Elsevier

Advances in Mathematics

Volume 375, 2 December 2020, 107407
Advances in Mathematics

3D quadratic NLS equation with electromagnetic perturbations

https://doi.org/10.1016/j.aim.2020.107407Get rights and content

Abstract

In this paper we study the asymptotic behavior of a quadratic Schrödinger equation with electromagnetic potentials. We prove that small solutions scatter. The proof builds on earlier work of the author for quadratic NLS with a non magnetic potential. The main novelty is the use of various smoothing estimates for the linear Schrödinger flow in place of boundedness of wave operators to deal with the loss of derivative.

As a byproduct of the proof we obtain boundedness of the wave operator of the linear electromagnetic Schrödinger equation on an L2 weighted space for small potentials, as well as a dispersive estimate for the corresponding flow.

Introduction

We consider a quadratic NLS equation with electromagnetic potential set on R3:{itu+Δu=i=13ai(x)iu+V(x)u+u2u(t=1)=u1. This general form of potential includes the classical Hamiltonian Schrödinger equation with electromagnetic potentials:{itu=HAuHA=(iA(x))2+VA(x)=(a1(x),a2(x),a3(x)), which corresponds to the usual Schrödinger equation with an external magnetic field B=curl(A) as well as an external electric field E=V. Its Hamiltonian isH(u)=12R3|(iA(x))u|2+V(x)|u|2dx. In our setting the potentials will be assumed to be small, and our goal is to study the asymptotic behavior of solutions to the equation (1.1).

Besides the physical interest of the problem, we are motivated by the fact that (1.1) is a toy model for the study of linearizations of dispersive equations around non-zero remarkable solutions (traveling waves or solitons for example). This is why we elected to work with the equation (1.1) and not a nonlinear version of (1.2).

The present article is a continuation of the author's earlier work [25], where the above equation was considered for ai=0. The main reason for adding the derivative term is that most models of physical relevance are quasilinear, and their linearizations will generically contain such derivative terms. Note that a more complete model would consist in treating a general quadratic nonlinearity Q(u,u): the present article leaves out the cases of nonlinearities u2 and |u|2. Our method would apply for the nonlinearity u2 (in fact this is a strictly easier problem). However the nonlinearity |u|2 is currently out of reach. In fact even in the flat case (no potentials are present), the problem has not been completely answered. We refer to the article of X. Wang [30] for more on this subject.

Regarding existing results on the behavior as t of solutions to equations of this type (nonlinearity with potential) we mention some works on the Strauss conjecture on non flat backgrounds: the equations considered have potential parts that lose derivatives, but the nonlinearity (of power type) typically has a larger exponent than what we consider in the present work. We can for example cite the work of K. Hidano, J. Metcalfe, H. Smith, C. Sogge, Y. Zhou ([16]) where the conjecture is proved outside a nontrapping obstacle. H. Lindblad, J. Metcalfe, C. Sogge, M. Tohaneanu and C. Wang ([26]) proved the conjecture for Schwarzschild and Kerr backgrounds. The proofs of these results typically rely on weighted Strichartz estimates to establish global existence of small solutions.

In this paper we prove that small, spatially localized solutions to (1.1) exist globally and scatter. We take the opposite approach to the works cited above. Indeed we deal with a stronger nonlinearity which forces us to take into account its precise structure. We rely on the space-time resonance theory of P. Germain, N. Masmoudi and J. Shatah (see for example [10]). It was developed to study the asymptotic behavior of very general nonlinear dispersive equations for power-type nonlinearities with small exponent (below the so-called Strauss exponent). In the present study the nonlinearity is quadratic (which is exactly the Strauss exponent in three dimension) hence the need to resort to this method. It has been applied to many models by these three authors and others. Without trying to be exhaustive, we can mention for example the water waves problem treated in various settings ([11], [13], [21], [22]), the Euler-Maxwell equation in [14], or the Euler-Poisson equation in [20]. Similar techniques have also been developed by other authors: S. Gustafson, K. Nakanishi and T.-P. Tsai studied the Gross-Pitaevskii equation in [15] using a method more closely related to J. Shatah's original method of normal forms [29]. M. Ifrim and D. Tataru used the method of testing against wave packets in [17] and [18] to study similar models, namely NLS and water waves in various settings. We also note that prior to the author's first work [25] which treated the case of a quadratic NLS equation with small time-dependent electric perturbations, the space-time resonance method had been adapted in the case of a large time-independent electric perturbation in [12]. However the quadratic nonlinearities considered are different, and the methods used also substantially differ.

