# Dynamics of a Gross-Pitaevskii Equation with Phenomenological Damping

Authors: Renato Colucci; Gerardo R. Chacón; Andrés Vargas
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We study the dynamical behavior of solutions of ann-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.


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## Body

## 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 coefficientsα and β, our purpose is to study the long-term dynamics of a general nonlinear Schrödinger evolution equation with potential of the form
(1)iut-αΔu+β|u|2u+f(x)u=v(x)·∇u,
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⊂ℝn (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 choosev(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
(2)v·∇u=i(Ω∧x)·∇u=iΩ·(x∧∇)u=-Ω·Lu,
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 ℝn 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 coefficientsα, 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
(3)iut=(1+iγ)[αΔu-β|u|2u-f(x)u+v(x)·∇u],
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 ℝn 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, Theorem4 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 vectorv∈ℂn with n⩾1, we define v1:=Re(v)∈ℝn and v2:=Im(v)∈ℝn. Let U⊂ℝn; 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∈ℂn and Δu:=Δu1+iΔu2∈ℂ. In addition, we extend bilinearly the Euclidean inner product “·” in ℝn to a symmetric inner product on ℂn denoted for simplicity by the same dot; that is, for vectors v,w∈ℂn, we have
(4)v·w:=(v1·w1-v2·w2)+i(v1·w2+v2·w1).
With this product, the standard Hermitian norm of v∈ℂn satisfies ∥v∥2=∥v1∥2+∥v2∥2=v·v-, where v-:=v1-iv2.Given coefficientsα,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⊂ℝn, for arbitrary n⩾1:
(5)iut-αΔu+β|u|pu+f(x)u=v(x)·∇u,u(x,0)=u0(x),
with unknown u(x,t):U×ℝ+→ℂ, a scalar function f(x):U→ℂ, and a vector-valued map v(x):U→ℂn. Our hypotheses on the constant coefficients are
(H1)α2>0,(H2)β2⩾p|β1|⩾0
and on the functional coefficients and initial data are
(H3)f(x)∈L∞(U),(H4)v(x)∈L∞(U,ℂn),(H5)u0(x)∈L2(U).
Additionally, we assume that
(H6)μ2:=essinfx∈Uf2(x)>0
and require that
(H7)4α2μ2>∥v∥∞2.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⊂ℝn, which is assumed to have regular boundary ∂U. Explicitly, we suppose that for all t⩾0,
(D1)u(x,t)|∂U=0,(D2)Δu(x,t)|∂U=0.
Alternatively we also consider U-periodic boundary conditions when the domain has the form U=(0,L)n⊂ℝn. 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 aj=0 and bj=L, it holds that
(P)∂ku(a,t)=∂ku(b,t),
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 ofU-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
(6)u∈𝒞b(ℝ+,H1(U)),ut∈𝒞b(ℝ+,H-1(U)),
where H-1 is the dual space of H1 and 𝒞b(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).


