LOW REGULARITY A PRIORI BOUNDS FOR THE MODIFIED KORTEWEG-DE VRIES EQUATION

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1 LOW REGULARITY A PRIORI BOUDS FOR THE ODIFIED KORTEWEG-DE VRIES EQUATIO ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU Abstract. We study the local well-osedness in the Sobolev sace H s (R) for the modified Korteweg-de Vries (mkdv) equation t u + xu 3 ± x u 3 = 0 on R. Kenig- Ponce-Vega [10] and Christ-Colliander-Tao [1] established that the data-to-solution ma fails to be uniformly continuous on a fixed ball in H s (R) when s < 1 4. In site of this, we establish that for 1 8 < s < 1 4, the solution satisfies global in time H s (R) bounds which deend only on the time and on the H s (R) norm of the initial data. This result is weaker than global well-osedness, as we have no control on differences of solutions. Our roof is modeled on recent work by Christ-Colliander- Tao [2] and Koch-Tataru [11] emloying a version of Bourgain s Fourier restriction saces adated to time intervals whose length deends on the satial frequency. 1. Introduction We study the well-osedness of the initial-value roblem for the modified Kortewegde Vries (mkdv) on R: (1.1) t u + 3 xu ± x u 3 = 0, u(0) = u 0 where u = u(x, t) R with (x, t) R 1+1. This equation has scaling u(x, t) λu(λx, λ 3 t) and the scale invariant homogeneous Sobolev norm is Ḣ 1 2. The equation is globally well-osed in H s for s 1. Secifically, given initial data in 4 Hs, a solution exists in C([0, + ); H s ) X, where X is a certain auxiliary function sace; this solution is unique among all solutions that reside in this function class; and for any T > 0, the data-to-solution ma from a fixed ball in H s to C([0, T ]; H s ) is uniformly continuous. The local result was roved by Kenig-Ponce-Vega [8] by the contraction method in a function sace where several disersive estimates for the linear flow hold. An alternate roof in the setting of the Fourier restriction norm saces was given later in Tao [15]. Colliander-Keel-Staffilani-Takaoka-Tao [3] roved that this local solution extends to a global solution by studying the almost conservation of of the norm of a high frequency-damed coy of the solution (the I-method). On the other hand, for s < 1, (1.1) on R is ill-osed in the sense that the data-to-solution ma fails to be 4 uniformly continuous on a fixed ball in H s. This was established by Kenig-Ponce- Vega [10] for the focusing equation (+ sign in front of the nonlinearity; Theorem 1

2 2 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU 1.3 on. 623 of their aer), and by Christ-Colliander-Tao [1] for the defocusing equation ( sign in front of the nonlinearity; Theorem 4 on of their aer) 1. This leaves oen the question as to whether or not there is a well-osedness result for s < 1 giving only the continuity (as oosed to uniform continuity) of the data-tosolution ma. One result in this direction is Kato [6], where global weak solutions for 4 s = 0 are constructed. We will here rove another result in this direction, giving an a riori bound in H s for 1 < s < 1 in terms of the 8 4 Hs norm of the initial data but establishing no continuity. Our method is analogous to that in Christ-Colliander-Tao [2] and Koch-Tataru [11] dealing with the nonlinear Schrödinger equation (LS) on R. The related roblem for the mkdv equation was considered by Liu [12]. Theorem 1.1. Let 1 < s < 1. Then for any R > 0 and T > 0 there exists2 8 4 C = C(R, T ) > 0 so that for any initial data u 0 S satisfying u 0 H s R, the unique solution u C([0, T ]; S) to (1.1) (focusing or defocusing) satisfies u L [0,T ] H s x C u 0 H s. We note that our roof also alies for s = 1, but with a C which deends on 8 the full H 1 8 frequency enveloe of u. This deendence is likely nonotimal, and it would simlify once the 1/8 threshold is crossed. We also note that in the rocess of establishing the above result we also rove that the solutions belong to a smaller sace X s defined later in the aer. An easy consequence of our result is the existence of weak solutions for H s data: Corollary 1.2. Given any initial data u 0 H s, there exists a global solution u to (1.1) which solves the equation in the sense of distributions and satisfies with C as in the theorem above. u(t) H s C(t, u 0 H s) The weak solution is constructed as a weak limit of strong solutions. The uniform local H s bound does not suffice in order to verify that the equation is verified in the sense of distributions. Instead, this is true due the uniform X s bound, which is also imlicit in the construction. We refer to these solutions as weak solutions as we currently do not have any uniqueness or continuous deendence result in H s for s < 1 4. Currently the analogous roblem for the eriodic mkdv ((1.1) with (x, t) T R) is better understood. The threshold of s = 1 4 for mkdv on R is relaced by s = The roof given by [1] holds for 1 4 < s < 1 4, but the authors remark that the restriction to s > 1 4 is likely an artifact of their method. 2 The roof actually yields C = max{1, R 8s 1+8s T s 1+8s } but this is very likely nonotimal.