The difficulty related to the strong nonlinearity was already present in [25]. However, in the present context, the derivative forces us to modify the approach developed in that paper since we must incorporate smoothing estimates into the argument. The inequalities we use in the present work were first introduced by C. Kenig, G. Ponce and L. Vega in [24] to prove local well-posedness of a large class of nonlinear Schrödinger equations with derivative nonlinearities. They allow to recover one derivative, which will be enough to deal with (1.1). Another estimate of smoothing-Strichartz type proved by A. Ionescu and C. Kenig in [19] will play an important role in the paper. Let us also mention that, in the case of magnetic Schrödinger equations with small potentials, a large class of related Strichartz and smoothing estimates have been proved by V. Georgiev, A. Stefanov and M. Tarulli in [9]. For large potentials of almost critical decay, Strichartz and smoothing estimates have been obtained by B. Erdoğan, M. Goldberg and W. Schlag in [6], [7], and a similar result was obtained recently by d'Ancona in [3]. A decay estimate for that same linear equation was proved by P. d'Ancona and L. Fanelli in [4] for small but rough potentials. A corollary of the main result of the present paper is a similar decay estimate under stronger assumptions on the potentials, but for a more general equation. Regarding dispersive estimates for the linear flow, we also mention the work of L. Fanelli, V. Felli, M. Fontelos and A. Primo in [8] where decay estimates for the electromagnetic linear Schrödinger equation are obtained in the case of particular potentials of critical decay. The proof is based on a representation formula for the solution of the equation. Finally, for the linear electromagnetic Schrödinger equation, a corollary of our main theorem is that its wave operator is bounded on a space that can heuristically be thought of as xL2H10, see the precise statement in Corollary 1.7 below.

To deal with the full equation, we must therefore use both smoothing and space-time resonance arguments simultaneously. The general idea is to expand the solution as a power series using Duhamel's formula repeatedly. This type of method has been used in the study of linear Schrödinger equations through Born series, see for example [28]. As we mentioned, this general plan was already implemented by the author for a less general equation in [25], where there is no loss of derivatives. The additional difficulty coming from high frequencies forces us to modify the approach, and follow a different strategy in the multilinear part of the proof (that is in the estimates on the iterated potential terms). Note that along the way we obtain a different proof of the result in [25]. In particular we essentially rely on Strichartz and smoothing estimates for the free linear Schrödinger equation, instead of the more stringent boundedness of wave operators. This would allow us to relax the assumptions on the potential in [25].

We start this section with some notations that will be used in the paper.

First we recall the formula for the Fourier transform:fˆ(ξ)=Ff(ξ)=R3eixξf(x)dx hence the following definition for the inverse Fourier transform:fˇ(x)=[F1f](x)=1(2π)3R3eixξf(ξ)dξ. Now we define Littlewood-Paley projections. Let ϕ be a smooth radial function supported on the annulus C={ξR3;11.04|ξ|1.04×1.1} such thatξR3{0},jZϕ(1.1jξ)=1. Notice that if jj>1 then 1.1jC1.1jC=.

We will denote Pk(ξ):=ϕ(1.1kξ) the Littlewood-Paley projection at frequency 1.1k.

Similarly, Pk(ξ) will denote the Littlewood-Paley projection at frequencies less than 1.1k.

It was explained in [25] why we localize at frequency 1.1k and not 2k. (See Lemma 7.8 in that paper.)

We will also sometimes use the notation fkˆ(ξ)=Pk(ξ)fˆ(ξ).

