Local Cauchy theory for the nonlinear Schrödinger equation in spaces of infinite mass
aa r X i v : . [ m a t h . A P ] J un Local Cauchy theory for the nonlinear Schrödingerequation in spaces of infinite mass
Simão CorreiaSeptember 18, 2018
Abstract
We consider the Cauchy problem for the nonlinear Schrödinger equation on R d , wherethe initial data is in H p R d q X L p p R d q . We prove local well-posedness for large ranges of p and discuss some global well-posedness results. Keywords : nonlinear Schrödinger equation; local well-posedness; global well-posedness,
AMS Subject Classification 2010 : 35Q55, 35A01.
In this work, we consider the classical nonlinear Schrödinger equation over R d : iu t ` ∆ u ` λ | u | σ u “ , u “ u p t, x q , p t, x q P R ˆ R d , λ P R , ă σ ă {p d ´ q ` and focus on the corresponding Cauchy problem u p q “ u P E , where E is a suitable functionspace. This model equation is the subject of more than fifty years of intensive research, whichmakes us unable to give a complete list of important references (we simply refer the monographs[2], [10], [11] and references therein). The usual framework one considers is E “ H p R d q , theso-called energy space, or more generally, E “ H s p R d q . A common property of these spaces isthat they are L -based. The reason for this constraint comes from the fact that the linear groupis bounded in L , but not in any other L p .In the sense of lifting the L constraint, we refer the papers [6], [7] and [4]. In the first paper,one considers local well-posedness on Zhidkov spaces E “ X k p R d q “ t u P L p R d q : ∇ u P H k ´ p R d qu . In the second, one takes the Gross-Pitaevskii equation and looks for local well-posedness on E “ t u P H loc p R d q : ∇ u P L p R d q , | u | ´ P L p R d qu . Finally, in the third work, one considers E “ H p R q ` X , where X is either a particular spaceof bounded functions with no decay or a subspace of L p R q (and not of L p R q ).The aim of this paper is to look for local well-posedness results over another class of spaces,namely E “ X p p R d q “ H p R d q X L p p R d q , ă p ă d {p d ´ q ` . In particular, we obtain local well-posedness in the most general energy space X σ ` p R d q andobtain global well-posedness over X p p R d q in the defocusing case λ ă for all p ď σ ` .1 emark . Our results can be extended to more general nonlinearities f p u q as in the H framework. We present our results for f p u q “ | u | σ u so not to complicate unnecessarily theproofs and deviate from the main ideas.We briefly explain the structure of this work: in Section 2, we derive the required groupestimates and show that the Schrödinger group is well-defined over X p p R d q . In Section 3, weshow local well-posedness for p ď σ ` , where the use of Strichartz estimates is available. Wealso prove global well-posedness for small σ (cf. Proposition 3.6). In Section 4, we deal with thecomplementary case p ą σ ` in dimensions d “ , . Notation.
The norm over L p p R d q will be denoted as }¨} p or }¨} L p , whichever is more convenient.The spatial domain R d will often be ommited. The free Schrödinger group in H p R d q is writtenas t S p t qu t P R . We write p ˚ “ dp {p d ´ p q ` . To avoid repetition, we hereby set ă p ă ˚ and ă σ ă {p d ´ q ` . We recall the essential Strichartz estimates. We say that p q, r q is an admissible pair if ď r ď ˚ , q “ d ˆ ´ r ˙ , r ‰ 8 if d “ . Lemma 2.1 (Strichartz estimates) . Given two admissible pairs p q, r q and p γ, ρ q , we have, forall sufficiently regular u and f and for any interval I Ă R , } S p¨q u } L q p I ; L r p R N qq À } u } (2.0) and ››››ż ă s ă t S p t ´ s q f p s q ds ›››› L q p I ; L r p R N qq À } f } L γ p I ; L ρ p R N qq . (2.1) Remark . The estimate (2.1) may be extended to other sets of admisssible pairs: see [5] and[12]. However, the linear estimate (2.1) is not valid for any other pairs and for u R L p R d q . Proposition 2.2 (Group estimates with loss of derivative) . Define k so that p k, p q is admissible.Then • (Linear estimate) For φ P S p R d q , } S p t q φ } p À } φ } p ` | t | ´ k } ∇ φ } , t P R . • (Non-homogeneous estimate) For f P C pr , T s ; S p R d qq and any p q, r q admissible, ››››ż ¨ S p¨ ´ s q f p s q ds ›››› L pp ,T q ; L p p R d qq À C p T q ´ } f } L pp ,T q ; L p p R d qq ` } ∇ f } L q pp ,T q ; L r p R d qq ¯ , (2.2) where C p¨q is a increasing bounded function over bounded intervals of R . Notice that, due to the scaling invariance of the Schrödinger equation, the polynomial growthin time in the linear estimate is unavoidable. 2 roof.
