1 Introduction

Let \(\Omega \subset \mathbb {R}^3\) be a bounded domain with smooth boundary \(\partial \Omega \), and let \(\alpha >0\) and \(\beta \ge 0\) be given constants. For \(t>0\), we consider the dimensionless form of the Navier–Stokes–Voigt (NSV) equations with memory and Ekman damping in the unknown velocity \(u=u(x,t)\) and pressure \(\pi =\pi (x,t)\)

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( u - \alpha \Delta u\right) - \displaystyle \int _0^\infty g(s) \Delta u(t-s)\mathrm{d}s + \beta u + (u\cdot \nabla ) u +\nabla \pi =f, \\ \text {div}\, u=0, \end{array}\right. } \end{aligned}$$
(1.1)

where f is an assigned external forcing term, subject to the nonslip boundary condition

$$\begin{aligned} \quad u_{|\partial \Omega }=0. \end{aligned}$$
(1.2)

The function \(g\,{:}\,[0,\infty )\rightarrow \mathbb {R}\), usually called memory kernel, is supposed to be convex nonnegative, smooth on \(\mathbb {R}^+=(0,\infty )\), vanishing at infinity and satisfying the normalization condition

$$\begin{aligned} \int _0^\infty g(s)\mathrm{d}s=1. \end{aligned}$$

In fact, in the last display, 1 may be replaced with any constant \(\gamma >0\) without any essential difference in the analysis that follows. The system is supplemented with the initial conditions

$$\begin{aligned} u(0)=u_0 \quad \text {and} \quad u(-s)_{|s>0}=\varphi _0(s), \end{aligned}$$
(1.3)

where \(u_0\) and \(\varphi _0(\cdot )\) are prescribed data. The initial velocity \(u_0\) and the past history of the velocity \(\varphi _0\), which need only be defined for almost every \(s>0\), will play a different role in the translation of (1.1) into an evolution system, as detailed in the subsequent Sect. 3. The drag/friction term \(\beta u\), widely used in geophysical hydrodynamics, is known as Ekman damping. The coefficient \(\beta \) is the Ekman pumping/dissipation constant, induced by the so-called Ekman layer appearing at the bottom of a rotating fluid (see Pedlosky 1987). The term \(\beta u\) is also referred to as the Rayleigh friction and employed in oceanic models such as the viscous Charney–Stommel barotropic ocean circulation model of the gulf stream. In the last decades, in connection with damped Euler and Navier–Stokes equations, several models involving dissipative terms of Ekman type have been considered in order to study the limit case of vanishing viscosity (see e.g. Chepyzhov et al. 2011; Constantin and Ramos 2007; Ilyin et al. 2004).

In literature the NSV system was introduced by Oskolkov (1973) as a model for the motion of a Kelvin–Voigt elastic incompressible fluid. The system reads as

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( u - \alpha \Delta u\right) - \Delta u + (u\cdot \nabla ) u +\nabla \pi =f, \\ \text {div}\, u=0. \end{array}\right. } \end{aligned}$$
(1.4)

The global well-posedness of (1.4), along with the existence of a finite dimensional global attractor, has been established in connection with numerical simulations of turbulent flows in statistical equilibrium, see Kalantarov and Titi (2009) and references therein. Letting the viscoelastic length scale \(\alpha \) go to zero we recover the three-dimensional Navier–Stokes system. In this regard, the recent contribution (Coti Zelati and Gal 2015) improves on the results of Kalantarov and Titi (2009) by proving the existence of a family of exponential attractors parameterized by the length scale \(\alpha \) which converges to the (weak) exponential attractor of the Navier–Stokes system as \(\alpha \rightarrow 0.\)

More recently, Gal and Tachim-Medjo have proposed in (2013) a NSV system incorporating hereditary effects of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( u - \alpha \Delta u\right) -\nu \Delta u-(1-\nu )\displaystyle \int _0^\infty g(s) \Delta u(t-s)\mathrm{d}s + (u\cdot \nabla ) u +\nabla \pi =f, \\ \text {div}\, u=0, \end{array}\right. } \end{aligned}$$
(1.5)

