Global well-posedness for a L^2-critical nonlinear higher-order Schrödinger equation
aa r X i v : . [ m a t h . A P ] O c t Global well-posedness for a L -criticalnonlinear higher-order Schr¨odinger equation Van Duong Dinh
Abstract
We prove the global well-posedness for a L -critical defocusing cubic higher-order Schr¨odingerequation, namely i∂ t u + Λ k u = −| u | u, where Λ = √− ∆ and k ≥ , k ∈ Z in R k with initial data u ∈ H γ , γ > γ ( k ) := k (4 k − k − . Let k ≥ , k ∈ Z . We consider the Cauchy problem for the defocusing cubic nonlinear higher-order Schr¨odinger equation posed on R k , namely (cid:26) i∂ t u ( t, x ) + Λ k u ( t, x ) = −| u ( t, x ) | u ( t, x ) , t ≥ , x ∈ R k ,u (0 , x ) = u ( x ) ∈ H γ ( R k ) , (NLS k )where Λ = √− ∆ is the Fourier multiplier by | ξ | . When k = 2, (NLS k ) corresponds to the well-known Schr¨odinger equation (see e.g. [2], [3], [4], [19], [24], [5], [7], [8], [9], [12], [13] and referencestherein). When k = 4, it is the fourth-order Schr¨odinger equation take into consideration therole of small fourth-order dispersion in the propagation of intense laser beams in a bulk mediumwith Kerr nonlinearity (see e.g. [17], [18], [21], [22]).It is worth noticing that the (NLS k ) is L -critical in the sense that if u is a solution to (NLS k )on ( − T, T ) with initial data u , then u λ ( t, x ) = λ − k/ u ( λ − k t, λ − x ) , (1.1)is also a solution of (NLS k ) on ( − λ k T, λ k T ) with initial data u λ (0) and k u λ (0) k L ( R k ) = k u k L ( R k ) . It is known (see e.g. [4], [10], [11]) that (NLS k ) is locally well-posed in H γ ( R k ) when γ > k u ( t ) k L ( R k ) = k u k L ( R k ) , (1.2)and H k/ solutions have the conserved energy,i.e. E ( u ( t )) := Z R k | Λ k/ u ( t, x ) | + 14 | u ( t, x ) | dx = E ( u ) . (1.3)The conservations of mass and energy combine with the persistence of regularity (see e.g. [11])immediately yield the global well-posedness for (NLS k ) with initial data in H γ ( R k ) when γ ≥ k/
2. Note also (see [10]) that one has the local well-posedness for (NLS k ) when initial data u ∈ L ( R k ) but the time of existence depends not only on the size but also on the profile ofthe initial data. In addition, if k u k L ( R k ) is small enough, then (NLS k ) is global well-posed andscattering in L ( R k ). It is conjectured that (NLS k ) is in fact globally well-posed for initial datain H γ ( R k ) with γ ≥
0. This paper concerns with the global well-posedness of (NLS k ) in H γ ( R k )1 Van Duong Dinh when 0 < γ < k/
2. Let us recall known results for the defocusing cubic Schr¨odinger equationin R ,i.e. (NLS ). The first attempt to this problem due to Bourgain in [2] where he used a“Fourier truncation” approach to prove the global existence for γ > /
5. It was then improvedfor γ > / E ( u ), which is not available when γ < E ( I N u ) with N ≫ I N is a smoothing operator whichbehaves like the identity for low frequencies | ξ | ≤ N and like a fractional integral operator oforder 1 − γ for high frequencies | ξ | ≥ N . Since I N u is not a solution to (NLS ), we may expectan energy increment. The key idea is to show that on the time interval of local existence, theincrement of the modified energy E ( I N u ) decays with respect to a large parameter N . Thisallows to control E ( I N u ) on time interval where the local solution exists, and we can iterate thisestimate to obtain a global in time control of the solution by means of the bootstrap argument.Fang-Grillakis then upgraded this result to γ ≥ / γ > / γ > /
3. Afterwards, Dodson established in [12]the global existence for (NLS ) when γ > /
4. The proof combines the almost conservation lawand an improved interaction Morawetz estimate. Recently, Dodson in [13] proved the globalwell-posedness and scattering for (NLS ) for initial data u ∈ L ( R ) using the bilinear estimateand a frequency localized interaction Morawetz estimate. We next recall some known resultsabout the global well-posedness below energy space for the fourth-order Schr¨odinger equation.In [16], the author considered the more general fourth-order Schr¨odinger equation, namely i∂ t u + λ ∆ u + µ ∆ u + ν | u | m u = 0 , and established the global well-posedness in H γ ( R n ) for γ > mn − √ (4 m − mn +7) +164 m underthe assumption 4 < mn < m + 2 and of course some conditions on λ, µ and ν . For the mass-critical fourth-order Schr¨odinger equation in high dimensions n ≥
5, Pausader-Shao proved in[23] that the L -solution is global and scattering under some conditions. Recently, Miao-Wu-Zhang in [20] showed the global existence and scattering below energy space for the defocusingcubic fourth-order Schr¨odinger equation in R n with n = 5 , ,
7. To our knowledge, there is noresult concerning the global existence (possibly scattering) for (NLS ).The purpose of this paper is to prove the global existence of (NLS k ) with k ≥ , k ∈ Z belowthe energy space H k/ ( R k ). Theorem 1.1.