For the linear estimate, write u “ S p t q φ . Then u P C p R ; H p R d qq satisfies iu t ` ∆ u “ , u p q “ φ. Multiplying the equation by | u | p ´ ¯ u , integrating over R d and taking the imaginary part, weobtain p ddt } u p t q} pp ď ˇˇˇˇ Im ż | u | p ´ ¯ u ∆ u ˇˇˇˇ ď p ´ ż | u | p ´ | ∇ u | ď p ´ } u p t q} p ´ p } ∇ u p t q} p Thus we have ddt } u p t q} p ď p p ´ q} ∇ u p t q} p . An integration between and t P R and the linear Strichartz estimate yield } u p t q} p À } φ } p ` ż t } ∇ u p s q} p ds À } φ } p ` | t | ´ k ˆż t } ∇ u p s q} kp ds ˙ k À } φ } p ` | t | ´ k } ∇ φ } . For the non-homogeneous estimate, set v p t q “ ´ i ş t S p t ´ s q f p s q ds . Then v P C pr , T s ; H p R d qq satisfies iv t ` ∆ v “ f, v p q “ . As for the previous estimate, we have p ddt } v p t q} pp À } v p t q} p ´ p } ∇ v p t q} p ` } v p t q} p ´ p } f p t q} p and so ddt } v p t q} p À } ∇ v p t q} p ` } v p t q} p } f p t q} p À } ∇ v p t q} p ` } v p t q} p ` } f p t q} p . The required estimate now follows by direct integration in p , t q , ă t ă T , and by the non-homogeneous Strichartz estimate. Lemma 2.3 (Local Strichartz estimate without loss of derivatives) . Given f P C pr , T s , S p R d qq , ››››ż ¨ S p¨ ´ s q f p s q ds ›››› L pp ,T q ,L p p R d qq À C p T, q q} f } L q pp ,T q ,L p p R d qq , q ą d ˆ ´ p ˙ . (2.3) Proof.
This estimate follows easily from the decay estimates of the Schrödinger group: indeed,given ă t ă T , ››››ż t S p t ´ s q f p s q ds ›››› L p p R d q À ż T } S p t ´ s q f p s q} L p p R d q ds À ż T | t ´ s | d p ´ p q } f p s q} L p p R d q ds À ˜ż T | t ´ s | qd p ´ p q ds ¸ q } f } L q pp ,T q ,L p p R d qq . We set X p p R d q “ L p p R d q X H p R d q . emark . From the Gagliardo-Nirenberg inequality, we have H p R d q ã Ñ X p p R d q . Proposition 2.4.