Now we come to the main norms used in the paper:

We introduce the following notation for mixed norms of Lebesgue type:fLxjpLxj˜q=f(,...,,xj,,..)Lx1,...,xj1,xj+1,...qLxjp. To control the profile of the solution we will use the following norm:fX=supkZξfkˆLx2 Roughly speaking, it captures the fact that the solution has to be spatially localized around the origin.

For the potentials, we introduce the following controlling norm:VY=VLx1+VLx+j=13|V|1/2Lxj˜Lxj2.

With these notations, we are ready to state our main theorem.

We prove that small solutions to (1.1) with small potentials exist globally and that they scatter. More precisely the main result of the paper is

Theorem 1.1

There exists ε>0 such that if ε0,δ<ε and if u1, V and ai satisfyVY+xVY+(1Δ)5VYδ,aiY+xaiY+(1Δ)5aiYδ,eiΔu1Hx10+eiΔu1Xε0, then (1.1) has a unique global solution. Moreover it satisfies the estimatesupt[1;)u(t)Hx10+eitΔu(t)X+supkZtuk(t)Lx6ε0. Moreover it scatters in Hx10: there exists uHx10 and a bounded operator W:Hx10Hx10 such thateitΔWu(t)uHx100 as t.

Remark 1.2

We have not strived for the optimal assumptions on the potentials or the initial data. It is likely that the same method of proof, at least in the case where ai=0, allows for potentials with almost critical decay (that is VLx3/2 and xVLx3 and similar assumptions on its derivative). Similarly the Hx10 regularity can most likely be relaxed.

Remark 1.3

Unlike in the earlier work [25], we cannot treat the case of time dependent potentials. This is mainly due to the identity (5.3) and its use in the subsequent proofs of our multilinear lemmas.

Remark 1.4

A similar scattering statement could be formulated in the space X although it is more technical. For this reason we have elected to work in Hx10 (see the proof in Appendix C).

As we mentioned above, the result proved in Theorem 1.1 has a direct consequence for the linear flow of the electromagnetic equation. We have the following corollary, which provides a decay estimate for the flow as well as a uniform in time boundedness of the profile of the solution on the space X.

Corollary 1.5

There exists ε>0 such that if δ<ε and if u1, V and ai satisfyVY+xVY+(1Δ)5VYδ,aiY+xaiY+(1Δ)5aiYδ,eiΔu1Hx10+eiΔu1X<+, then the Cauchy problem{itu+Δu=i=13ai(x)iu+V(x)uu(t=1)=u1 has a unique global solution u(t) that obeys the estimatesupt[1;)u(t)Hx10+eitΔu(t)X+supkZtuk(t)Lx6eiΔu1Hx10+eiΔu1X.

Remark 1.6

As we noted above for the nonlinear equation, the assumptions made on the regularity and decay of the potentials are far from optimal. We believe that minor changes in the proof would lead to much weaker conditions in the statement above.

Now we write a corollary of the above estimate (1.5) in terms of the wave operator for the linear electromagnetic Schrödinger operator corresponding to (1.2).

Indeed the X part of this estimate can be written in that settingsupt[1;)eitΔu(t)X=supt[1;)eitΔei(t1)HAu1X where we recall that HA=(iA(x))2+V. We directly deduce the following

Corollary 1.7

Let W denote the wave operator of HA. There exists ε>0 such that for every δ<ε, if the potentials A, V and the initial data u1 satisfyVY+xVY+(1Δ)5VYδ,AiY+xAiY+(1Δ)5AiYδ,(Ai)2Y+x(Ai)2Y+(1Δ)5(Ai)2Yδ,eiΔu1Hx10+eiΔu1X<+, then we haveWu1Hx10+Wu1XeiΔu1Hx10+eiΔu1X.

We work with the profile of the solution f(t)=eitΔu(t).