Let k ≥ , k ∈ Z . The initial value problem (NLS k ) is globally well-posed in H γ ( R k ) for any k/ > γ > γ ( k ) := k (4 k − k − . Moreover, the solution satisfies k u ( T ) k H γ ( R k ) ≤ C (1 + T ) (4 k − k − γ )2((14 k − γ − k (4 k − + , for | T | → ∞ , where the constant C depends only on k u k H γ ( R k ) . The proof of this theorem is based on the I -method similar to [5] (see also [16]). We shallconsider a modified I -operator and show a suitable “almost conservation law” for the higher-order Schr¨odinger equation. The global well-posedness then follows by a usual scheme as in [5].This paper is organized as follows. In Section 2, we recall some linear and bilinear estimatesfor the higher-order Schr¨odinger equation, and also a modified I -operator together with its basicproperties. We will show in Section 3 an almost conservation law and a modified local well-posedresult. The proof of Theorem 1 . A . B to denote an estimate of the form A ≤ CB for some absolute constant C . The notation A ∼ B means that A . B and B . A . We write A ≪ B to denote A ≤ cB for some smallconstant c >
0. We also use the Japanese bracket h a i := p | a | ∼ | a | and a ± := a ± ǫ with some universal constant 0 < ǫ ≪ lobal well-posedness L -critical higher-order Schr¨odinger Let ϕ be a smooth, real-valued, radial function in R k such that ϕ ( ξ ) = 1 for | ξ | ≤ ϕ ( ξ ) = 0for | ξ | ≥
2. Let M = 2 k , k ∈ Z . We denote the Littlewood-Paley operators by \ P ≤ M f ( ξ ) := ϕ ( M − ξ ) ˆ f ( ξ ) , \ P >M f ( ξ ) := (1 − ϕ ( M − ξ )) ˆ f ( ξ ) , [ P M f ( ξ ) := ( ϕ ( M − ξ ) − ϕ (2 M − ξ )) ˆ f ( ξ ) , where ˆ · is the spatial Fourier transform. We similarly define P Lemma 2.1. Let γ ≥ and ≤ p ≤ q ≤ ∞ . k P ≥ M f k L px . M − γ k Λ γ P ≥ M f k L px , k P ≤ M Λ γ f k L px . M γ k P ≤ M f k L px , k P M Λ ± γ f k L px ∼ M ± γ k P M f k L px , k P ≤ M f k L qx . M k/p − k/q k P ≤ M f k L px , k P M f k L qx . M k/p − k/q k P M f k L px . Let γ, b ∈ R . The Bourgain space X γ,bτ = | ξ | k is the closure of space-time Schwartz space S t,x underthe norm k u k X γ,bτ = | ξ | k := k h ξ i γ (cid:10) τ − | ξ | k (cid:11) b ˜ u k L τ L ξ , where ˜ · is the space-time Fourier transform,i.e.˜ u ( τ, ξ ) := Z Z e − i ( tτ + xξ ) u ( t, x ) dtdx. We shall use X γ,b instead of X γ,bτ = | ξ | k when there is no confusion. We recall a following specialproperty of X γ,b space (see e.g. [24, Lemma 2.9]). Lemma 2.2. Let γ, γ , γ ∈ R and Y be a Banach space of functions on R × R k . If k e itτ e it Λ k f k Y . k f k H γx , for all f ∈ H γx and all τ ∈ R , then k u k Y . k u k X γ, / , for all u ∈ S t,x . Moreover, if k [ e itτ e it Λ k f ][ e itζ e it Λ k f ] k Y . k f k H γ k f k H γ x , for all f ∈ H γ x , f ∈ H γ x and all τ, ζ ∈ R , then k u u k Y . k u k X γ , / k u k X γ , / , for all u , u ∈ S t,x . Van Duong Dinh Throughout this paper, a pair ( p, q ) is called admissible in R k if( p, q ) ∈ [2 , ∞ ] , ( q, p ) = (2 , ∞ ) , p + 1 q = 12 . (2.4)We recall the following Strichartz estimate (see e.g. [10], [21]). Proposition 2.3. Let k ≥ , k ∈ Z . Suppose that u is a solution to i∂ t u ( t, x ) + Λ k u ( t, x ) = F ( t, x ) , u (0 , x ) = u ( x ) , ( t, x ) ∈ R × R k . Then for all ( p, q ) and ( a, b ) admissible pairs, k u k L pt L qx . k u k L x + k F k L a ′ t L b ′ x . Here ( a, a ′ ) and ( b, b ′ ) are H¨older exponents. A direct consequence of Lemma 2 . . X γ,b space. Corollary 2.4. Let ( p, q ) be an admissible pair. Then k u k L pt L qx . k u k X , / , (2.5) for all u ∈ S t,x . We also have the following bilinear estimate in R k . Proposition 2.5. Let k ≥ , k ∈ Z and M , M ∈ Z be such that M ≤ M . Then k [ e it Λ k P M u ][ e it Λ k P M v ] k L t L x . ( M /M ) ( k − / k u k L x k v k L x . Proof. We refer the reader to [2] for the standard case k = 2. The proof for k > M ∼ M , the result follows easily from the Strichartz estimate, k [ e it Λ k P M u ][ e it Λ k P M v ] k L t L x ≤ k e it Λ k P M u k L t L x k e it Λ k P M v k L t L