The Schrödinger group t S p t qu t P R over H p R d q defines, by continuous exten-sion, a one-parameter continuous group on X p p R d q .Proof. Given any φ P H p R d q , we have } S p t q ∇ φ } “ } ∇ φ } . Together with Proposition 2.2, thisimplies that } S p t q φ } X p À p ` | t | ´ k q { } φ } X p , t P R . Therefore, for each fixed t P R , S p t q may be extended continuosly to X p . By density, it followseasily that S p t ` s q “ S p t q S p s q , t, s P R , and S p q “ I on X p . Finally, we prove continuity at t “ : given φ P X p p R d q and ǫ ą , take φ ǫ P H p R d q such that } φ ǫ ´ φ } X p ă ǫ. Then lim sup t Ñ } S p t q φ ´ φ } X p ď lim sup t Ñ ` } S p t qp φ ´ φ ǫ q} X p ` } S p t q φ ǫ ´ φ ǫ } X p ` } φ ǫ ´ φ } X p ˘ À lim sup t Ñ ´ p ` | t | ´ k q { } φ ´ φ ǫ } X p ` } S p t q φ ǫ ´ φ ǫ } H ¯ À ǫ. Remark . Fix d “ . Using the same ideas, one may easily observe that the Schrödingergroup is well-defined on the Zhidkov space X p R q “ t u P L p R q : ∇ u P H p R qu . Indeed, for any ď p ď 8 a direct integration of the equation gives ddt } u p t q} p ď } ∆ u p t q} p . Hence, choosing k so that p k, p q is an admissible pair, } u p t q} p ď } u } p ` ż t } ∆ u p s q} p ds ď } u } p ` Ct ´ k } ∆ u } L k pp ,t q ,L p q ď } u } p ` Ct ´ k } ∆ u } , where C is a constant independent on p (this comes from the fact that such a constant may beobtained via the interpolation between L t L x and L t L x ). Then, taking the limit p Ñ 8 , weobtain } u p t q} À } u } ` t } ∆ u } , t ą , u P H p R q . For higher dimensions, a similar procedure may be applied, at the expense of some derivatives(one must use Sobolev injection to control L p , with p large). As one might expect, this argumentdoes not provide the best possible estimate: in [6], one may see that } u p t q} À p ` t q p} u } ` } ∇ u } q , t ą , u P H p R q . Remark . One may ask if the required regularity is optimal: can we define the Schrödingergroup on X sp p R d q : “ H s p R d q X L p p R d q ? What is the optimal s ? Taking into consideration theprevious remark, we conjecture that it should be possible to lower the regularity assumption.This entails a deeper analysis of the Schrödinger group, as it was done in [6].4 Local well-posedness for p ď σ ` In order to clarify what do we mean by a solution of (NLS), we give the following
Definition 3.1 (Solution over X p p R d q ) . Given u P X p p R d q , we say that u P C pr , T s , X p p R d qq is a solution of (NLS) with initial data u if the Duhamel formula is valid: u p t q “ S p t q u ` iλ ż t S p t ´ s q| u p s q| σ u p s q ds, t P r , T s . Throughout this section, let p γ, ρ q and p q, r q be admissible pairs such that r “ p σ ` q ρ “ max t σ ` , p u . (3.1)It is easy to check that such pairs are well-defined for p ď σ ` . Proposition 3.2 (Uniqueness over X p p R d q ) . Suppose that p ď σ ` . Let u , u P C pr , T s , X p p R d qq be two solutions of (NLS) with initial data u P X p p R d q . Then u ” u .Proof. Taking the difference between the Duhamel formula for u and u , u p t q ´ u p t q “ iλ ż t S p t ´ s q p| u p s q| σ u p s q ´ | u p s q| σ u p s qq ds Then, for any interval J “ r , t s Ă r , T s , since X p p R d q ã Ñ L r p R d q , } u ´ u } L q p J,L r q À }| u | σ u ´ | u | σ u } L γ p J,L ρ q À }p} u } σr ` } u } σr q} u ´ u } r } L γ p J q À ` } u } L pr ,T s ,X p p R d qq ` } u } L pr ,T s ,X p p R d qq ˘ } u ´ u } L γ p J,L r q À C p T q} u ´ u } L γ p J,L r q The claimed result now follows from [2, Lemma 4.2.2].
