Well posedness and smoothing effect of Schrödinger-Poisson equation

In this work we take under consideration the Cauchy problem for the Schrodinger-Poisson type equation i∂tu=−∂x2u+V(u)u−f(∣u∣2)u, where f represents a local nonlinear interaction (we take into account both attractive and repulsive models) and V is taken as a suitable solution of the Poisson equation V=1∕2∣x∣∗(C−∣u∣2), C∊Cc∞ is the doping profile or impurities. We show that this problem is locally well posed in the weighted Sobolev spaces Hs≔{φ∊Hs(R):∫(1+x2)1∕2∣φ∣2<∞} with s⩾1, which means the local existence, uniqueness, and continuity of the solution with respect to the initial data. Moreover, under suitable assumptions on the local interaction, we show the existence of global solutions. Finally, we establish that for s⩾1 local in time and space, smoothing effects are present in the solution; more precisely, in this problem there is locally a gain of half a derivative.


I. INTRODUCTION
In this paper we are mainly concerned with a kind of smoothing effect which is present in the solutions of the one-dimensional ͑1D͒ Schrödinger-Poisson equation.In addition, and since our local existence analysis is developed under weaker assumptions on the decay at infinity of the solutions, we will also consider the related well-posedness problem.
Our starting point is the one-dimensional ͑1D͒ ͑unscaled͒ Schrödinger-Poisson problem Here, C͑x͒ denotes the fixed positively charged background ions or impurities ͑will be referred to as the doping profile in the sequel͒ and it is assumed to be a ͑positive͒ regular function with compact support ͑for further details in semiconductor models, see Ref. 12 and references therein͒.The term f͉͑u͉ 2 ͒u represents a local interaction which is intended to take into account the Pauli exclusion principle for fermions ͑see Refs.18, 11, and 6͒.Among the several models proposed to include the exchange effects of charged particles ͑say, electrons͒, we may cite the Schrödinger-Poisson-X␣ model f͉͑u͉ 2 ͒u = ␣͉u͉ 2/N u ͑notice that in dimension 1, the X␣ model yields the "focusing cubic nonlinear Schrödinger" ͑NLS͔͒; actually, one rather considers ͉u͉ q u with a subcritical exponent q ͑which in one dimension reads 0 Ͻ q Ͻ 4͒ ͑for further details on the Schrödinger-Poisson-X␣ model, see Refs. 13 and 1 and references therein͒.Before going into the details on the coupling with the Poisson equation, let us mention that our results allow rather general interactions, say, f C ϱ ͑R͒ ͑in the focusing case, we also impose a subcritical exponent condition͒.Let W͑x͒ be any fundamental solution of the Poisson equation in Eq. ͑1.3͒; thus the solution must be written as a Hartree-type potential V = W͑x͒ ‫ء‬ ͑C − ͉u͉ 2 ͒.Equation ͑1.1͒ becomes i‫ץ‬ t u = − ‫ץ‬ x 2 u + ͑W͑x͒ ‫ء‬ ͑C − ͉u͉ 2 ͒͒u − f͉͑u͉ 2 ͒u, x ⑀ R.