for some parameter \(\nu \in (0,1)\). This model has been derived by considering a Cauchy stress tensor that takes both the Newtonian and the viscoelastic contributions into account. From the physical viewpoint, the integrodifferential term is introduced to describe non-Newtonian fluids. In Gal and Tachim-Medjo (2013) the assumption \(\nu \in (0,1)\) leads to a model where instantaneous and memory-type viscous terms coexist (see also Gatti et al. 2005). When \(\nu =1\), the system (1.5) collapses into (1.4), while the limit case \(\nu =0\) corresponds to the situation where the concentration of polymers totally prevails over the solvent. Also, the formal choice of the memory kernel g equal to the Dirac mass at \(0^+\) in (1.5) corresponds to the NSV system (1.4). The well-posedness and the asymptotic properties of (1.5) have been achieved in Gal and Tachim-Medjo (2013). The main result is the existence of finite dimensional exponential attractors for the related dynamical system. In addition, replacing the memory kernel g with a family of suitable rescalings \(g_\varepsilon \) converging to the Dirac mass (see point III of Sect. 9), the corresponding family of \(\varepsilon \)-exponential attractors is proved to be robust in the limit \(\varepsilon \rightarrow 0\), where the system (1.4) is recovered. Such a stability property has been previously exploited in Gatti et al. (2005), where the asymptotic behavior of a two-dimensional Jeffreys model is shown to converge to its Navier–Stokes (formal) limit.

The dissipative nature of the above-mentioned NSV type models is relatively easy to understand. The energy conservation law associated to (1.4) yields the cumulative dissipation integral

$$\begin{aligned} \int _0^T \int _\Omega |\nabla u|^2 \,\mathrm{d}x \mathrm{d}t \end{aligned}$$
(1.6)

which allows to keep the high modes of the solution, and in turn the nonlinear terms of critical character, under control when showing the existence of an absorbing set in the phase space. When instantaneous and hereditary kinematic viscosity coexist as in (1.5), a weaker hereditary dissipation term (see (4.2) below) involving the high modes of the past history of the velocity averaged over time appears in the energy balance in addition to (1.6).

The key novel feature of the model (1.1), and of the analysis we will perform herein, is that we are able to establish dissipativity and asymptotic well-posedness—in the dynamical systems sense, namely, existence of an absorbing set and of a finite dimensional attractor—even when the instantaneous viscosity term leading to the dissipation integral (1.6) is absent, and therefore no instantaneous control on the high modes is assumed, provided that the much weaker, and physically relevant, Ekman-type damping term \(\beta u\) is taken into account together with the hereditary dissipation integral (4.2).

The fact that the coupled effect of the hereditary analogue of (1.6) with the Ekman term is enough to guarantee asymptotic well-posedness of the system is, in our view, rather striking. In fact, to our knowledge, there are no theoretical results regarding nonlinear systems whose high modes dissipation features are purely hereditary as those of (1.1). In order to explain the general picture in the realm of incompressible fluid models, consider the following more general system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( u - \alpha \Delta u\right) - \gamma \displaystyle \int _0^\infty g(s) \Delta u(t-s)\mathrm{d}s +\beta u + (u\cdot \nabla ) u +\nabla \pi =f, \\ \text {div}\, u=0. \end{array}\right. } \end{aligned}$$
(1.7)

Note that (1.1) corresponds to \(\gamma =1\) in (1.7). We can distinguish three further cases in (1.7):

  • Navier–Stokes system with memory (\(\alpha =0, \beta =0, \gamma >0\)). Well-posedness, in particular uniqueness, is out of reach for this system, even in dimension two. The two-dimensional case, when instantaneous viscosity is present and therefore (1.6) appears in the energy balance, has been dealt with in Gatti et al. (2005).

  • Euler–Voigt system with damping (\(\alpha>0, \beta >0, \gamma =0\)). This model is a regularization of the Euler system. The well-posedness in the natural phase space together with the existence of strong solutions has been proved in Larios and Titi (2010) when \(\beta =0\). Nevertheless, a careful inspection of the techniques of Larios and Titi (2010) shows that the addition of the low modes damping/driving term \(\beta u\) would not suffice to establish the dissipative nature of the system.

  • Navier–Stokes–Voigt system with memory (\(\alpha>0, \beta =0, \gamma >0\)). The well-posedness in the natural phase space can be easily proved as in the previous case. Here the memory term provides an integrated in time control of \(\nabla u\). However, the dissipativity seems to be an insurmountable task unless dissipation for the low modes (that is, the Ekman term) is taken into account.