Theorem 3.3 (Local well-posedness on X p p R d q , p ď σ ` ) . Given u P X p p R d q , there exists T “ T p} u } X p q ą and an unique solution u P C pr , T q , X p p R d qq X L γ pp , T q , W ,ρ p R d qq X L q pp , T q , W ,r p R d qq of (NLS) with initial data u . One has u ´ S p¨q u P C pr , T s .L p R d qq X L q pp , T q , L r p R d qq X L γ pp , T q , L ρ p R d qq . (3.2) Moreover, the solution depends continuously on the initial data and may be extended in an uniqueway to a maximal time interval r , T ˚ p u qq . If T ˚ p u q ă 8 , then lim t Ñ T ˚ p u q } u p t q} X p “ `8 . Remark . The property (3.2) is a type of nonlinear "smoothing" effect: the integral term inDuhamel’s formula turns out to have more integrability than the solution itself (a similar propertywas seen in [7]). This insight allows the use of Strichartz estimates at the zero derivatives level.Without this possibility, one would be restricted to the estimate (2.3) and the possible ranges of σ and p would be significantly smaller. 5 roof. Step 1. Define S “ L pp , T q , L q X L q pp , T q , L r q X L γ pp , T q , L ρ q . and S “ L pp , T q , H q X L q pp , T q , W ,r q X L γ pp , T q , W ,ρ q . Consider the space E “ ! u P L pp , T q , X p q X L γ pp , T q , W ,ρ q X L q pp , T q , W ,r q : ~ u ~ : “ } u } L pp ,T q ,L p q ` } u ´ S p¨q u } S ď M ) . endowed with the distance d p u, v q “ } u ´ v } S . It is not hard to check that p E , d q is a complete metric space: indeed, if t u n u n P N is a Cauchysequence in E , then t u n ´ S p¨q u u n P N is a Cauchy sequence in S . Then there exists u P D pr , T sˆ R d q such that u n ´ S p¨q u Ñ u ´ S p¨q u in S . By [2, Theorem 1.2.5], this convergence impliesthat u ´ S p¨q u P S , } u ´ S p¨q u } S ď lim inf } u n ´ S p¨q u } S Finally, it follows from the Gagliardo-Nirenberg inequality that, for some ă θ ă , } u n ´ u } L pp ,T q ,L p q À } u n ´ u } ´ θL pp ,T q ,L q } ∇ u n ´ ∇ u } θL pp ,T q ,L q Ñ and so u n Ñ u in L pp , T q , L p q . Step 2.
Define, for any u P E , p Φ u qp t q “ S p t q u ` iλ ż t S p t ´ s q| u p s q| σ u p s q ds, ă t ă T. It follows from the definition of r (see (3.1)) that X p p R d q ã Ñ L r p R d q . Then } Φ u ´ S p¨q u } S À }| u | σ u } L γ pp ,T q ,W ,ρ q À }} u } σr p} u } r ` } ∇ u } r q} L γ p ,T q À ››› } u } σX p p} u } X p ` } ∇ p u ´ S p¨q u q} r ` } S p¨q ∇ u } r ››› L γ p ,T q À T γ } u } σ ` L pp ,T q ,X p q ` T γ ´ q } u } σL pp ,T q ,X p q } ∇ p u ´ S p¨q u q} L q pp ,T q ,L r q ` T γ ´ q } u } σL pp ,T q ,X p q } S p¨q ∇ u } L q pp ,T q ,L r q À ´ T q ` T q ´ q ¯ p M σ ` ` } u } σ ` X p q . It follows that, for M „ } u } X p and T sufficiently small, we have Φ : E ÞÑ E . Step 3.