͑1.4͒
The H 1 theory of the Cauchy problem given by Eq. ͑1.4͒ is widely developed in the work of Cazenave. 3However, the general result presented there ͑see Theorem 3.3.1͒does not apply in lower dimensions ͑1 or 2͒ due to the fact that Green's function W͑x͒ is unbounded and therefore both the kernel W͑x͒ and the related "external" potential ͑W ‫ء‬ C͒͑x͒ are not contained in L p + L ϱ for any p ͑see Examples 3.2.1 and 3.2.8͒.
The well posedness of the 1D Schrödinger-Poisson equation without the doping profile C͑x͒ was given first by Steinrück 19 ͓who also neglected the local interaction f͉͑u͉ 2 ͒u͔ and then by Stimming 20 ͑who adapted the proof given by the former in order to include the exchange potential͒ by means of semigroup theory using ⌺ ª ͕ H 1 : x L 2 ͖ as a work space.In addition they discussed the related semiclassical limit ͑which falls into the Wigner-functions approach, which is out of the scope of our work͒.Nevertheless, any kind of smoothing effect is taken under consideration.
Finally, the choice of Green's function W͑x͒ deserves some comments.Following Ref. 3, Example 3.2.8, the kernel of the Hartree-type potential is chosen to be an even function and this leads to Despite the fact that other choices are also used in the literature, ͓for instance, Zhang et al. 24 took W͑x͒ = x + ͉x͉͔, in our work this symmetry is crucial in the choice of both the operator and the work space, and therefore in the improvement of the decay at infinity assumption.More precisely, from decomposition V = b͑͒͑1+x 2 ͒ 1/2 + V ϱ ͓where b͑͒ is a constant which depends on the size of initial data ͔, the ͑linear͒ operator is taken as Since for b͑͒ Ͼ 0 the associated quadratic form is positive, this operator indeed generates a semigroup.Moreover, both ʈ • ʈ H 1 and the related norm are equivalent, from where H 1 appears as the energy space associated to this operator.This shows that H 1 could be seen as a natural space for the problem in Eqs.͑1.1͒, ͑1.2͒, and ͑1.5͒.Furthermore, we are also interested in showing a kind of smoothing effect which, roughly speaking, can be expressed as to gain half a derivative, so we need to investigate the well posedness of this problem in the spaces H s ª ͕ H s : ͐͑1+x 2 ͒ 1/2 ͉͑x͉͒ 2 dx Ͻϱ͖, with s ജ 1.This means the local existence, uniqueness, and continuity of the solution with respect to the initial data.
Let us now turn to the smoothing effect.From the mathematical point of view, the Schrödinger equation appears as a delicate problem, since it possesses a mixture of properties of parabolic and hyperbolic equations.Indeed, it is almost reversible and it has conservation laws and also some dispersive properties such as the Klein-Gordon equation, but it has infinite speed of propagation.On the other hand, the Schrödinger equation has a kind of smoothing effect shared by parabolic problems but the time reversibility prevents it from generating an analytic semigroup.Despite the fact that the expression smoothing effect is used when referring to the loss of singularity ͑e.g., Strichartz estimates 21 ͒, in this work, will denote the gain of derivatives.The first result in this direction ͑concerning dispersive equations͒ was given by Kato, 9 who showed that the solutions of the 1D Korteweg-de Vries equation ‫ץ‬ t w + ‫ץ‬ x 3 w + w‫ץ‬ x w = 0 satisfy which means that the solution w͑x , t͒ gains ͑locally in time and space͒ one derivative.The corresponding version of the above estimate for the free Schrödinger group ͕e was simultaneously established by Constantin and Saut, 5 Sjölin, 17 and Vega. 22A sharp version for the 1D case can be found in Ref. 10.We refer also to Ref. 2 for further developments.Actually, similar results for the Benjamin-Ono equation and the derivative NLS equation has been shown by Ponce 14 and Rial, 16 respectively.Anyway, in the linear Schrödinger equation, there is locally a gain of half a derivative and this is the kind of smoothing effect we will extend to other typical nonlinearities.

Main results
We

II. BASIC RESULTS
In this section we fix some notations and give a list of results which are useful in the development of our work.
• L p ͑I , X͒, L p functions from some interval I to some Banach space X.
and both notations will be used.