## 1.1. Precise Setting

To ease notation, for any complex vectorv∈ℂn with n⩾1, we define v1:=Re(v)∈ℝn and v2:=Im(v)∈ℝn. Let U⊂ℝn; 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∈ℂn and Δu:=Δu1+iΔu2∈ℂ. In addition, we extend bilinearly the Euclidean inner product “·” in ℝn to a symmetric inner product on ℂn denoted for simplicity by the same dot; that is, for vectors v,w∈ℂn, we have
(4)v·w:=(v1·w1-v2·w2)+i(v1·w2+v2·w1).
With this product, the standard Hermitian norm of v∈ℂn satisfies ∥v∥2=∥v1∥2+∥v2∥2=v·v-, where v-:=v1-iv2.Given coefficientsα,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⊂ℝn, for arbitrary n⩾1:
(5)iut-αΔu+β|u|pu+f(x)u=v(x)·∇u,u(x,0)=u0(x),
with unknown u(x,t):U×ℝ+→ℂ, a scalar function f(x):U→ℂ, and a vector-valued map v(x):U→ℂn. Our hypotheses on the constant coefficients are
(H1)α2>0,(H2)β2⩾p|β1|⩾0
and on the functional coefficients and initial data are
(H3)f(x)∈L∞(U),(H4)v(x)∈L∞(U,ℂn),(H5)u0(x)∈L2(U).
Additionally, we assume that
(H6)μ2:=essinfx∈Uf2(x)>0
and require that
(H7)4α2μ2>∥v∥∞2.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⊂ℝn, which is assumed to have regular boundary ∂U. Explicitly, we suppose that for all t⩾0,
(D1)u(x,t)|∂U=0,(D2)Δu(x,t)|∂U=0.
Alternatively we also consider U-periodic boundary conditions when the domain has the form U=(0,L)n⊂ℝn. 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 aj=0 and bj=L, it holds that
(P)∂ku(a,t)=∂ku(b,t),
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 ofU-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
(6)u∈𝒞b(ℝ+,H1(U)),ut∈𝒞b(ℝ+,H-1(U)),
where H-1 is the dual space of H1 and 𝒞b(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 attractor𝒜⊂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 setB⊂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 inL2(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 semigroupS(t) possesses an absorbing ball in L2(U) as
(7)∥S(t)u0∥L2⩽r0ift⩾τ0,wherer0:=1 and τ0:=(1/K)log∥u0∥L2, with K:=μ2-(1/4α2)∥v∥∞2;(ii)
the semigroupS(t) possesses an absorbing ball in H01(U) as
(8)∥S(t)u0∥H01⩽r1ift⩾τ1,wherer1:=(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(9)ℬ:={u∈L2(U):∥u∥L2⩽r0,∥∇u∥L2⩽r1}⊂H1(U)
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]).
Letf,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,
(10)∫tt+ϱ‍f(s)ds⩽a1,∫tt+ϱ‍g(s)ds⩽a2,∫tt+ϱ‍h(s)ds⩽a3.
Then it holds that(11)f(t+ϱ)⩽(a1ϱ+a3)ea2,∀t⩾t0.


### 2.1. Absorbing Ball inL2(U)

Multiplying (5) by u-, integrating over U⊂ℝn using integration by parts with any set of the proposed boundary conditions, and taking the imaginary part,
(12)12ddt∥u∥L22+α2∥∇u∥L22+β2∥u∥Lp+2p+2+∫U‍f2(x)|u|2dx=Im(∫U‍(v·∇u)u-dx)⩽∥v·∇u∥L2∥u∥L2⩽∥∇u∥L2∥v∥∞∥u∥L2⩽α2∥∇u∥L22+14α2∥v∥∞2∥u∥L22.
After eliminating the third term to the left-hand side of the previous calculation, we get
(13)ddt∥u∥L22⩽-2K∥u∥L22,
where
(14)K=K(α2,f2,v):=μ2-14α2∥v∥∞2.
Then, applying Gronwall’s lemma and since (H7) implies that K>0, we obtain
(15)∥u∥L22⩽∥u0∥L22e-2Kt.
Setting τ0:=(1/K)log∥u0∥L2, we have that
(16)∥u(t)∥L2⩽1=:r0,ift⩾τ0.
This proves the existence of an absorbing ball in L2(U).


