On global attractors for 2D damped driven nonlinear Schr\"odinger equations

Well-posedness and global attractor are established for 2D damped driven nonlinear Schr\"odinger equation with almost periodic pumping in a bounded region. The key role is played by a novel application of the energy equation.

We impose the Dirichlet boundary condition ψ(x, t) = 0, x ∈ ∂Ω, t ∈ R. (1.2) Our analysis of the nonautonomous Schrödinger equation is motivated by mathematical problems of laser and maser coherent radiation. In particular, the maser action is described by the damped driven nonlinear Maxwell-Schrödinger equations in a bounded cavity, and the Dirichlet boundary condition (1.2) holds with a high precision in the case of a solid-state gain medium [17]. We will identify complex numbers z ∈ C with two-dimensional real vectors z = (Re z, Im z). We assume that the nonlinear term is potential, i.e., Here U (ψ) ∈ C 3 (R 2 ) is a real function, and the derivatives mean the derivatives with respect to ψ 1 = Re ψ and ψ 2 = Im ψ, i.e., f (ψ) := (∂ 1 U (ψ), ∂ 2 U (ψ)), ψ ∈ R 2 . . Similarly, f ′ (ψ) and f ′′ (ψ) are tensors: i, j, k = 1, 2. Let us comment on previous works in these directions. In the case p(x, t) ≡ 0 (and γ = 0) the wellposedness for nonlinear Schrödinger equations of type (2.1) was established in [7]. The approach [7] relies essentially on the conservation of charge and energy.
Compact attractors in the energy space H 1 were constructed for weakly damped nonlinear Schrödinger equation i) on the circle Ω = R/Z or a bounded interval Ω ⊂ R by Ghidaglia [12], ii) on a bounded region Ω ⊂ R 2 by Abounouh and Goubet [1,2] and iii) on the entire space Ω = R N with N ≤ 3 by Laurençot [21]. Similar results were obtained by Ghidaglia and Heron [13] for the Ginzburg-Landau equations on a bounded region Ω ⊂ R n with n = 1, 2. The methods of these papers rely on the Ball ideas [5]. Note that the pumping terms (or 'external force') in [1,2,13,21] do not depend on time. In [12], bounded absorbing sets are established for the pumping term depending on time, while the attractor is constructed for the autonomous case and for time-periodic pumping. The papers [14,15,16] are concerned with smoothness properties of functions from attractors of 1D nonlinear Schrödinger equations.
These methods and results were extended i) to 1D nonautonomous Schrödinger equations on the circle by Wang [27] and ii) to nonautonomous KdV and 2D Navier-Stokes equations [23].
Tao [26] established the existence of a global attractor for radial solutions to nonlinear autonomous defocusing Schrödinger equation without damping in R n with n ≥ 11. The paper [25] concerns the wellposendess and decay of solutions to 2D damped autonomous Schrödinger equation in a bounded region.
Recently Cazenave and Han established the long-time bechavior for nonlinear Schrödinger equation in R n with a nonlinear subcritical dissipation [8].
Let us comment on our approach. First, we prove the existence of a weak solution ψ(x, t), using the standard Galerkin approach. Our choice of the dimension two is caused by the Trudinger inequality, which provides the uniqueness of the weak solution by Theorem 3.6.1 of [7].
Further, we prove that ψ(x, t) is indeed a strong solution. This is the first result for nonautonomous 2D Schrödinger equation in a bounded region. Note that in our case when p(x, t) ≡ 0 and γ > 0, the charge and energy conservation do not hold for the nonautonomous equation (1.1). Respectively, the approach of [7] is not applicable here. This is why we introduce a novel method based on the energy equation (3.1). This method allows us to substitute the role of charge and energy conservations from [7] in proving that i) ψ(x, t) is the strong solution and ii) the solution continuously depends on the initial state and the pumping.
To prove the convergence in H 1 -norm of all finite energy solutions to a compact attractor, we develop the Ball ideas introduced in the context of autonomous equations [5], and their extensions to nonautonomous 1D Schrödinger and KdV equations [27,23]. The almost-periodicity of the pumping plays a crucial role in our approach.