Now we show a contraction estimate: given u, v P E , d p Φ p u q , Φ p v qq À }| u | σ u ´ | v | σ v } L γ p L ρ q À }p} u } σr ` } v } σr q} u ´ v } r } L γ p ,T q À T γ ´ q ´ } u } σL pp ,T q ,X p p R d q ` } u } σL pp ,T q ,X p p R d q ¯ } u ´ v } L q pp ,T q ,L r q À T q ´ q ´ M σ ` } u } σX p ¯ d p u, v q . T “ T p} u } X p q small enough, the mapping Φ : E ÞÑ E is a strict contractionand so, by Banach’s fixed point theorem, Φ has a unique fixed point over E . This gives the localexistence of a solution u P C pr , T s , X p p R d qq of (NLS) with initial data u . From the uniquenessresult, such a solution can then be extended to a maximal interval of existence p , T ˚ p u qq .If such an interval is bounded, then necessarily one has } u p t q} X p Ñ 8 as t Ñ T ˚ p u q . Thecontinuous dependence on the initial data follows as in the H case (see, for example, the proofof [2, Theorem 4.4.1]) Remark . The condition p ď σ ` is necessary for one to use Strichartz estimates with noderivatives. Indeed, when one applies Strichartz to the integral term of the Duhamel formula,one has ››››ż ¨ S p¨ ´ s q| u p s q| σ u p s q ds ›››› L q pp ,T q ,L r q À } u } σ ` L γ σ ` q pp ,T q ,L ρ σ ` q q for any admissible pairs p q, r q and p γ, ρ q . Since the solution u only lies on spaces with spatialintegrability larger or equal than p , one must have p ď ρ p σ ` q ď σ ` (because ρ ě ). Proposition 3.4 (Persistence of integrability) . Suppose that ˜ p ă p ď σ ` . Given u P X ˜ p p R d q ,consider the X p p R d q -solution u P C pr , T ˚ p u qq , X p q of (NLS) with initial data u . Then u P C pr , T ˚ p u qq , X ˜ p q .Proof. By the local well-posedness result over X ˜ p p R d q and by the uniqueness over X p p R d q , thereexists a time T ą such that u P C pr , T s , X ˜ p p R d qq . Thus the statement of the proposition isequivalent to saying that u does not blow-up in X ˜ p p R d q at a time T ă T ă T ˚ p u q . Since u isbounded in X p over r , T s , it follows from the local existence theorem that } u ´ S p¨q u } L pp ,T q ,H q ă 8 . Then, for any ă t ă T , } u } L pp ,t q ,X ˜ p q À } S p¨q u } L pp ,t q ,X ˜ p q ` } u ´ S p¨q u } L pp ,t q ,X ˜ p q À } u } X ˜ p ` } u ´ S p¨q u } L pp ,t q ,H q ă 8 , which implies that u does not blow-up at time t “ T . Proposition 3.5 (Conservation of energy) . Suppose that p ď σ ` . Given u P X p p R d q , thecorresponding solution u of (NLS) with initial data u satisfies E p u p t qq “ E p u q : “ } ∇ u } ´ λσ ` } u } σ ` σ ` , ă t ă T p u q . Consequently, if λ ă , then T ˚ p u q “ 8 . Moreover, if λ ą and T ˚ p u q ă 8 , then lim t Ñ T ˚ p u q } ∇ u p t q} “ lim t Ñ T ˚ p u q } u p t q} σ ` “ 8 . (3.3) Proof.
Since the conservation law is valid for u P H p R d q , through a regularization argument,the same is true for any u P X p p R d q . If λ ă , one has } u p t q} X σ ` p R d q À E p u q , ă t ă T ˚ p u q . By the blow-up alternative, this implies that u , as a X σ ` p R d q solution, is globally defined.By persistence of integrability, this implies that u is global in X p p R d q . If λ ą , suppose bycontradiction that (3.3) is not true. Then, by conservation of energy, u is bounded in X σ ` p R d q and therefore it is globally defined (as an X σ ` p R d q solution, but also as an X p p R d q solution, bypersistence of integrability). 7 roposition 3.6. Fix λ ă . If σ ` p d ` q σ ď , then, for any u P X σ ` p R d q , the corresponding solution u of (NLS) is globally defined. Remark . Notice that the condition on σ implies that σ ă min t? , {p d ` qu ă { d . Proof.
