Dynamics of a Gross-Pitaevskii Equation with Phenomenological Damping

Abstract

We study the dynamical behavior of solutions of an n-dimensional nonlinear Schrödinger equation with potential and linear derivative terms under the presence of phenomenological damping. This equation is a general version of the dissipative Gross-Pitaevskii equation including terms with first-order derivatives in the spatial coordinates which allow for rotational contributions. We obtain conditions for the existence of a global attractor and find bounds for its dimension.

1. Introduction

Nonlinear Schrödinger equations have enjoyed a considerable amount of attention during several decades due to their frequent appearance in the modeling of interesting physical phenomena in many different areas (e.g., optics, fluid mechanics, condensed matter, etc.). Additionally, the rigorous mathematical treatment of these equations has produced a great deal of new insights and techniques accompanied by a voluminous bibliography. We refer the reader to excellent sources as [1–4] for a detailed account of these aspects.

Here, for complex coefficients 𝛼 and 𝛽, our purpose is to study the long-term dynamics of a general nonlinear Schrödinger evolution equation with potential of the form 𝑖𝑢_𝑡−𝛼Δ𝑢+𝛽|𝑢|²𝑢+𝑓(𝑥)𝑢=𝑣(𝑥)⋅∇𝑢,(1) where 𝑢(𝑥,𝑡) and 𝑓(𝑥) are complex-valued functions and 𝑣(𝑥) is a complex vector-valued map, all defined on a bounded open domain 𝑈⊂ℝ^𝑛 (for 𝑛⩾1) with regular boundary 𝜕𝑈. We prove the existence of a global attractor for (1), under Dirichlet and 𝑈-periodic boundary conditions, assuming Im(𝛼)>0 and some additional requirements on 𝑓(𝑥) and 𝑣(𝑥) for which existence and uniqueness of solutions are guaranteed. Our choice of signs for the terms in this equation was made such that the hypotheses needed for our analysis make, if necessary, the imaginary parts of constant and functional coefficients positive.

Particular instances of (1) have been extensively considered in the literature, specially in two and three dimensions. In fact, when the coefficients 𝛼, and 𝛽 and the function 𝑓(𝑥) are real and assuming 𝑣(𝑥)=0, this is the celebrated time-dependent Gross-Pitaevskii equation (GP) employed to model nonlinear behavior in several physical systems. Of special relevance is the mean-field description of Bose-Einstein condensation achieved in dilute atomic gases, a state of matter which exhibits peculiar phenomena characteristic of the superfluid nature of the system.

If for the previous situation in three dimensions we choose 𝑣(𝑥)=𝑖Ω∧𝑥∈𝑖ℝ³, where Ω∈ℝ³ is an angular velocity vector, we recover the GP equation with angular momentum rotational term. Indeed, the right-hand side of (1) becomes 𝑣⋅∇𝑢=𝑖(Ω∧𝑥)⋅∇𝑢=𝑖Ω⋅(𝑥∧∇)𝑢=−Ω⋅𝐿𝑢,(2) where 𝐿∶=−𝑖(𝑥∧∇) is precisely the quantum mechanical angular momentum operator. This equation has been used to model physical experiments where the condensate is set into rotation with an angular frequency ‖Ω‖, making it known as the Gross-Pitaevskii equation for rotating Bose gases. In this situation, the observation of quantized vortices has been a topic of considerable experimental and theoretical studies; the reader is referred to [5] and references therein for details on vortices in Bose-Einstein condensates and the recent review [6] for a treatment that includes numerical aspects (see also [7] for the simulation of rotating condensates). Existence and uniqueness results for solutions of GP equation with rotation in ℝ^𝑛 for dimensions 𝑛=2 and 𝑛=3 and under different assumptions can be found in [8–10]. Note that rotating and nonnecessarily nonlinear Schrödinger equations appear also in several other contexts related to wave propagation (see references in [11]).

The general case of fully complex coefficients 𝛼, 𝛽 and complex-valued functions 𝑓(𝑥), 𝑣(𝑥) in (1) can be thought of as a mixture between the aforementioned GP equation and a gradient flow which accounts for dissipation in the system proportional to the rate of change of the wave function 𝑢(𝑥,𝑡). This is indeed the case if (1) has the particular form 𝑖𝑢_𝑡=1+𝑖𝛾𝛼Δ𝑢−𝛽|𝑢|²𝑢−𝑓(𝑥)𝑢+𝑣(𝑥)⋅∇𝑢,(3) for real constants 𝛼, 𝛽, 𝛾, a real-valued function 𝑓(𝑥), and an imaginary vector field 𝑣(𝑥) as in the rotational term explained earlier. In the physical literature phenomenological damping has been introduced to study Bose-Einstein condensates (both in the rotating and nonrotating cases) affected by complicated sources of dissipation related to their interaction with the thermal cloud around the condensate. For example, [12] studies the case of trapped atomic condensates and [13] analyzes dark matter wave solitons in the same context, while [14–16] study the formation of vortex lattices in rotating condensates, including numerical simulations. A theoretical explanation of the phenomenological damping terms can be found in [17] (see also [18, 19]).