B. Results
Throughout this work we will make use of the following lemmas.Lemma 2.1: Gagliardo-Nirenberg inequality.See Cazenave, 3 Theorem 1.3.7.
Let 1 ഛ p , q , r ഛϱ, and j , m N such that 0 ഛ j Ͻ m.If ͓j / m ,1͔ is such that then there exists C = C͑ , p , q , r , j , m , N͒ such that , and q = r =2, standard estimates lead to
Lemma 2.4: See Folland, 7 Lemma 6.16.Let S. Then there exists C = C͑͒ such that Lemma 2.5: Assume s ജ 1.Then there exists a bounded operator B B͑L 2 ͒ such that Proof: It suffices to show that the operator given by F͑͑‫ץ‬ and b belongs to The following result plays a major role in our present work since it allows to lift results from s =1 to s Ͼ 1.The proof is given for lack of a suitable reference.
Proof: Let us rely on an induction argument, since ͓J r ; ͔ = J r−1 ͓J ; ͔ + ͓J r−1 ; ͔J.The inductive step is treated as follows.Let L be the linear operator given by Lª͓J ; ͔.From Lemmas 2.5, and 2.3 and standard interpolation results, it follows that L : H → H is a bounded operator, from where ʈJ ͓J ; ͔ʈ 0 ഛ Cʈʈ holds.On the other hand, it remains to show that ͓J r ; ͔ B͑L 2 ͒ for small values of r.Then take 0 ഛ r ഛ 1 and let B be the pseudodifferential operator with symbol b͑k͒ = k −1 ͑͑1+k 2 ͒ r/2 −1͒.Since J r ‫ץ+1=‬ x B and b satisfies the assumptions of Lemma 2.3, the result is proved.
Lemma 2. 9: For each (fixed) A Ͼ 0 the Schrödinger group ͕e it͑‫ץ‬ x 2 −A͒ ͖ tR is well defined in H s and satisfies ; a direct computation yields the conser- This leads to the estimate ʈU A ͑t͒ʈ H 1 2 ഛ C͑A͒ʈʈ H 1 2 , and the result is true for s =1.
As it was stated before, Lemma 2.6 will be used to lift previous result to s Ͼ 1. Setting ª U A ͑t͒, one has From Lemma 2.6, the estimate, ʈ͓J s ; ͔ʈ 0 ഛ Cʈʈ s−1 follows, which leads to ‫ץ‬ t ʈʈ s 2 ഛ CAʈʈ s ʈʈ s−1 and therefore to ‫ץ‬ t ʈʈ s ഛ CAʈʈ s−1 .The result follows from standard inductive argument.
Consider now the case A = 0 and the related group U 0 ª e it‫ץ‬ x 2 .

Remark 2.2: In this case, the L
2 norm control given by ʈU A ͑t͒ʈ Lemma 2.10: U 0 ͑t͒ is well defined in H s and verifies • ʈU 0 ͑t͒ʈ s = ʈʈ s , valid for all s R, and Proof: The first assertion follows immediately from ͓J s ; ‫ץ‬ x 2 ͔ = 0. From the second claim follows.
Finally, the continuous dependence on the initial data requires some continuity of the family U A with respect to the parameter A which is given by the following lemma.
Lemma Taking into account the identity U A ͑t͒ − U B ͑t͒ = U B ͑t͒g͑t͒, the general result follows from a /3 argument.

III. WELL POSEDNESS OF THE CAUCHY PROBLEM
This section is concerned with the local existence of the Cauchy problem posed in the weighted Sobolev Spaces H s = H s പ L 2 with s ജ 1, where the local interaction satisfies f C ϱ ͑R͒, the doping profile C C c ϱ , and V͑u͒ is given by the potential V shall be written as Some remarks are listed below.Remark 3.1: Since H s L 2 , both potentials V ϱ and V are well defined.Remark 3.2: Since V ϱ L ϱ (see Lemma 3.1 below) if the initial data are such that A͑͒ 0, then V͑u͒ = O͉͑x͉͒ for ͉x͉ → + ϱ.
In the sequel we will consider initial data such that A͑͒ ജ 0. The special case given by A͑͒ = 0 leads to the identity V ϱ = V; therefore such potential is bounded.Both instances A͑͒ Ͼ 0 and A͑͒ = 0 will be called, respectively, the subcritical and critical cases.