As evidenced by the failure of the standard techniques for fully hereditary analogues of even very well understood systems like the two-dimensional Navier–Stokes case, the mathematical treatment of (1.1) becomes much harder in absence of the dissipation integral (1.6). We overcome the intrinsic difficulty of reconstructing the necessary control on the high modes by nontrivially combining suitable energy functionals with a recent Gronwall-type lemma wih parameter (see Pata 2011). In fact, we are able to further weaken the nature of the low modes dissipation term \(\beta u\) provided the forcing f is taken to be regular enough: a detailed formulation is given in (3.2). We believe that the technique here proposed will provide new insight in the study other systems with nonlinear terms of critical growth and very weak dissipation properties.

1.1 Structure of the Article

After recalling the relevant mathematical framework for Navier–Stokes type systems with memory in Sect. 2, (1.1) is recast in Sect. 3 as an abstract evolution system in the Dafermos history space setting (Dafermos 1970). We prove that such an evolution system generates a strongly continuous semigroup with continuous dependence on the initial data. The more challenging part of our entire analysis is found in Sect. 4. Therein, we prove our main result which is the dissipative character of the semigroup. If no forcing term is present, the dissipative estimate remains true even if \(\beta =0\). This is the content of Sect. 5. In the presence of memory terms, which are intrinsically hyperbolic, finite time regularization of the solution cannot occur. As a substitute, in Sect. 6 we show that the trajectories are exponentially attracted by more regular bounded sets. This property is relied upon in Sect. 7, where the existence of a regular exponential attractor for the semigroup of finite fractal dimension in the phase space is established. The global attractor is thus recovered as a byproduct. Precise statements are given in Theorem 7.2 and Corollary 7.3, and their proofs are carried out in Sect. 8. Finally, we send to Sect. 9 for several directions of future investigation.

2 Mathematical Setting and Notation

Throughout this work, \(L^2(\Omega ), H_0^1(\Omega ), H^r(\Omega )\) will be the standard Lebesgue–Sobolev spaces on \(\Omega \). In particular, the external force f will be assumed to be a (constant-in-time) vector of \([L^2(\Omega )]^3\).

We introduce the Hilbert space \(({\varvec{H}},\langle \cdot ,\cdot \rangle ,\Vert \cdot \Vert )\) given by

$$\begin{aligned} {\varvec{H}}=\lbrace u \in [L^2(\Omega )]^3: \text {div}\, u=0, \ u\cdot n_{\vert \partial \Omega }=0 \rbrace , \end{aligned}$$

n being the outward normal to \(\partial \Omega \), along with the Hilbert space

$$\begin{aligned} {\varvec{V}}=\big \{u\in \left[ H_0^1(\Omega )\right] ^3: \text {div}\, u=0 \big \}, \end{aligned}$$

with inner product and norm

$$\begin{aligned} \langle u,v\rangle _1= \langle \nabla u,\nabla v\rangle \quad \text {and} \quad \Vert u\Vert _{1}=\Vert \nabla u\Vert . \end{aligned}$$

We denote by \({\varvec{V^\prime }}\) its dual space. We will also encounter the space

$$\begin{aligned} {\varvec{W}}={\varvec{V}}\cap [H^2(\Omega )]^3. \end{aligned}$$

Calling \(P: [L^2(\Omega )]^3 \rightarrow {\varvec{H}}\) the Leray orthogonal projection onto \({\varvec{H}}\), we consider the Stokes operator on \({\varvec{H}}\)

$$\begin{aligned} A= -P \Delta \quad \text {with domain}\quad {\mathfrak D}(A)={\varvec{W}}. \end{aligned}$$

It is well known that A is a positive selfadjoint operator with compact inverse (see e.g. Temam 2001). This allows us to define the scale of compactly nested Hilbert spaces

$$\begin{aligned} V^r={\mathfrak D}(A^{\frac{r}{2}}),\quad r\in \mathbb {R}, \end{aligned}$$

endowed with the inner products and norms

$$\begin{aligned} \langle u, v \rangle _{r}=\langle A^{\frac{r}{2}}u, A^{\frac{r}{2}} v \rangle \quad \text {and} \quad \Vert u\Vert _{r}=\Vert A^{\frac{r}{2}} u\Vert . \end{aligned}$$

In particular,

$$\begin{aligned} V^{-1}={\varvec{V^\prime }},\quad V^0={\varvec{H}}, \quad V^1={\varvec{V}}, \quad V^2={\varvec{W}}. \end{aligned}$$

