Local and Global Well-Posedness for the Critical Schrodinger-Debye System

We establish local well-posedness results for the Initial Value Problem associated to the Schr\"odinger-Debye system in dimensions $N=2, 3$ for data in $H^s\times H^{\ell}$, with $s$ and $\ell$ satisfying $\max \{0, s-1\} \le \ell \le \min\{2s, s+1\}$. In particular, these include the energy space $H^1\times L^2$. Our results improve the previous ones obtained in \cite{Bidegaray1}, \cite{Bidegaray2} and \cite{Corcho-Linares}. Moreover, in the critical case (N=2) and for initial data in $H^1\times L^2$, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned in \cite{Fibich-Papanicolau} concerning the formation of singularities for these solutions.


Introduction
We consider the Initial Value Problem (IVP) for the Schrödinger-Debye system where u = u(x, t) is a complex-valued function, v = v(x, t) is a real-valued function and ∆ is the Laplacian operator in the spacial variable. This model describes the propagation of an electromagnetic wave through a nonresonant medium whose material response time is relevant. See Newell and Moloney [16] for a more complete discussion of this model.
For sufficiently regular data, the mass of the solution u of the system (1.1) is invariant. More precisely, Other conservation laws for this system are not known, but the following pseudo-Hamiltonian structure holds: The system (1.1) can be decoupled by solving the second equation with respect to v, to obtain the integro-differential equation The rest of this introduction is organized as follows: in section 1.1 we review the previous existing results regarding the local and global theory for (1.1). In section 1.2, we describe our new results in dimensions N = 2, 3.
We begin with a review of the local and global theory for the Cauchy problem (1.1) with initial data (u 0 , v 0 ) in Sobolev spaces H s (R N ) × H ℓ (R N ), N = 1, 2, 3.
These results were obtained by a fixed-point procedure applied to the Duhamel formulation for the integro-differential equation (1.7), using the Strichartz estimates for the unitary Schrödinger group Following the same approach, Corcho and Linares ( [9]) improved the results stated in Theorem 1.1 in the one-dimensional case. More precisely, they established the following assertions: The new ingredients used in the proof of Theorem 1.2 are commutator estimates for fractional Sobolev spaces and the smoothing effect for the Schrödinger group deduced by Kenig, Ponce and Vega (see [14,15]). Furthermore, the authors also showed that, although the fixed point procedure is performed only on the function u, equation (1.6) can be used to obtain the persistence property of the solution v in H s (R N ) in the cases described in Theorem 1.1.
Concerning global existence, it was also proved in [9] that the local-in-time results for the solution u of the integro-differential equation (1.7), given in Theorem 1.1 (c) and Theorem 1.2 (b) and (c), can be extended to all positive times. However, the method used does not provide control of the evolution in time of the H s -norm of the corresponding solution v. Indeed, contrarily to the NLS equation, (1.1) does not possess a Hamiltonian structure, hence the extension to any positive times of the local-in-time solutions (u, v) is not straighforward.
Recently, however, Corcho and Matheus (see [10]) studied the case N = 1 in the framework of Bourgain spaces and obtained the following local and global well-posedness results for the system: is locally Lipschitz. In addition, in the case ℓ = s with −3/14 < s ≤ 0, the local solutions can be extended to any time interval [0, T ].
The global results in Theorem 1.3 are based on a good control of the L 2 -norm of the solution v, which provides global well-posedness in L 2 × L 2 . Global well-posedness below L 2 regularity is then obtained via the I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao in [6].
Regarding the formation of singularities in the critical case (N = 2), Fibich and Papanicolaou ( [11]) studied this system in the focusing case using the lens transformations, but did not derive any result as to the blow-up of the solutions. On the other hand, from a numerical point of view, Besse and Bidégaray ([1]) used two different methods suggesting that blow-up occurs for initial data u 0 (x, y) = e −(x 2 +y 2 ) and v 0 = λ|u 0 | 2 . However, prior to the present paper, the blow-up problem remained open.
In this paper we give a negative answer to the question of existence of blow-up solutions for initial data in H 1 (R 2 ) × L 2 (R 2 ) (see Theorem 3.1). Note that this result is not in contradiction with the numerical simulations in [1]. Indeed, in the latter, the suggested blow-up occurs for the norm u(·, t) L ∞ which, in two dimensions, is not controlled by u(·, t) H 1 . Also, contrarily to the NLS case, we prove that the blow-up occurs neither in the defocusing nor in the focusing case. This is due to the delay induced by the term µv t in the left-hand-side of the second equation of (1.1), which prevents the solution from concentrating critically. As expected, this behavior does not depend on the size of µ, as long as this parameter stays positive. This was already remarked in [1]: In order to prove our global result and overcome the difficulty caused by the absence of conservation of the energy of (1.1), we use a careful control of its derivative (1.4) for . This method requires the availability of a local theory in this space, a case which is not covered in the previous literature and had to be derived here as well. More precisely, we prove local well-posedness in dimensions N = 2, 3, for initial data in H s × H ℓ with s and ℓ satisfying max{0, s − 1} ≤ ℓ ≤ min{2s, s + 1} (see Theorem 2.1 and Figure 1.2).