x . k u k L x k v k L x . Note that (4 , 4) is an admissible pair. Let us consider the case M ≪ M . By duality, it sufficesto prove (cid:12)(cid:12)(cid:12) Z Z R k × R k G ( −| ξ | k − | η | k , ξ + η ) \ P M u ( ξ ) \ P M v ( η ) dξdη (cid:12)(cid:12)(cid:12) . ( M /M ) ( k − / k G k L τ L ξ k ˆ u k L ξ k ˆ v k L ξ . (2.6)By renaming the components, we can assume that | ξ | ∼ | ξ | ∼ M and | η | ∼ | η | ∼ M , where ξ = ( ξ , ξ ) , η = ( η , η ) with ξ, η ∈ R k − . We make a change of variables τ = −| ξ | k −| η | k , ϑ = ξ + η and dτ dϑ = Jdξ dη . An easy computation shows that J = | k ( | η | k − η − | ξ | k − ξ ) | ∼ | η | k − ∼ M k − . The Cauchy-Schwarz inequality with the fact that | ξ | . M then yieldsLHS(2 . 6) = (cid:12)(cid:12)(cid:12) Z Z Z R × R k × R k − G ( τ, ϑ ) \ P M u ( ξ ) \ P M v ( η ) J − dτ dϑdξ (cid:12)(cid:12)(cid:12) ≤ k G k L τ L ξ Z R k − (cid:16) Z Z R × R k | \ P M u ( ξ ) | | \ P M v ( η ) | J − dτ dϑ (cid:17) / dξ ≤ k G k L τ L ξ M ( k − / (cid:16) Z Z Z R × R k × R k − | \ P M u ( ξ ) | | \ P M v ( η ) | J − dτ dϑdξ (cid:17) / ≤ k G k L τ L ξ M ( k − / (cid:16) Z Z Z R × R k × R k − | \ P M u ( ξ ) | | \ P M v ( η ) | J − dξdη (cid:17) / . k G k L τ L ξ ( M /M ) ( k − / k \ P M u k L ξ k \ P M v k L ξ . This proves (2 . lobal well-posedness L -critical higher-order Schr¨odinger . . Corollary 2.6. Let k ≥ , k ∈ Z and u , u ∈ X , / be supported on spatial frequencies | ξ | ∼ M , M respectively. Then for M ≤ M , k u u k L t L x . ( M /M ) ( k − / k u k X , / k u k X , / . (2.7) A similar estimate holds for u u or u u . I -operator For 0 < γ < k/ N ≫ 1, we define the Fourier multiplier I N by d I N f ( ξ ) := m N ( ξ ) ˆ f ( ξ ) , (2.8)where m is a smooth, radially symmetric, non-increasing function such that m N ( ξ ) := (cid:26) | ξ | ≤ N, ( N − | ξ | ) γ − if | ξ | ≥ N. (2.9)For simplicity, we shall drop the N from the notation and write I and m instead of I N and m N .The operator I is the identity on low frequencies | ξ | ≤ N and behaves like a fractional integraloperator of order k/ − γ on high frequencies | ξ | ≥ N . We recall some basic properties of the I -operator in the following lemma. Lemma 2.7. Let q ∈ (1 , ∞ ) and γ ∈ (0 , k/ . Then k If k L qx . k f k L qx , (2.10) k f k H γx . k If k H k/ x . N k/ − γ k f k H γx . (2.11) Proof. The estimate (2 . 10) follows from the fact that m satisfies the H¨ormander multipliercondition. For (2 . k f k H γx . Z | ξ |≤ N h ξ i γ | c If ( ξ ) | dξ + Z | ξ |≥ N h ξ i γ ( N − | ξ | ) k/ − γ ) | c If ( ξ ) | dξ . Z | ξ |≤ N h ξ i k | c If ( ξ ) | dξ + Z | ξ |≥ N h ξ i k | c If ( ξ ) | dξ . k If k H k/ x . This gives the first estimate in (2 . k If k H k/ x . Z | ξ |≤ N h ξ i k | ˆ f ( ξ ) | dξ + Z | ξ |≥ N h ξ i k ( N − | ξ | ) γ − k/ | ˆ f ( ξ ) | dξ . Z | ξ |≤ N h ξ i k/ − γ ) h ξ i γ | ˆ f ( ξ ) | dξ + Z | ξ |≥ N N k/ − γ ) h ξ i γ | ˆ f ( ξ ) | dξ . N k/ − γ ) (cid:16) Z | ξ |≤ N h ξ i γ | ˆ f ( ξ ) | dξ + Z | ξ |≥ N h ξ i γ | ˆ f ( ξ ) | dξ (cid:17) . N k/ − γ ) k f k H γx . The proof is complete. As mentioned in the introduction, the equation (NLS k ) is locally well-posed in H γ for any γ > 0. Moreover, the time of existence depends only on the H γx -norm of the initial data. Thus,the global well-posedness will follows from a global L ∞ t H γx bound of the solution by the usualiterative argument. For H γ solution with γ ≥ k/ 2, one can obtain easily the L ∞ t H γx bound ofsolution using the persistence of regularity and the conserved quantities of mass and energy. But Van Duong Dinh it is not the case for H γ solution with γ < k/ . 