A priori estimates for smooth solutions
Here we prove a priori estimates for sufficiently smooth solutions to the Schrödinger equation (1.1). The estimates provide the existence of bounded absorbing sets in the energy norm. We will get rid of the smoothness assumption in the next section.
Nonlinearity. We will assume that the potential U (ψ) ∈ C 3 (C), satisfies the following estimates with all κ l > 0 and all b l ∈ R. Let us recall that f ′ (ψ) in (2.5) and (2.6) denotes the matrix (1.5). For example, the potentials U (ψ) = a 2 |ψ| 4 + a 1 |ψ| 2 + a 0 (2.8) satisfy all these conditions if a k ∈ R and a 2 > 0. Any smooth potential which differs from (2.8) only in a bounded domain |ψ| ≤ C also satisfies these conditions.
Denote H s := H s (Ω), L p := L p (Ω). Let us introduce the Hilbert phase space where · stands for the norm in L 2 . Then E * = H −1 .
The pumping. We will assume that By definition (see Appendix of [18] for the case of scalar functions), a function p ∈ C b (R, E) is uniformly almost periodic if for any sequence t k ∈ R there is a subsequence t k * such that the translations p k * (t) := p(t + t k * ) are uniformly converging, (2.11)

Energy estimates
First we prove a priori estimates in the norm of the energy space E.

Well-posedness
In this section, we prove the well-posedness of the Cauchy problem for the Schrödinger equation (2.1) in the energy class. The key role is played by the energy equation (2.1).

Weak solutions
First, we construct 'weak solutions'.
Proof. The existence of a weak solution ψ(t) ∈ C(R, E w ) ∩ L ∞ loc (R, E) ∩ W 1,∞ loc (R, E * ) follows by the Galerkin approximations. We recall this construction in Appendix A since we will use it in the proof of (3.1). The uniqueness of this solution in the case p(x, t) ≡ 0 and γ = 0 is deduced in Theorem 3.6.1 of [7] from the Trudinger inequality [3]. The proof of the uniqueness for γ > 0 and p(x, t) satisfying (2.10) remains almost unchanged: we give the required modifications in Appendix B. The uniqueness implies that the convergence (A.5) holds actually for the entire sequence of Galerkin approximations: for any s < 1 The uniform bounds (A.2) hold for the Galerkin approximations ψ m . Hence the bounds (2.12) hold also for their limit ψ(t).

Energy equation
Let us prove the energy equation (3.1).
Proof. The identity (3.1) for smooth solutions can be rewritten as Equivalently, We will deduce (3.4) by the limit transition in the corresponding equation for the Galerkin approximations. Namely, the Galerkin equations (A.1) imply for any T ∈ R The right-hand side of (3.5) converges to the same of (3.4).
b) The left-hand side of (3.4) is dominated by the limit of the left-hand side of (3.5).
Proof. It suffices to consider the case T > 0. As a result, we obtain in the limit the energy inequality: the left-hand side of (3.4) is dominated by its right-hand side for every T ∈ R.
On the other hand, ψ(t) ∈ C(R, E w ) by Lemma 3.2, and hence, ψ(T ) ∈ E. Therefore, the same lemma implies the existence of a solutionψ(t) ∈ C(R, E w ) ∩ L ∞ loc (R, E) ∩ W 1,∞ loc (R, E * ) starting from the initial stateψ(T ) = ψ(T ) ∈ E. The same arguments of Lemma 3.4 show that the right-hand side of (3.4), withψ instead of ψ, is dominated by its left-hand side.
At last, the uniqueness of a solution in Lemma 3.2 implies thatψ(t) = ψ(t) for t ∈ R. Hence the integral energy equations (3.4) holds for t ∈ R. Finally, (3.1) holds for a.a. t ∈ R and in the sense of distributions. The proposition is proved.

