NONLINEAR EVOLUTION EQUATIONS SHORT NOTES WEEK #1

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1 NONLNEA EVOLUTON EQUATONS SHOT NOTES WEEK #. Free Schrödinger Equation Let Bx 2 `...`Bx 2 n denote the Laplacian on d with d ě. The initial-value problem for the free Schrödinger equation (in d space dimensions) is given by " ibt u u, (SE) up0, xq ϕpxq, u : ˆ d Ñ C. Here ϕ : d Ñ C is some given function, which represents the initial datum for the u upt, xq at time t 0. The free Schrödinger equation (and its linear and nonlinear generalizations discussed in the following sections) is one of the most fundamental equations in physics, as it describes the time evolution of a (non-relativistic) free quantum particle. Furthermore, equations of Schrödinger type also arise in nonlinear optics, ferromagnetism, water wave models, and geometry. t will be a non-trivial point to clarify in what sense upt, xq solves the equation. Of course, if upt, xq has a classical derivatives in x and t up to order two and one, respectively, it is clear what it means that upt, xq is a solution. However, the notion of classical derivatives is too limited and (actually not suitable) for the analysis of the Schrödinger equation and its initial-value problem. As we will see, a more natural class of functions is provided by the Sobolev spaces H s p d q, where we regard solutions u upt, xq sucht that upt, q P H s p d q for each time t P. For notational convenience, we shall often write uptq upt, q in what follows. We first study the following basic mathematical issues and properties for the free Schrödinger equation: Find a solution formula and a suitable class of function spaces. Dispersive properties for the Schrödinger propagator. Conservation laws and symmetries. nfinite speed of propagation and asymptotic behavior as t Ñ 8. The approach we follow to study the free Schrödinger equation will be essentially based on Fourier methods and some distribution theory; we refer to [3] for a quick review of these topics... Schrödinger Propagator. Let us assume that initial datum ϕ P Sp d q is a Schwartz function and we suppose that uptq P Sp d q for all t P is a solution of (SE). By taking the Fourier transform of upt, xq with respect to x P d, we obtain that ib t pupt, ξq 4π 2 ξ 2 pupt, ξq for all t P and ξ P d. For each ξ P d fixed, this is just an ordinary differential equation in t, which can be easily integrated to find that (.) pupt, ξq e 4π2 ξ 2 it pϕpξq, using that pup0, ξq pϕpξq due to the initial condition. For each t P fixed, the function ξ ÞÑ e 4π2 ξ 2 it is smooth with bounded derivatives of any order. Hence, the expression on right-hand side of equation (.) belongs to Sp d q for any t P Date: Version..

2 2 NONLNEA EVOLUTON EQUATONS: WEEK # (which is consistent with our assumption that uptq P Sp d q.) Thus we can take the inverse Fourier transform and use the convolution theorem for the Fourier transform to obtain the formula e 4π2 (.2) upt, xq F ξ 2 it pϕ pxq pk t ϕqpxq, where pf gqpxq ş fpx yqgpyq dy denotes convolution on d and the function d (.3) K t pxq : F e i4π2 ξ 2 t pxq 2 {4t ei x for t 0 and x P d. p4πitqd{2 The explicit formula for K t can be obtained using some complex analysis. Note that we take the branch of the complex square root with e? z ě 0 for e z ě 0. t is useful to write equation (.2) in terms of the short-hand notation (.4) uptq Tptqϕ where Tptq : Sp d q Ñ Sp d q for t 0 denotes the so-called free Schrödinger propagator in d defined as Tptqϕ : pk t ϕq. Furthermore, using the properties of K t pxq, it is not hard to verify that K t Ñ δ (the delta distribution) in S p d q as t Ñ 0, and hence lim tñ0 ptptq ϕq pδ ϕq ϕ provided that ϕ P Sp d q, which is in accordance with the initial condition for (SE) above. Thus, for t 0, we have that Tp0q. Moreover, it is easy to see that the map Tptq : Sp d q Ñ Sp d q is linear, i. e., Tptqpaϕ ` bψq atptqϕ ` btptqψ for all a, b P C and all ϕ, ψ P Sp d q. On the other hand, if we are given an initial datum ϕ P Sp d q, we can define the function upt, xq by means of the formula (.4) and thus obtain a solution to the initial-value problem (SE). t is not hard to check that, for upt, xq defined by (.4), we have u P C 8 p; Sp d qq, i. e. for each t P the map t ÞÑ uptq is smooth from the reals into the Schwartz space. Next, we wish to relax the assumption that ϕ P Sp d q. n fact, we can extend the map Tptq to S p d q by duality; i. e., for f P S p d q given, we define Tptqf P S p d q by ptptqfqpψq : fptptqψq for all ψ P Sp d q. For s P given, we recall the definition of the Sobolev space (.5) H s p d q tf P S p d q : }f} H s ă `8u, endowed with the Sobolev norm (.6) }f} H s }p ` ξ 2 q s{2 p f}l 2. Note that H s p d q is Banach space; in fact, H s p d q is a Hilbert space with the scalar product given by xf, gy H s fpξqpgpξqp p ` ξ 2 q s dξ. d Clearly, we have the inclusion H s p d q Ă H s p d q for s ě s. For negative s ă 0, the elements of H s p d q are only distributions in general (and hence the notation u upt, xq is slightly inappropriate when uptq P H s p d q for s ă 0). But for s ě 0, any element f P H s p d q automatically belongs to L 2 p d q and therefore the notation u upt, xq is well-defined. n fact, we will mostly consider solutions uptq P H s p d q with nonnegative s ě 0.

3 NONLNEA EVOLUTON EQUATONS: WEEK # 3 Since H s p d q Ă S p d q for any s P, it makes sense to let Tptq act on elements in H s p d q and thus we can consider initial data ϕ P H s p d q with Sobolev index s P. That this class of initial data is natural can be inferred from the following result. Proposition.. Let s P. Then the family ttptqu tp is a continuous oneparameter unitary group on H s p d q. That is, we have the following properties. (i) Group property: t holds that Tpt ` t q TptqTpt q for all t, t P and Tp0q. (ii) Unitarity: For every t P, the map Tptq : H s p d q Ñ H s p d q is unitary, i. e., Tptq is bijective and an isometry, which means that }Tptqϕ} H s }ϕ} H s for all ϕ P H s p d q. (iii) Continuity: For every ϕ P H s p d q, the map t ÞÑ Tptqϕ is continuous from into H s p d q. That is, we have Tp qϕ P Cp; H s p d qq. emarks.. The Sobolev spaces H 0 p d q L 2 p d q and H p d q, will be of particular importance, as they turn out to be naturally related to important conservation laws for the Schrödinger equation. Proof. To show (i), it suffices to consider Tptqϕ with ϕ P Sp d q; by duality, the identities in (i) then extend to Tptq acting on S p d q. f we take the Fourier transform, we obtain F `Tpt ` t qϕ e 4π2 ξ 2 ipt`t q pϕ e 4π2 ξ 2 it e 4π2 ξ 2 it pϕ. From this we conclude that (i) holds by taking the inverse Fourier transform. To show that Tptq is an isometry on H s p d q, we observe }Tptqϕ} H s }p ` ξ 2 q s{2 e 4π2 ξ 2 it pϕ} L 2 }p ` ξ 2 q s{2 pϕ} L 2 }ϕ} H s, where we used that e 4π2 ξ 2 it for all t P and ξ P d. Moreover, from (i) we easily see that Tptq is bijective on H s p d q with inverse Tptq Tp tq. This shows (ii). Finally, we prove (iii) by using the dominated convergence theorem which yields lim }ptptq tñt Tpt qqϕ} 2 H lim s tñt d lim d tñt 0 p ` ξ 2 q s pϕpξq 2 dξ 0, d whence the proof is complete. e 4π2 ξ 2 it e 4π2 ξ 2 it p ` ξ 2 q s pϕpξq 2 dξ e 4π2 ξ 2 it e 4π2 ξ 2 it p ` ξ 2 q s pϕpξq 2 dξ n view of the results in Proposition., we introduce the intuitive (and commonly used) notation e it Tptq for the free Schrödinger propagator in d. Let us now introduce a notion of solution for problem (SE) that involves the associated integral equation with the propagator.