3 LOW REGULARITY BOUDS FOR KDV 3 for mkdv on T. Kaeler-Toalov [5] construct, via inverse scattering theory, global solutions in L 2. Tsutsumi-Takaoka [14] construct solutions for data in H s for 3 < 8 s < 1 via Fourier restriction norm estimates and a nonlinear ansatz. Both of these 2 results assert the continuity of the data-to-solution ma. Regarding our result, we believe that in rincile, by adding another correction term (or maybe more) to the modified energy in 5, we could imrove the lower threshold to s 1 since the trilinear 6 l2 U s,2 A estimate in 4 is valid down to this threshold. It seems that to ush to s < 1 would require a better understanding 6 of diagonal or resonant frequency interactions. We do not know if there is any significance to the number s = 1 in regard to the actual behavior of solutions or 6 whether it is just an artifact of our method. An outline of the aer is as follows. In 2, we define the function saces emloyed in the analysis. We use the U and V saces, originally introduced to this subject in unublished work of Tataru and then in Koch-Tataru [11], since they are ideally suited to time-truncations. In 3, we discuss the fundamental disersive estimates emloyed in the roofs of the trilinear estimate and the energy bound. These include the Strichartz estimates, local smoothing and maximal function estimates, and Bourgain s bilinear refined Strichartz estimates. In 4, the trilinear estimate is roved along the lines of Christ-Colliander-Tao [2] and Koch-Tataru [11]. In 5, an energy bound is obtained on a high-frequency-damed energy functional. The method here is essentially an adatation of the I-method of Colliander-Keel-Staffilani-Takaoka-Tao [3]. Our method does not establish any analogue of this energy bound for differences of solutions, which is the reason we cannot obtain a full well-osedness result in H s, 1 < s < 1. Finally, in 6, the comonents are brought together to give a roof of 8 4 Theorem 1.1. In the conclusion of the introduction we give a heuristic that exlains why, when s < 1, we exect a iece of the solution at frequency 1 to roagate according 4 to linear dynamics for at least a time 4s 1 1. Solutions to the linear equation satisfy the Strichartz estimate (see Lemmas 3.3, 3.4 below) (1.2) D 1/6 x e t 3 x φ L 6 t L 6 x φ L 2. ow suose u is a solution to (1.1) which is localized at frequency 1, and suose u e t 3 x φ on [0, T ], with φ H s 1. In the integral equation, we need to have (1.3) t u(t) = e t 3 x φ e (t t ) x 3 x u(t ) 3 dt, t 0 0 L e (t t ) x 3 x u(t ) 3 dt 1. [0,T ] Hs x

4 4 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU We estimate this term as t L e (t t ) x 3 x u(t ) 3 dt 1+s u 3 L 1 [0,T ] L T x 1+s u 3 L. 6 0 [0,T ] Hs [0,T ] L6 x x aking the heuristic substitution u(t) e t 3 x φ and alying the Strichartz estimate (1.2), u L 6 [0,T ] L 6 x e t 3 x φ L 6 [0,T ] L 6 x 1 6 φ L s, we see that to achieve (1.3), we need T 4s 1. otivated by this, our main function saces X s defined in the next section are constructed by using linear tye norms at frequency on the timescale 4s Acknowledgments..C. was suorted in art by SF grant DS , J.H. was suorted in art by SF grant DS and a fellowshi from the Sloan foundation and D.T. was suorted in art by SF grant DS and by the iller Foundation. 2. Function saces We first recall from Koch-Tataru [11] (see also the careful exosition in Hadac- Herr-Koch [4, 2]) the sace-time function saces U (I) (atomic-sace) and V (I) (sace of functions of bounded -variation), 1. These are defined on a time interval I = [a, b), where a < b + and take values in L 2 (R) or any other Hilbert sace. Given a artition a = t 0 < t 1 < < t K = b of I and a sequence {φ k } K 1 k=0 L2 x such that φ 0 = 0 and K φ k 1 L = 1, the function 2 x K a(t) = φ k 1 χ [tk 1,t k )(t) is called a U (I) atom. The sace U (I) is then the collection of functions u(t) on I of the form (2.1) u(t) = where a l are U (I) atoms, with norm u(t) U (I) = + l=0 λ l a l, + inf λ l. reresentations (2.1) l=0 It follows that elements u(t) of U (I) are right-continuous and satisfy the boundary conditions (2.2) u(a) = lim t a u(t) = 0 and u(b) def = lim t b u(t) exists.

5 LOW REGULARITY BOUDS FOR KDV 5 To define the sace V (I), we consider functions v : I L 2 x such that (2.3) v(a) = lim v(t) exists and v(b) def = lim v(t) = 0, t a t b and for such functions v(t) define the norm ( K 1/ v V (I) = su v(t k ) v(t k 1 ) L 2 {t k x), } where the suremum is taken over artitions a = t 0 < < t K = b. The fact that the requirement (2.3) is reserved in the limit under the V (I) norm follows from [4, Pro 2.4(i)]. ote that for I = [a, b), < a < b <, we have u U (I) = χ I u U ([,+ )) rovided u(a) = 0. If u(a) 0, then the left-side is not defined (i.e. u / U (I)), while the right-side is defined. Also, v V (I) + v(a) L 2 x = χ I v V ([,+ )) rovided v(b) = 0. If v(b) 0, then the left-side is not defined (i.e. v / V (I)), while the right-side is defined. ote that a consequence of (2.4) is that for any v with v(b) = 0, we have (2.4) χ I v V ([,+ )) 2 v V (I). Lemma 2.1 (U-V embeddings). Fix an interval I = [a, b). (1) If 1 q <, then u U q u U and u V q u V. (2) If 1 < and u(b) = 0, then u V u U. (3) If 1 < q <, u(a) = 0, and u V is right-continuous, then u U q u V. (4) Suose that 1 < q <, and T is a linear oerator with the boundedness roerties: T u E C q u U q A, for some Banach sace E. Then T u E C u U A, with 0 < C C q, T u E ln C q C u V A, with imlicit constant deending only on the roximity of q and. The first three statements are from Koch-Tataru [11], while the last originates in Hadac Herr Koch [4]. The recise references in [4] for all four arts are: for (1), see Pro. 2.2(ii) and Pro. 2.4(iv); for (2), see Pro. 2.4(iii); for (3), see Cor. 2.6; for (4) Pro We emhasize that in (3), (4), we have strict inequality < q. We also remark that (4) should be thought of as a quantitative version of (3).