From the mathematical side, equations similar to (3), but with 𝑣=0, have been used to study the behavior of Ginzburg-Landau vortices in [20] and, in the case of low dissipation, the asymptotics for perturbation of constant solutions in [21]. Long-time behavior, establishing the existence of attractors under several assumptions, has also been the topic of several studies; see, for example, [22] for Ginzburg-Landau equations on bounded domains, [23] for weakly damped nonlinear Schrödinger equations in ℝ^𝑛 with 𝑛⩽3, and for dissipative nonlinearities see [24] and references therein. To our knowledge attractors for the dissipative Gross-Pitaevskii equation with rotation have not been explicitly considered.

Since the arguments needed for our analysis work in the fully complex coefficients situation, we adopt this broader setting and, in addition, do not necessarily assume that the right-hand side of (1) has the particular form of an angular momentum contribution. This situation corresponds to the case where the ratio between imaginary and real parts of 𝛼, 𝛽, 𝑓(𝑥), and 𝑣(𝑥) in (3) is not a fixed constant 𝛾. In fact, this can include cases where there are sources of dissipation which are proportional to some of the individual terms in the equation, like the nonlinearity or the gradient terms. Such contributions are important, for example, to study exploding and collapsing Bose-Einstein condensates (see [25] and references therein for examples of pure nonlinear dissipation in that context, as well as [26] for simulation of collapse and explosion).

We point out that numerical studies in [27] show that cubic or quintic damping terms prevent blowup of solutions to focusing nonlinear Schrödinger equations without “rotational” term, while just linear damping is not enough below a certain threshold. This agrees with Theorem 8 on existence of attractors (that covers such situation if one assumes an arbitrarily small amount of linear damping) and provides some justification for the hypotheses under which this result is proved.

Finally, to summarize the content of the following sections we mention that, under appropriate assumptions introduced later, Theorem 4 establishes the existence and uniqueness of solutions to the dissipative GP equation on bounded open domains, Theorem 8 shows the existence of a compact global attractor, and Theorem 11 provides bounds for its Hausdorff and fractal dimensions.

1.1. Precise Setting

To ease notation, for any complex vector 𝑣∈ℂ^𝑛 with 𝑛⩾1, we define 𝑣₁∶=Re(𝑣)∈ℝ^𝑛 and 𝑣₂∶=Im(𝑣)∈ℝ^𝑛. Let 𝑈⊂ℝ^𝑛; if 𝑢(𝑥,𝑡)=𝑢₁(𝑥,𝑡)+𝑖𝑢₂(𝑥,𝑡)∶𝑈×ℝ⁺→ℂ is a complex-valued function, the gradient ∇𝑢 and Laplacian Δ𝑢 (both with respect to spatial coordinates) are given by ∇𝑢∶=∇𝑢₁+𝑖∇𝑢₂∈ℂ^𝑛 and Δ𝑢∶=Δ𝑢₁+𝑖Δ𝑢₂∈ℂ. In addition, we extend bilinearly the Euclidean inner product “⋅” in ℝ^𝑛 to a symmetric inner product on ℂ^𝑛 denoted for simplicity by the same dot; that is, for vectors 𝑣,𝑤∈ℂ^𝑛, we have 𝑣⋅𝑤∶=𝑣₁⋅𝑤₁−𝑣₂⋅𝑤₂+𝑖𝑣₁⋅𝑤₂+𝑣₂⋅𝑤₁.(4) With this product, the standard Hermitian norm of 𝑣∈ℂ^𝑛 satisfies ‖𝑣‖²=‖𝑣₁‖²+‖𝑣₂‖²=𝑣⋅𝑣, where 𝑣∶=𝑣₁−𝑖𝑣₂.