The local wellposedness of (1.1) in Hx10 follows from the estimate proved (in a much more general setting) by S. Doi in [5]:fLt([1;T])Hx10f1Hx10+(T1)fLt([1;T])Hx102

The proof of global existence and decay relies on a bootstrap argument: we assume that for some T>1 and for ε1=Aε0 (A denotes some large number) the following bounds holdsupt[1;T]f(t)Xε1,supt[1;T]f(t)Hx10ε1, and then we prove that these assumptions actually imply the stronger conclusionssupt[1;T]f(t)Xε12,supt[1;T]f(t)Hx10ε12. The main difficulty is to estimate the X norm. To do so we expand ξfˆ as a series by essentially applying the Duhamel formula recursively. The difference with [25] is that, for high output frequencies, iterating the derivative part will prevent the series from converging if we use the same estimates as in that paper. To recover the derivative loss we use smoothing estimates which allow us to gain one derivative back at each step of the iteration. It is at this stage (the multilinear analysis) that the argument from [25] must be modified. Instead of relying on the method of M. Beceanu and W. Schlag [1], [2], the estimations are done more in the spirit of K. Yajima's paper [31].

Our discussion here and in the next subsection will be carried out for a simpler question than that tackled in this paper. However it retains the main novel difficulty compared to the earlier paper [25], namely the loss of derivative in the potential part. More precisely, we see how to estimate the Lx2 norm of fˆ(t,ξ). First, we explain the way we generate the series representation of fˆ(t,ξ). We consider the Duhamel formula for fˆ: The potential part has the form0tR3eis(|ξ|2|η1|2)W1ˆ(ξη1)α1(η1)fˆ(s,η1)dη1ds, where W1 denotes either ai or V and α1(η1)=1 if W1=V and η1,i if W1=ai.

The general idea is to integrate by parts in time in that expression iteratively to write fˆ as a series made up of the boundary terms remaining at each step. Roughly speaking we will obtain two types of terms, corresponding either to the potential part or the bilinear part of the nonlinearity:γ=1n1Wγˆ(ηγ)αγ(ηγ)|ξ|2|ηγ|2ηnWnˆ(ηn1ηn)αn(ηn)eit(|ξ|2|ηn|2)fˆ(t,ηn)dηndη andγ=1n1Wγˆ(ηγ)αγ(ηγ)|ξ|2|ηγ|2ηneit(|ξ|2|ηn1ηn|2|ηn|2)fˆ(t,ηn1ηn)fˆ(t,ηn)dηndη.

Now we prove that the series obtained in the previous section converges in LtLx2. We prove estimates like(1.8),(1.9)LtLx2Cnδnε1 for some universal constant C and where the implicit constant does not depend on n. Heuristically, each V factor contributes a δ in the estimate.

First, we write that in physical space we have, roughly speaking:(1.8)=0τ1eiτ1ΔW1D1˜0τ2τ1ei(τ2τ1)ΔW2D2˜...×0τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ1...dτn, where we denoted Di˜ the operator equal to 1 if Wi=V and xj if Wi=aj.

Then, using Strichartz estimates, we can write(1.8)Lx2W1D1˜0τ2τ1ei(τ2τ1)ΔW2D2˜...×0τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ2...dτnLτ12Lx6/5. Next if W1=V, then we can use Strichartz estimates again and write(1.8)Lx2Cδ0τ2τ1ei(τ2τ1)ΔW2D2˜...×0τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ2...dτnLτ12Lx6CδW2D2˜...0τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ3...dτnLτ22Lx6/5. If W1=ai then in this case we use smoothing estimates to write that(1.8)Lx2Cδxi0τ2τ1ei(τ2τ1)ΔW2D2˜...×0τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ2...dτnLxiLτ1,xi˜2CδW2D2˜...0τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ3...dτnLxi1Lτ2,xi˜2. Then we continue this process recursively to obtain the desired bound: if we encounter a potential without derivative, we use Strichartz estimates, and if the potential carries a derivative, then we use smoothing estimates.

Say for example that in the expression above, D2˜=1. Then we write that(1.8)Lx2C2δ20τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ3...dτnLτ22Lx6. Otherwise, if D2˜=xk then we obtain(1.8)Lx2C2δ20τnτn1ei(τnτn1)ΔWnDn˜eiτnΔfdτ3...dτnLxkLτ2,xk˜2. To close the estimates when Wn=V, we write that, using Strichartz inequalities, we have:(1.8)Lx2Cn1δn1VeiτnΔfZCnδneiτnΔfLτn2Lx6CnδnfLx2, where Z denotes either Lτn2Lx6/5 or Lxi1Lτn,xi˜2 depending on whether Wn1=V or ai. The case where Wn=ak is treated similarly, except that we use smoothing instead of Strichartz estimates.