6 6 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU We now define the sace DU 2 (I) = { t u u U 2 (I) }, where the derivative is taken in the sense of distributions. Given f DU 2 (I), a u U 2 (I) such that t u = f is in fact unique (recall u(a) = 0). Hence we can define f DU 2 (I) = u U 2 (I), which makes DU 2 (I) a Banach sace. For examle, if u is an atom, i.e. u = K φ k 1χ [tk 1,t k ) with a = t 0 < < t K = b, φ 0 = 0 and K φ k 1 2 L = 1, 2 x then K f = t u = (φ k φ k 1 )δ tk, def (where δ tk is the Dirac mass at t k and we take φ K = 0) is an element of DU 2 (I) with f DU 2 (I) = 1. ote that in this f, there is no Dirac mass at osition a but there is one at osition b (namely φ K 1 δ b ). Lemma 2.2 (DU-V duality). We have (DU 2 (I)) = V 2 (I) with resect to the usual airing f, v = b f(t), v(t) a x dt = b f v dx dt. a x Proof. First, we show that if u U 2 is such that t u = f, u(a) = 0, then f, v χ I u U 2 (I) v V 2 (I) for all v V 2 (I). Indeed, it suffices to show this for u an atom, i.e u = K φ k 1χ [tk 1,t k ), where a = t 0 < < t K = b and φ 0 = 0 and K φ k 2 L = 2 x 1. Since u(a) = 0 and v(b) = 0, we have f, v = t u, v = u, t v = = K K φ k 1, (v(t k ) v(t k 1 )) b a χ [tk 1,t k ) φ k 1, t v x By Cauchy-Schwarz, ( K ) 1/2 ( K 1/2 f, v φ k 1 2 L v(t 2 x k ) v(t k 1 ) 2 Lx) v 2 V 2. ext we show that su f DU 2 (I) 1 f, v = v V 2 (I). Pick a artition a = t 0 < < t K = b and define φ 0 = 0, and for 2 k K define φ k 1 = ( K v(t k ) v(t k 1 ) j=2 v(t j) v(t j 1 ) 2 L 2 x ) 1/2

7 LOW REGULARITY BOUDS FOR KDV 7 Then, defining u = K φ k 1χ [tk 1,t k ) and f = t u and arguing as above, u is an atom and ( K 1/2 f, v = v(t j ) v(t j 1 ) 2 Lx). 2 j=2 Taking the suremum over all artitions and using that lim t a v(t) = v(a), we obtain the claim. Finally, we must show that if ṽ (DU 2 (I)), then there exists v V 2 (I) such that ṽ(f) = f, v for all f DU 2 (I). Fix a < t < b, and we first define w(t) as follows. The functional φ ṽ(φ δ t ) (where δ t is the Dirac mass at t) is a bounded linear maing L 2 x C. Hence there exists w(t) L 2 x such that φ, w(t) x = ṽ(φ δ t ). It follows from [4, Pro. 2.4(i)] that w(a) def = lim t a w(t) exists and w(b) def = lim t b w(t) exists. Set v(t) = w(t) w(b). Then if u is an atom in U 2 def (I) (taking φ K = 0 for notational convenience in the summations) and f = t u, K K K f, v = (φ k φ k 1 )δ tk, v = (φ k φ k 1 ), v(t k ) x = (φ k φ k 1 ), w(t k ) x = ( K K ) ṽ((φ k φ k 1 )δ tk ) = ṽ (φ k φ k 1 )δ tk = ṽ(f) ow we use the U and V saces defined above to construct similar saces adated to the Airy flow. As base Hilbert saces in which functions in U and V take values, we will use L 2, H s, as well as a different norm H s on Hs defined by φ H s = ( ξ 2 + ) s 2 ˆφ L 2, 1 Finally, for a ositive smooth even symbol a satisfying a ξ (ξ) a(ξ) we define the sace H a with norm φ 2 Ha = φ, a(d)φ If the L 2 sace in the definition of U (I), V (I) and DU 2 saces is relaced by another Hilbert sace H {L 2, H s, H s, Ha }, we denote the corresonding saces by U 2 (I; H), V 2 (I; H), resectively DU 2 (I; H). Finally, ulling back by the Airy grou e t 3 x gives the saces u U A (I;H) def = e t 3 x u U (I;H), u V A (I;H) def = e t 3 x u V (I;H), def u DU 2 A (I;H) = e t 3 x u DU 2 (I;H) The roerties in Lemmas 2.1,2.2 are easily transferred to this setting. Consider a dyadic artition of frequencies ( = 2 k for some k = 0, 1,...), E = {ξ : /2 ξ 2 }, and let E 0 = [ 1, 1]. Fix consideration to the time interval [0, 1). Consider a smooth Littlewood-Paley artition of unity in frequency 1 = P

8 8 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU where each multilier P is localized to the corresonding set E. For H as above let [ ( def 2 ] 1/2 u l 2 L [0,1) H = P u(t) L [0,1) H). Clearly u L [0,1) H u l 2 L [0,1) H, but the converse is not true. To measure the solutions to the mkdv equation we define the saces X s with the norm ( def u X s = su χ I P u 2 U + ) 1/2 su χ 2 I = 4s 1 A Hs I P u 2 U, 2 > I = 4s 1 A Hs where 3 the suremum is taken over all half-oen subintervals I = [a, b) [0, 1) of length 1 4s. To measure the nonlinearity in the mkdv equation we define the saces Y s with the norm f Y s def = ( su P f 2 DU + ) 1/2 su P 2 I = 4s 1 A Hs f 2 DU, 2 > I = 4s 1 A (I;Hs ) Similarly we define the sace X a and Y a. 3. Basic estimates Lemma 3.1. Suose t u + 3 xu = f on [0, 1). Then u X s u l 2 L [0,1) Hs + f Y s Proof. Reduce to the case of a single frequency by alying P to the equation, and then consider a fixed time interval I = [t 0, t 1 ). We need to show But t [e t 3 x u(t)] = e t 3 x f(t), and thus Hence χ I u U 2 A H u(t 0 ) H + f DU 2 A (I;H). f DU 2 A (I;H) = e t 3 x f(t) DU 2 (I;H) = χ I (e t 3 x u(t) u(a)) U 2 H. χ I u U 2 A H = χ I e t 3 x u(t) U 2 H χ I (e t 3 x u(t) u(t0 )) U 2 H + χ I u(t 0 ) U 2 H = u(t 0 ) H + f DU 2 A (I;H). Lemma 3.2 (Bernstein inequality). For 1 q, P f L q 1 q 1 f L 3 ote that here we have written χ I P u U s,2 and not P u A U s,2 A (I). aturally, we are not assuming u vanishes at the left endoint of each of these intervals.