11) that the H γx -norm of the solution u can be controlled by the H k/ x -normof Iu . It leads to consider the following modified energy functional E ( Iu ( t )) := 12 k Iu ( t ) k H k/ x + 14 k Iu ( t ) k L x . (3.12)Since Iu is not a solution to (NLS k ), we can expect an energy increment. We have the following“almost conservation law”. Proposition 3.1. Let k ≥ , k ∈ Z . Given k/ > γ > γ ( k ) := k (4 k − k − , N ≫ , and initialdata u ∈ C ∞ ( R k ) with E ( Iu ) ≤ , then there exists a δ = δ ( k u k L x ) > so that the solution u ∈ C ([0 , δ ] , H γ ( R k )) of (NLS k ) satisfies E ( Iu ( t )) = E ( Iu ) + O ( N − γ ( k )+ ) , (3.13) where γ ( k ) := k (6 k − k − for all t ∈ [0 , δ ] . Remark 3.2. This proposition tells us that the modified energy E ( Iu ( t )) decays with respectto the parameter N . We will see in Section 4 that if we can replace the increment N − γ ( k )+ inthe right hand side of (3 . 13) with N − γ ( k )+ for some γ ( k ) > γ ( k ), then the global existencecan be improved for all γ > k k + γ ( k )) . In particular, if γ ( k ) = ∞ , then E ( Iu ( t )) is conserved,and the global well-posedness holds for all γ > . 1, we recall the following interpolation result (see [6, Lemma12.1]). Let η be a smooth, radial, decreasing function which equals 1 for | ξ | ≤ | ξ | − for | ξ | ≥ 2. For N ≥ α ∈ R , we define the spatial Fourier multiplier J αN by d J αN f ( ξ ) := ( η ( N − ξ )) α ˆ f ( ξ ) . (3.14)The operator J αN is a smoothing operator of order α , and it is the identity on the low frequencies | ξ | ≤ N . Lemma 3.3 (Interpolation [6]) . Let α > and n ≥ . Suppose that Z, X , ..., X n are translationinvariant Banach spaces and T is a translation invariant n -linear operator such that k J α T ( u , ..., u n ) k Z . n Y i =1 k J α u i k X i , for all u , ..., u n and all ≤ α ≤ α . Then one has k J αN T ( u , ..., u n ) k Z . n Y i =1 k J αN u i k X i , for all u , ..., u n , all ≤ α ≤ α , and N ≥ , with the implicit constant independent of N . Using this interpolation lemma, we are able to prove the following modified version of theusual local well-posed result. Proposition 3.4. Let γ ∈ ( γ ( k ) , k/ and u ∈ H γ ( R k ) be such that E ( Iu ) ≤ . Then thereis a constant δ = δ ( k u k L x ) so that the solution u to (NLS k ) satisfies k Iu k X k/ , / δ . . (3.15) Here X γ,bδ is the space of restrictions of elements of X γ,b endowed with the norm k u k X γ,bδ := inf {k w k X γ,b | w | [0 ,δ ] × R k = u } . (3.16) see Theorem 1.1 for the definition of γ ( k ). lobal well-posedness L -critical higher-order Schr¨odinger Proof. We recall the following estimates involving the X γ,b spaces which are proved in theAppendix. Let γ ∈ R and ψ ∈ C ∞ ( R ) be such that ψ ( t ) = 1 for t ∈ [ − , k ψ ( t ) e it Λ k u k X γ,b . k u k H γx , (3.17) (cid:13)(cid:13)(cid:13) ψ δ ( t ) Z t e i ( t − s )Λ k F ( s ) ds (cid:13)(cid:13)(cid:13) X γ,b . δ − b − b ′ k F k X γ, − b ′ , (3.18)where ψ δ ( t ) := ψ ( δ − t ) provided 0 < δ ≤ < b ′ < / < b, b + b ′ < . (3.19)Note that the implicit constants are independent of δ . This implies for 0 < δ ≤ b, b ′ as in(3 . 19) that k e it Λ k u k X γ,bδ . k u k H γx , (3.20) (cid:13)(cid:13)(cid:13) Z t e i ( t − s )Λ k F ( s ) ds (cid:13)(cid:13)(cid:13) X γ,bδ . δ − b − b ′ k F k X γ, − b ′ δ . (3.21)By the Duhamel principle, we have k Iu k X k/ ,bδ = (cid:13)(cid:13)(cid:13) e it Λ k Iu + Z t e it Λ k I ( | u | u )( s ) ds (cid:13)(cid:13)(cid:13) X k/ ,bδ . k Iu k H k/ x + δ − b − b ′ k I ( | u | u ) k X k/ , − b ′ δ . By the definition of restriction norm (3 . k Iu k X k/ ,bδ . k Iu k H k/ x + δ − b − b ′ k I ( | w | w ) k X k/ , − b ′ , where w agrees with u on [0 , δ ] × R k and k Iu k X k/ ,bδ ∼ k Iw k X k/ ,b . Let us assume for the moment that k I ( | w | w ) k X k/ , − b ′ . k Iw k X k/ ,b . (3.22)This implies that k Iu k X k/ ,bδ . k Iu k H k/ x + δ − b − b ′ k Iu k X k/ ,bδ . Note that k Iu k H k/ x ∼ k Iu k ˙ H k/ x + k Iu k L x ≤ k u k L x . As k Iu k X k/ ,bδ is continuous in the δ variable, the bootstrap argument (see e.g. [24, Section 1.3])yields k Iu k X k/ ,bδ . . This proves (3 . . . 3. Note that the I -operator defined in (2 . 8) is equal to J αN defined in (3 . 14) with α = k/ − γ . Thus, by Lemma 3 . 3, (3 . 