Given coefficients 𝛼,𝛽∈ℂ and a constant 𝑝⩾2, we consider the following dissipative (fully complex) GP equation with nonlinearity of order (𝑝+1) on a bounded open domain 𝑈⊂ℝ^𝑛, for arbitrary 𝑛⩾1: 𝑖𝑢_𝑡−𝛼Δ𝑢+𝛽|𝑢|^𝑝(𝑥)(𝑥)𝑢()𝑢+𝑓𝑢=𝑣⋅∇𝑢,𝑥,0=𝑢₀(𝑥),(5) with unknown 𝑢(𝑥,𝑡)∶𝑈×ℝ⁺→ℂ, a scalar function 𝑓(𝑥)∶𝑈→ℂ, and a vector-valued map 𝑣(𝑥)∶𝑈→ℂ^𝑛. Our hypotheses on the constant coefficients are 𝛼₂()𝛽>0,H1₂⩾𝑝𝛽₁⩾0(H2) and on the functional coefficients and initial data are 𝑓(𝑥)∈𝐿^∞(𝑈),()𝑣(𝑥)H3∈𝐿^∞𝑈,ℂ^𝑛,()𝑢H4₀(𝑥)∈𝐿²(𝑈).(H5) Additionally, we assume that 𝜇₂∶=essinf_𝑥∈𝑈𝑓₂(𝑥)>0(H6) and require that 4𝛼₂𝜇₂>‖𝑣‖²_∞.(H7)

Remark 1. Notice that hypothesis (H2) allows for 𝛽=0 in which case there is no nonlinearity term in (5). Independently, hypotheses (H1) and (H6) already imply (H7) when 𝑣(𝑥)=0 and this includes the standard GP equation (i.e., no rotational term) but with dissipation.

Remark 2. As a consequence of hypothesis (H6), (5) has a nonzero linear damping term of the form 𝑖𝑓₂(𝑥)𝑢. Obviously, such term is always present in the case of GP with phenomenological damping (3) and is not the only source of dissipation.

For (5) we will consider Dirichlet boundary conditions on the domain 𝑈⊂ℝ^𝑛, which is assumed to have regular boundary 𝜕𝑈. Explicitly, we suppose that for all 𝑡⩾0, 𝑢(𝑥,𝑡)|_𝜕𝑈()|=0,D1Δ𝑢(𝑥,𝑡)_𝜕𝑈=0.(D2) Alternatively we also consider 𝑈-periodic boundary conditions when the domain has the form 𝑈=(0,𝐿)^𝑛⊂ℝ^𝑛. This means that for any pair of points 𝑎,𝑏∈𝑈=[0,𝐿]^𝑛 which are equal except at an arbitrary fixed component 𝑗∈{1,…,𝑛} for which 𝑎_𝑗=0 and 𝑏_𝑗=𝐿, it holds that 𝜕^𝑘𝑢(𝑎,𝑡)=𝜕^𝑘𝑢(𝑏,𝑡),(P) for all 𝑡⩾0 and admissible integers 𝑘⩾0. In this expression, 𝜕⁰𝑢∶=𝑢 and 𝜕^𝑘𝑢 denotes any 𝑘-order partial (weak) derivative admissible for 𝑢.

Remark 3. In the case of 𝑈-periodic boundary conditions, it is actually assumed that points of the form 𝑎,𝑏∈[0,𝐿]^𝑛 described earlier are identified, so in fact any function defined on 𝑈 must take the same values on those points but, in contrast to Dirichlet conditions, the precise value does not have to be fixed.
Existence and uniqueness of solutions to (5) with Dirichlet (D1)-(D2) or 𝑈-periodic (P) boundary conditions hold from classical results.

Theorem 4. Given hypotheses (H1)–(H7), under Dirichlet or 𝑈-periodic boundary conditions, there exists a unique solution 𝑢 of (5) such that 𝑢∈𝒞_𝑏ℝ⁺,𝐻¹(𝑈),𝑢_𝑡∈𝒞_𝑏ℝ⁺,𝐻⁻¹(𝑈),(6) where 𝐻⁻¹ is the dual space of 𝐻¹ and 𝒞_𝑏(ℝ⁺,𝐻¹(𝑈)) is the set of continuous bounded functions from ℝ⁺ into 𝐻¹(𝑈).

The proof of the previous theorem follows by application of the standard Faedo-Galerkin method with small modifications to the proof of [28, Theorems IV.5.1 and IV.6.1]. The necessary estimates are similar to the ones given later to show the existence of an absorbing set in 𝐿²(𝑈). It is important to notice that the estimates use only the 𝐿²-norm of 𝑢₀, which exists by (H5).

2. Existence of a Global Attractor

In this section we prove the existence of a compact global attractor 𝒜⊂𝐿²(𝑈). That is, a compact subset of 𝐿²(𝑈) such that if 𝑆(𝑡) denotes the semigroup generated by solutions of (5), then(i)𝑆(𝑡)𝒜=𝒜 for all 𝑡⩾0;(ii)for every bounded set 𝐵⊂𝐿²(𝑈), it holds that the 𝐿²-distance 𝑑(𝑆(𝑡)𝐵,𝒜)→0 as 𝑡→∞.