4 4 NONLNEA EVOLUTON EQUATONS: WEEK # Definition.. Let s P and ϕ P H s p d q. We say that u P Cp; H s p d qq is a (strong) H s -solution of problem (SE) if we have uptq e it ϕ for all t P. emark.. At this point, the above definition may look weird. However, when we turn to nonlinear evolution problems it will become clear how the concept of strong and weak H s -solutions turns out to be useful. For the linear Schrödinger equation, however, the notions of strong and weak H s -solution coincide. 2. n general, the map t ÞÑ Tptqϕ is only known to be continuous from to H s p d q. The derivative ib t uptq uptq does not belong H s p d q for any t in general, but rather in the larger space H s 2 p d q. n fact, we can conclude that a strong solution u P Cp; H s p d qq automatically belongs to the space u P C 0 p; H s p d qq X C p; H s 2 p d qq. 3. As a trivial consequence of Proposition., we have the following uniqueness result: f u P Cp; H s p d qq and ũ P Cp; H s p d qq are strong H s -solutions of (SE) with upt 0 q ũpt 0 q for some t 0 P, then uptq ũptq for all t P. 4. Since H s p d q Ă H s p d q when s ě s, a strong H s -solution is also a strong H s -solution provided that s ě s. Next, we turn to a useful result about e it which is referred to as a dispersive estimate for the Schrödinger propagator. Lemma. (Dispersive Estimates). Suppose ď p ď 2 and let 2 ď p ď 8 be its dual exponent, i. e., {p ` {p. Then it holds }e it ϕ} L p ď 4πt d 2 p p p q }ϕ} L p for t 0. n particular, we have the L 8 -L -estimate }e it ϕ} L 8 ď 4πt d 2 }ϕ}l for t 0. emark. Let ϕ P L p d q X L 2 p d q with ϕ ı 0. Note that }e it ϕ} L 2 }ϕ} L 2 is constant in time, whereas }e it ϕ} L p Ñ 0 as t Ñ 8 for any p ą 2. This behavior reflects the dispersive character of the Schrödinger equation. Proof. Let us first take p 2, which implies that p 2. By Proposition. (ii), (.7) }e it ϕ} L 2 }ϕ} L 2. Next, we consider p and thus we have p 8. ecalling that e it ϕ pk t ϕq, we find (.8) }e it ϕ} L 8 ď sup K t px yq ϕpyq dy ď xp d 4πt }ϕ} d d{2 L, where we used the bound K t p q ď 4πt d{2 for t 0. We now complete the proof by invoking the iesz-thorin interpolation theorem (see, e. g., [3]) which says the following: Suppose T is bounded linear operator from L pi to L qi with i 0, such that }T f} L q 0 ď M 0 }f} L p 0 and }T f} L q ď M }f} L p, with some constants M 0 ě 0 and M ě 0. Then T is also a bounded linear map from L p ϑ to L q ϑ such that }T f} L q ϑ ď M0 ϑ M ϑ }f} L p ϑ, where ϑ ` ϑ, ϑ ` ϑ, ϑ P p0, q. p ϑ p 0 p q ϑ q 0 q ecalling the bounds (.7) and (.8), we now apply the iesz-thorin theorem to T e it with pp 0, q 0 q p2, 2q and pp, q q p, 8q, which finishes the proof.