22) is proved once there is α > k J α ( | w | w ) k X k/ , − b ′ . k J α w k X k/ ,b , for all 0 ≤ α ≤ α . Splitting w to low and high frequency parts | ξ | . | ξ | ≫ J α , it suffices to show k| w | w k X γ, − b ′ . k w k X γ,b , (3.23) Van Duong Dinh for all γ ∈ [ γ ( k ) , k/ . 23) follows from (cid:12)(cid:12)(cid:12) Z Z R × R k ( h Λ i γ w ) w w w dtdx (cid:12)(cid:12)(cid:12) . k w k X γ,b k w k X γ,b k w k X γ,b k w k X ,b ′ . (3.24)Note that the last term should be precise as k w k X ,b ′ τ = −| ξ | k but it does not effect our estimate.Using H¨older’s inequality, we can bound the left hand side of (3 . 24) asLHS(3 . ≤ k h Λ i γ w k L t L x k w k L t L x k w k L t L x k w k L t L x . Since (4 , 4) is an admissible pair, Corollary 2 . k h Λ i γ w k L t L x . k w k X γ,b , k w k L t L x . k w k X ,b ≤ k w k X γ,b . Similarly, Sobolev embedding and Corollary 2 . k w k L t L x . k h Λ i k/ w k L t L x . k w k X k/ ,b ≤ k w k X γ,b . The last estimate comes from the fact that γ > γ ( k ) > k/ 6. Finally, we interpolate between k w k L t L x = k w k X , and k w k L t L x . k w k X , / to get k w k L t L x . k w k X ,b ′ . Combing these estimates, we have (3 . . Proof of Proposition . . By the assumption E ( Iu ) ≤ 1, Proposition 3 . δ = δ ( k u k L x ) such that the solution u to (NLS k ) satisfies (3 . ddt E ( u ( t )) = Re Z R k ∂ t u ( t, x )( | u ( t, x ) | u ( t, x ) + Λ k u ( t, x )) dx = Re Z R k ∂ t u ( t, x )( | u ( t, x ) | u ( t, x ) + Λ k u ( t, x ) + i∂ t u ( t, x )) dx = 0 . Similarly, we have ddt E ( Iu ( t )) = Re Z R k I∂ t u ( t, x )( | Iu ( t, x ) | Iu ( t, x ) + Λ k Iu ( t, x ) + i∂ t Iu ( t, x )) dx = Re Z R k I∂ t u ( t, x )( | Iu ( t, x ) | Iu ( t, x ) − I ( | u ( t, x ) | u ( t, x ))) dx. Here the second line follows by applying I to both sides of (NLS k ). Integrating in time andapplying the Parseval formula, we obtain E ( Iu ( t )) − E ( Iu ) = Re Z δ Z P j =1 ξ j =0 (cid:16) − m ( ξ + ξ + ξ ) m ( ξ ) m ( ξ ) m ( ξ ) (cid:17) d I∂ t u ( ξ ) c Iu ( ξ ) c Iu ( ξ ) c Iu ( ξ ) dt. Here R P j =1 ξ j =0 denotes the integration with respect to the hyperplane’s measure δ ( ξ + ... + ξ ) dξ ...dξ . Using that iI∂ t u = − Λ k Iu − I ( | u | u ), we have | E ( Iu ( t )) − E ( Iu ) | ≤ Term + Term , where Term = (cid:12)(cid:12)(cid:12) Z δ Z P j =1 ξ j =0 µ ( ξ , ξ , ξ ) [ Λ k Iu ( ξ ) c Iu ( ξ ) c Iu ( ξ ) c Iu ( ξ ) dt (cid:12)(cid:12)(cid:12) , lobal well-posedness L -critical higher-order Schr¨odinger = (cid:12)(cid:12)(cid:12) Z δ Z P j =1 ξ j =0 µ ( ξ , ξ , ξ ) \ I ( | u | u )( ξ ) c Iu ( ξ ) c Iu ( ξ ) c Iu ( ξ ) dt (cid:12)(cid:12)(cid:12) , with µ ( ξ , ξ , ξ ) := 1 − m ( ξ + ξ + ξ ) m ( ξ ) m ( ξ ) m ( ξ ) . Our purpose is to prove Term + Term . N − γ ( k )+ . Let us consider the first term (Term ). To do so, we decompose u = P M ≥ P M u =: P M ≥ u M with the convention P u := P ≤ u and write Term as a sum over all dyadic pieces.By the symmetry of µ in ξ , ξ , ξ and the fact that the bilinear estimate (2 . 7) allows complexconjugations on either factors, we may assume that M ≥ M ≥ M . Thus,Term . X M ,M ,M ,M ≥ M ≥ M ≥ M A ( M , M , M , M ) , where A ( M , M , M , M ) := (cid:12)(cid:12)(cid:12) Z δ Z P j =1 ξ j =0 µ ( ξ , ξ , ξ ) \ Λ k Iu M ( ξ ) [ Iu M ( ξ ) [ Iu M ( ξ ) [ Iu M ( ξ ) dt (cid:12)(cid:12)(cid:12) . For simplifying the notation, we will drop the dependence of M , M , M , M and write A insteadof A ( M , M , M , M ). In order to have Term . N − γ ( k )+ , it suffices to prove A . N − γ ( k )+ M − . (3.25)To show (3 . N with M j . It is worth to notice that M . M due to the fact that P j =1 ξ j = 0. Case 1. N ≫ M . In this case, we have | ξ | , | ξ | , | ξ | ≪ N and | ξ + ξ + ξ | ≤ N , hence m ( ξ + ξ + ξ ) = m ( ξ ) = m ( ξ ) = m ( ξ ) = 1 and µ ( ξ , ξ , ξ ) = 0 . Thus (3 . 25) holds trivially. Case 2. M & N ≫ M ≥ M . Since P j =1 ξ j = 0, we get M ∼ M . We also have from themean value theorem that | µ ( ξ , ξ , ξ ) | = (cid:12)(cid:12)(cid:12) − m ( ξ + ξ + ξ ) m ( ξ ) (cid:12)(cid:12)(cid:12) . |∇ m ( ξ ) · ( ξ + ξ ) | m ( ξ ) . M M . The pointwise bound, H¨older’s inequality, Plancherel theorem and bilinear estimate (2 . 