9 LOW REGULARITY BOUDS FOR KDV Strichartz, local smoothing, and maximal function estimates. A air (, q) of Hölder exonents will be called admissible if (3.1) q = 1 2, 4, 2 q. In articular, we note that the following airs (, q) of indices are admissible: (, 2), (6, 6), (4, ). Lemma 3.3 (Strichartz estimates). Let (, q) satisfy the admissibility condition (3.1). Then (3.2) D 1 x e t 3 x φ L t Lq x φ L 2. In articular, we have, for 1, P e t 3 x φ L t L 2 x φ L 2, P e t 3 x φ L 6 t L 6 x 1 6 φ L 2, P e t 3 x L 4 t L x 1 4 φ L 2. Proof. In Kenig-Ponce-Vega [7] Lemma 2.4 / Kenig-Ponce-Vega [8] Lemma 3.18(i), the estimate D 1 4 x e t 3 x φ L 4 t L φ x L 2 x is roved. On the other hand, we have trivially, e t 3 x φ L t L 2 = φ x L 2. x ow we can aly Stein s theorem on analytic interolation [13] to obtain (3.2). Lemma 3.4 (Local smoothing/maximal function estimates). Let (, q) satisfy the admissibility condition (3.1). Then (3.3) D 1 5 x e t 3 x φ L x L q t φ L 2. In articular, we note the following estimates, for 1: P e t 3 x φ L x L 2 t c 1 φ L 2, P e t 3 x φ L 6 x L 6 t c 1 6 φ L 2, P e t 3 x φ L 4 x L t c 1 4 φ L 2. Proof. The local smoothing estimate (Kenig-Ponce-Vega [8], Theorem 3.5(i)) is x e t 3 x φ L x L 2 t φ L 2. It is basically reducible to Plancherel in t. On the other hand, we have the maximal function estimate (Kenig-Ponce-Vega [8], Theorem 3.7(i) on. 556) D 1 4 x e t 3 x φ L 4 x L t φ L 2.

10 10 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU It is roved by reducing by duality and a T T argument to an estimate that is roved by the theorem on fractional integration and a ointwise Airy function estimate. We now aly Stein s theorem on analytic interolation [13] to obtain (3.3). The next two corollaries are consequences of these estimates, and relate the Strichartz sace-time norms to the Airy-atomic norm U s,2 A norm of any function u(x, t) (not necessarily a solution to the linear Airy equation). Corollary 3.5. If I = [a, b) is any interval, and u = u(x, t) any function, then for (, q) satisfying the admissibility condition (3.1), we have, for 1, (3.4) P u L I Lq x 1 χi u U A L2, and we have the dual relation for > 2 (3.5) P u DU 2 A (I;L 2 ) 1 u L, I Lq x where (, q ) denotes the Hölder dual air. Proof. To rove (3.4), it suffices to assume I = [, + ), since χ I can be inserted. It also suffices to consider a U A -atom K K (3.6) u(t, x) = χ [tk 1,t k )(t)e t 3 x φk 1 (x), φ k 1 L = 1, φ 2 0 = 0, x and rove that (3.7) P u L t Lq x 1. But (3.7) follows directly from (3.2), as follows: P u = K χ L t Lq [tk 1,t x k )(t)p e t 3 x φk 1 L t Lq x 1 K φ k 1 L 2 x = 1. To rove (3.5), note that since (DU 2 (I; L 2 )) = V 2 (I; L 2 ), we have P u DU 2 A (I;L 2 ) = su v V 2 (I;L 2 ) 1 A P u v dx dt. But P u, v u L I Lq x I x P v L I Lq x, and by (3.4) and Lemma 2.1(3) (alied on the interval [, + )), we have, for > 2, P v L I Lq x χ Iv U A L2 χ Iv V 2 A L 2. Aly (2.4) ( χ I v V 2 A L 2 2 v V 2 A (I;L2 )) to comlete the roof.

11 LOW REGULARITY BOUDS FOR KDV 11 Corollary 3.6. If (, q) is admissible according to (3.1) and, q r, then (3.8) P u L x L q I 5 1 χ I u U r A L 2. for any interval I = [a, b). We also have the dual relation for q > 2, (3.9) P u DU 2 A (I;L 2 ) 5 1 u L, x L q I where (, q ) is the Hölder dual air. Proof. As we argued in the roof of Cor. 3.5, it suffices to rove (3.8) for u an atom of the form (3.6) (with relaced by q) on I = [, + ). For such u we write u = u k, u k = χ [tk 1,t k )(t)p e t 3 x φk 1 Alying (3.3) for each u k, it remains to show that u r L xl q t k u k r L xl q t or equivalently u r L rx L q r t u k r L rx L q r t k But u k have disjoint suorts therefore u r = u k r and the last relation follows by the triangle inequality. For (3.9), we note that since (DU A (I; L 2 )) = VA 2(I; L2 ) P u DU 2 A (I;L 2 ) = su v V 2 A (I;L 2 ) =1 I x P u v dx dt. But by Hölder, P u v dx dt u L I x and by (3.8) and for q > 2, we have I Lq x P v L I Lq x, P v L I Lq x χ Iv U q A L2 χ Iv V 2 A L 2. Finally aly (2.4) to obtain the bound by v V 2 A (I;L 2 ) Bilinear estimate. Lemma 3.7 (Bilinear estimate). Suose E 1, E 2 R and 1, 2 > 0 are dyadic values (no restriction to 1) such that ξ 1 E 1 and ξ 2 E 2, ξ 1 + ξ 2 1 and ξ 1 ξ 2 2. Let P j be the x-frequency rojection oerators defined as P j f(ξ) = χ Ej (ξ) ˆf(ξ) for a function f = f(x). Then, (3.10) P 1 e t 3 x φ P2 e t 3 x ψ L 2 t L 2 x ( 1 2 ) 1 2 P1 φ L 2 P 2 ψ L 2.