In order to prove the existence of the attractor, we will show first the existence of an absorbing set in 𝐿²(𝑈), that is, a set ℬ such that any bounded set of 𝐿²(𝑈) enters into ℬ after a certain time.

Theorem 5. Assume that hypotheses (H1)–(H7) hold for (5) under Dirichlet or 𝑈-periodic boundary conditions. Then there exist positive real constants 𝜏₀ and 𝜏₁ such that(i) the semigroup 𝑆(𝑡) possesses an absorbing ball in 𝐿²(𝑈) as 𝑆(𝑡)𝑢_0𝐿2⩽𝑟₀𝑖𝑓𝑡⩾𝜏₀,(7) where 𝑟₀∶=1 and 𝜏₀∶=(1/𝐾)log‖𝑢₀‖_𝐿2, with 𝐾∶=𝜇₂−(1/4𝛼₂)‖𝑣‖²_∞;(ii) the semigroup 𝑆(𝑡) possesses an absorbing ball in 𝐻¹₀(𝑈) as 𝑆(𝑡)𝑢_0𝐻10⩽𝑟₁𝑖𝑓𝑡⩾𝜏₁,(8) where 𝑟₁∶=(1/𝛼₂)𝑒^{(1/2)‖𝑣‖2∞}(‖𝑣‖²_∞+𝛼²₂‖𝑓‖²_∞+1)^1/2 and 𝜏₁∶=𝜏₀+𝛼₂.

From the previous theorem we have that the set ℬ∶=𝑢∈𝐿²(𝑈)∶‖𝑢‖_𝐿²⩽𝑟₀,‖∇𝑢‖_𝐿²⩽𝑟₁⊂𝐻¹(𝑈)(9) is absorbing for all the bounded sets of 𝐿²(𝑈) provided that 𝑡⩾𝜏₁.

Proof. We divide the proof into two steps. Before that, we recall the following inequality for the convenience of the reader (see [28, Lemma III.1.1] for a proof).

Lemma 6 (uniform Gronwall’s lemma [29]). Let 𝑓,𝑔, and ℎ be locally integrable functions on (𝑡₀,∞) such that 𝑓^′ is locally integrable on (𝑡₀,∞) and 𝑓^′⩽𝑓𝑔+ℎ, for all 𝑡⩾𝑡₀. Assume additionally that there are real positive constants 𝑎₁, 𝑎₂, 𝑎₃ such that for every 𝑡⩾𝑡₀, _𝑡^𝑡+𝜚𝑓(𝑠)𝑑𝑠⩽𝑎₁,_𝑡^𝑡+𝜚𝑔(𝑠)𝑑𝑠⩽𝑎₂,_{𝑡𝑡+𝜚}ℎ(𝑠)𝑑𝑠⩽𝑎₃.(10)
Then it holds that 𝑓⩽𝑎𝑡+𝜚¹𝜚+𝑎₃𝑒^𝑎2,∀𝑡⩾𝑡₀.(11)

2.1. Absorbing Ball in 𝐿²(𝑈)

Multiplying (5) by 𝑢, integrating over 𝑈⊂ℝ^𝑛 using integration by parts with any set of the proposed boundary conditions, and taking the imaginary part, 12𝑑‖𝑢‖𝑑𝑡^2𝐿2+𝛼₂‖∇𝑢‖²_𝐿2+𝛽₂‖𝑢‖_𝐿^𝑝+2_𝑝+2+_𝑈𝑓₂(𝑥)|𝑢|²𝑑𝑥=Im_𝑈(𝑣⋅∇𝑢)⩽‖‖𝑢𝑑𝑥𝑣⋅∇𝑢_𝐿²‖𝑢‖_𝐿²⩽‖∇𝑢‖_𝐿²‖𝑣‖_∞‖𝑢‖_𝐿²⩽𝛼₂‖∇𝑢‖²_𝐿2+14𝛼₂‖𝑣‖²_∞‖𝑢‖²_𝐿2.(12) After eliminating the third term to the left-hand side of the previous calculation, we get 𝑑‖𝑢‖𝑑𝑡^2𝐿2⩽−2𝐾‖𝑢‖²_𝐿2,(13) where 𝐾=𝐾𝛼₂,𝑓₂,𝑣∶=𝜇₂−14𝛼₂‖𝑣‖²_∞.(14) Then, applying Gronwall’s lemma and since (H7) implies that 𝐾>0, we obtain ‖𝑢‖²_𝐿2⩽𝑢₀²_𝐿2𝑒^−2𝐾𝑡.(15) Setting 𝜏₀∶=(1/𝐾)log‖𝑢₀‖_𝐿2, we have that ‖𝑢(𝑡)‖_𝐿²⩽1=∶𝑟₀,if𝑡⩾𝜏₀.(16) This proves the existence of an absorbing ball in 𝐿²(𝑈).

