Cauchy problem for the nonlinear Schrödinger equation coupled with the Maxwell equation

In this paper, we study the nonlinear Schrödinger equation coupled with the Maxwell equation. Using energy methods, we obtain a local existence result for the Cauchy problem.


Introduction
In this paper, we consider the following nonlinear Schrödinger equation coupled with Maxwell equation stated in R + × R 3 : iψ t + ∆ψ = eφψ + e 2 |A| 2 ψ + 2ie∇ψ · A + ieψ div A − g(|ψ| 2 )ψ, (1.1) where ψ : R + × R 3 → C, A : R + × R 3 → R 3 , φ : R + × R 3 → R, e ∈ R and i denotes the unit complex number, that is, i 2 = −1. In this setting, ψ is an electrically charged field and (φ, A) represents a gauge potential of an electromagnetic field. System (1.1)-(1.3) describes the interaction of this Schrödinger wave function ψ with the Maxwell gauge potential. The constant e represents the strength of the interaction. For more details and physical backgrounds, we refer to [Fel98].
Since we are interested in the Cauchy Problem, let us consider the following set of initial data: where the regularity of each functions is given in Theorem 1.1. It is known that System (1.1)-(1.3) has a so-called gauge ambiguity. Namely if (ψ, A, φ) is a solution of (1.1)-(1.3), then (exp(ieχ)ψ, A + ∇χ, φ − χ t ) is also a solution of (1.1)-(1.3) for any smooth function χ : R + × R 3 → R. To push out this ambiguity, we adopt in the sequel the Coulomb gauge: (1.5) div A = 0, which is propagated by the set of Equations (1.1)-(1.3). Indeed, if initially div A(0, · ) = div A t (0, · ) = 0, then (1.5) holds for all t > 0. (See e.g. [CW16] for the proof.) In this setting, the last Equation (1.3) can be solved explicitly and the solution is given by which imposes that From (1.5), we also observe that (1.1) can be written as where V is the non-local potential: V (x) = e 2 2 (−∆) −1 |ψ| 2 and L A is the magnetic Schrödinger operator which is defined by A = (A 1 , A 2 , A 3 ) and (1.7) In this context, the two conserved quantities of the Schrödinger-Maxwell system are the charge Q and the energy E: where G(t) = t 0 g(s)ds. To prove that (1.8) is formally conserved, one has to multiply Equation (1.1) by ψ, integrate over R 3 and take the imaginary part of the resulting equation. In a similar way, the conservation of (1.9) can be proved by multiplying (1.1)-(1.3) by ∂ t ψ, ∂ t A and ∂ t φ respectively. This conserved quantities play a fundamental role if one wants to investigate the stability properties of such system, which is one of our main motivations. Indeed, in a previous paper [CW17], we have showed that for small e > 0, System (1.1)-(1.3) admits a unique orbitally stable ground state of the form: In order to investigate the stability of such standing waves (ψ e,ω , A e,ω , φ e,ω ) as performed in [CW17], it is necessary to prove that the Cauchy Problem (1.1)-(1.3) is almost locally well-posed around (ψ e,ω , A e,ω , φ e,ω ). In a previous paper [CW16], we have proved the local existence of solutions for the nonlinear Klein-Gordon-Maxwell system in Sobolev spaces of high regularity. The method was to convert the Klein-Gordon-Maxwell system into a symmetric hyperbolic system and apply the standard energy estimate. Although our Schrödinger-Maxwell system (1.1)-(1.3) looks similar, especially Equation (1.2) is completely the same, the usual reduction tools does not lead us to a symmetric hyperbolic system, which causes the necessity of a new strategy.
Let us also introduce results concerning the solvability of the Cauchy problem related to (1.1)-(1.3). In [BT09], [NW07], the linear Schrödinger equation (g ≡ 0) coupled with the Maxwell equations has been studied. Using the Strichartz estimate, the authors obtained the global well-posedness in the energy space. Recently in [ADM17], it was shown, by using the Strichartz estimate obtained in [NW07], that the system (1.1)-(1.3) is locally well-posed in H 2 × H 3 2 × H 1 2 and the global existence holds for finite energy weak solutions, when the nonlinear term g is defocusing (namely the case with +|ψ| p−1 ψ in (1.1)). We also mention the paper [NT86], where the Cauchy problem of the Schrödinger-Maxwell system in the Lorentz gauge has been studied by using the energy method. On the other hand, a huge attention has been paid in the magnetic Schrödinger equation (1.6). Especially in [Mic08], the local well-posedness for (1.6) in the energy space has been established in the case V ≡ 0. However, in this situation, the magnetic potential A is given and was assumed to be C ∞ , which cannot be expected a priori in our case. We also refer to [DFVV10] for the Strichartz estimate for the magnetic Schrödinger operator (1.7) in the case A ∈ L 2 loc (R 3 ).