By contradiction, assume that u blows-up at time t “ T . The previous proposition thenimplies that lim t Ñ T } ∇ u p t q} “ 8 . The first step is to obtain a corrected mass conservation estimate: indeed, by direct integrationof the equation, ddt } u p t q ´ S p t q u } “ Im ż | u p t q| σ u p t qp u p t q ´ S p t q u q “ ´ Im ż | u p t q| σ u p t q S p t q u À } u p t q} σ ` σ ` } S p t q u } σ ` . Integrating on p , t q , } u p t q ´ S p t q u } À ż t } u p s q} σ ` σ ` } S p s q u } σ ` ds À } S p¨q u } L pp ,T q ,L σ ` q ż t } u p s q} σ ` σ ` ds À ż t } u p s q} σ ` σ ` ds All of these formal computations can be justified by a suitable regularization and approximationargument. The next step is to use the conservation of energy and the Gagliardo-Nirenberginequality to obtain a bound on } ∇ u p t q} . } ∇ u p t q} “ E p u q ` σ ` } u p t q} σ ` σ ` À ` } u p t q ´ S p t q u } σ ` σ ` ` } S p t q u } σ ` σ ` À ` } ∇ p u p t q ´ S p t q u q} dσ } u p t q ´ S p t q u } ´p d ´ q σ À ` } ∇ p u p t q ´ S p t q u q} dσ ˆż t } u p s q} σ ` σ ` ds ˙ ´p d ´ q σ For t close to T , } ∇ p u p t q ´ S p t q u q} „ } ∇ u p t q} and, by conservation of energy, } u p t q} σ ` σ ` À } ∇ u p t q} σ ` σ ` Thus } ∇ u p t q} ´ dσ À ` ˆż t } ∇ u p s q} σ ` σ ` ds ˙ ´p d ´ q σ . which, together with the condition on σ , implies that g p t q : “ } ∇ u p t q} σ ` σ ` ď } ∇ u p t q} p ´ dσ q ´p d ´ q σ À ` ż t } ∇ u p s q} σ ` σ ` ds À ` ż t g p s q ds. The desired contradiction now follows from a standard application of Gronwall’s lemma.8 emark L -subcritical regime) . As it is well-known, theglobal existence in H p R d q for σ ă { d follows easily from the conservation of mass and energyand from the Gagliardo-Nirenberg inequality. In Proposition 3.6, we managed to perform asimilar argument by using the corrected mass M p t q “ } u p t q ´ S p t q u } . However, the range of exponents for which the result is valid still leaves much to be desired. Weare left with some questions: Is there another choice for "corrected mass" that allows a largerrange of exponents? Is it possible that the large tails of the initial data contribute to blow-upbehaviour?
Remark L -(super)critical regime) . One may ask whether the known blow-up results for σ ě { d can be extended to initial data in X p p R d q which do not lie in L p R d q .First of all, notice that X p p R d q X L p| x | dx q ã Ñ L p R d q . Thus, in order to obtain blow-up outside L , one must first show blow-up in H without thefinite variance assumption. This is an open problem, which has been solved in [9] under radialhypothesis and relying heavily on the conservation of mass (which is unavailable on X p p R d q ).For the nonradial case, recent works (see, for example, [8]) only manage to prove unboundednessof solutions of negative energy. The problem of blow-up solutions strictly in X p p R d q is an evenharder problem, requiring a better control on the tails of the solution. Remark . It is useful to understand how scalings affect the X p p R d q norm: recalling that the (NLS) is invariant under the scaling u λ p t, x q “ λ { σ u p λ t, λx q , we have } u λ p t q} p “ λ σ ´ dp } u p λ t q} p , } ∇ u λ p t q} “ λ σ ` ´ d } ∇ u p λ t q} . Thus the (NLS) is X p p R d q -subcritical for σ ă p { d . In this situation, global existence for smalldata is equivalent to global existence for any data. Recall, however, that, for σ ě { d , existenceof blow-up phenomena is known for special initial data in H p R d q ã Ñ X p p R d q . Therefore, it isimpossible to obtain a global existence result for small data for { d ď σ ă p { d . Notice that inthe energy case p “ σ ` , one has σ ă p { d for any