Local well-posedness in dimensions N = 2, 3
In this section we obtain local well-posedness for the system (1.1) following the approach used by Ginibre, Tsutsumi and Velo in [13] for the Zakharov system. As in [13] we measure the solutions in appropriate Bourgain space, or Fourier restriction, norms. More specifically, we consider the solution (u, v) of the system (1 is the Fourier restriction norm associated to the Schrödinger group S(t) defined in (1.8).
In these definitions f (ξ, τ ) denotes the space-time Fourier transform of f (x, t) and |ξ| is the Euclidean norm of the frequency vector ξ ∈ R N . We recall that Now we state the main local well-posedness result.
, with s and ℓ satisfying the conditions: for suitable b and c close to 1 2 + (ψ T denotes, as usual, a cutoff function for the time interval

Preliminary Estimates.
In the sequel, we use the following notation. For λ ∈ R and we denote by λ± a number slightly larger, respectively smaller, than λ. The bracket · is defined as · = 1 + | · |. We introduce the variables In terms of these variables, the resonance relation for system (1.1) is the following: 1 and a, a 1 and a 2 be non-negative numbers satisfying the conditions Let m be such that with strict inequality on the left of (2.18) if equality holds on the right of (2.16) or if hold.

Bilinear Estimates.
It is well known that in the framework of Bourgain spaces local well-posedness results can usually be reduced to the proof of adequate k-linear estimates. In the present case, to prove Theorem 2.1 it suffices to establish the following two bilinear estimates: holds provided c = 1 2 + ε, b 1 = 1 2 − ε 1 and b 2 = 1 2 + ε 2 for an adequate selection of the parameters 0 ≤ ε, ε 1 , ε 2 ≪ 1.
Proposition 2.4. Let s ≥ 0, ℓ ≤ min{2s, s + 1} and the functions u and w be supported in time in the region |t| ≤ CT . Then, the bilinear estimate for an adequate selection of the parameters 0 ≤ ε, ε 3 ≪ 1.
The proofs of the Propositions 2.3 and 2.4 follow similar arguments as the ones used in [13] to prove Lemmas 3.4 and 3.5, for the Zakharov system in all dimensions. Thus, we only present here the proof of Proposition 2.3, corresponding to Lemma 3.4 of [13] in our context (dimensions N = 2, 3), followed by a brief sketch of the proof of Proposition 2.4.

2.4.
Proof of the Proposition 2.4. Following the same ideas as in the proof of Proposition 2.3, in this case we need to estimate the following functional: Unlike the Zakharov system, here we do not have the presence of the derivative term (|ξ|) in the numerator of W . Thus, we estimate (2.39) in the same way as in the proof of Lemma 3.5 in [13] (N = 2, 3), replacing |ξ| ξ ℓ by ξ ℓ , which is equivalent to changing ℓ + 1 by ℓ in the internal computations.
which follow in the same manner as in the proof of Lemmas 3.6 and 3.7 in [13].