More generally (see Boyer and Fabrie 2013),

$$\begin{aligned} V^r&={\varvec{V}}\cap [H^r(\Omega )]^3, \quad 1\le r\le 2,\\ V^r&\subset {\varvec{V}}\cap [H^r(\Omega )]^3, \quad r>2. \end{aligned}$$

We also recall the Poincaré inequality

$$\begin{aligned} \sqrt{\lambda _1} \Vert u\Vert \le \Vert u\Vert _1, \quad \forall \, u \in {\varvec{V}}, \end{aligned}$$
(2.1)

where \(\lambda _1>0\) is the first eigenvalue of A. The symbol \(\langle \cdot ,\cdot \rangle \) will also stand for the duality product between \(V^r\) and its dual space \(V^{-r}\).

As customary, we write the trilinear form on \({\varvec{V}}\times {\varvec{V}}\times {\varvec{V}}\)

$$\begin{aligned} b(u,v,w)=\int _{\Omega } (u \cdot \nabla ) v \cdot w \mathrm{d}x= \sum _{i,j=1}^3 \int _{\Omega } u_i \frac{\partial v_j}{\partial x_i} w_j \mathrm{d}x, \end{aligned}$$

satisfying the relation

$$\begin{aligned} b(u,v,v)=0. \end{aligned}$$

The associated bilinear form \(B\,{:}\,{\varvec{V}}\times {\varvec{V}}\rightarrow {\varvec{V^\prime }}\) is defined as

$$\begin{aligned} \langle B(u,v),w\rangle = b(u,v,w). \end{aligned}$$

Next, we turn our attention to the memory kernel. Defining

$$\begin{aligned} \mu (s)=-g'(s), \end{aligned}$$

the prime being the derivative, we suppose \(\mu \) nonnegative, absolutely continuous, decreasing (hence \(\mu '\le 0\) almost everywhere), and summable on \(\mathbb {R}^+\) with total mass

$$\begin{aligned} \kappa =\int _0^\infty \mu (s)\ \mathrm{d}s>0. \end{aligned}$$

The classical Dafermos condition will be also assumed (see Dafermos 1970), namely,

$$\begin{aligned} \mu '(s)+\delta \mu (s)\le 0, \end{aligned}$$
(2.2)

for some \(\delta >0\) and almost every \(s>0\). Then, we introduce the \(L^2\)-weighted Hilbert space on \(\mathbb {R}^+\)

$$\begin{aligned} {\mathcal M}=L^2_{\mu }(\mathbb {R}^+;{\varvec{V}}), \end{aligned}$$

with inner product and norm

$$\begin{aligned} \langle \eta , \xi \rangle _{{\mathcal M}}=\int _0^\infty \mu (s)\langle \eta (s),\xi (s)\rangle _{1} \mathrm{d}s \quad \text {and} \quad \Vert \eta \Vert _{{\mathcal M}}=\bigg (\int _0^\infty \mu (s)\Vert \eta (s)\Vert _{1}^2\mathrm{d}s\bigg )^\frac{1}{2}, \end{aligned}$$

along with the infinitesimal generator of the right-translation semigroup on \({\mathcal M}\)

$$\begin{aligned} T\eta =-\partial _s \eta \quad \text {with domain}\quad {\mathfrak D}(T)=\big \{\eta \in {\mathcal M}:\, \partial _s \eta \in {\mathcal M},\,\,\eta (0)=0\big \}. \end{aligned}$$

Here, \(\partial _s \eta \) is the distributional derivative of \(\eta (s)\) with respect to the internal variable s. Finally, we define the extended memory space

$$\begin{aligned} {\mathcal H}={\varvec{V}}\times {\mathcal M}\end{aligned}$$

endowed with the product norm

$$\begin{aligned} \Vert (u,\eta )\Vert _{{\mathcal H}}^2=\alpha \Vert u\Vert ^2_{1}+\Vert u\Vert ^2 +\Vert \eta \Vert ^2_{{\mathcal M}}. \end{aligned}$$

Due to the Poincaré inequality (2.1), this norm is equivalent to the natural one on \({\mathcal H}\).