Strong solutions
Now we can prove that ψ(t) ∈ C(R, E) ∩ C 1 (R, E * ). Namely, ψ(t) ∈ C(R, E w ) ∩ L ∞ loc (R, E) by Lemma 3.2. On the other hand, the energy equation (3.4) holds for all T ∈ R, and the right-hand side of (3.4) is a continuous function of T ∈ R by the same arguments as in the proof of Lemma 3.4. Hence (3.4) implies that the norm ψ(t) E is also a continuous function of t ∈ R. Therefore, ψ(t) ∈ C(R, E), and the equation (2.1) together with Lemma A.1, ii) and condition (2.10) imply thatψ(t) ∈ C(R, E * ).

Continuous dependence
The continuity of the map (3.2) follows by similar arguments. Namely, let ψ(0), ψ n (0) ∈ E, p, p n ∈ C(R, E) and Consider the corresponding unique solutions ψ n (t) ∈ C(R, E) ∩ C 1 (R, E * ) to (2.1). Similarly to (3.3), we obtain for any s < 1 and any T > 0 Hence the Banach theorem on weak compactness implies that On the other hand, the energy equation (3.4) implies the norm-convergence by the same arguments as in the proof of Lemma 3.4, using the uniform bounds (2.12) and the uniform convergence (3.8). Hence Now Theorem 3.1 is proved.

Weak continuity of the process
In conclusion, we prove a technical lemma which will be important in the next section. Consider the corresponding unique solutions ψ n (t) ∈ C(R, E) ∩ C 1 (R, E * ) to The limit functionψ(t) ∈ C([0, T ], H s ) satisfies the Schrödinger equation (2.1). This follows by the limit transition in the equation (3.10) taking into account Lemma A.1, i) and the uniform convergence of p n in (3.9). Moreover,ψ(0) = ψ(0) by the convergence of ψ n (0) in (3.9). Hence the uniqueness arguments as above imply thatψ(t) ≡ ψ(t) for all t ∈ [0, T ]. Therefore, the continuity of S(T, 0) follows from (3.11) since the limit does not depend on the subsequence n ′ .

Global attractor
In this section, we prove the convergence in the H 1 -norm of all finite energy solutions to a compact attractor. Proposition 3.1, i) allows us to define the continuous process S p (t, τ ) in E acting by Now we prove the existence of a uniform compact attractor in H 1 . Let us recall the definition of the uniform attractor for the nonautonomous equations, see, e.g., [19], [10, Definition A2.3], [11, Definition (7)]. The term 'uniformly attracting' means that, for any bounded subset B ⊂ E, Proof. Using (2.4) and (2.5), we obtain Applying the Hölder inequality and the Sobolev embedding theorem, we obtain for any s ≥ 1/2 for any s ≥ 0.

Strong convergence
The following proposition is the key step in the proof of Theorem 4.2.
Proposition 4.4. For any sequences t k , τ k → ∞ with t k − τ k → ∞ and φ k ∈B, there is a subsequence k * and φ ∈ E such that Proof. The sequence S p (t k , τ k )φ k is bounded in E by (2.12). Let us fix an arbitrary s ∈ [1/2, 1). The inclusion E ⊂ H s is compact by the Sobolev embedding theorem. Hence there is a subsequence strongly converging in H s : Now to prove (4.7) it suffices to check that for a subsequence {k * } ⊂ {k ′ } lim sup Recall that t k − τ k → ∞. Hence similarly to (4.8), for any T > 0, there exists a subsequence {k ′′ } ⊂ {k ′ } and an element φ(−T ) ∈ E such that We set p k ′′ (t) := p(t k ′′ − T + t) and denote In particular, for t = T the convergence (4.8) implies that Now we are going to apply Lemma 3.5 iv). At this moment we need the almost-periodicity (2.11) of the pumping p(t). It implies that for a subsequence {k * } ⊂ {k ′′ } Hence (4.10), (4.11) and the continuity of the map (3.2) imply the convergence for t ∈ [0, T ], (4.14) In particular, for t = T the convergence (4.12) gives that Now we apply the integral identity (4.6) to solutions ψ k * (t) and ψ * (t) of the equation (2.1) with the pumping p k * (t) and p * (t) respectively. We obtain (4.17) where the integrals withṗ k * (τ ) andṗ(τ ) are definite according to (4.5). Making k * → ∞ in (4.16) we obtain lim sup by the following arguments: By definitions (1.7) and (4.3), Now Lemma 4.3 together with (4.8), (4.12) and (4.15) imply that Hence (4.19) becomes According to the definition (4.12), we can replace ψ k * (T ) by S p (t k * , τ k * )φ k * . Now making T → ∞, we get (4.9) since γ > 0.