### 2.2. Absorbing Ball inH01(U)

Now we multiply (5) by Δu-, integrate as before over U⊂ℝn using boundary conditions (after integration by parts) to simplify, and take the imaginary part to get
(17)12ddt∥∇u∥L22+α2∥Δu∥L22+Im(∫U‍(v·∇u)Δu-dx)=Im(∫U‍f(x)uΔu-dx)+Im(β∫U‍|u|puΔu-dx).
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
(18)Im(β∫U‍|u|puΔu-dx)=-β2∫U‍|u|p∥∇u∥2dx-pIm(β∫U‍|u|p-2(u∇u-)·Re(u∇u-)dx)=-β2∫U‍|u|p∥∇u∥2dx-pβ1∫U‍|u|p-2Im(u∇u-)·Re(u∇u-)dx-pβ2∫U‍|u|p-2∥Re(u∇u-)∥2dx⩽-β2∫U‍|u|p∥∇u∥2dx+p|β1|∫U‍|u|p-2∥u∇u-∥2dx⩽-(β2-p|β1|)∫U‍|u|p∥∇u∥2dx,
where the inequality appears after eliminating the last term since β2⩾0. Replacing this in the original equation, we can write
(19)12ddt∥∇u∥L22+α2∥Δu∥L22+(β2-p|β1|)∫U‍|u|p∥∇u∥2dx⩽Im(∫U‍f(x)uΔu-dx)-Im(∫U‍(v·∇u)Δu-dx)⩽∥f∥∞∥u∥L2∥Δu∥L2+∥v∥∞∥∇u∥L2∥Δu∥L2⩽12α2∥f∥∞2∥u∥L22+α22∥Δu∥L22+12α2∥v∥∞2∥∇u∥L22+α22∥Δu∥L22.
Simplifying, using β2⩾p|β1| and working with t⩾τ0 so that ∥u∥L2⩽r0, we arrive at
(20)ddt∥∇u∥L22⩽1α2(r02∥f∥∞2+∥v∥∞2∥∇u∥L22).
To apply the uniform Gronwall lemma, it is necessary to have an upper bound on ∫tt+ϱ‍∥∇u∥L22ds. From (12), we have
(21)12ddt∥u∥L22+α2∥∇u∥L22+β2∥u∥Lp+2p+2+∫U‍f2(x)|u|2dx⩽∥∇u∥L2∥v∥∞∥u∥L2⩽α22∥∇u∥L22+12α2∥v∥∞2∥u∥L22.
Now, eliminating the last two positive terms at the left-hand side, using t⩾τ0 again, and integrating, we find that
(22)∫tt+ϱ‍∥∇u∥L22ds⩽r02(ϱα22∥v∥∞2+1α2)=:a1.
There are also obvious bounds given by
(23)∫tt+ϱ‍1α2∥v∥∞2ds⩽ϱα2∥v∥∞2=:a2,∫tt+ϱ‍1α2r02∥f∥∞2ds⩽ϱα2r02∥f∥∞2=:a3.
With this, Lemma 6 immediately implies
(24)∥∇u(t+ϱ)∥L22⩽r02(1α22∥v∥∞2+1α2ϱ+ϱα2∥f∥∞2)e(ϱ/α2)∥v∥∞2.
Choosing ϱ=α2 and assuming t⩾τ1≔τ0+α2, the previous inequality becomes
(25)∥∇u(t)∥L2⩽r0α2e(1/2)∥v∥∞2(1+∥v∥∞2+α22∥f∥∞2)1/2=:r1.
This proves the existence of an absorbing ball in H01(U).To show the existence of a compact global attractor inL2(U), we will make use of the following result (see [28, Theorem I.1.1]) with H=L2(U).Theorem 7.
Suppose thatS(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
(26)ω(ℬ):=⋂s⩾0‍⋃t⩾s‍S(t)ℬ¯
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 Theorem7. 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
(27)ℬ:={u∈L2(U):∥u∥L2⩽r0,∥∇u∥L2⩽r1}⊂H1(U),
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
(28)⋃t⩾τ1‍S(t)B
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
(29)limt→τ*∥∇u(t)∥L2=∞.
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.


## 2.1. Absorbing Ball inL2(U)

Multiplying (5) by u-, integrating over U⊂ℝn using integration by parts with any set of the proposed boundary conditions, and taking the imaginary part,
(12)12ddt∥u∥L22+α2∥∇u∥L22+β2∥u∥Lp+2p+2+∫U‍f2(x)|u|2dx=Im(∫U‍(v·∇u)u-dx)⩽∥v·∇u∥L2∥u∥L2⩽∥∇u∥L2∥v∥∞∥u∥L2⩽α2∥∇u∥L22+14α2∥v∥∞2∥u∥L22.
After eliminating the third term to the left-hand side of the previous calculation, we get
(13)ddt∥u∥L22⩽-2K∥u∥L22,
where
(14)K=K(α2,f2,v):=μ2-14α2∥v∥∞2.
Then, applying Gronwall’s lemma and since (H7) implies that K>0, we obtain
(15)∥u∥L22⩽∥u0∥L22e-2Kt.
Setting τ0:=(1/K)log∥u0∥L2, we have that
(16)∥u(t)∥L2⩽1=:r0,ift⩾τ0.
This proves the existence of an absorbing ball in L2(U).