5 NONLNEA EVOLUTON EQUATONS: WEEK # 5 As a consequence of Lemma., we remark that e it with t 0 is not a bounded operator from L q p d q in L q p d q if q 2. To see this, we argue by contradiction as follows. Let q ą 2 and assume that }e it ϕ} L q ď C}ϕ} L q for all ϕ P L q p d q with some constant C. Now we take ϕ P L q p d q X L q p d q Ă L 2 p d q with ϕ ı 0. Using Lemma. with q p and Proposition., we find }ϕ} L q }e it e it ϕ} L q ď C}e it ϕ} L q ď C 4πt α }ϕ} L q with some α ą 0. By taking the limit t Ñ 8, we conclude that }ϕ} L q 0, which is a contradiction. The (similar) proof that e it with t 0 is not bounded from L q p d q into itself when ď q ă 2 is left to the reader..2. Symmetries and Conservation Laws. The free Schrödinger equation has a number of obvious and less obvious symmetries, which are as follows. Suppose upt, xq solves ib t u u in the sense that uptq e it ϕ for some ϕ P L 2 p d q. Then the following are also solutions of the Schrödinger equation: Phase shift: For ϑ P, Time translations: For t 0 P, Space translations: For x 0 P d, otations: For A P Opnq, Galilean boosts: For v P d, e iϑ upt, xq. upt t 0, xq. upt, x x 0 q. upt, Axq. e ip 2 x v 4 v2 tq upt, x vtq. Scaling transformations: For λ ą 0 fixed, λ d{2 upλ 2 t, λxq. Pseudo-conformal transformations: For α, ϑ, γ, ω P with αϑ ωγ,. ˆ 2 γ ` ϑt eiω x {p4pα`ωtqq u pα ` ωtqd{2 α ` ωt, x α ` ωt The verification of these symmetries for the Schrödinger equation are left as an exercise. n particular, we mention that the pseudo-conformal symmetry of the Schrödinger equation is a very peculiar property. We can summarize the symmetries listed above by saying that the (SE) has the following symmetry group Upq ˆ Galileip d q ˆ ą0 ˆ SL 2 pq, where the Galilean group Galileip d q contains the symmetries of time and space translations, rotations in d, and boost symmetry. t is easy to see Galileip d q has dimension 2d ` ` d 2 pd q. The factor SL 2pq (the group of real 2 ˆ 2 matrices with determinant ) represents the pseudo-conformal symmetry. Associated to the symmetries above, we obtain the following (formal) conserved quantities and identities satisfied by solutions u upt, xq of the Schrödinger equation:

6 6 NONLNEA EVOLUTON EQUATONS: WEEK # L 2 -Mass: Energy : Mruptqs : upt, xq 2 dx const. d Eruptqs : 2 d x upt, xq 2 dx const. (Linear) Momentum: P k ruptqs : m upt, xqb xk upt, xq dx const. d with k,..., d. (Angular) Momentum: Ω kl ruptqs : m upt, xq px k B xl x l B xk q upt, xq dx const. d with k, l,..., d. (Note that Ω kl rus Ω lk rus is anti-symmetric.) Center-of-Mass Motion: ˆ d x k upt, xq 2 dx 2P k rup0qs. dt d with k,..., d. Dilation dentity: ˆ d 2 m upt, xqx x upt, xq dx 2Erup0qs dt d Pseudo-Conformal Conservation Law: d px ` 2it x qupt, xq 2 dx const. dt d These conservation laws and identities require regularity and/or spatial decay in order to make sense. For instance, the dilation identity requires that uptq P L 2 p d q and xuptq P L 2 p d q. At this point, we omit the technical issue about verifying whether such properties of upt, xq follow from imposing such conditions on the initial datum. (Further below, we will return to this technical point.) However, a rigorous proof for the conservation of L 2 -mass, energy, and linear momentum are easily obtained for strong H s -valued solutions u P C 0 p; H s p d qq, provided that s P is chosen such that the quantities are well-defined. Lemma.2. Let s P and suppose u P C 0 p; H s p d qq is a strong H s -solution of (SE). Then we have (i) f s ě 0, then Mruptqs Mrϕs for all t P. (ii) f s ě, then (iii) f s ě {2, then Eruptqs Erϕs for all t P. P k ruptqs P k rϕs for all t P and k,..., d. The factor is a useful convention. 2