We mention that if we look for the standing wave (1.10), we are led to the following non-local elliptic problem: which is referred as the Schrödinger-Poisson(-Slater) equation. The existence of ground states of (1.11) as well as their orbital stability have been widely studied (see [AP08], [BF14], [BS11], [CDSS13], [Kik07] and references therein). Finally, the orbital stability of standing waves for the magnetic Schrödinger equation (1.6) has been considered in [CE88], [GR91]. Our study on the solvability of the Cauchy problem for (1.1)-(1.3) and the result established in [CW17] enable us to generalize these previous results to the full Schrödinger-Maxwell system. Before stating the main result of this paper, we introduce the following notations. As usual, L p (R 3 ) denotes the usual Lebesgue space: We define the Sobolev space H s (R 3 ) as follows: where F(u)(ξ) is the Fourier transform of u. We also introduce the homogeneous Sobolev spaceḢ 1 (R 3 ) as being the completion of C ∞ 0 (R 3 , C) for the norm u → |ξ|F(u)(ξ) L 2 (R 3 ) . Recall that the spaceḢ 1 (R 3 ) is continuously embedded into L 6 (R 3 ). Finally let C(I, E) be the space of continuous functions from an interval I of R to a Banach space E. For 1 j 3, we set ∂ x 3 u and for a non-negative integer s, D s denotes the set of all partial space derivatives of order s. Different positive constants might be denoted by the same letter C. We also denote by Re(u) and Im(u) the real part and the imaginary part of u respectively.
We assume that g satisfies (1.12) g ∈ C m+1 (R, R) and g(0) = 0, for some m ∈ N with m 2, so that the function W : Some typical examples of the nonlinear term g are the power nonlinearity g(s) = ±s 2m + 3 ([p] denotes the integer part of p), or the cubic-quintic nonlinearity g(s) = s − λs 2 for λ > 0, which frequently appears in the study of solitons in physical literatures. (See [RV08] for example.) In this setting, we prove the following result.
Theorem 1.1. -Let s be any integer larger than 3 2 and assume that The proof of Theorem 1.1 is based on energy estimates and particularly, on the strategies developed in [Col02] and [CC04]. Note also that to overcome the loss of derivatives embedded in Equation (1.2), we use the original idea of Ozawa and Tsutsumi presented in [OT92].
The paper is organized as follows. In Section 2, we transform System (1.1)-(1.3) into a system to which we can apply the usual energy method. Section 3 is devoted to the proof of Theorem 1.1.

Transformation of the equations
In this section, we transform the original System (1.1)-(1.3) into a new symmetric system to which we can apply an energy method. In order to overcome the loss of derivatives contained in Equations (1.1)-(1.2), we introduce the following new unknowns (see [OT92]): Ψ = ∂ t ψ and Φ = ∂ t φ. Let us first derive equations for Ψ and Φ. Differentiating Equation (1.1) with respect to t, one obtains Taking advantage of the new unknown Ψ, we also transform Equation (1.1) into an elliptic version Moreover, we derive an equation for Φ by applying ∂ t on Equation (1.3): Next, in order to ensure the Coulomb condition on A for all t > 0, we introduce the projection operator P on divergence free vector fields : so that if div A = 0, then PA = A. Thus applying P on Equation (1.2), we derive Note that any solution to (2.1) satisfying div A(0, · ) = 0 and div A t (0, · ) = 0, obviously satisfies div A(t, · ) = 0 for all t > 0.
At this step, we have transformed System (1.1)-(1.3) into In order to take advantage of elliptic regularity properties, we transform Equations (2.2) by adding −αψ (α > 0 will be chosen in Lemma 3.3 below) to both sides of the equation to obtain: For simplicity, introduce U = (φ, Φ) and rewrite Equations (2.5) and (2.6) as It is then necessary to work with A t as new unknown. We recall first that A = (a 1 , a 2 , a 3 ). To properly write the equations on A and A t , for j = 1, 2, 3, k = 1, 2, 3 and = 1, 2, 3, we introduce We also need to give some details on the projection operator P. For that purpose, we introduce the Riesz transform R j from L 2 (R 3 ) to L 2 (R 3 ) which is given by Then, P can be rewritten as P = (P j,m ) 1 j,m 3 where Now we compute the equations for each components of A j . First by the definitions of A j , one finds that Next from Equation (2.4), we have where h 1 j,k , h 2 j,k, , h 3 j,k are non-local functions defined as follows: Finally one has Computing separately each term of the right-hand side of the previous equation, we obtain ∂ t (ψ∂ xm ψ) = Ψ∂ xm ψ + ψ∂ xm Ψ, ∂ t (|ψ| 2 a m ) = (Ψψ + ψΨ)a m + |ψ| 2 r m .