## 2.2. Absorbing Ball inH01(U)

Now we multiply (5) by Δu-, integrate as before over U⊂ℝn using boundary conditions (after integration by parts) to simplify, and take the imaginary part to get
(17)12ddt∥∇u∥L22+α2∥Δu∥L22+Im(∫U‍(v·∇u)Δu-dx)=Im(∫U‍f(x)uΔu-dx)+Im(β∫U‍|u|puΔu-dx).
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
(18)Im(β∫U‍|u|puΔu-dx)=-β2∫U‍|u|p∥∇u∥2dx-pIm(β∫U‍|u|p-2(u∇u-)·Re(u∇u-)dx)=-β2∫U‍|u|p∥∇u∥2dx-pβ1∫U‍|u|p-2Im(u∇u-)·Re(u∇u-)dx-pβ2∫U‍|u|p-2∥Re(u∇u-)∥2dx⩽-β2∫U‍|u|p∥∇u∥2dx+p|β1|∫U‍|u|p-2∥u∇u-∥2dx⩽-(β2-p|β1|)∫U‍|u|p∥∇u∥2dx,
where the inequality appears after eliminating the last term since β2⩾0. Replacing this in the original equation, we can write
(19)12ddt∥∇u∥L22+α2∥Δu∥L22+(β2-p|β1|)∫U‍|u|p∥∇u∥2dx⩽Im(∫U‍f(x)uΔu-dx)-Im(∫U‍(v·∇u)Δu-dx)⩽∥f∥∞∥u∥L2∥Δu∥L2+∥v∥∞∥∇u∥L2∥Δu∥L2⩽12α2∥f∥∞2∥u∥L22+α22∥Δu∥L22+12α2∥v∥∞2∥∇u∥L22+α22∥Δu∥L22.
Simplifying, using β2⩾p|β1| and working with t⩾τ0 so that ∥u∥L2⩽r0, we arrive at
(20)ddt∥∇u∥L22⩽1α2(r02∥f∥∞2+∥v∥∞2∥∇u∥L22).
To apply the uniform Gronwall lemma, it is necessary to have an upper bound on ∫tt+ϱ‍∥∇u∥L22ds. From (12), we have
(21)12ddt∥u∥L22+α2∥∇u∥L22+β2∥u∥Lp+2p+2+∫U‍f2(x)|u|2dx⩽∥∇u∥L2∥v∥∞∥u∥L2⩽α22∥∇u∥L22+12α2∥v∥∞2∥u∥L22.
Now, eliminating the last two positive terms at the left-hand side, using t⩾τ0 again, and integrating, we find that
(22)∫tt+ϱ‍∥∇u∥L22ds⩽r02(ϱα22∥v∥∞2+1α2)=:a1.
There are also obvious bounds given by
(23)∫tt+ϱ‍1α2∥v∥∞2ds⩽ϱα2∥v∥∞2=:a2,∫tt+ϱ‍1α2r02∥f∥∞2ds⩽ϱα2r02∥f∥∞2=:a3.
With this, Lemma 6 immediately implies
(24)∥∇u(t+ϱ)∥L22⩽r02(1α22∥v∥∞2+1α2ϱ+ϱα2∥f∥∞2)e(ϱ/α2)∥v∥∞2.
Choosing ϱ=α2 and assuming t⩾τ1≔τ0+α2, the previous inequality becomes
(25)∥∇u(t)∥L2⩽r0α2e(1/2)∥v∥∞2(1+∥v∥∞2+α22∥f∥∞2)1/2=:r1.
This proves the existence of an absorbing ball in H01(U).To show the existence of a compact global attractor inL2(U), we will make use of the following result (see [28, Theorem I.1.1]) with H=L2(U).Theorem 7.
Suppose thatS(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
(26)ω(ℬ):=⋂s⩾0‍⋃t⩾s‍S(t)ℬ¯
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 Theorem7. 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
(27)ℬ:={u∈L2(U):∥u∥L2⩽r0,∥∇u∥L2⩽r1}⊂H1(U),
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
(28)⋃t⩾τ1‍S(t)B
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
(29)limt→τ*∥∇u(t)∥L2=∞.
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 infinitesimalm-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⊂ℝn 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 basisφ1(t),…,φm(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⊂ℝn but independent of the size |U|, the value m, and of the chosen orthonormal basis, such that
(30)∑i=1m‍∥∇φi∥L22⩾1|U|2/n(c1m1+2/n-c2m).