σ ` ă ˚ . Remark . The main obstacle in proving global existencefor small data turns out to be the linear part of the Duhamel formula S p t q u , since there isn’t,to our knowledge, a way to bound uniformly this term over X p p R d q . The other possibility is toleave the linear term with a space-time norm: indeed, for some powers σ ą { d , it is well-knownthat, if u P H p R d q is such that } S p¨q u } L a pp , ,L σ ` p R d qq is small , a “ σ p σ ` q ´ σ p d ´ q , then the corresponding solution of (NLS) is globally defined (see [3]). It is not hard to checkthat the result can be extended to u P X σ ` p R d q . p ą σ ` As it was observed in Remark 3.2, the condition p ď σ ` was necessary in order to useStrichartz estimates with no loss in regularity. For p ą σ ` , in order to estimate L t L px , one9ust turn to estimate (2.2), which has a loss of one derivative. Therefore the distance one definesfor the fixed-point argument must include norms with derivatives. This implies the need of alocal Lipschitz condition || u | σ ∇ u ´ | v | σ ∇ v | À C p| u | , | v | , | ∇ u | , | ∇ v |q p| u ´ v | ` | ∇ p u ´ v q|q , which we can only accomplish for σ ě .Because of the restriction σ ě , one must have ă p ă ˚ , which excludes any dimensiongreater than three. For d “ , it turns out that no range of p ą σ ` can be considered. Indeed,if one uses (2.2) with f “ | u | σ u , ››››ż ¨ S p¨ ´ s q| u p s q| σ u p s q ds ›››› L pp ,T q ,L p q À } u } σ ` L σ ` pp ,T q ,L p p σ ` q q ` } ∇ p| u | σ u q} L q pp ,T q ,L r q . We focus on the first norm on the right hand side. To control such a term, either X p ã Ñ L p p σ ` q x and } u } L σ ` pp ,T q ,L p p σ ` q À T σ ` } u } L pp ,T q ,X p q , or, setting r ě so that ´ r “ ´ p p σ ` q , one estimates } u } L σ ` pp ,T q ,L p p σ ` q q À } ∇ u } L σ ` pp ,T q ,L r q À T σ ` ´ q } ∇ u } L q pp ,T q ,L r q . In the first case, one needs ă p p σ ` q ă ˚ “ . In the second, one must impose σ ` ă q .A simple computation yields p p σ ` q ă , which is again impossible, since p p σ ` q ą . Theorem 4.1 (Local well-posedness on X p p R d q for d “ , ) . Given u P X p p R d q , there exists T “ T p} u } X p q ą and an unique solution u P C pr , T s , X p p R d qq of (NLS) with initial data u . The solution depends continuously on the initial data and may beextended uniquely to a maximal interval r , T ˚ p u qq . If T ˚ p u q ă 8 , then lim t Ñ T ˚ p u q } u p t q} X p “ `8 . Proof.
Consider the space E “ ! u P L pp , T q , X p q : ~ u ~ : “ } u } L pp ,T q ,X p q ď M ) . endowed with the natural distance d p u, v q “ ~ u ´ v ~ . The space p E , d q is clearly a complete metric space. If u, v P E , then }| u | σ u ´ | v | σ v } L pp ,t q ,L p q À ż t ´ } u } σp p σ ` q ` } v } σp p σ ` q ¯ } u ´ v } p p σ ` q ds X p p R d q ã Ñ L p p σ ` q p R d q , }| u | σ u ´ | v | σ v } L pp ,t q ,L p q À T ´ } u } σL pp ,t q .X p q ` } v } σL pp ,t q .L p q ¯ } u ´ v } L pp ,t q .X p q . (4.1)Choose an admissible pair p γ, ρ q with ρ sufficiently close to 2. We have } ∇ p| u | σ u ´ | v | σ v q } L γ pp ,T q ,L ρ q À ››` | u | σ ´ ` | v | σ ´ ˘ p| u ´ v || ∇ v | ` | v || ∇ p u ´ v q|q ›› L γ pp ,T q ,L ρ q . As an example, we treat the term | u | σ ´ | u ´ v || ∇ v | : }| u | σ ´ | u ´ v || ∇ v |} ρ À } u } σ ´ σρ ´ ρ } u ´ v } σρ ´ ρ } ∇ v } À } u } σX p } u ´ v } X p , Therefore } ∇ p| u | σ u ´ | v | σ v q } L γ pp ,T q ,L ρ q À T γ ´ } u } σL pp ,T q ,X p q ` } v } σL pp ,T q ,X p q ¯ } u ´ v } L pp ,T q ,X p q À T γ M σ d p u, v q . (4.2)For u P E , define Φ p u qp t q “ S p t q u ` iλ ż t S p t ´ s q| u p s q| σ u p s q ds, ď t ď T. The estimates (4.1) and (4.2), together with (2.2) and Strichartz’s