12 12 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU Proof. and thus [P 1 e t 3 x φ P2 e t 3 x ψ] (ξ, t) = [P 1 e t 3 x φ P2 e t 3 x ψ] (ξ, τ) = where, in the last line, (ξ 1, ξ 2 ) is the solution to ξ 1 E 1 e itξ3 1 ˆφ(ξ1 )e itξ3 2 ˆψ(ξ2 ) ξ 2 E 2 ξ=ξ 1 +ξ 2 ξ 1 E 1 δ(τ ξ1 3 ξ2) 3 ˆφ(ξ 1 ) ˆψ(ξ 2 ) ξ 2 E 2 ξ=ξ 1 +ξ 2 = χ E 1 (ξ 1 )χ E2 (ξ 2 ) ˆφ(ξ 1 ) ˆψ(ξ 2 ) 3(ξ 2 1 ξ 2 2) τ = ξ ξ 3 2, ξ = ξ 1 + ξ 2. [In fact, there could be 0, 1, or 2 solutions (ξ 1, ξ 2 ) deending uon the articular (ξ, τ); a roer argument would exhibit these regions searately, etc.] The Jacobian for the change of variable (ξ, τ) (ξ 1, ξ 2 ) is dτdξ = 3 ξ 2 1 ξ 2 2 dξ 1 dξ 2. The result then follows from Plancherel s theorem and this change of variable. Corollary 3.8. Under the hyothesis of Lemma 3.7, if u = u(x, t), v = v(x, t) are any functions, then 4 (3.11) P 1 u P 2 v L 2 I L 2 x ( 1 2 ) 1 2 χi P 1 u U 2 A L 2 χ IP 2 v U 2 A L 2 (3.12) P 1 u P 2 v L 2 I L 2 x ( 1 2 ) 1 2 ln 2 1 χ I P 1 u V 2 L2 χ IP A 2 v V 2 A L 2 2 Proof. It clearly suffices to rove the estimates for I = [, + ), since we can insert χ I cutoffs on u and v. We begin noting that if we fix u = e t 3 x ψ, and v a U 2 A atom, i.e. K K v(x, t) = χ [tk 1,t k )(t)e t 3 x φk 1, φ 0 = 0, φ k 1 2 L = 1, 2 x then it follows from Lemma 3.7 that (3.13) P 1 u P 2 v L 2 I L 2 x ( 1 2 ) 1 2 ψ L 2. By linearity in u, we obtain the estimate (3.11) when both u and v are UA 2 atoms. The general case of (3.11) follows by linearity and density. The estimate (3.12) follows 4 ote the use of the truncation functions χ I and then evaluation in U 0,2 A ([, + )) or V 0,2 0,2 0,2 A ([, + )) on the right-side. We are not using the norms UA (I) or VA (I), since they require vanishing at the left and right endoints of I, resectively. We do not want to imose such a condition for finite-length intervals I.

13 LOW REGULARITY BOUDS FOR KDV 13 from (3.11) by the argument in [4, Cor. 2.18] which aeals to their Pro (our Lemma 2.1(4)). 4. Trilinear estimate Proosition 4.1 (Trilinear estimate). For all 1 7 < s < 1 4 and 1 we have x (u 1 u 2 u 3 ) Y s u 1 X s u 2 X s u 3 X s. Proof. We insert frequency rojections P j, P where, j. Denoting the truncated functions by u j = P j u j for j > while u j = P < u j for j =, we reduce matters to roving, for an interval J = 4s 1 with >, a bound of the tye (4.1) P x (u 1 u 2 u 3 ) DU 2 A (J;H s ) α(, 1, 2, 3 ) 3 j=1 su I j = 4s 1 j χ Ij u j U 2 A H s as well as the similar bound with P relaced by P <. This can be rewritten as P x (u 1 u 2 u 3 ) DU 2 A (J;L 2 ) α(, 1, 2, 3 ) s 1 s 2 s 3 s 3 j=1 su I j = 4s 1 j χ Ij u j U 2 A L 2 Here α should have certain summability roerties. As a general rule, we need at least that α(, 1, 2, 3 ) 1, and in some cases, need a slight ower decay in and/or j to insure the summation with resect to all indices. Case 1. 1, 2, 3. We can assume that In this case, all I j have length J and can be neglected. We distribute the derivative, which in the worst case alies to u 3. By (3.5) and (3.8), P (u 1 u 2 x u 3 ) DU 2 A (J;L 2 ) u 1 u 2 x u 3 L 1 J L 2 x J 1 2 u1 u 2 x u 3 L 2 J L 2 x 2s 1 2 u1 L 4 x L J u 2 L 4 x L J xu 3 L x L 2 J 2s χ J u 1 U 2 A L χ J u 2 U 2 A L 2 χ Ju 3 U 2 A L 2 Thus we have (4.1) with α = 2s s s 2, which suffices for all s. Case The u 2, u 3 terms need to be evaluated in norms restricted to intervals I of size I = 4s 1 3. We divide J into J / I = ( 3 /) 1 4s 1 intervals

14 14 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU of size I = 3 4s 1. For u VA 2(J; L2 ) we estimate by duality (Lemma 2.2) ( 3 u 1 u 2 u 3 u dx dt u 1 u 2 u 3 u dx dt J x ) 1 4s su I J I = 4s 1 3 I ( 3 ) 1 4s su u 1 u 2 L 2 I L u 2 x u 3 L 2 I L. 2 x I J I = 4s 1 3 Using the bilinear estimate (3.11),(3.12) we bound the above by ( 3 ) 1 4s 2 3 ln 2 3 su χ I u 1 U 2 A L 2 χ Iu 2 U 2 A L 2 χ Iu 3 U 2 A L 2 χ Iu V 2 A L. 2 I J I = 4s 1 3 Finally, we aly (2.4) ( χ I P u V 2 A 2 P u V 2 A (J)). Adding a factor of to account for the derivative in (4.1) we obtain α = 3 1 6s 5s 1 s ln 2 3 so this case is handled if s 1 6. Case = 3. We can assume that ξ 1, ξ 2 have the same sign and that ξ 3 has the oosite sign. [Indeed, if 1 2, then this is achieved by ermuting 2 and 3 if necessary, and if 1 2 3, then this can be arranged by ermuting the indices.] ote that then obviously we have ξ 1 ξ 3 3, but also since 2 3, we have ξ 1 + ξ 3 = ξ + ξ 2 3 and ξ ξ 2 3. We again argue by duality (Lemma 2.2) and divide into subintervals of size I = 3 4s 1. For v VA 2(J; L2 ), t J x u 1 u 2 u 3 u dx dt ( 3 ) 1 4s su I J I = 4s 1 3 t I x x u 1 u 2 u 3 u u dx dt ( 3 ) 1 4s su u u 2 L 2 I L u 2 x 1 u 3 L 2 I L. 2 x I J I = 4s 1 3 We then aly the bilinear estimate (3.11), (3.12) to bound the above by ( 3 ) 1 4s 2 3 ln 2 3 su χ I u 1 U 2 A L 2 χ Iu 2 U 2 A L 2 χ Iu 3 U 2 A L 2 χ Iu V 2 A L 2 I J I = 4s 1 3 Finally, we aly (2.4). Thus we have α = 1 6s 3 5s s 1, which is satisfactory if s > 1 7.