2.2. Absorbing Ball in 𝐻¹₀(𝑈)

Now we multiply (5) by Δ𝑢, integrate as before over 𝑈⊂ℝ^𝑛 using boundary conditions (after integration by parts) to simplify, and take the imaginary part to get 12𝑑‖‖𝑑𝑡∇𝑢²_𝐿2+𝛼₂‖Δ𝑢‖²_𝐿2+Im_𝑈(𝑣⋅∇𝑢)Δ𝑢𝑑𝑥=Im_𝑈𝑓(𝑥)𝑢Δ𝑢𝑑𝑥+Im𝛽_𝑈|𝑢|^𝑝𝑢Δ𝑢𝑑𝑥.(17) Integrating by parts the last term of the previous equality and using that for any 𝑧,𝑤∈ℂ it holds that Im(𝑧𝑤)=𝑧₁𝑤₂+𝑧₂𝑤₁, we get Im𝛽_𝑈|𝑢|^𝑝𝑢Δ𝑢𝑑𝑥=−𝛽_2𝑈|𝑢|^𝑝‖∇𝑢‖²𝛽𝑑𝑥−𝑝Im_𝑈|𝑢|^𝑝−2(𝑢∇𝑢)⋅Re(𝑢∇𝑢)𝑑𝑥=−𝛽_2𝑈|𝑢|^𝑝‖∇𝑢‖²𝑑𝑥−𝑝𝛽_1𝑈|𝑢|^𝑝−2Im(𝑢∇𝑢)⋅Re(𝑢∇𝑢)𝑑𝑥−𝑝𝛽_2𝑈|𝑢|^𝑝−2‖Re(𝑢∇𝑢)‖²𝑑𝑥⩽−𝛽_2𝑈|𝑢|^𝑝‖∇𝑢‖²𝛽𝑑𝑥+𝑝_1𝑈|𝑢|^𝑝−2‖𝑢∇𝑢‖²𝛽𝑑𝑥⩽−₂−𝑝𝛽_1𝑈|𝑢|^𝑝‖∇𝑢‖²𝑑𝑥,(18) where the inequality appears after eliminating the last term since 𝛽₂⩾0. Replacing this in the original equation, we can write 12𝑑‖‖𝑑𝑡∇𝑢²_𝐿2+𝛼₂‖Δ𝑢‖²_𝐿2+𝛽₂−𝑝𝛽_1𝑈|𝑢|^𝑝‖∇𝑢‖²𝑑𝑥⩽Im_𝑈𝑓(𝑥)𝑢Δ𝑢𝑑𝑥−Im_𝑈(𝑣⋅∇𝑢)Δ⩽𝑓𝑢𝑑𝑥_∞‖𝑢‖_𝐿²‖Δ𝑢‖_𝐿²+‖𝑣‖_∞‖∇𝑢‖_𝐿²‖Δ𝑢‖_𝐿²⩽12𝛼₂𝑓²_∞‖𝑢‖²_𝐿2+𝛼₂2‖Δ𝑢‖²_𝐿2+12𝛼₂‖𝑣‖²_∞‖∇𝑢‖²_𝐿2+𝛼₂2‖Δ𝑢‖²_𝐿2.(19) Simplifying, using 𝛽₂⩾𝑝|𝛽₁| and working with 𝑡⩾𝜏₀ so that ‖𝑢‖_𝐿²⩽𝑟₀, we arrive at 𝑑‖‖𝑑𝑡∇𝑢²_𝐿2⩽1𝛼₂𝑟²₀𝑓²_∞+‖𝑣‖²_∞‖∇𝑢‖²_𝐿2.(20) To apply the uniform Gronwall lemma, it is necessary to have an upper bound on ∫_𝑡^𝑡+𝜚‖∇𝑢‖²_𝐿2𝑑𝑠. From (12), we have 12𝑑‖𝑢‖𝑑𝑡^2𝐿2+𝛼₂‖∇𝑢‖²_𝐿2+𝛽₂‖𝑢‖_𝐿^𝑝+2_𝑝+2+_𝑈𝑓₂(𝑥)|𝑢|²⩽‖‖𝑑𝑥∇𝑢_𝐿²‖𝑣‖_∞‖𝑢‖_𝐿²⩽𝛼₂2‖∇𝑢‖²_𝐿2+12𝛼₂‖𝑣‖²_∞‖𝑢‖²_𝐿2.(21) Now, eliminating the last two positive terms at the left-hand side, using 𝑡⩾𝜏₀ again, and integrating, we find that _𝑡^𝑡+𝜚‖∇𝑢‖²_𝐿2𝑑𝑠⩽𝑟²₀𝜚𝛼²₂‖𝑣‖²_∞+1𝛼₂=∶𝑎₁.(22) There are also obvious bounds given by _𝑡^𝑡+𝜚1𝛼₂‖𝑣‖²_∞𝜚𝑑𝑠⩽𝛼₂‖𝑣‖²_∞=∶𝑎₂,_{𝑡𝑡+𝜚}1𝛼₂𝑟²₀𝑓²_∞𝜚𝑑𝑠⩽𝛼₂𝑟²₀𝑓²_∞=∶𝑎₃.(23) With this, Lemma 6 immediately implies ∇𝑢𝑡+𝜚²_𝐿2⩽𝑟²₀1𝛼²₂‖𝑣‖²_∞+1𝛼₂𝜚+𝜚𝛼₂𝑓²_∞𝑒^{(𝜚/𝛼2)‖𝑣‖2∞}.(24) Choosing 𝜚=𝛼₂ and assuming 𝑡⩾𝜏₁∶=𝜏₀+𝛼₂, the previous inequality becomes ‖∇𝑢(𝑡)‖_𝐿²⩽𝑟₀𝛼₂𝑒^{(1/2)‖𝑣‖2∞}1+‖𝑣‖²_∞+𝛼²₂𝑓²_∞^1/2=∶𝑟₁.(25) This proves the existence of an absorbing ball in 𝐻¹₀(𝑈).