7) yield A . M M k Λ k Iu M Iu M k L t L x k Iu M Iu M k L t L x . M M (cid:16) M M (cid:17) ( k − / (cid:16) M M (cid:17) ( k − / M k Y j =1 k Iu M j k X , / . M M (cid:16) M M (cid:17) ( k − / (cid:16) M M (cid:17) ( k − / M k/ M k/ h M i k/ h M i k/ Y j =1 k Iu M j k X k/ , / = (cid:16) M N (cid:17) / (cid:16) M M (cid:17) / (cid:16) NM (cid:17) k − N − ( k − / M − Y j =1 k Iu M j k X k/ , / . N − ( k − / M − Y j =1 k Iu M j k X k/ , / . (3.26)0 Van Duong Dinh Using (3 . 15) and the fact that γ ( k ) < k − / k ≥ , k ∈ Z , we have (3 . Case 3. M ≥ M & N . In this case, we simply bound | µ ( ξ , ξ , ξ ) | . m ( ξ ) m ( ξ ) m ( ξ ) m ( ξ ) . Here we use that m ( ξ ) & m ( ξ ) and m ( ξ ) ≤ m ( ξ ) ≤ M . M and M ≥ M . Subcase 3a. M ≫ M & N . We see that M ∼ M since P j =1 ξ j = 0. The pointwise bound,H¨older’s inequality, Plancherel theorem and bilinear estimate (2 . 7) again give A . m ( M ) m ( M ) m ( M ) m ( M ) k Λ k Iu M Iu M k L t L x k Iu M Iu M k L t L x . m ( M ) m ( M ) m ( M ) m ( M ) (cid:16) M M (cid:17) ( k − / (cid:16) M M (cid:17) ( k − / M k/ M k/ M k/ h M i k/ Y j =1 k Iu M j k X k/ , / . Thanks to (3 . m ( M ) m ( M ) m ( M ) m ( M ) (cid:16) M M (cid:17) ( k − / (cid:16) M M (cid:17) ( k − / M k/ M k/ M k/ h M i k/ . N − γ ( k )+ M − . (3.27)Remark that the function m ( λ ) λ α is increasing, and m ( λ ) h λ i α is bounded below for any α + γ − k/ > m ( λ ) λ α ) ′ = (cid:26) αλ α − if 1 ≤ λ ≤ N,N k/ − γ ( α + γ − k/ λ α + γ − k/ − if λ ≥ N. We shall shortly choose an appropriate value of α , says α ( k ), so that m ( M ) h M i α ( k ) & , m ( M ) M α ( k )3 & m ( N ) N α ( k ) = N α ( k ) . (3.28)Using that m ( M ) ∼ m ( M ), we haveLHS(3 . . M α ( k ) − / h M i α ( k ) − / M / m ( M ) M α ( k )3 m ( M ) h M i α ( k ) M k − / . N α ( k ) M k − α ( k )2 (cid:16) M M (cid:17) α ( k ) − / (cid:16) h M i M (cid:17) α ( k ) − / (cid:16) M M (cid:17) / . N − ( k − α ( k ))+ M − . Therefore, if we choose α ( k ) so that γ ( k ) = k − α ( k ) or α ( k ) = k − γ ( k ) = k (2 k − k − , then weget (3 . α ( k ) + γ ( k ) − k/ ≥ k ≥ , k ∈ Z , hence (3 . 28) holds. Subcase 3b. M ∼ M & N . In this case, we see that M . M . Arguing as in Subcase 3a,we obtain A . m ( M ) m ( M ) m ( M ) m ( M ) k Λ k Iu M Iu M k L t L x k Iu M Iu M k L t L x . m ( M ) m ( M ) m ( M ) m ( M ) (cid:16) M M (cid:17) ( k − / (cid:16) M M (cid:17) ( k − / h M i k/ M k/ M k/ h M i k/ Y j =1 k Iu M j k X k/ , / . As in Subcase 3a, our aim is to prove m ( M ) m ( M ) m ( M ) m ( M ) (cid:16) M M (cid:17) ( k − / (cid:16) M M (cid:17) ( k − / h M i k/ M k/ M k/ h M i k/ . N − γ ( k )+ M − . (3.29) lobal well-posedness L -critical higher-order Schr¨odinger . 28) to getLHS(3 . . m ( M ) m ( M ) m ( M ) m ( M ) h M i / M k − / . m ( M ) M α ( k )2 h M i α ( k ) − / m ( M ) M α ( k )2 m ( M ) M α ( k )3 m ( M ) h M i α ( k ) M k − α ( k ) − / . N α ( k ) (cid:16) M M (cid:17) α ( k ) (cid:16) h M i M (cid:17) α ( k ) − / M k − α ( k )3 . N − ( k − α ( k ))+ M − . Choosing α ( k ) as in Subcase 3a, we get (3 . ). We again decompose u in dyadic frequencies, u = P M ≥ u M . By the symmetry, we can assume that M ≥ M ≥ M . We can assume furtherthat M & N since µ ( ξ , ξ , ξ ) vanishes otherwise. Thus,Term . X M ,M ,M ,M ≥ M ≥ M ≥ M B ( M , M , M , M ) , where B ( M , M , M , M ) := (cid:12)(cid:12)(cid:12) Z δ Z P j =1 ξ j =0 µ ( ξ , ξ , ξ ) \ P M I ( | u | u )( ξ ) [ Iu M ( ξ ) [ Iu M ( ξ ) [ Iu M ( ξ ) dt (cid:12)(cid:12)(cid:12) . As for the Term , we will use the notation B instead of B ( M , M , M , M ). Using the trivialbound | µ ( ξ , ξ , ξ ) | . m ( M ) m ( M ) m ( M ) m ( M ) , H¨older’s inequality and Plancherel theorem, we bound B . m ( M ) m ( M ) m ( M ) m ( M ) k P M I ( | u | u ) k L t L x k Iu M k L t L x k Iu M k L t L x k Iu M k L ∞ t L ∞ x . Lemma 3.5. We have k P M I ( | u | u ) k L t L x . h M i k/ k Iu k X k/ , / , (3.30) k Iu M j k L t L x . h M j i k/ k Iu M j k X k/ , / , j = 2 , , (3.31) k Iu M k L ∞ t L ∞ x . k Iu M k X k/ , / . (3.32) Proof. The estimate (3 . 