15 LOW REGULARITY BOUDS FOR KDV Energy bound For exositional convenience, in this section, we will assume that we are in the more difficult case s 0. We study the almost conservation of the H s norm using a variant of the I-method of Colliander-Keel-Staffilani-Takaoka-Tao [3]. The main result of this section is as follows: Proosition 5.1 (Energy bound). For all 1 s 0, > 0 and u solving (1.1) 8 we have the following bound in the time interval [0, 1]: (5.1) u 2 l 2 L H s c( u 4 l 2 L H s + u 6 X s ) Due to the l 2 dyadic summation on the left we cannot simly obtain a uniform in time bound for the H s norm of u. Instead for small ɛ > 0 we introduce a class S of real smooth ositive even symbols a(ξ) which have the following roerties: (i) a(ξ) is constant for ξ. (ii) Regularity: (5.2) α ξ a(ξ) c α a(ξ) ξ α (iii) Decay roerties (5.3) 1 2 d log a(ξ) d log(1 + ξ 2 ) 0 The latter roerty imlies that a(ξ) is nonincreasing but decays no faster than 5 ξ 1 2. For a S we will rove the uniform bound (5.4) u 2 L H a u(0) 2 H a + c( u 2 L H s u 2 L H a + u 4 X s u 2 X a ) which imlies the desired bound (5.1). consider a symbol a S such that a (ξ) def = To see this, for each dyadic we { 2s if ξ s ξ 1 2 if ξ 2. Then (5.1) follows from (5.4) alied to a due to the obvious relations u 2 l 2 L H s u 2 L H a, u 2 X s u 2 X a It remains to rove the bound (5.4). We define the energy functional E 0 (u) def = A(D)u, u = u 2 H a 5 In effect decay rates u to ξ 1 are still accetable, but not needed here.

16 16 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU and comute its derivative along the flow. Since a(ξ) is even and u is real, A(D)u is real. Also, A(D) is self-adjoint since a(ξ) is real. Thus, substituting (1.1), d dt E 0(u) = R 4 (u) def = ±2 A(D) x u, u 3. Using the fact that u is a real valued function, which imlies that û( ξ) = û(ξ), we write R 4 as a multilinear oerator in Fourier sace: R 4 (u) = ±2 iξ 1 a(ξ 1 ) û(ξ 1 )û(ξ 2 )û(ξ 3 )û(ξ 4 ) dσ, P 4 where P 4 = { (ξ 1, ξ 2, ξ 3, ξ 4 ) R 4 ξ 1 + ξ 2 + ξ 3 + ξ 4 = 0 }. This exression for R 4 can be symmetrized as R 4 (u) = ± 1 i(ξ 1 a(ξ 1 ) + ξ 2 a(ξ 2 ) + ξ 3 a(ξ 3 ) + ξ 4 a(ξ 4 ))û(ξ 1 )û(ξ 2 )û(ξ 3 )û(ξ 4 ) dσ. 2 P 4 We seek to cancel this term by erturbing the energy to E 0 + E 1, where E 1 has the form E 1 (u) = b 4 (ξ 1, ξ 2, ξ 3, ξ 4 )û(ξ 1 )û(ξ 2 )û(ξ 3 )û(ξ 4 ) dσ. P 4 To determine the roer choice for b 4, we comute d dt E 1(u) = b 4 (ξ 1, ξ 2, ξ 3, ξ 4 )i(ξ1 3 + ξ2 3 + ξ3 3 + ξ4) 3 û(ξ 1 )û(ξ 2 )û(ξ 3 )û(ξ 4 ) dσ + R 6 (u), P 4 where R 6 (u) has the form (if we for convenience go ahead and assume that b 4 is symmetric under exchange of any air from ξ 1, ξ 2, ξ 3 and ξ 4 ) R 6 (u) = 1 b 4 (ξ 1, ξ 2, ξ 3, 4 ξ)iξû3 (ξ)û(ξ 1 )û(ξ 2 )û(ξ 3 ) dσ. P 4 ow we see that the roer choice of b 4 to cancel the term R 4 is b 4 (ξ 1, ξ 2, ξ 3, ξ 4 ) = ± 1 ξ 1 a(ξ 1 ) + ξ 2 a(ξ 2 ) + ξ 3 a(ξ 3 ) + ξ 4 a(ξ 4 ) 2 ξ1 3 + ξ2 3 + ξ3 3 + ξ4 3 In conclusion, we have d dt (E 0 + E 1 )(u) = R 6 (u). Hence in order to rove (5.4) we need to establish the following two bounds: (5.5) E 1 (u) u 2 H s u 2 H a resectively (5.6) 1 0 R 6 (u(t))dt u 4 X s u 2 X a In order to do this we need to study the size and regularity of b 4.