In this work, we will also make use of higher order memory spaces. To this end, we define

$$\begin{aligned} {\mathcal M}^1=L^2_{\mu }(\mathbb {R}^+;{\varvec{W}}), \end{aligned}$$

with inner product and norm analogous to those of \({\mathcal M}\), and the corresponding higher order extended memory space

$$\begin{aligned} {\mathcal H}^1={\varvec{W}}\times {\mathcal M}^1 \end{aligned}$$

with norm

$$\begin{aligned} \Vert (u,\eta )\Vert _{{\mathcal H}^1}^2=\alpha \Vert u\Vert _2^2+\Vert u\Vert _1^2 +\Vert \eta \Vert ^2_{{\mathcal M}^1}. \end{aligned}$$

2.1 General Agreement

Throughout the paper, the symbols \(C>0\) and \({\mathcal Q}(\cdot )\) will denote a generic constant and a generic increasing positive function, respectively, depending only on the structural parameters of the problem, but independent of f (unless otherwise specified). Moreover, given a Banach space \(\mathcal {X}\) and \(R>0\), we denote by

$$\begin{aligned} \mathbb {B}_{\mathcal {X}}(R)=\lbrace x\in \mathcal {X}: \Vert x\Vert _{\mathcal {X}}\le R \rbrace \end{aligned}$$

the ball of \(\mathcal {X}\) of radius R about zero.

3 The Dynamical System

As anticipated in the Introduction, the original problem (1.1)–(1.2) is translated into an evolution system in the so-called Dafermos past history framework (Dafermos 1970). First, we observe that, without any loss of generality, we can assume that \(f\in {\varvec{H}}\) which amounts to changing \(\pi \) (by adding \(Pf-f\) to \(\nabla \pi \)). Hence we apply the projection P to the first equation (1.1) and we transform (1.1)–(1.2) into

$$\begin{aligned} \partial _t\left( u + \alpha Au\right) +\int _0^\infty g(s) A u(t-s)\mathrm{d}s + \beta u + B(u,u)=f. \end{aligned}$$
(3.1)

In more generality, we will replace the damping \(\beta u\) with a term of the form \(\beta A^{-\vartheta }u\), for some \(\vartheta \ge 0\). Then, we introduce the past history variable

$$\begin{aligned} \eta ^t(s)=\int _0^s u(t-\sigma )\mathrm{d}\sigma , \end{aligned}$$

that satisfies the differential identity

$$\begin{aligned} \partial _t\eta ^t(s) = -\partial _s\eta ^t(s) + u(t). \end{aligned}$$

At this point, recalling the definition of the operator T, a formal integration by parts leads to the differential problem in the unknown variables \(u=u(t)\) and \(\eta =\eta ^t(\cdot )\)

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\left( u + \alpha Au\right) + \displaystyle \int _0^\infty \mu (s) A \eta (s)\mathrm{d}s + \beta A^{-\vartheta } u + B(u,u)=f, \\ \partial _t\eta = T \eta + u, \end{array}\right. } \end{aligned}$$
(3.2)

for some \(\vartheta \ge 0\), where \(\mu =-g'\). As we said, this is actually a generalization of equation (3.1), which corresponds to the case \(\vartheta =0\) (the strongest damping within this class). In turn, the initial conditions (1.3) transform into

$$\begin{aligned} u(0)=u_0 \quad \text {and} \quad \eta ^0=\eta _0, \end{aligned}$$

having set

$$\begin{aligned} \eta _0(s)=\int _0^s \varphi _0(\sigma )\mathrm{d}\sigma . \end{aligned}$$

We begin with the definition of weak solution.

Definition 3.1

Given \(U_0=(u_0,\eta _0)\in {\mathcal H}\), a function \(U=(u,\eta )\in \mathcal {C}([0,\infty ), {\mathcal H})\), with \(\partial _t u\in L^2(0,\tau ;{\varvec{V}})\) for every \(\tau >0\), is a weak solution to (3.2) with initial datum

$$\begin{aligned} U(0)=(u(0),\eta ^0)=U_0 \end{aligned}$$

if for every test function \(v\in {\varvec{V}}\) and almost every \(t>0\)

$$\begin{aligned} \langle \partial _t u,v \rangle + \alpha \langle \partial _t u,v \rangle _1+ \int _0^\infty \mu (s) \langle \eta (s), v \rangle _1 \mathrm{d}s +\beta \langle A^{-\vartheta }u, v \rangle + b(u,u,v)=\langle f,v \rangle , \end{aligned}$$

where \(\eta \) fulfills the explicit representation

$$\begin{aligned} \eta ^t(s)= {\left\{ \begin{array}{ll} \displaystyle \int _0^s u(t-\sigma ) \mathrm{d}\sigma , &{} 0<s\le t ,\\ \eta ^0(s-t)+\displaystyle \int _0^t u(t-\sigma ) \mathrm{d}\sigma , &{} s>t. \end{array}\right. } \end{aligned}$$
(3.3)