A. Local existence. Critical and subcritical cases
Since the results of this subsection are obtained by means of the fixed point techniques, some estimates are needed.
Take j H s പ L 1 and consider 1 / 2 ª ͓͉͐x − y ͉ −͑x͔͒j͑y͒dy.Taking derivatives with respect to x yields Since similar results hold for x → + ϱ, one has the estimate ʈ‫ץ‬ x V͑j͒ʈ L 2 ഛ Cʈjʈ L 1 .In order to prove the Lipschitz property, consider s ജ 1.From Lemma 2.5 there exists a pseudodifferential operator B-which satisfies the assumptions of Lemma 2.3 such that J s =1 + B‫ץ‬ x J s−1 .It follows that J s ‫ץ͑‬ x V͑j͒͒ = ‫ץ‬ x V͑j͒ + BJ s−1 ͑j− 1 2 Љ͐ yR j͑y͒dy͒, and this yields The third and fourth claims are obtained by taking, respectively, The following conservation law will be useful in the sequel.Lemma 3.2: Charge conservation.Let N be a real function and u C͑͑a , b͒ , Proof: Taking the time derivative, it follows that Since N is real we get Re͑iN͑u͒͒ = 0. On the other hand, since s ജ 1 the boundary term in the integration by parts vanishes.
In the sequel we introduce the following auxiliary problem where A ജ 0 is fixed: Property 3.1: Assume that f C ϱ ͑R͒ and let s ജ 1.If H s , then there exist T = T͑͒ Ͼ 0 and a unique maximal solution u C͓͑0,T͑͒͒ , H s ͒ പ C 1 ͓͑0,T͑͒͒ , H s−2 ͒ of the problem in Eqs.(3.8)  and (3.9).u is maximal in the following sense: if T͑͒ Ͻ + ϱ then ʈu͑t͒ʈ H s → + ϱ as t → T͑͒.

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Well posedness and smoothing effect J. Math Proof: Let U A ͑t͒ be the unitary group generated by the operator i‫ץ‬ x 2 − iA and let F A be the operator yielded by Duhamel's formula Thus the fixed point problem is given by F A ͑u͒ = u.In the sequel, it will be shown that F is well defined and is a contraction in H s ͑from now on the subscript will be omitted͒.
A suitable arrangement of terms leads to and using Lemma 3.1, one might bound each term separately as follows: Similar estimates as above lead to

and this gives
Therefore, collecting estimates ͑3.13͒ and ͑3.14͒, the Lipschitz property of the nonlinear term V ϱ ͑u͒u − f͉͑u͉ 2 ͒u is shown.
The following lemma yields the continuous dependence of the iteration process given by F A with respect to the parameter A. Lemma  Theorem 3.1: Let f C ϱ ͑R͒ and let s ജ 1.For every H s , such that ʈʈ 0 2 ഛ ʈCʈ L 1, there exists T = T͑͒ Ͼ 0 and there exists a unique, maximal solution u C͓͑0,T͑͒͒ , H s ͒ പ C 1 ͓͑0,T͑͒͒ , H s−2 ͒ of the problem in Eqs.(3.1) and (3.2). u is maximal in the sense that if T͑͒ Ͻ + ϱ, then ʈu͑t͒ʈ H s → + ϱ as t → T͑͒.
Proof: Let be such that A͑͒ ജ 0 and take A = 1 2 A͑͒.Applying Proposition 3.1 we get a solution u C͓͑0,T͑͒͒ , H s ͒ പ C 1 ͓͑0,T͑͒͒ , H s−2 ͒ of the problem in Eqs.͑3.8͒ and ͑3.9͒.From the conservation law of Lemma 3.2, it follows that A͑u͒ = A͑͒; finally, the definition of A͑u͒ and V ϱ ͑u͒ ͓see Eqs.͑3.5͒ and ͑3.4͔͒ show that V͑u͒ = A + V ϱ ͑u͒, from where we conclude that u is a solution of the problem in Eqs.͑3.1͒ and ͑3.2͒.Remark 3.4: Since A͑•͒ depends only on the initial data, uniqueness follows directly from the uniqueness of the auxiliary problem in Eqs.(3.8) and (3.9).On the other hand, the continuous dependence of u with respect to the initial data follows from Lemma 3.3.

B. Conservation laws
The problem in Eqs.͑3.1͒ and ͑3.2͒ presents a conserved quantity, which is given by the energy functional ͑see Cazenave where FЈ = f and F͑0͒ =0.Lemma 3.4: Conservation of energy.Let A ജ 0, s ജ 1, and This shows that E is a well defined functional in H s .Taking into consideration identity ͑3.3͒ and calling U͑x͒ = ͉x͉ ‫ء2/‬ C, one might write A straightforward computation yields On the other hand, since ͐ xR f͉͑u͉ 2 ͒Re͑u‫ץ‬ t u ¯͒dx = ͕f͉͑u͉ 2 ͒u ; ‫ץ‬ t u͖, one may write The first term vanishes since u solves the equation; the remaining terms are equal since ͉x͉ is an even function.This shows that E is constant along the trajectory u͑t͒.