estimates then imply that ~ Φ p u q~ À} u } X p ` ››››ż ¨ S p¨ ´ s q| u p s q| σ u p s q ds ›››› L pp ,T q ,L p q ` ››››ż ¨ S p¨ ´ s q| u p s q| σ u p s q ds ›››› L pp ,T q , H q À } u } X p ` ´ }| u | σ u } L pp ,T q ; L p q ` } ∇ p| u | σ u q} L γ pp ,T q ; L ρ q ¯ À } u } X p ` ´ T ` T γ ¯ M σ ` and d p Φ p u q , Φ p v qq À ´ }| u | σ u ´ | v | σ v } L pp ,T q ; L p q ` } ∇ p| u | σ u q ´ ∇ p| v | σ v q} L γ pp ,T q ; L ρ q ¯ À ´ T ` T γ ¯ M σ d p u, v q . (4.3)Choosing M „ } u } X p , for T “ T p} u } X p q small enough, it follows that Φ : E ÞÑ E is a strictcontraction. Banach’s fixed point theorem now implies that Φ has a unique fixed point over E ,which is the unique solution u of (NLS) with initial data u on the interval p , T q . This solutionmay then be extended uniquely to a maximal interval of existence p , T p u qq . The blow-upalternative follows by a standard continuation argument. Finally, if u, v are two solutions withinitial data u , v P X p p R d q , as in (4.3), one has d p u, v q “ d p Φ p u q , Φ p v qq À } u ´ v } X p ` ´ T ` T γ ¯ M σ d p u, v qÀ } u ´ v } X p ` ´ T ` T γ ¯ ` max t} u } X p , } v } X p u ˘ σ d p u, v q Thus, for T “ T p} u } X p , } v } X p q small, d p u, v q À } u ´ v } X p , and continuous dependence follows. 11 roposition 4.2 (Persistence of integrability) . Fix d “ , and p ą ˜ p . Given u P X ˜ p p R d q ,consider the X p p R d q -solution u P C pr , T ˚ p u qq , X p q of (NLS) with initial data u . Then u P C pr , T ˚ p u qq , X ˜ p q .Proof. As in the proof of Proposition 3.4, given T ă T ˚ p u q , one must prove that the L ˜ p normof u is bounded over p , T q . Applying (2.2) to the Duhamel formula of u , } u } L pp ,T q ,L ˜ p q À } u } X ˜ p ` }| u | σ u } L pp ,T q ,L ˜ p q ` }| u | σ | ∇ u |} L γ pp ,T q ,L ρ q , for any admissible pair p γ, ρ q . The penultimate term is treated using the injection X p p R d q ã Ñ L ˜ p p σ ` q : }| u | σ u } L pp ,T q ,L ˜ p q “ } u } σ ` L σ ` pp ,T q ,L ˜ p p σ ` q q À T } u } σ ` L pp ,T q ,X p p R d q ă 8 . Choose ρ sufficiently close to 2 so that X p p R d q ã Ñ L σρ ´ ρ p R d q . Then }| u | σ | ∇ u |} L γ pp ,T q ,L ρ q À ›››› } u } σ σρ ´ ρ } ∇ u } ›››› L γ p ,T q À T γ } u } σ ` L pp ,T q ,X p q ă 8 . Therefore } u } L pp ,T q ,L ˜ p q is finite and the proof is finished. In light of the results we have proven, we highlight some new questions that have risen:1. Local well-posedness: In dimensions d ě , the local well-posedness in the case p ą σ ` remains open. Is this optimal? As we have arqued in Remark 3.2, this case requires newestimates for the Schrödinger group.2. Global well-posedness: this problem is completely open for p ą σ ` . Even if the energy iswell-defined, there are still several cases where global well-posedness (even for small data)remains unanswered.3. New blow-up behaviour: in the opposite perspective, is it possible to exhibit new blow-upphenomena? This would be especially interesting either for the defocusing case or for the L -subcritical case, where blow-up behaviour in H is impossible.4. Stability of ground-states: in the H framework, the work of [1] has shown that the ground-states are orbitally stable under H perturbations. Does the result still hold if we consider X p perturbations? The author was partially suported by Fundação para a Ciência e Tecnologia, through the grantsUID/MAT/04561/2013 and SFRH/BD/96399/2013.12 eferences [1] T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinearSchrödinger equations.
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Simão Correia
CMAF-CIO and FCULCampo Grande, Edifício C6, Piso 2, 1749-016 Lisboa (Portugal) [email protected]@fc.ul.pt