Remark 3.2

Given \(u\in \mathcal {C}([0,\infty ), {\varvec{V}})\), the function \(\eta =\eta ^t(\cdot )\) satisfies the representation formula above if and only if it is a mild solution (in the sense of Pazy 1983) to the nonhomogeneous linear equation

$$\begin{aligned} \partial _t\eta = T \eta + u. \end{aligned}$$

Theorem 3.3

For every initial datum \(U_0\in {\mathcal H}\), system (3.2) admits a unique solution \(U(t)=(u(t),\eta ^t)\).

Accordingly, the problem generates a dynamical system, otherwise called strongly continuous semigroup,

$$\begin{aligned} S(t):{\mathcal H}\rightarrow {\mathcal H},\quad t\ge 0, \end{aligned}$$

acting by the formula

$$\begin{aligned} S(t)U_0=U(t). \end{aligned}$$

This is a one-parameter family of maps S(t) on \({\mathcal H}\) satisfying the properties:

  • \(S(0)=\mathrm{Id}_{\mathcal H}\);

  • \(S(t+\tau )=S(t)S(\tau )\), for every \(t,\tau \ge 0\);

  • \(t\mapsto S(t)U_0\in {\mathcal C}([0,\infty ),{\mathcal H})\), for every \(U_0\in {\mathcal H}\).

Given an initial datum \(U_0\in {\mathcal H}\), the corresponding energy at time \(t\ge 0\) reads

$$\begin{aligned} {\mathcal E}(t)=\frac{1}{2}\Vert S(t)U_0\Vert _{\mathcal H}^2=\frac{1}{2}\Big [\Vert u(t)\Vert ^2+\alpha \Vert u(t)\Vert _1^2 +\Vert \eta ^t\Vert ^2_{{\mathcal M}}\Big ]. \end{aligned}$$

The existence of a weak solution is carried out via a Galerkin approximation scheme, by exploiting standard energy estimates (much more immediate than the uniform ones of the next Sect. 4), and then passing to the limit in the usual way. Indeed (see the energy equality (4.3)), we easily get that, for any given \(\tau > 0\) and \(R\ge 0\),

$$\begin{aligned} \Vert U(t)\Vert _{{\mathcal H}}\le {\mathcal Q}(R+\tau ), \end{aligned}$$
(3.4)

for all \(t\le \tau \) and all initial data \(U(0)\in {\mathbb B}_{{\mathcal H}}(R)\). In particular, the increasing function \({\mathcal Q}\) does not depend on \(\beta \ge 0\). We refer the interested reader to Conti et al. (2016) for more details on the Galerkin scheme in connection with equations with memory.

Remark 3.4

It is worth noting that the argument does not require \(\beta \) to be strictly positive, nor inequality (2.2) is needed at this level. Indeed, both \(\beta >0\) and (2.2) will come into play in connection with the dissipative properties of the semigroup.

Uniqueness is a consequence of the following continuous dependence result.

Proposition 3.5

For every \(\tau > 0\) and \(R\ge 0\), any two solutions \(U_1(t)\) and \(U_2(t)\) to equation (3.2) fulfill the estimate

$$\begin{aligned} \Vert U_1(t)-U_2(t)\Vert _{{\mathcal H}}\le {\mathcal Q}(R+\tau )\Vert U_1(0)-U_2(0)\Vert _{{\mathcal H}}, \end{aligned}$$
(3.5)

for all \(t\le \tau \) and all initial data \(U_i(0)\in {\mathbb B}_{{\mathcal H}}(R)\).