Compactness
Let us denote by A the set of all points φ from (4.7). This set is obviously closed. Now Theorem 4.2 will follow from the next lemma.
Lemma 4.5. i) The set A is compact in E, and ii) the set A is uniformly attracting in E.
Proof. i) Let us consider a sequence φ n ∈ A. Then, for each n, as k → ∞, where φ nk ∈B. Hence there exists a sequence k(n) such that as n → ∞. However, Proposition 4.4 implies that for a subsequence n ′ Hence φ ∈ A, and (4.21) implies that φ n ′ E − −→ φ. Therefore, the first assertion of the lemma is proved.
ii) Let us assume the contrary. Then there exists a sequence S(t k , τ k )φ k such that On the other hand, Proposition 4.4 implies that for a subsequence k ′ , This contradiction proves the second assertion of the lemma.
Here P m stands for the orthogonal projection of L 2 onto E m , and p m (t) := P m p(t). Applying the calculations from the proof of Lemma 2.1 to the equations (A.1), we get the uniform estimates of type (2.12) for ψ m with the same constants: Hence, the Galerkin approximations ψ m (t) exist globally in time.
Let us show that these uniform estimates imply the existence of a limiting function ψ(t) of the Galerkin approximations. For this purpose we will use the known continuity property of the nonlinear term: , the nonlinearity N : ψ(·) → f (ψ(·)) is the continuous map H s → L r .
iii) N is the continuous map E 1 → L 2 .
Proof. i) Condition (2.5) implies that Hence, the Hölder inequality and the Sobolev embedding theorem give, for r ≥ 1, ii) In particular, (A.3) implies that N is continuous H 2/3 → L 2 . It remains to note that E ⊂ H 2/3 , while iii) The map N : E 1 → C(Ω) is continuous by the Sobolev embedding theorem.
Further, the Galerkin approximations are equicontinuous in H s with any s < 1 by the Dubinsky 'Theorem on Three Spaces' ( [22, Theorem 5.1]). Namely, this equicontinuity follows from the interpolation inequality: for any δ > 0 Here the first term on the right is small for sufficiently small δ > 0, since sup m sup t≥0 ψ m (t) E < ∞ by (A.2), while the second term is small for |t 1 − t 2 | ≪ 1, since sup m sup t≥0 ψ m (t) E * < ∞ by the Galerkin equations (A.1) together with Lemma A.1, ii) and (2.10). Now the Arzelà-Ascoli theorem implies that there exists a subsequence ψ m ′ converging in C(R, H s ): The limit function ψ(t) ∈ C(R, H s ) satisfies the estimates (2.12) by the uniform bounds (A.2), and hence, ψ(t) ∈ C(R, E w ) ∩ L ∞ loc (R, E). The limit function ψ(t) satisfies the Schrödinger equation (2.1) in the sense of distributions. This follows from the convergence (A.5) with s < 1 close to 1 by limit transition in the Galerkin equations (A.1) taking into account Lemma A.1. Finally, the convergence ψ m (0) → ψ(0) as m → ∞ implies that ψ(0) = ψ(0).