In the case of the nonlinear term in f (1.9), the f that was present in (1.8) is replaced by the quadratic nonlinearity. As a result, the same strategy essentially reduces to estimating that quadratic term in L2. This takes us back to a situation that is handled by the classical theory of space-time resonances: such a term was already present in the work of P. Germain, N. Masmoudi and J. Shatah ([10]).

Of course, in reality, the situation is more complicated: here we were imprecise as to which smoothing effects we were using. Moreover we have mostly ignored the difficulty to combine the above smoothing arguments with the classical space-time resonance theory. In the actual proof we must resort to several smoothing estimates, see Section 2 for the complete list.

In the actual proof we must keep the X-norm of the profile under control. The situation is more delicate than for the L2 norm, but the general idea is similar and was implemented in our previous paper [25]. We recall it in this section for the convenience of the reader.

To generate the series representation we cannot merely integrate by parts in time since when the ξ-derivative hits the phase, an extra t factor appears. Roughly speaking we are dealing with terms like0tR3seis(|ξ|2|η1|2)W1ˆ(ξη1)α1(η1)fˆ(s,η1)dη1ds for the potential part.

The idea is to integrate by parts in frequency to gain additional decay, and then perform the integration by parts in time. For the term above this yields an expression like0tR3η1|η1|2eis(|ξ|2|η1|2)W1ˆ(ξη1)α1(η1)η1fˆ(s,η1)dη1ds+{easier terms}. Then we can integrate by parts in time to obtain terms like:R3η1|η1|2(|ξ|2|η1|2)eit(|ξ|2|η1|2)W1ˆ(ξη1)α1(η1)η1fˆ(t,η1)dη1+0tR3η1|η1|2(|ξ|2|η1|2)eis(|ξ|2|η1|2)W1ˆ(ξη1)α1(η1)sη1fˆ(s,η1)dη1ds+{easier terms}. The boundary term will be the first term of the series representation. Then we iterate this process on the integral part. This is indeed possible since η1fˆ and fˆ satisfy the same type of equation tfˆeitΔ(Vu+u2) up to lower order terms and with different potentials (essentially V and xV respectively).

After generating the series, the next step is to prove a geometric sequence type bound for the L2 norm of its terms. If we are away from space resonances, namely if the multiplier η1/|η1|2 is not singular, then we are essentially in the same case as in the previous section on the L2 estimate of the solution. However if the added multiplier is singular, then we cannot conclude as above. The scheme we have described here is only useful away from space resonances.

We can modify this approach and choose to integrate by parts in time first. We obtain boundary terms with an additional t factor compared to the previous section:R31|ξ|2|η1|2eit(|ξ|2|η1|2)W1ˆ(ξη1)α1(η1)tfˆ(t,η1)dη1. The key observation here is that if we are away from time resonances, that is if the multiplier (|ξ|2|η1|2)1 can be seen as a standard Fourier multiplier, then we can use the decay of eitΔf in L6 to balance the t factor.

Overall we have two strategies: one that works well away from space resonances, and the other away from time resonances. Since the space-time resonance set is reduced to the origin (that is the multipliers (|ξ|2|η1|2)1 and η1/|η1|2 are both singular simultaneously only at the origin) we use the appropriate one depending on which region of the frequency space we are located in. This general scheme was developed by the author in [25].

We start by recalling some known smoothing and Strichartz estimates in Section 2. We then prove easy corollaries of these that are tailored to our setting. The next three sections are dedicated to the main estimate (1.6) on the X norm of the solution: Section 3 is devoted to expanding the derivative of the Fourier transform of the profile as a series. In Section 4 we estimate the first iterates. As we pointed out above, this is a key step since our multilinear approach essentially reduces the estimation of the n-th iterates to that of the first iterates. Finally we prove in Section 5 that the Lx2 norm of the general term of the series representation of ξfˆ decays fast (at least like δn for some δ<1). This allows us to conclude that the series converges. We start in Section 5.1 by developing our key multilinear lemmas that incorporate the smoothing effect of the linear Schrödinger flow in the iteration. They are then applied to prove the desired bounds on the n-th iterates.