Proof

Let \(\tau >0\) be fixed, and let \(U_1=(u_1,\eta _1), U_2=(u_2,\eta _2)\) be two solutions to (3.2) on the time-interval \([0,\tau ]\) such that \(U_i(0)\in {\mathbb B}_{{\mathcal H}}(R)\). Then, the difference

$$\begin{aligned} \bar{U}=(\bar{u},\bar{\eta })=U_1-U_2 \end{aligned}$$

solves

$$\begin{aligned} \partial _t\left( \bar{u} + \alpha A\bar{u}\right) + \displaystyle \int _0^\infty \mu (s) A \bar{\eta }(s)\mathrm{d}s + \beta A^{-\vartheta } \bar{u} + B(u_1,\bar{u})+B(\bar{u},u_2)=0. \end{aligned}$$

Testing the equation by the admissible test function \(\bar{u}\), we easily find

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Gamma + \langle \bar{\eta }, \bar{u}\rangle _{{\mathcal M}} \le - b(\bar{u},u_2,\bar{u}), \end{aligned}$$

where

$$\begin{aligned} \Gamma (t) = \frac{1}{2} \Vert \bar{u}(t)\Vert ^2 + \frac{\alpha }{2}\Vert \bar{u}(t)\Vert _1^2. \end{aligned}$$

In light of (3.4),

$$\begin{aligned} - b(\bar{u},u_2,\bar{u})\le C \Vert u_2\Vert _1\Vert \bar{u}\Vert _1^2\le {\mathcal Q}(R+\tau )\Gamma , \end{aligned}$$

and an integration in time yields

$$\begin{aligned} \Gamma (t) + \int _0^t \langle \bar{\eta }^y, \bar{u}(y)\rangle _{{\mathcal M}} \mathrm{d}y \le \Gamma (0)+ {\mathcal Q}(R+\tau )\int _0^t \Gamma (y) \mathrm{d}y, \quad \forall \, t\le \tau . \end{aligned}$$

Observing that \(\bar{\eta }\) fulfills the representation (3.3), from Conti et al. (2016, Theorem 5.1) we learn that

$$\begin{aligned} \frac{1}{2} \Vert \bar{\eta }^t\Vert _{{\mathcal M}}^2-\frac{1}{2} \Vert \bar{\eta }^0\Vert _{{\mathcal M}}^2\le \int _0^t \langle \bar{\eta }^y, \bar{u}(y)\rangle _{{\mathcal M}} \mathrm{d}y. \end{aligned}$$

In summary,

$$\begin{aligned} \Vert \bar{U}(t)\Vert _{{\mathcal H}}^2\le \Vert \bar{U}(0)\Vert _{{\mathcal H}}^2+ {\mathcal Q}(R+\tau )\int _0^t \Vert \bar{U}(y)\Vert _{{\mathcal H}}^2 \mathrm{d}y, \end{aligned}$$

and the claim follows from the integral Gronwall lemma.\(\square \)

In particular, we draw from Proposition 3.5 that S(t) fulfills the further continuity property

  • \(S(t)\in {\mathcal C}({\mathcal H},{\mathcal H})\), for every \(t\ge 0\).

4 Dissipativity

The main result of this section provides a uniform-in-time a priori estimate on the solutions \(U(t)=S(t)U_0\). To this end, we introduce the function

$$\begin{aligned} \phi (\vartheta )= \vartheta -\frac{1}{2}, \end{aligned}$$

and we strengthen the assumptions on the external force by requiring

$$\begin{aligned} f\in V^\varrho \quad \text {for some}\quad \varrho >\phi (\vartheta ). \end{aligned}$$
(4.1)

Since \(\phi (\vartheta )<\vartheta \), there is no harm in assuming also \(\varrho \le \vartheta \). When \(\phi (\vartheta )<0\), which is the same as saying \(\vartheta < 1/2\), we only require \(\varrho = 0\), i.e.

$$\begin{aligned} f\in V^0={\varvec{H}}, \end{aligned}$$

which does not add anything to our general assumptions made at the beginning. As expected, \(\phi \) is increasing, since the influence of the damping term \(A^{-\vartheta }\) becomes weaker when \(\vartheta \) is larger.

Remark 4.1

We highlight the fact that \(f\in {\varvec{H}}\) in the physically relevant case \(\vartheta =0\), corresponding to the Ekman damping.

Theorem 4.2

Let \(\beta >0\) be fixed, and let (4.1) hold. Then, for any initial datum \(U_0\in {\mathcal H}\), the corresponding energy fulfills the estimate

$$\begin{aligned} {\mathcal E}(t)\le {\mathcal Q}({\mathcal E}(0))\mathrm{e}^{-\omega t} +C\Vert f\Vert ^2_\varrho , \end{aligned}$$

where \(\omega >0\) is a universal constant depending only on the structural parameters of the problem.