30) is in turn equivalent to k h Λ i k/ P M I ( | u | u ) k L t L x . k Iu k X k/ , / . Since h Λ i k/ I obeys a Leibniz rule, it suffices to prove k P M (( h Λ i k/ Iu ) u u ) k L t L x . Y j =1 k Iu j k X k/ , / . (3.33)The Littlewood-Paley theorem and H¨older’s inequality implyLHS(3 . . k h Λ i k/ Iu k L t L x k u k L t L x k u k L t L x . Van Duong Dinh We have from Strichartz estimate (2 . 5) that k h Λ i k/ Iu k L t L x . k h Λ i k/ Iu k X , / = k Iu k X k/ , / . Combining Sobolev embedding and Strichartz estimate (2 . 5) yield k u k L t L x . k h Λ i k/ u k L t L / x . k h Λ i k/ u k X , / . k Iu k X k/ , / , where the last estimate follows from (2 . k u k L t L x . This shows (3 . . 31) follows easily from Strichartz estimate. For (3 . k Iu M k L ∞ t L ∞ x . k h Λ i k/ Iu M k L ∞ t L x . k h Λ i k/ Iu M k X , / = k Iu M k X k/ , / . The proof is complete.We use Lemma 3 . B . m ( M ) m ( M ) m ( M ) m ( M ) 1 h M i k/ h M i k/ h M i k/ k Iu k X k/ , / Y j =2 k Iu M j k X k/ , / , with M ≥ M ≥ M and M & N . Using (3 . . 25) follows once we have m ( M ) m ( M ) m ( M ) m ( M ) 1 h M i k/ h M i k/ h M i k/ . N − γ ( k )+ M − . (3.34)We now break the frequency interactions into two cases: M ∼ M and M ∼ M since P j =1 ξ j =0. Case 1. M ∼ M , M ≥ M ≥ M and M & N . We see thatLHS(3 . ∼ m ( M )( m ( M )) m ( M ) 1 h M i k/ h M i k . m ( M ) N α ( k ) m ( M ) h M i k/ h M i k − α ( k ) . N α ( k ) m ( M ) h M i k − α ( k ) . N α ( k ) M k − α ( k )2 . N − ( k − α ( k ))+ M − . Here we use that m ( M ) h M i α ( k ) ≥ m ( N ) N α ( k ) = N α ( k ) , m ( M ) . h M i k/ and that m ( y ) h x i α ( k ) & ≤ y ≤ x . Case 2. M ∼ M , M ≥ M ≥ M and M & N . We haveLHS(3 . . m ( M ) m ( M ) 1 h M i k h M i k/ . m ( M ) h M i α ( k ) m ( M ) h M i α ( k ) h M i k − α ( k ) h M i k/ − α ( k ) . N − ( k − α ( k ))+ M − . Here we use again m ( M ) h M i α ( k ) , m ( M ) h M i α ( k ) & 1. By choosing α ( k ) as in Subcase 3a,we prove (3 . . (cid:3) Remark 3.6. Let us now comment on the choices of α ( k ) and γ ( k ). As mentioned in Remark3 . 2, if the increment of the modified energy is N − γ ( k ) , then we can show (see Section 4, after(4 . H γ ( R k ) with γ > k k + γ ( k )) =: γ ( k ). Welearn from (3 . 26) that γ ( k ) ≤ k − / 2, hence γ ( k ) ≥ k k − . On the other hand, in Subcase 3a,we need α ( k ) + γ − k/ > α ( k ) = k − γ ( k ). Since γ > γ ( k ), we have α ( k ) + γ − k/ >α ( k ) + γ ( k ) − k/ ≥ α ( k ) + k k − − k . We thus choose α ( k ) := k − k k − = k (2 k − k − , hence γ ( k ) = k − α ( k ) = k (6 k − k − . lobal well-posedness L -critical higher-order Schr¨odinger . We now are able to show the global existence given in Theorem 1 . 1. We only consider positivetime, the negative one is treated similarly. The conservation of mass and Lemma 2 . k u ( t ) k H γx . k Iu ( t ) k H k/ x ∼ k Iu ( t ) k H k/ x + k Iu ( t ) k L x . E ( Iu ( t )) + k u k L x . (4.35)By density argument, we may assume that u ∈ C ∞ ( R k ). Let u be a global solution to (NLS k )with initial data u . As E ( Iu ) is not necessarily small, we will use the scaling (1 . 1) to makethe energy of rescaled initial data small in order to apply the almost conservation law given inProposition 3 . 1. Let λ > u λ be as in (1 . E ( Iu λ (0)) = 12 k Iu λ (0) k H k/ x + 14 k Iu λ (0) k L x . (4.36)We then estimate k Iu λ (0) k H k/ x . N k/ − γ ) k u λ (0) k H γx = N k/ − γ ) λ − γ k u k H γx , and k Iu λ (0) k L x . k u λ (0) k L x = λ − k k u k L x . λ − k k u k H γx . Note that γ > γ ( k ) ≥ k/ . 36) gives for λ ≫ E ( Iu λ (0)) . ( N k/ − γ ) λ − γ + λ − k )(1 + k u k H γx ) ≤ C N k/ − γ ) λ − γ (1 + k u k H γx ) . We now choose λ := N k/ − γγ (cid:16) C (cid:17) − γ (1 + k u k H γx ) γ (4.37)so that E ( Iu λ (0)) ≤ / 2. We then apply Proposition 3 . u λ (0). Note that we may reapplythis proposition until E ( Iu λ ( t )) reaches 1, that is at least C N γ ( k ) − times. Therefore, E ( Iu λ ( C N γ ( k ) − δ )) ∼ . (4.38)Now given any T ≫ 1, we choose N ≫ T ∼ N γ ( k ) − λ k C δ. Using (4 . T ∼ N γ k )+ k ) γ − k γ − . (4.39)Here γ > γ ( k ) = k γ ( k )+ k ) , hence the power of N is positive and the choice of N makes sensefor arbitrary T ≫ 1. Next, using (1 . E ( Iu ( t )) = λ k E ( Iu λ ( λ k t )) . Thus, we have from (4 . . 