2.5.
Proof of the Theorem 2.1. The proof follows the, now standard, contraction method applied to a localized in time cut-off integral formulation associated to the system (1.1), in the Bourgain spaces defined by the norms (2.10) and (2.11) (see [13], for example, for complete details of a similar proof). As is well known, the success of this method relies almost exclusively on the availabilty of certain multilinear estimates in these norms for the nonlinear terms of the equations. In our case these estimates are the ones obtained in Propositions 2.3 and 2.4. We start with the following integral system 1 Here S(t) is given by (1.8), ψ 1 ∈ C ∞ (R; R + ) is an even function, such that 0 ≤ ψ 1 ≤ 1 and Observe that for the v equation (2.41), we are not using the standard Duhamel, or variation of parameters, form as in (1.6). Both forms could be used here, but our choice makes computations slightly simpler.
The contraction condition can be obtained in a similar way and the proof is finished.
Remark 2.6. In the case b = c = 1/2 the proof of the Theorem 2.1 follows by similar arguments, but using the following modified norms |u | s := u X s,1/2 + ξ s τ + 1 2 |ξ| 2 −1 u(ξ, τ ) in order to obtain the immersions in the spaces C

Global well-posedness for the critical model
As mentioned in the introduction, in dimension N = 2, the system (1.1) is a perturbation of the scaling-critical cubic NLS equation (1.2). In this section we derive a priori estimates in the energy space H 1 (R 2 ) × L 2 (R 2 ) for the focusing and defocusing cases of (1.1), which allow us to extend the local solutions obtained in the previous section to all positive times.
to the Initial Value Problem (1.1).
Proof. In view of the local well-posedness result detailed in the last section and the conservation of the L 2 -norm of u, we only need to obtain an a priori bound for the function We begin by estimating v(·, t) 2 L 2 . Using the explicit representation for v, given by and applying the Minkowski and Gagliardo-Nirenberg inequalities we get where β is the constant from the Gagliardo-Nirenberg inequality. Now, we use Hölder's inequality to obtain, from (3.46) v(·, t) 2 (3.47) On the other hand, from (1.5), we have (3.48) We begin by treating the term 2v|u| 2 dx, which, in view of (3.45), can be rewritten in the form where We now proceed with the estimates of the terms A(t) and B(t). To estimate A(t) we use the Hölder and Gagliardo-Nirenberg inequalities to obtain (3.50) Now we estimate B(t). By the Hölder, Minkowski and Gagliardo-Nirenberg inequalities we get for all t ∈ [0, T µ ]. Next, we estimate the growth of E(t). Using (1.4), we have , from which follows Finally, we combine (3.47) with (3.54) to obtain Hence, by Gronwall's lemma, Since the time T µ = µ 4β 4 u 0 2 L 2 depends only on the conserved quantity u 0 L 2 , we can iterate this procedure in order to extend this solution to all positive times. Note, however, that the solution can blow-up at infinity.

4.1.
Global well-posedness in L 2 × L 2 (N = 2, 3). We observe that following the same ideas outlined in Remark 5.5 in [10] we can extend our local results in L 2 × L 2 to any positive time T .
Furthermore, differentiating (3.45) with respect to x and taking the L 2 -norm yields where we have used the Sobolev embedding H 1 (R) ֒→ L ∞ (R). Finally, we obtain We conclude as in Theorem 3.1.

4.3.
On the blow-up in H 1 × H 1 (N = 2). Since, in two dimensions, u H 1 does not control u L ∞ , the considerations in the previous Remark do not apply. However, our global result for initial data (u 0 , v 0 ) ∈ H 1 × L 2 shows that a possible blow-up in H 1 × H 1 can only occur for ∇v L 2 .

Comparison between the cubic NLS and the Schrödinger-Debye equations.
In the next table, we summarize all known results concerning the local and global wellposedness for these equations. It illustrates the regularization induced by the delay µ in the Schrödinger-Debye system.