To show the existence of a compact global attractor in 𝐿²(𝑈), we will make use of the following result (see [28, Theorem I.1.1]) with 𝐻=𝐿²(𝑈).

Theorem 7. Suppose that 𝑆(𝑡) is a continuous semigroup acting on a metric space ℋ such that there exists a bounded absorbing set ℬ and for every bounded set 𝐵⊂ℋ there exists 𝑡₀>0 such that ⋃_𝑡⩾𝑡0𝑆(𝑡)𝐵 is relatively compact in ℋ. Then the 𝜔-limit set 𝜔(ℬ)∶=_{𝑠⩾0𝑡⩾𝑠}𝑆(𝑡)ℬ(26) is a compact global attractor for the semigroup 𝑆(𝑡) in ℋ.

Theorem 8. Under hypotheses (H1)–(H7) for (5), supplemented with Dirichlet or 𝑈-periodic boundary conditions, there exists a compact global attractor 𝒜⊂𝐿²(𝑈).

Proof. We will make use of Theorem 7. Since we have already shown the existence of the inertial set, we only need to prove that for any bounded set 𝐵∈𝐿²(𝑈), there exists 𝑡₀>0 such that the set ⋃_𝑡⩾𝑡0𝑆(𝑡)𝐵 is relatively compact in 𝐿²(𝑈). As a consequence of Theorem 5, we have proved the existence of a ball ℬ∶=𝑢∈𝐿²(𝑈)∶‖𝑢‖_𝐿²⩽𝑟₀,‖∇𝑢‖_𝐿²⩽𝑟₁⊂𝐻¹(𝑈),(27) such that for any bounded set 𝐵⊂𝐿²(𝑈), it holds that 𝑆(𝑡)𝐵⊂ℬ for 𝑡⩾𝜏₁. Since ℬ is bounded in 𝐻¹(𝑈) and the embedding of 𝐻¹(𝑈) in 𝐿²(𝑈) is compact, we conclude that the set _𝑡⩾𝜏1𝑆(𝑡)𝐵(28) is relatively compact in 𝐿²(𝑈). This finishes the proof.

Remark 9. We point out here that when no dissipation is assumed in (5), which requires in particular 𝛼₂=𝛽₂=0, 𝑓₂(𝑥)=0, and 𝑣₁(𝑥)=0, a very different situation may occur. For example, for the nondissipative Gross-Pitaevskii equation with rotation, it has been proved in [10, Theorem 2.2] that there are conditions on 𝑓₁(𝑥) and 𝑣₂(𝑥) such that (5) has initial data 𝑢₀(𝑥) for which solutions present finite time blowup in dimensions 𝑛=2 or 𝑛=3; that is, there is a finite time 𝜏_∗>0 such that lim_{𝑡→𝜏∗}‖∇𝑢(𝑡)‖_𝐿²=∞.(29) This can happen, for example, with an axial-symmetric parabolic potential for 𝑛=2,3 in the case of a focusing (𝛽₁>0) and energy subcritical nonlinearity of appropriate order (more precisely 4/𝑛⩽𝑝<4/(𝑛−2) with 𝑝 even). In our situation, the hypotheses employed (in particular the kind of dissipation) eliminate the possibility of such phenomena.