IV. GLOBAL EXISTENCE IN H s
This section is devoted to establishing a global existence result for the problem in Eqs.͑3.1͒ and ͑3.2͒.
Since the existence of global solutions is strongly related with the critical exponent ͑see Weinstein 23 ͒, some control on the local interaction must be made.This suggests the following two basic assumptions: Remark 4.1: Since f Ͻ 0 means repulsive interaction, no further assumption shall be made.On the opposite, since the local interaction is given by f͉͑u͉ 2 ͒u, the case =2 corresponds to the critical exponent (see Refs. 8 and 23).Therefore, in the attractive model, assumption (A1) means subcritical exponent.Proof: Take s = 1 and let u C͑͑−T , T͒ , H 1 ͒ പ C 1 ͑͑−T , T͒ , H −1 ͒ be the ͑local͒ solution of the problem in Eqs.͑3.1͒ and ͑3.2͒.Using that both terms ͕͉͑x͉ ‫ء2/‬ ͉u͉ 2 ͒u ; u͖ and A͕u ; u͖ are nonnegative, one has By means of estimate ͑2.1͒ ͑see the Gagliardo-Nirenberg inequality͒, this yields which, together with Lemma 3.1, leads to Taking Ͻ1 / 2 and using the conservation law of Lemma 2 one has the following estimate: On the other hand, following assumption ͑A2͒, the term 1 / 2͐ xR F͉͑u͉ 2 ͒ is non-negative and therefore the previous estimate also holds in this case. Take From the inductive step, one has ʈu͑t͒ʈ k ഛ C͑t , ʈ 0 ʈ H k͒ and, therefore, the result proceeds from Gronwall's lemma.

V. SMOOTHING EFFECT
In this section we establish the kind of smoothing effect which is present in the solution of the Cauchy problem Proof: The proof will be given in several steps.
and this shows that one is allowed to restrict to functions C c ϱ such 0 ഛ ഛ 1 and ϵ 1 in ͓−R , R͔.
The following identity will be useful in the sequel: Take now ⍀ C ϱ such that ª ‫ץ‬ x ⍀.Let H be the Hilbert transform ͑see Lemma 2.2͒ and let P ± ª 1/2͑1±iH͒ be the projection operators ͑see Rial 16 ͒.We also consider the following identity: 1 2 ‫ץ‬ t ͗⍀J s P ± u;J s P ± u͘ = ͕i⍀J s ‫ץ‬ x 2 P ± u;J s P ± u͖ − ͕i⍀J s P ± V͑u͒u;J s P ± u͖ + ͕i⍀J s P ± f͑u͒u;J s P ± u͖.

͑5.4͒
On the other hand, the linear term can be written as follows: ͕i⍀J s ‫ץ‬ x 2 P ± u;J s P ± u͖ = ͕i͓⍀;‫ץ‬ x ‫ץ͔‬ x J s P ± u;J s P ± u͖ + ͕i⍀‫ץ‬ x J s P ± u;‫ץ‬ x J s P ± u͖ = − ͕i‫ץ‬ x J s P ± u;J s P ± u͖.
Remark 5.4: From Lemma 2.2 it follows that B 1 is a bounded operator in L 2 .Furthermore, one has the following estimate:

͑5.10͒
Second step.In this step it will be proven that if the result is true for ͑both͒ projection operator P ± u, then it is true for u.

Theorem 3.1: Assume
begin with the local well posedness in H s of the Cauchy problem in Eqs.͑1.1͒ and ͑1.2͒.f C ϱ ͑R͒ and let s ജ 1. Assume further H s is such that ʈʈ L 2 2 ഛ ʈCʈ L 1. Then there exist T * = T * ͑͒ Ͼ 0, T * = T * ͑͒ Ͼ 0, and a unique maximal solution u C͑͑−T * , T * ͒ , H s ͒ പ C 1 ͑͑−T * , T * ͒ , H s−2 ͒ of the problem in Eqs.(1.1) and (1.2). u is maximal in the following sense: if T * Ͻ + ϱ or T * Ͻ + ϱ, then ʈu͑t͒ʈ H s → + ϱ as t → −T * or t → T * .We then present a global result which is valid for local interactions with subcritical exponent.