We end the paper with the easier energy estimate (1.7) in Section 6, which concludes the proof.

Acknowledgments: The author is very thankful to his PhD advisor Prof. Pierre Germain for the many enlightening discussions that led to this work. He also wishes to thank Prof. Yu Deng for very interesting discussions on related models.

Section snippets

Known results

In this section we recall some smoothing and Strichartz estimates from the literature. In this paper we will use easy corollaries of these estimates (see next subsection) to prove key multilinear Lemmas in Section 5.

We start with the classical smoothing estimates of C. Kenig, G. Ponce and L. Vega ([24], theorem 2.1, corollary 2.2, theorem 2.3). Heuristically the dispersive nature of the Schrödinger equation allows, at the price of space localization, to gain one half of a derivative in the

Expansion of the solution as a series

In this section and the next two, our goal is to prove (1.6). To do so we start as in [25] by expanding ξlfˆ as a power series. This is done through integrations by parts in time. The full details are presented in this section.

Bounding the terms from the first expansion

In this section we bound the terms from the first expansion (see the various estimates announced in Section 3.1).

We distinguish in the first subsection the estimates that are done without the use of smoothing estimates, from the ones that require recovering derivatives (terms of potential type) in the second subsection.

Multilinear terms

In this section we prove Proposition 3.5. The estimates are based on key multilinear lemmas proved in the first subsection. We then use them to bound the iterates in the following subsection.

Energy estimate

Here we prove the H10 estimate on the solution. The method is, as in the proof of (1.6), to expand the solution as a series. This case however is simpler, in the sense that only integrations by parts in time are required. In other words the series is genuinely obtained by repeated applications of the Duhamel formula. The terms of the series are then estimated using lemmas from Section 5.1.

First recall that the bilinear part of the Duhamel formula has already been estimated in [25], Lemma 8.1:

Lemma 6.1

References (31)

  • C. Kenig et al.

    Small solutions to nonlinear Schrödinger equations

    Ann. Inst. Henri Poincaré, Anal. Non Linéaire

    (1993)
  • M. Beceanu

    Structure of wave operators for a scaling-critical class of potentials

    Am. J. Math.

    (2014)
  • M. Beceanu et al.

    Structure formulas for wave operators

  • P. D'Ancona

    On large potential perturbations of the Schrödinger, wave and Klein-Gordon equations

  • P. D'Ancona et al.

    Decay estimates for the wave and Dirac equations with a magnetic potential

    Commun. Pure Appl. Math.

    (2007)
  • S. Doi

    On the Cauchy problem for Schrödinger type equations and the regularity of solutions

    J. Math. Kyoto Univ.

    (1994)
  • B. Erdoğan et al.

    Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in R3

    J. Eur. Math. Soc.

    (2008)
  • B. Erdoğan et al.

    Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions

    Forum Math.

    (2009)
  • L. Fanelli et al.

    Time decay of scaling critical electromagnetic Schrödinger flows

    Commun. Math. Phys.

    (2013)
  • V. Georgiev et al.

    Smoothing Strichartz estimates for the Schrödinger equation with small magnetic potential

    Discrete Contin. Dyn. Syst.

    (2007)
  • P. Germain et al.

    Global solutions for the 3D quadratic Schrödinger equations

    Int. Math. Res. Not.

    (2009)
  • P. Germain et al.

    Global solutions for the gravity water waves equation in dimension 3

    Ann. Math. (2)

    (2012)
  • P. Germain et al.

    Nonlinear resonances with a potential: multilinear estimates and an application to NLS

    Int. Math. Res. Not.

    (2015)
  • P. Germain et al.

    Global existence for capillary water waves

    Commun. Pure Appl. Math.

    (2015)
  • Y. Guo et al.

    Global solutions of the Euler-Maxwell two-fluid system in 3D

    Ann. Math.

    (2016)
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