7 NONLNEA EVOLUTON EQUATONS: WEEK # 7 Proof. Let u P Cp; H s p d qq be a strong H s -solution of (SE). f s ě 0, then u is also a strong H 0 -solution. Since Mruptqs }uptq} 2 H }uptq} 2 0 L, assertion 2 (i) directly follows from Proposition. with s 0. Next, from the fact that z x fpξq 2πiξfpξq p and Plancherel s identity we get Eruptqs } x uptq} 2 L 2 } 2πξ y uptq} 2 L 2 } 2πξ e 4π ξ 2 it pϕpξq} 2 L 2 } 2πξ pϕ}2 L 2 } xϕ} 2 L 2 Erϕs. using also that e 4π ξ 2it for any t P and ξ P d. For the proof of (iii), we first note that P k ruptqs m upt, xqb xk upt, xq dx xuptq, ib xk uptqy. d Now we use that xf, gy xf, p pgy (Parseval s identity) where xf, gy ş fpxqgpxq dx d is the scalar product on L 2 p d q. Thus, P k ruptqs xuptq, y ξ B k 2π ξ uptqy y e 4π2 ξ 2 it pϕ, ξ F k e 4π 2 ξ 2 it pϕ 2π B pϕ, ξ F k 2π e 4π2 ξ 2 it e 4π 2 ξ 2 it pϕ x pϕ, ξ k 2π pϕy xϕ, ib xk ϕy P k rϕs. When we discuss the nonlinear Schrödinger equation further below, we will make strong use of these conversation laws..3. Strichartz Estimates. We now come to more technical point about refined mapping properties of the Schrödinger propagator e it, which are referred to as Strichartz estimates. n fact, these are powerful space-time estimates that will play an essential rôle in the analysis of nonlinear Schrödinger equations (which we will turn to below). ntegration in Banach Spaces: Bochner ntegrals. As a preparation, we introduce the following construction due to S. Bochner: Let Ă be an open interval and suppose that X is a Banach space with norm } }. We shall be rather brief here; a more detailed discussion of this subject matter can be found in []. We say that a function f : Ñ X is measurable if there exists a sequence of functions pf n q npn Ă C c p; Xq (the set of continuous functions from to X with compact support in ) such that lim f nptq fptq for almost every t P. nñ8 ecall that for almost every t P means that t P zn, where N Ă is set of (Lebesgue) measure zero. t is easy to see that if f : Ñ X is measurable, then }f} : Ñ is measurable. Next, we define the notion of integrability in the sense of Bochner. First, if g P C c p; Xq is a continuous function with compact support and values in X, we define its integral gptq dt in the usual iemann sense. Next, we say that a measurable function f : Ñ X is integrable (in the sense of Bochner) if there exists a sequence pf n q npn Ă C c p; Xq such that (.9) lim }f n ptq fptq} dt 0. nñ8

8 8 NONLNEA EVOLUTON EQUATONS: WEEK # For integrable f : Ñ X, it can be shown that the limit fptq dt : lim f n ptq dt nñ8 exists in X and that it does not depend on the particular choice of the sequence pf n q npn Ă C c p; Xq satisfying (.9). Note that integrals ş f nptq dt on the right side are already defined since f n P C c p; Xq. We call ş fptq dt the (Bochner) integral of f. Moreover, a theorem due to Bochner asserts that if f : Ñ X is measurable, then f : Ñ X is integrable if and only if }f} : Ñ is integrable, in which case we have fptq dt ď }fptq} dt. Bochner s theorem enables us to deal with integrable functions f : Ñ X in the same manner as we deal with real-valued integrable functions. n particular, we have the following dominated convergence theorem: Let pf n q npn be a sequence of integrable functions f n : Ñ X, let g P L pq, and let f : Ñ X. Assume that " }fn ptq} ď gptq for a. e. t P and all n P N, lim nñ8 f n ptq fptq for a. e. t P. Then f : Ñ X is integrable and we have f n ptq dt fptq dt lim nñ8 with strong convergence in X. For p P r, 8s, we denote by L p p; Xq the set of (equivalence classes) of measurable functions f : Ñ X such that t ÞÑ }fptq} belongs to L p pq. For f P L p p; Xq, we define the norm $ ˆ & }fptq} dt{p p for p ă 8, }f} L p p;xq % ess sup tp }fptq} for p 8. Next, we list some basic properties and results. L p p; Xq is a Banach space for ď p ď 8. Cc 8 p; Xq is dense in L p p; Xq for ď p ă 8. Duality: Let X be a reflexive Banach space and let X be its dual space. Then pl p p; Xqq L p p; X q for ď p ă 8 with {p ` {p. n particular, the space L p p; Xq is reflexive for ă p ă 8. Generalized Hölder s inequality: f f P L p p; Xq and ψ P L q pq with {p ` {q {r and r P r, 8s, then fψ P L r p; Xq with }fψ} Lr p;xq ď }f} Lp p;xq}ψ} L p pq. n particular, if Ă is a bounded interval and p ď q, then }f} L p p;xq ď q p pq }f}l q p;xq for all f P L q p; Xq. Let X, Y be Banach spaces and A P LpX, Y q, i. e., A : X Ñ Y is a linear and bounded map. f f P L p p; Xq then Af P L p p; Y q. Moreover, if f P L p; Xq, then ˆ A fptq dt Afptq dt. The space C 0 p; Xq of continuous functions f : Ñ X is a Banach space when equipped with the norm }f} C 0 p;xq : }f} L 8 p;xq.