3. Dimension of the Attractor

In this section we provide an estimate for the dimension of the attractor by the method of evolution of infinitesimal 𝑚-dimensional volume elements in the phase space 𝐿²(𝑈) under the flow induced by the action of the semigroup 𝑆(𝑡) of Theorem 5 (see [28, Chapter V] for details). For this estimate we will explicitly assume that the bounded open domain 𝑈⊂ℝ^𝑛 is at least of class 𝒞¹; that is, 𝑈 has a boundary 𝜕𝑈 with a structure of an (𝑛−1)-dimensional manifold of class 𝒞¹.

Given an orthonormal basis 𝜑₁(𝑡),…,𝜑_𝑚(𝑡) for a time-dependent 𝑚-dimensional subspace of 𝐿²(𝑈), it is possible (cf. [28, Appendix, Corollary 4.1]) to find constants 𝑐₁,𝑐₂⩾0, depending on 𝑛, 𝑘 and the shape of 𝑈⊂ℝ^𝑛 but independent of the size |𝑈|, the value 𝑚, and of the chosen orthonormal basis, such that ^𝑚_𝑖=1∇𝜑_𝑖²_𝐿2⩾1|𝑈|^2/𝑛𝑐₁𝑚^1+2/𝑛−𝑐₂𝑚.(30) Moreover, if the functions 𝜑_𝑖∈𝐻¹₀(𝑈), it is possible to choose 𝑐₂=0. This holds for Dirichlet boundary conditions.

In order to study the evolution of infinitesimal volume elements, we linearize (5) around a solution 𝑢(𝑡)=𝑆(𝑡)𝑢₀, with 𝑢₀∈𝒜 as follows: 𝑖𝜙_𝑡=𝐹^′(𝑢)𝜙∶=𝛼Δ𝜙−𝛽𝑝+1|𝑢|^𝑝(𝑥)𝜙(0)𝜙−𝑓𝜙+𝑣⋅∇𝜙,=𝜉.(31) Then, noting that Re(𝜙_𝑡)=Im(𝐹^′(𝑢)𝜙), the evolution of the 𝑚-dimensional volume elements is given by (cf. [30, Section 13.2]) 𝑉_𝑚(𝑡)=𝑉_𝑚(0)expIm^𝑡₀Tr𝐹^′(𝑢)𝑃_𝑚(𝑠)𝑑𝑠,(32) where 𝑃_𝑚 is the orthogonal projection of 𝐿²(𝑈) over the subspace spanned by the first 𝑚 eigenvectors of the Laplacian. The asymptotic growth rate of 𝑉_𝑚(𝑡) is given by lim_𝑡→∞1𝑡^𝑡₀ImTr𝐹^′(𝑢)𝑃_𝑚(𝑠)𝑑𝑠.(33) Since we need the maximal growth rate, we compute the supremum over all 𝑢₀∈𝒜 and over all the 𝑚-dimensional projectors as follows: sup_𝑢0∈𝒜sup_𝑃𝑚limsup_𝑡→+∞1𝑡^𝑡₀ImTr𝐹^′(𝑢)𝑃_𝑚(𝑠)𝑑𝑠=∶Tr_𝑚(𝒜).(34)

