 X 34, b u X 3 4,b u 2 X 3 4,b, (3.6) Let û = χ TN (τ, ξ), ˆū 2 = χ RN (τ, ξ) and û 3 = χ BN (τ, ξ), where R N, T N is defined in the above and B N = {(τ, ξ) : τ + 4N 3 ξ 2, ξ }. So we have that for large N: (û ˆū 2 )(τ, ξ), (τ, ξ) A, where A = {(τ, ξ) : τ + 4N 3 ξ 0, ξ 0 }. Since A supp û 3, we get On the other hand, û 3, û û 2 C A. (3.7) u 3 X 34, b [ supp û 3 N 6( b)] 2 N 3( b), (3.8) u X 3 4,b, u 2 X 3 4,b N 3 4. (3.9) If the estimate (3.6) holds, then we must have by (3.7)-(3.8): CN 3( b) (N 3 4 ) 2 = CN 3( 2 b), (3.20) where C is a positive constant independent of N. Since N is an arbitrary large positive number, we conclude by (3.20) that b 2. 8

19 Hence, it remains only to exclude the case b = 2. Let η be a sufficiently small positive number independent of N. We choose û = χ TN (τ, ξ), ˆū 2 = χ RN (τ, ξ) and û 3 = ( + τ) χ YN (τ, ξ), where R N, T N and Y N is defined in the above. Therefore: û 3, û ˆū 2 ηn 3 ( + τ) τ 4N 3 + η 4N 3 τ 4N 3 η 4N 3 dξdτ ηn 3 N 3 ( + τ) dτ N 3 log N. (3.2) On the other hand, since ξ C for (τ, ξ) supp û 3, so we have τ 4N 3 + η 4N 3 ( ηn 3 u 3 X 34 ( + τ) 2 ξ 3, 2 τ 4N 3 η 4N 3 ( N 3 ηn 3 ) 2 ( + τ) 2 dτ u X 3, u 4, 2 2 X 3 4, 2 ) 2 τ ξ 4 εξ 2 2 dξdτ N 3 2 (log N) 2 (3.22) N 3 4. (3.23) Hence, if the estimate (3.6) holds with b = 2, we must have by (3.2)-(3.23): N 3 log N CN 3 2 (log N) 2 (N 3 4 ) 2 = CN 3 (log N) 2, (3.24) where C is a positive constant independent of N. Therefore, we let N in (3.24) to obtain a contradiction, which completes that when s = 3 4 and b = 2, the estimate (3.6) fails. 4 Appendix This appendix is devoted to the proof of the rationality of the assumption (2.6) and (2.7). To this end, we firstly give a fundamental lemma. Lemma 4.. For all λ, λ 2, λ 3 R, we have 2 and similarly for all ξ, ξ 2 R, we have 0 χ λ τ 2(τ)χ λ2 τ 2(τ)χ λ3 +2τ 2(τ) dτ 2, 0 0 χ ξ ξ 2(ξ)χ ξ2 +ξ 2(ξ) dξ 2. 9

20 Now we give the proof for the assumption (2.6) for the estimate (3.), precisely, we have the following proposition 4.. Proposition 4.. ξ s ξ 2 s ξ 3 s [3;R R] λ b λ 2 b λ 3 b ξ s ξ 2 s 3 j= χ λ j 2 [3;R R] ξ 3 s λ b λ 2 b λ 3 b. Proof. Applying Lemma 4. to the right hand side of the above, we have ξ s ξ 2 s ξ 3 s [3;R R] λ b λ 2 b λ 3 b ξ s ξ 2 s ξ 3 s 2 λ b λ 2 b λ 3 b χ λ τ 2(τ)χ λ2 τ 2(τ)χ λ3 +2τ 2(τ) dτ 0 [3;R R] = ξ s ξ 2 s 2 λ τ b λ 2 τ b λ 3 + 2τ b ξ 3 s 0 λ b λ 2 b λ 3 b χ λ τ 2(τ)χ λ2 τ 2(τ)χ λ3 +2τ 2(τ) λ τ b λ 2 τ b λ 3 + 2τ b dτ (4.) [3;R R] Since if τ M, then Therefore (4.) 2 0 ξ s ξ 2 s λ τ M λ. ξ 3 s χ λ τ 2(τ)χ λ2 τ 2(τ)χ λ3 +2τ 2(τ) λ τ b λ 2 τ b λ 3 + 2τ b dτ [3;R R] ξ s ξ 2 s ξ 3 s χ λ 2χ λ2 2χ λ3 2 λ b λ 2 b λ 3 b [3;R R]. (4.2) We use time-translation invariance (see Lemma 3.4 (8) in Tao [20]) of the [k, Z]-multiplier norm in the last inequality. This completes the proof. Finally, we give the proof of rationality of the assumption (2.7) for the estimate (3.), precisely, we have the following Proposition 4.2. Proposition 4.2. ξ s ξ 2 s 3 j= χ ξ j 2 [3;R R] ξ 3 s λ b λ 2 b λ 3 b ξ s ξ 2 s 2 j= χ ξ j 2 [3;R R] ξ 3 s λ b λ 2 b λ 3 b. 20

21 Proof. Applying Lemma 4. to the right hand side of the above, we have Since and ξ s ξ 2 s 3 j= χ ξ j 2 [3;R R] ξ 3 s λ b λ 2 b λ 3 b ξ s ξ 2 s 3 j= χ 0 ξ j 2 ξ 3 s λ b λ 2 b λ 3 b χ ξ ξ 2(ξ)χ ξ2 +ξ 2(ξ) dξ 0 [3;R R] 0 ξ s ξ 2 s τ q(ξ ξ) b τ 2 q(ξ 2 + ξ) b 2 j= χ ξ j 2 0 ξ ξ s ξ 2 + ξ s λ b λ 2 b ξ ξ s ξ 2 + ξ s χ ξ ξ 2χ ξ2 +ξ 2 ξ 3 s τ q(ξ ξ) b τ 2 q(ξ 2 + ξ) b λ 3 b dξ [3;R R]. (4.3) ξ j ξ ξ j, ξ 0, τ j q(ξ j ξ) τ j q(ξ j ) = λ j, ξ 0, ξ j 2, where q(ξ) = ξ 4 + εξ 2 and j =, 2. So (4.3) 0 0 ξ ξ s ξ 2 + ξ s ξ 3 s χ ξ ξ 2χ ξ2 +ξ 2 τ q(ξ ξ) b τ 2 q(ξ 2 + ξ) b λ 3 b dξ [3;R R] ξ s ξ 2 s 2 j= χ ξ j 2 [3;R R] ξ 3 s λ b λ 2 b λ 3 b. We make use of the space-translation invariance (see Lemma 3.4 (8) in Tao [20])of the [k, Z]-multiplier norm in the last inequality. This completes the proof. References [] M. Beals, Self-Spreading and strength of Singularities for solutions to semilinear wave equations, Annals of Math 8 (983), [2] M. Ben-Artzi, H. Koch and J.C. Saut. Dispersion estimates for fourth order Schrödinger equations, C.R.A.S., 330, Serie, (2000), [3] I. Bejenaru, T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233, (2006), 228 C

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