9 NONLNEA EVOLUTON EQUATONS: WEEK # 9 Space-Time Lebesgue Norms. Let Ă be an open interval; in particular, we could take. We now apply the previous general theory of Bochner integrals to the case X L r p d q. For ď p, r ď 8, we consider the Banach space L p p; L r p d qq endowed with the norm } } L p p;l r p qq. Note that if both p ă 8 and r ă 8, we d have the explicit formula ˆ dt{p ˆ p{r }u} L p p;l r p d qq }uptq} p upt, xq r dx dt {p L r p d q d Furthermore we record the following useful fact: For and p ă 8 and r ă 8, the space of Schwartz functions on space-time Sp `d q Ă L p p; L r p d qq is dense. t is convenient to make the following definition. Definition.2. The pair pp, rq P r2, 8s ˆ r2, 8s is (Strichartz) admissible if 2 ˆ p d 2 r and 2 ď r ď 2d d 2 for d ě 3, 2 ď r ă 8 for d 2, 2 ď r ď 2 for d. 2d Note that p8, 2q is always admissible. The pair p2, d 2q is admissible for d ě 3. Lemma.3 (Strichartz Estimates). For any pp, rq admissible pair, there is constant C ą 0 such that the following bounds hold. (i) Homogeneous Strichartz estimate: For any ϕ P L 2 p d q, }e it ϕ} L p p;l r p d qq ď C}ϕ} L 2. (ii) nhomogeneous Strichartz estimate: For any 8 ď t 0 ă 8 and any f P L p p; L r p d qq, we have t e ipt sq fpsq ds ď C}f} L p p,l r p d qq, t 0 L p p,l r p d qq where p and r denote the dual indices of p and r, respectively. emark. Let ϕ P L 2 p d q. Note that assertion (i) implies that e it ϕ P L r p d q for a. e. t P and any admissible pair pp, rq. For instance for d dimension and the choice pp, rq p4, 8q, we obtain }e it ϕ} 4 L dt ď 8 C}ϕ}4 L 2. Thus e it ϕ P L 8 for a. e. t P, i. e., the function e it ϕ is (essentially) bounded on for almost every time t P. However, one can show that, for suitable ϕ P L 2 pq, the function e it ϕ does not belong to L 8 pq for all t 0. See lecture/exercise for a simple counterexample. Proof. Let pp, rq be admissible, where we make the additional assumption that 2 ď r ă 2d d 2 The proof for the endpoint case r 2d here; see Keel & Tao. d 2 for d ě 3. requires some extra effort, which we omit

10 0 NONLNEA EVOLUTON EQUATONS: WEEK # We first prove assertion (ii), from which (i) will be deduced later. Step : nhomogeneous Strichartz Estimate. Let pp, r q be such that {p `{p and {r `{r. Suppose that f P Spˆ d q is a Schwartz function on space-time. Clearly, we have that f P L p p, L r p d qq. For 8 ď a ă b ď `8, we estimate b e ipt sq fpsq ds ď }e ipt sq fpsq} Lr p d q ds (.0) a L r p d q ď c t s d 2 p{r {rq }fpsq} L r p d q ds, with some constant c cprq ą 0, where we used the dispersive estimate from Lemma. (with choosing p r there). Next, we recall the Hardy Littlewood Sobolev inequality (see, e. g., [3]), which in d dimension states that (.) gpsq t s λ ds ď c p,λ }g} Lb pq, L a pq provided that a ` b ` and 0 ă λ ă. λ Applying this estimate with λ d 2 p{r {rq, where we verify that 0 ă λ ă holds for our choice of pp, rq, we find t e ipt sq fpsq ds t 0 L p p,l r p d qq L ď C t s d 2 p{r {rq }fpsq} L r p d q ds ď C}f} L p p;l r p d qq. p pq with some constant C Cpp, rq ą 0. Going back to (.0) and using that Sp `d q is dense in L p p; L r p d qq, we conclude that (ii) holds. Step 2: Homogeneous Strichartz Estimates. ecall the duality (.2) L p p; L r p d qq L p p; L r p d qq, provided that L r p d q is reflexive, which forces us to assume that r ă 8 holds. (For r 8, which is only possible in d dimension, one can prove the homogeneous by a separate argument, which we skip here.) Let ϕ P Sp d q. By the duality (.2) and the density of Sp `d q Ă L p p; L r p d qq, we conclude that }e it ϕ} L p p;l r p d qq sup ˇ e it ϕpxqfpt, xq dx dt ˇ }f} L p p;l r p d qq d (.3) fpsp `d q sup }f} L p p;l r p d qq fpsp `d q ď }ϕ} L2 p d q ˆ ˇ ϕpxq d sup }f} L p p;l r p d qq fpsp `d q e it fpt, xq dt e it fpt, q dt dx ˇ L 2 p d q where we used Fubini s theorem and the Cauchy-Schwarz inequality. Since Sp d q Ă L 2 p d q is dense, the above estimate extends to all ϕ P L 2 p d q.,

11 NONLNEA EVOLUTON EQUATONS: WEEK # Next, we claim that, for f P Sp `d q, we have e it fpt, q dt ď c}f} L p p;l r p d qq. L 2 p d q ndeed, we use a so-called T T -argument to show this bound. Observe that 2 ˆ ˆ e it fpt, q dt e it fpt, xq dt e is fps, xq ds dx L 2 p d q d ˆ fpt, xq e ipt sq fps, xq ds dt dx d ˆ fpt, xq e ipt sq fps, xq ds dx dt d ď }f} L p p;l r p d qq e ipt sq fps, q ds ď C}f} 2 L p p;l r p d qq. Lp p;l r p d qq where we used Fubini s theorem, e is g e is g, and in the last step we made use of the bound shown in Step for a 8 and b `8. Plugging this estimate into (.3), we deduce that assertion (i) holds. (The remaining special case r 8 that can only occur in d space dimension is left to the reader.) We complete the discussion of Strichartz estimates for e it with the following useful result, which generalizes the inhomogeneous Strichartz estimate. Corollary.. Let pp, r q and pp 2, r 2 q be two admissible pairs. Then for any 8 ď t 0 ă 8 and f P L p 2 p; L r 2 p d qq, we have t e ipt sq fpsq ds ď C}f} L p 2 p;l r 2 p t d qq 0 L p p;l r p d qq with some constant C Cpd, p, p 2 q ą 0. emark. Note that Lemma.3 (ii) is the special case when pp, q q pp 2, q 2 q. Proof. We refer to the proof of [3, Corollary 4.]; the minor modifications are left to the reader. eferences [] T. Cazenave und A. Haraux: An introduction to semilinear evolution equations, Volume 3 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, New York (998). [2] T. Cazenave: Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 0. New York University, Courant nstitute of Mathematical Sciences, AMS (2003). [3] F. Linares und G. Ponce: ntroduction to Nonlinear Dispersive Equations, Universitext, Springer, New York (2009).