In order to have exponential decay, we look for the smallest 𝑚 such that the number Tr_𝑚(𝒜) is negative. In detail, let 𝜑₁(𝑡),…,𝜑_𝑚(𝑡) be an orthonormal basis for 𝑃_𝑚𝐿²(𝑈); then ImTr𝐹^′(𝑢)𝑃_𝑚=^𝑚_𝑖=1Im𝐹^′(𝑢)𝜑_𝑖,𝜑_{𝑖𝐿2(𝑈)},(35) with Im𝐹^′(𝑢)𝜑_𝑖,𝜑_{𝑖𝐿2(𝑈)}=−𝛼₂∇𝜑_𝑖²_𝐿2−𝛽₂𝑝+1_𝑈|𝑢|^𝑝𝜑_𝑖²−𝑑𝑥_𝑈𝑓₂(𝑥)𝜑_𝑖²𝑑𝑥+Im_𝑈𝑣⋅∇𝜑_𝑖𝜑_𝑖𝑑𝑥.(36) Now, we estimate each of the last three terms separately. For the first, since 𝛽₂⩾0, we can simply write −𝛽₂𝑝+1_𝑈|𝑢|^𝑝𝜑_𝑖²𝑑𝑥⩽0,(37) for the second, we have −_𝑈𝑓₂(𝑥)𝜑_𝑖²𝑑𝑥⩽−𝜇₂𝜑_𝑖²_𝐿2=−𝜇₂<0,(38) and for the last term, Im_𝑈𝑣⋅∇𝜑_𝑖𝜑_𝑖𝑑𝑥⩽_𝑈𝑣⋅∇𝜑_𝑖𝜑_𝑖⩽𝑑𝑥_𝑈‖𝑣‖∇𝜑_𝑖𝜑_𝑖⩽‖𝑣‖𝑑𝑥_∞𝑈∇𝜑_𝑖𝜑_𝑖⩽‖𝑣‖𝑑𝑥_∞∇𝜑_𝑖𝐿2𝜑_𝑖𝐿2⩽𝛼₂2∇𝜑_𝑖²_𝐿2+‖𝑣‖²_∞2𝛼₂.(39) Replacing these bounds in (35), we get for the imaginary part of the trace ImTr𝐹^′(𝑢)𝑃_𝑚𝛼⩽−²2^𝑚_𝑖=1∇𝜑_𝑖²_𝐿2−𝑚𝜇₂‖𝑣‖+𝑚^2∞2𝛼₂𝛼⩽−²𝑐₁2|𝑈|^2/𝑛𝑚^1+2/𝑛𝛼+𝑚²𝑐₂2|𝑈|^2/𝑛+‖𝑣‖²_∞2𝛼₂−𝜇₂⩽−𝜅₁𝑚^1+2/𝑛+𝜅₂,(40) where we have used (30) and a Young inequality to obtain the required constants 𝜅_𝑖 given by 𝜅₁𝛼∶=²𝑐₁|𝑈|𝑛+2^−2/𝑛,𝜅₂2|𝑈|∶=⁻¹()𝛼𝑛+2²𝑐₁2^𝑛/2𝛼₂𝑐₂2|𝑈|^2/𝑛+‖𝑣‖²_∞2𝛼₂−𝜇₂^1+𝑛/2.(41)

Remark 10. Notice, however, that in (40) the use of a Young inequality requires 𝜇₂<𝛼₂𝑐₂2|𝑈|^2/𝑛+‖𝑣‖²_∞2𝛼₂,(42) which produces a well-defined positive real constant 𝜅₂. In particular, for Dirichlet boundary conditions (where 𝑐₂ can be taken to be zero), in addition to hypothesis (H7), this requirement reads ‖𝑣‖_∞>2𝛼₂𝜇₂. It is also possible to replace 𝜇₂ by zero in the expression for 𝜅₂ and the bound is still fine but not as sharp. See Remark 12 if (42) does not hold.
This provides an estimate for the dimension of the attractor 𝒜. In fact, similarly to what is done in [28, Section VI.7] using Proposition V.2.1 and Theorem V.3.3 there (see also [30, Theorem 2.4]), we have the following theorem.

Theorem 11. Under the assumptions of Theorem 8 and if (42) holds, let 𝑚∈ℕ be such that 𝑚−1<2𝜅₂𝜅₁^{𝑛/(𝑛+2)}⩽𝑚.(43) Then the infinitesimal 𝑚-dimensional volume elements in phase space 𝐿²(𝑈) decay exponentially as 𝑡→∞. Moreover, the global attractor 𝒜 has finite Hausdorff dimension dim_𝐻(𝒜)⩽𝑚 and finite fractal dimension dim_𝐹(𝒜)⩽2𝑚.

Explicitly, we have that 𝑚∼2^{𝑛/(𝑛+2)}𝛼₂𝑐₁4|𝑈|^{(𝑛−2)/(𝑛+2)}𝛼₂𝑐₂2|𝑈|^2/𝑛+‖𝑣‖²_∞2𝛼₂−𝜇₂^𝑛/2.(44)

Remark 12. Actually, a bound on the Hausdorff dimension is already obtained by finding the minimum value of 𝑚∈ℕ in (40), such that Im[Tr(𝐹^′(𝑢)𝑃_𝑚)]<0 (see aforementioned references). Therefore, dim_𝐻(𝒜) is strictly smaller than (44) without the 2^{𝑛/(𝑛+2)} factor. This also shows that if (42) does not hold, the Hausdorff dimension is strictly less than 1.
As a final conclusion, we stress the fact that the vector coefficient 𝑣(𝑥) of the “rotational” term does indeed increase the dimension of the attractor in a factor roughly proportional to ‖𝑣‖^𝑛_∞, where 𝑛 is the dimension of the ambient space. This should be compared with the role played by this term in proving finite time blowup for the nondissipative focusing case in [10], alluded in Remark 9.

Acknowledgments

The authors are indebted to the referee for indicating relevant references concerning numerical studies of Bose-Einstein condensates. This work was partially supported by Research Project “Dynamics of a nonlinear Schrödinger equation”, ID-PROY 004850 of the Faculty of Sciences of Pontificia Universidad Javeriana, Bogotá, Colombia.