Moreover from (2.3) and (2.6), one finds that from which we conclude that The equation on A j can be written as a symmetric system of the form where H j = t (0, 0, 0, 0, h 1 j,k , h 2 j,k, , h 3 j,k , h 4 j,k ), M j (∇) = 3 k=1 M j ∂ x k are 24 × 24 symmetric matrices. Recalling that A j = (a j , p j , q j , r j , λ j , µ j , ν j , τ j ), where a j , q j , r j are scalar functions, p j , λ j , ν j and τ j are functions with values in R 3 and µ j is a function with values in R 9 , M j can be simply written by blocks: Note that M j are 24 × 24 symmetric matrices whose components are all constants. Thus from (2.7), (2.8) and (2.9), we have transformed Equations (1.1)-(1.3) into the following system: Ψ, A, R). (2.13)

Solvability of the Cauchy Problem
The aim of this section is to prove Theorem 1.1. To this end, we use a fix-point argument on a suitable version of System (2.10)-(2.13). In this procedure, the necessary estimates follow from the application of the usual energy methods.
The proof is divided into 6 steps. We first recall the following classical lemma. (See e.g. [AG91, Proposition 2.1.1, p. 98] for the proof.) Step 1: Solving the elliptic equation where C 1 , C 2 , C 3 and C 4 are positive constants depending only on R.
Next for 0 k s, we apply D k+1 to the first line of (3.3), multiply the resulting equation by D k+1 φ and make an integration by parts to obtain Using the Leibniz rule, Lemma 3.1 and the Schwarz inequality, one has Summing up the inequalities (3.9) from k = 0 to s and recalling the fact that ψ L ∞ ([0,T ];H s ) R, we obtain where C 1 (R) is a constant depending only on R. Finally, the Sobolev embedding W 1,6 (R 3 ) → L ∞ (R 3 ) provides that from which we deduce that there exists a constant C 2 (R) depending only on R such that which ends the proof of (3.7). The proof of estimates (3.8) is similar and we omit the details.
As a consequence, there exists a unique solution χ(t, · ) ∈ H 1 (R 3 , C) to (3.4) and there exists a constant C 5 (R) such that Proof. -First we note that b is hermitian by the condition div A = 0. Indeed, one has u). The continuity is a direct consequence of the Cauchy-Schwarz inequality and the fact that This shows that b is elliptic on Then the Lax-Milgram theorem ensures the existence of a unique solution χ to (3.4) in H 1 (R 3 ). Using the elliptic regularity theory and recalling that where C 5 (R) is a constant depending only on R. This ends the proof of Lemma 3.3.
Lemma 3.4. -Let R ε be the unique solution of Equation (3.11). Then there exist constants C 6 (R), C 7 (R) independent of ε such that Proof. -We first begin with the L 2 -estimate. We multiply (3.11) by R ε and integrate over R 3 . Since J is skew-symmetric, one obtains (3.12) ∂ ∂t For j = 1, 2, 3, we have from ∂ x j a j H s R that Since Ψ, R ∈ H s and A ∈ H s+1 , using Lemmas 3.2-3.3, one can also compute as follows : Collecting (3.12)-(3.14), we derive By the Gronwall inequality and from Next we perform the H s -estimate. We apply D s on (3.11), multiply the resulting equation by D s R ε , integrate over R 3 and use the Gronwall inequality. We limit our attention to non-trivial terms. Recalling that χ ∈ C([0, T ]; H s+2 ) and using Lemma 3.3, we obtain Moreover, one gets Arguing similarly as above, one finds that 1 2 , which ends the proof of Lemma 3.4. Now we argue as in [BdBS95], [CG01] and we perform the limit ε → 0. By Lemma 3.4, we know that R ε is uniformly bounded in L ∞ ([0, T ], H s ). From (3.11), one also has This implies that for all t 0 and ε ∈ (0, 1].
Thus passing to a subsequence, we may assume that From (3.11), one can see that R is a solution of Equation (3.10) and satisfies Moreover since R ε (0) → (Re Ψ 0 , Im Ψ 0 ), we get R(0) = (Re Ψ 0 , Im Ψ 0 ). We then deduce the existence of a solution Q to Equation (3.5) satisfying Step 4: Solving the symmetric system (3.6) First we note that it is straightforward to prove the existence of a unique solution B j to Equation (3.6). (We refer to [AG91, Proposition 1.2, p. 115] for the proof.) Furthermore, by using the Fourier transform F, one has directly for j = 1, 2, 3, from which we deduce that R j and hence P j,m are bounded from L 2 (R 3 ) to L 2 (R 3 ). As a consequence, using the fact Ψ ∈ H s , A ∈ H s+2 , R ∈ H s+1 and χ ∈ H s+2 , one can prove that Thus applying the energy estimate to (3.6), recalling that M j (∇) = 3 k=1 M j ∂ x k is symmetric and using the fact M j consists of constant elements, we get Then by the Gronwall inequality, we obtain the following estimate.
Lemma 3.5. -Let B j be the unique solution of (3.6). Then there exists a constant C 8 (R) > 0 such that  Collecting (3.15) and (3.16), we can state the following result.
Proposition 3.6. -There existsT > 0 such that for 0 < T T , S maps B(R) into itself.
Step 5: Contraction mapping Now we establish the following result.
Proof. -The proof is based on the fact that s > 3 2 and on the fact that all the functions of Equations (3.3)-(3.6) are Lipschitz with respect to their arguments. The proof is classical and we omit the details.