17 LOW REGULARITY BOUDS FOR KDV 17 Lemma 5.2. Let a S. Then there exists a symbol b 4 in R 4 so that (5.7) ξ 1 a(ξ 1 ) + ξ 2 a(ξ 2 ) + ξ 3 a(ξ 3 ) + ξ 4 a(ξ 4 ) = b 4 (ξ ξ ξ ξ 3 4) on P 4 with the following size and regularity in dyadic regions { ξ j j > } resectively {ξ j j = }: (5.8) α 1 ξ 1 α 2 ξ 2 α 3 ξ 3 α 4 ξ 4 b 4 (ξ 1, ξ 2, ξ 3, ξ 4 ) c α b 4 ( 1, 2, 3, 4 ) α 1 1 α 2 2 α 3 3 α 4 4, where b 4 ( 1, 2, 3, 4 ) = a( 2 ) 2 4 when Proof. On P 4, we have the factorization ξ ξ ξ ξ 3 4 = (ξ 1 + ξ 2 )(ξ 1 + ξ 3 )(ξ 2 + ξ 3 ). Let j denote the dyadic zone of ξ j (as before the dyadic zone includes all frequencies below ). On P 4 we necessarily have 3 4. If all ξ j, then the left hand side of (5.7) is zero since a(ξ) = const for ξ, we have that. Therefore, we take b 4 = 0 there and assume 4 in the remainder of the roof. We consider several cases. Case Then we define ξ 1 a(ξ 1 ) + ξ 2 a(ξ 2 ) b 4 (ξ) = (ξ 1 + ξ 2 )(ξ 1 + ξ 3 )(ξ 1 + ξ 4 ) 1 (ξ 1 + ξ 3 )(ξ 1 + ξ 4 ) ξ 3 a(ξ 3 ) + ξ 4 a(ξ 4 ) ξ 3 + ξ 4 Since ξ 1 + ξ 2 2 and ξ 1 + ξ 3, ξ 1 + ξ 4 4, the conclusion easily follows by taking advantage of the cancellation in the last fraction when ξ 3 + ξ 4 = 0. Case Then we have ξ 1 + ξ 3, ξ 2 + ξ 3, ξ 1 + ξ 2 + ξ 3 4. Hence we define (5.9) b 4 (ξ) = ξ 1a(ξ 1 ) + ξ 2 a(ξ 2 ) + ξ 3 a(ξ 3 ) (ξ 1 + ξ 2 + ξ 3 )a( ξ 1 ξ 2 ξ 3 ) (ξ 1 + ξ 2 )(ξ 1 + ξ 3 )(ξ 2 + ξ 3 ) and the only difficulty comes from the division by ξ 1 + ξ 2. We rewrite b 4 as 1 b 4 (ξ) = (ξ 1 + ξ 3 )(ξ 2 + ξ 3 ) (g(ξ 1, ξ 2 ) g(ξ 3, ξ 1 ξ 2 ξ 3 )) where the function g is defined by g(ξ, η) = ξa(ξ) + ηa(η). ξ + η Since a is even and satisfies (5.2), it follows that g is smooth on the dyadic scale and has size a() when ξ η. The conclusion again follows. Case Using a artition of unit on the scale and ermuting the indices we can assume that we localized the roblem to a region where

18 18 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU ξ 2 + ξ 3, ξ 1 + ξ 2 + ξ 3. Then we define b 4 using again (5.9), and rewrite it in the form 1 g(ξ 1, ξ 2 ) g(ξ 3, ξ 1 ξ 2 ξ 3 )) b 4 (ξ) = ξ 2 + ξ 3 (ξ 1 + ξ 3 ) ow the first factor is ellitic, and in the second factor the numerator vanishes on {ξ 1 + ξ 3 = 0} therefore we have again a smooth division on the dyadic scale. The next result imlies the bound (5.5): Corollary 5.3. Let a S and b 4 as in Lemma 5.2. Then (5.10) E 1 (u) u 2 H a u 2 H 1 2 Proof. Given the exression of b 4, it suffices to rove this when û is ositive and b 4 is estimated ointwise by (5.8). Using again the notation u = P u for > and u = P u, by Bernstein s inequality we have E 1 (u) a( 2 ) u 2 1 u 2 u 3 u 4 L a( 2 ) ( a(2 ) 1 = a( 1 ) u 1 L 2 u 2 L 2 u 3 L 2 u 4 L 2 ) u 1 H a u 2 H a u 3 Ḣ 1 2 u 4 Ḣ 1 2 and the summation with resect to the i s is now straightforward. We conclude the roof of Proosition 5.1 with Proof of the estimate 5.6. Writing ξ = ξ 4 + ξ 5 + ξ 6 as the frequency decomosition in the cubic roduct we write R 6 (u) in the form R 6 (u) = 1 iξb 4 (ξ, ξ 1, ξ 2, ξ 3 )û(ξ 1 )û(ξ 2 )û(ξ 3 )û(ξ 4 )û(ξ 5 )û(ξ 6 ) dσ. 4 P 6 where P 6 = {ξ 1 + ξ 2 + ξ 3 + ξ 4 + ξ 5 + ξ 6 = 0}. For b 4 we use the extension given by Lemma 5.2. Since this extension is smooth in all variables on the dyadic scales, without any restriction we can searate variables and reduce the roblem to the case when b 4 is of roduct tye. Then we can return to the hysical sace and rewrite R 6 (u) = b 4 (, 1, 2, 3 ) u 1 u 2 u 3 P (u 4 u 5 u 6 )dx, u i = P i u, 2,, 7 where the factors in b 4 are harmlessly included in the sectral rojectors. This is allowed because L 2 bounded multiliers are also bounded in U 2 A saces.

19 LOW REGULARITY BOUDS FOR KDV 19 By symmetry we can assume that 1 2 3, as well as We also take an increasing rearrangement { 1, 2, 3, 4, 5, 6 } = { 1, 2, 3, 4, 5, 6 } where we must always have 5 6. Our next contention is that we can harmlessly discard the rojector P by searating variables. To see this we use the Fourier reresentation of the symbol (ξ 1 + ξ 2 + ξ 3 ) = e iλξ 1 e iλξ 2 e iλξ 3 f(λ)dλ, f (λ) = e iλξ (ξ)dξ The comlex exonentials are bounded symbols and thus bounded on UA 2L2, while f L 1 1 uniformly in. Assuming now that we have searated variables, we can sum the coefficient in R 6 with resect to b 4 (, 1, 2, 3 ) a( 2 )3 1 a( 2 )3 1 3 and we are left with having to estimate I = a( 2 ) = R u 1 u 2 u 3 u 4 u 5 u 6 dxdt We divide the time interval [0, 1] in 6 1 4s subintervals of size 6 1+4s corresonding to the highest frequency factor. We estimate the integral in each such subinterval, taking a loss of 6 1 4s due to the interval summation. Deending on how many frequency 6 factors there are we slit into several cases: Case (a) Then we can use two bilinear L 2 bounds for the roducts u 3 u 5 and u 4 u 6 and Bernstein to derive a ointwise bound for u 1 and u 2. We obtain I (i)(a) 1 5 = 6 a( 2 ) s = 6 ( a(2 ) 1 a( 1 ) 2 2 su ) s 6 s 1 8s 3 3 s 4 s 6 I = 1+4s 6 6 χ I u j U 2 A L 2 j=1 2 u j X a j=1 6 u j X s where the factors were reorganized to make clear the summation with resect to the j s. It is also transarent here that the total balance of exonents can only be favorable if s 1. 8 Case (b) The same argument as above alies after observing that two of the frequencies ξ 4, ξ 5 and ξ 6 must have an 6 searation, therefore the bilinear L 2 estimate can be alied. j=3