Moreover, if the functions φi∈H01(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:
(31)iϕt=F′(u)ϕ:=αΔϕ-β(p+1)|u|pϕ-f(x)ϕ+v·∇ϕ,ϕ(0)=ξ.
Then, noting that Re(ϕt)=Im(F′(u)ϕ), the evolution of the m-dimensional volume elements is given by (cf. [30, Section 13.2])
(32)Vm(t)=Vm(0)exp[Im(∫0t‍Tr(F′(u)Pm(s))ds)],
where Pm is the orthogonal projection of L2(U) over the subspace spanned by the first m eigenvectors of the Laplacian. The asymptotic growth rate of Vm(t) is given by
(33)limt→∞1t∫0t‍Im[Tr(F′(u)Pm(s))]ds.
Since we need the maximal growth rate, we compute the supremum over all u0∈𝒜 and over all the m-dimensional projectors as follows:
(34)supu0∈𝒜supPmlimsupt→+∞1t∫0t‍Im[Tr(F′(u)Pm(s))]ds=:Trm(𝒜).In order to have exponential decay, we look for the smallestm such that the number Trm(𝒜) is negative. In detail, let φ1(t),…,φm(t) be an orthonormal basis for PmL2(U); then
(35)Im[Tr(F′(u)Pm)]=∑i=1m‍Im[〈F′(u)φi,φi〉L2(U)],
with
(36)Im[〈F′(u)φi,φi〉L2(U)]=-α2∥∇φi∥L22-β2(p+1)∫U‍|u|p|φi|2dx-∫U‍f2(x)|φi|2dx+Im(∫U‍(v·∇φi)φ-idx).
Now, we estimate each of the last three terms separately. For the first, since β2⩾0, we can simply write
(37)-β2(p+1)∫U‍|u|p|φi|2dx⩽0,
for the second, we have
(38)-∫U‍f2(x)|φi|2dx⩽-μ2∥φi∥L22=-μ2<0,
and for the last term,
(39)Im∫U‍(v·∇φi)φ-idx⩽∫U‍|v·∇φi||φi|dxIm∫U‍(v·∇φi)φ-idx⩽∫U‍∥v∥∥∇φi∥|φi|dxIm∫U‍(v·∇φi)φ-idx⩽∥v∥∞∫U‍∥∇φi∥|φi|dxIm∫U‍(v·∇φi)φ-idx⩽∥v∥∞∥∇φi∥L2∥φi∥L2Im∫U‍(v·∇φi)φ-idx⩽α22∥∇φi∥L22+∥v∥∞22α2.
Replacing these bounds in (35), we get for the imaginary part of the trace
(40)Im[Tr(F′(u)Pm)]⩽-α22∑i=1m‍∥∇φi∥L22-mμ2+m∥v∥∞22α2⩽-α2c12|U|2/nm1+2/n+m(α2c22|U|2/n+∥v∥∞22α2-μ2)⩽-κ1m1+2/n+κ2,
where we have used (30) and a Young inequality to obtain the required constants κi given by
(41)κ1:=(α2c1n+2)|U|-2/n,κ2:=2|U|-1(n+2)(α2c12)n/2(α2c22|U|2/n+∥v∥∞22α2-μ2)1+n/2.Remark 10.
Notice, however, that in (40) the use of a Young inequality requires
(42)μ2<α2c22|U|2/n+∥v∥∞22α2,
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 attractor𝒜. 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 Theorem8 and if (42) holds, let m∈ℕ be such that
(43)m-1<(2κ2κ1)n/(n+2)⩽m.
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 dimH(𝒜)⩽m and finite fractal dimension dimF(𝒜)⩽2m.Explicitly, we have that(44)m~2n/(n+2)[(α2c14|U|)(n-2)/(n+2)(α2c22|U|2/n+∥v∥∞22α2-μ2)]n/2.Remark 12.
Actually, a bound on the Hausdorff dimension is already obtained by finding the minimum value ofm∈ℕ in (40), such that Im[Tr(F′(u)Pm)]<0 (see aforementioned references). Therefore, dimH(𝒜) is strictly smaller than (44) without the 2n/(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 coefficientv(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.



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