38) and (4 . 39) that E ( Iu ( T )) = λ k E ( Iu λ ( λ k T )) = λ k E ( Iu λ ( C N γ ( k ) − δ )) ∼ λ k ≤ N k ( k/ − γ ) γ ∼ T k ( k − γ )2( γ k )+ k ) γ − k + . This shows that there exists C = C ( k u k H γx , δ ) such that E ( Iu ( T )) ≤ C T k ( k − γ )2( γ k )+ k ) γ − k + , for T ≫ 1. This together with (4 . 35) show that k u ( T ) k H γx . C T k ( k − γ )2(2( γ k )+ k ) γ − k + + C , where C , C depend only on k u k H γx . The proof of Theorem 1 . Van Duong Dinh A Linear estimate in X γ,b spaces In this section, we will give the proof of linear estimates (3 . 17) and (3 . 18) which is essentiallygiven in [15]. The estimate (3 . 17) follows from the fact that k u k X γ,b = k e − it Λ k u k H bt H γx . (A.40)Indeed, we have k ψ ( t ) e it Λ k u k X γ,b = k e − it Λ k ψ ( t ) e it Λ k u k H bt H γx = k ψ k H bt k u k H γx . k u k H γx . For (3 . (cid:13)(cid:13)(cid:13) ψ δ ( t ) Z t g ( s ) ds (cid:13)(cid:13)(cid:13) H bt . δ − b − b ′ k g k H − b ′ t . (A.41)In fact, using ( A. (cid:13)(cid:13)(cid:13) ψ δ ( t ) Z t G ( s ) ds (cid:13)(cid:13)(cid:13) H bt H γx . k G k H − b ′ t H γx . (A.42)We now apply ( A. 41) for g ( s ) = b G ( s, ξ ) with ξ fixed to have (cid:13)(cid:13)(cid:13) ψ δ ( t ) Z t b G ( s, ξ ) ds (cid:13)(cid:13)(cid:13) H bt . δ − b − b ′ k b G ( t, ξ ) k H − b ′ t , (A.43)where b · is the spatial Fourier transform. If we denote H ( t, x ) := ψ δ ( t ) Z t G ( s, x ) ds, then ( A. 43) becomes k b H ( t, ξ ) k H bt . δ − b − b ′ k b G ( t, ξ ) k H − b ′ t . Squaring the above estimate, multiplying both sides with h ξ i γ and integrating over R k , weobtain ( A. A. ψ δ ( t ) Z t g ( s ) ds = ψ δ ( t ) Z R (cid:16) Z t e iτs ds (cid:17) ˆ g ( τ ) dτ = ψ δ ( t ) Z R e itτ − iτ ˆ g ( τ ) dτ = ψ δ ( t ) X k ≥ t k k ! Z | δτ |≤ ( iτ ) k − ˆ g ( τ ) dτ − ψ δ ( t ) Z | δτ |≥ ( iτ ) − ˆ g ( τ ) dτ + ψ δ ( t ) Z | δτ |≥ ( iτ ) − e itτ ˆ g ( τ ) dτ =: I + II + III. Let us consider the first term. The Cauchy-Schwarz inequality gives k I k H bt ≤ X k ≥ k ! k t k ψ δ k H bt δ − k k g k H − b ′ t (cid:16) Z | δτ |≤ h τ i b ′ dτ (cid:17) / . Using that t k ψ δ ( t ) = δ k ϕ k ( δ − t ) where ϕ k ( t ) = t k ψ ( t ), we have k t k ψ δ k H bt = δ k k ϕ k ( δ − t ) k H bt = δ k (cid:16) Z R h τ i b δ | ˆ ϕ k ( δτ ) | dτ (cid:17) / . δ k δ / − b k ϕ k k H bt . We also have Z | δτ |≤ h τ i b ′ dτ = Z | τ |≤ (cid:10) δ − τ (cid:11) b ′ δ − dτ . δ − − b ′ , lobal well-posedness L -critical higher-order Schr¨odinger b ′ < / 2. This implies k I k H bt . X k ≥ k ! δ k δ / − b δ − k k g k H − b ′ t δ − / − b ′ . δ − b − b ′ k g k H − b ′ t . Similarly, we have k II k H bt . k ψ δ k H bt k g k H − b ′ t (cid:16) Z | δτ |≥ | τ | − h τ i b ′ dτ (cid:17) / . δ − b − b ′ k g k H − b ′ t , by using that k ψ δ k H bt . δ / − b k ψ k H bt . δ / − b and Z | δτ |≥ | τ | − h τ i b ′ dτ = Z | τ |≥ | δ − τ | − (cid:10) δ − τ (cid:11) b ′ δ − dτ ≤ δ − b ′ Z | τ |≥ | τ | − h τ i b ′ dτ . δ − b ′ . Here b ′ < / − b ′ ) > J ( t ) := Z | δτ |≥ ( iτ ) − ˆ g ( τ ) e itτ dτ. We see that ˆ J ( ζ ) = Z | δτ |≥ ( iτ ) − ˆ g ( τ ) δ ( ζ − τ ) dτ, where δ is the Dirac delta function. This yields that k J k H bt = (cid:16) Z h ζ i b | ˆ J ( ζ ) | dζ (cid:17) / = (cid:16) Z | δτ |≥ h τ i b | τ | − | ˆ g ( τ ) | dτ (cid:17) / ≤ k g k H − b ′ t sup | δτ |≥ | τ | − h τ i b + b ′ . δ − b − b ′ k g k H − b ′ t . Similarly, k J k L t . δ − b ′ k g k H − b ′ t . Thus, the Young’s inequality gives k III k H bt = k h τ i b ( ˆ ψ δ ⋆ ˆ J ) k L τ . k| τ | b ˆ ψ δ k L τ k ˆ J k L τ + k ˆ ψ δ k L τ k h τ i b ˆ J k L τ . δ − b − b ′ k g k H − b ′ t . 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