20 20 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU Case (c) As in the revious case we can aly the L 2 bilinear estimate for two of the high frequency factors, say u 5 u 6. Then we use the L 4 xl t bound for u 2 and u 3, the L x L 2 t for u 4 as well as the L bound for u 1. We obtain I (i)(a) = = = = 6 a( 2 ) s su ( a(2 ) 1 a( 1 ) 2 ) 1 2 ( 2 3 ) s 6 I = 1+4s 6 Again the summability with resect to j s is straightforward. 6. Proof of Theorem u j X a j=1 6 χ I u j U 2 A L 2 j=1 6 u j X s For exositional convenience, in this section, we will assume again that we are in the more difficult case s < 0. We first establish a short time small data result: Proosition 6.1. Let 1 and 1 4 s < 0. For any initial data u 0 S with u 0 H s 1, the unique solution u C([0, 1]; S) to (1.1) (focusing or defocusing) satisfies u L [0,1] H s C u 0 H s. Proof. For h [0, 1] let u h be the global solution to (1.1) with initial data u 0h = hu 0. By Lemma 3.1 and the trilinear estimate (Pro. 4.1), (6.1) u h X s u h l 2 L [0,1) Hs + u h 3 X s. By the energy bound in Pro. 5.1 we have (6.2) u h 2 l 2 L [0,1) Hs u h0 2 H s + u h 4 X s + u h 6 X s. Combining (6.1) and (6.2), we obtain u h X s C(h u 0 H s + u h 3 X s ) Since u 0 H s 1 and u h X s is a continuous function of h vanishing at h = 0, we conclude via a continuity argument that Returning to (6.2), it follows that u h X s h u 0 H s, h [0, 1] j=3 u L H s u 0 H s The roof is concluded.

21 LOW REGULARITY BOUDS FOR KDV 21 Given Proosition 6.1, we can conclude the roof of Theorem 1.1 using a scaling argument. Let 0 s > 1 8 and u 0 H s with u 0 H s R. Then we have u 0 H 1 8 R 1 8 s, 1 Let u 0λ (x) = λu 0 (λx) and u λ (x, t) = λu(λx, λ 3 t). Then u λ solves (1.1) with initial data u 0λ. We consider u λ on the time interval [0, 1), with λ to be chosen below. We have u λ0 H 1 8 λ λ 3 8 u0 H 1 8 λ 3 8 R 1 8 s,, λ > 1 Taking λ such that λ 3 8 R 1 8 s 1 we can aly Proosition 6.1 to conclude that u λ L [0,1] H 1 8 λ u 0λ 1. H 8 λ Scaling back to the interval [0, T ] with T = λ 3 we obtain u L [0,T ] H 1 8 u 0 1, T 1 8 H 8 R 1 8 s 1 The last restriction gives a bound from below on, (R, T ) def = (RT 1 8 ) ( 1 8 +s) 1 Taking a weighted square sum with resect to such in the revious relation we obtain u L [0,T ] H s (R,T ) u 0 H s (R,T ) This in turn shows that concluding the roof of the theorem. u L [0,T ] H s (R, T ) s u 0 H s References [1]. Christ, J. Colliander, T. Tao, Asymtotics, frequency modulation, and low regularity illosedness for canonical defocusing equations. Amer. J. ath. 125 (2003), no. 6, [2]. Christ, J. Colliander, T. Tao, A riori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev saces of negative order, J. Funct. Anal. 254 (2008), no. 2, [3] J. Colliander,. Keel, G. Staffilani, H. Takaoka, T. Tao, Shar global well-osedness for KdV and modified KdV on R and T. J. Amer. ath. Soc. 16 (2003), no. 3, [4]. Hadac, S. Herr, H. Koch, Well-osedness and scattering for the KP-II equation in a critical sace, Ann. Inst. H. Poincar Anal. on Linaire 26, no. 3, (2009). [5] T. Kaeler, P. Toalov, Global well-osedness of mkdv in L 2 (T, R). Comm. Partial Differential Equations 30 (2005), no. 1-3, [6] T. Kato, On the Korteweg-de Vries equation, anuscrita ath., 28 (1979), [7] C. Kenig, G. Ponce, L. Vega, On the (generalized) Korteweg-de Vries equation. Duke ath. J. 59 (1989), no. 3,

22 22 ICHAEL CHRIST, JUSTI HOLER, AD DAIEL TATARU [8] C. Kenig, G. Ponce, L. Vega, Well-osedness and scattering results for the generalized Kortewegde Vries equation via the contraction rincile. Comm. Pure Al. ath. 46 (1993), no. 4, [9] C. Kenig, G. Ponce, L. Vega, A bilinear estimate with alications to the KdV equation. J. Amer. ath. Soc. 9 (1996), no. 2, [10] C. Kenig, G. Ponce, L. Vega, On the ill-osedness of some canonical disersive equations. Duke ath. J. 106 (2001), no. 3, [11] H. Koch and D. Tataru, A riori bounds for the 1D cubic LS in negative Sobolev saces, Int. ath. Res. ot. IR 2007, no. 16, Art. ID rnm053, 36 [12] B. Liu, A-riori bounds for KdV equation below H 3/4, arxiv: [13] E. Stein, Interolation of linear oerators. Trans. Amer. ath. Soc. 83 (1956), [14] H. Takaoka, Y. Tsutsumi, Well-osedness of the Cauchy roblem for the modified KdV equation with eriodic boundary condition. Int. ath. Res. ot. 2004, no. 56, [15] T. Tao, ultilinear weighted convolution of L 2 -functions, and alications to nonlinear disersive equations. Amer. J. ath. 123 (2001), no. 5, University of California, Berkeley address: mchrist@math.berkeley.edu Brown University address: holmer@math.brown.edu University of California, Berkeley address: tataru@math.berkeley.edu