Remark 4.3

It is worth mentioning that a dissipative estimate of this kind seems to be out of reach when \(\beta =0\).

The main consequence of Theorem 4.2 is that, when \(\beta >0\), the dynamical system S(t) is dissipative, namely the trajectories originating from any given bounded set belong uniformly in time to an absorbing set. By definition, this is a bounded set \(\mathcal {B}_0\subset {\mathcal H}\) with the following property: for any bounded set \(\mathcal {B}\subset {\mathcal H}\) of initial data there is an entering time \(t_\mathrm{e}=t_\mathrm{e}(\mathcal {B})\ge 0\) such that

$$\begin{aligned} S(t)\mathcal {B}\subset \mathcal {B}_0,\quad \forall \, t\ge t_\mathrm{e}. \end{aligned}$$

It is then immediate to see that one can take

$$\begin{aligned} \mathcal {B}_0={\mathbb B}_{{\mathcal H}}(R_0), \end{aligned}$$

for any fixed \(R_0> \sqrt{2C}\Vert f\Vert _{\rho }\), with the constant C of the previous statement.

The remaining of the section is devoted to the proof of Theorem 4.2. This will require a number of steps.

4.1 Technical Lemmas

The main tool needed in the proof is a Gronwall-type lemma from Pata (2011).

Lemma 4.4

Let \(\Lambda _\varepsilon \) be a family of absolutely continuous nonnegative functions on \([0,\infty )\) satisfying for every \(\varepsilon >0\) small and some \(\varkappa >0\) and \(M\ge 0\) the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda _\varepsilon +\varkappa \varepsilon \Lambda _\varepsilon \le C\varepsilon ^p\Lambda _\varepsilon ^q+\frac{M}{\varepsilon ^r}, \end{aligned}$$

where the nonnegative parameters p, q, r fulfill

$$\begin{aligned} p-1>(q-1)(1+r)\ge 0. \end{aligned}$$

Moreover, let E be a continuous nonnegative function on \([0,\infty )\) such that

$$\begin{aligned} \frac{1}{m}\,E(t)\le \Lambda _\varepsilon (t)\le m E(t) \end{aligned}$$

for every \(\varepsilon >0\) small and some \(m\ge 1\). Then, there exists \(\omega >0\) such that

$$\begin{aligned} E(t)\le {\mathcal Q}(E(0))\mathrm{e}^{-\omega t}+CM. \end{aligned}$$

We also recall a basic interpolation result (see e.g. Lions and Magenes 1972).

Lemma 4.5

Let \(a<b<c\). Then

$$\begin{aligned} \Vert u\Vert _b\le \Vert u\Vert _c^\varpi \Vert u\Vert _a^{1-\varpi },\quad \forall \, u\in V^c, \end{aligned}$$

with

$$\begin{aligned} \varpi =\frac{b-a}{c-a}. \end{aligned}$$

4.2 Energy Functionals

Let now

$$\begin{aligned} U(t)=(u(t), \eta ^t) \end{aligned}$$

be the solution to (3.2) originating from a given \(U_0\in {\mathcal H}\). In what follows, we will use several times without explicit mention the Young, Hölder and Poincaré inequalities. We will also perform several formal computations, justified within a suitable regularization scheme.

We introduce the (nonnegative) functional

$$\begin{aligned} \Pi (t)=-\frac{1}{2} \int _0^\infty \mu '(s)\Vert \eta ^t(s)\Vert ^2_{1}\mathrm{d}s, \end{aligned}$$
(4.2)

satisfying the equality (see e.g. Grasselli and Pata 2002)

$$\begin{aligned} \Pi =-\langle T \eta , \eta \rangle _{{\mathcal M}}. \end{aligned}$$

Hence, recalling that \(b(u,u,u)=0\), the basic multiplication of (3.2) by U in \({\mathcal H}\) gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} {\mathcal E}+ \beta \Vert u\Vert ^2_{-\vartheta } + \Pi = \langle f, u \rangle . \end{aligned}$$
(4.3)

Next, in order to handle the possible singularity of \(\mu \) at zero, borrowing an idea from (Pata 2006) we fix \(s_*>0\) such that

$$\begin{aligned} \int _{0}^{s_*}\mu (s)\mathrm{d}s \le \frac{\kappa }{2}. \end{aligned}$$

Setting

$$\begin{aligned} \mu _*(s)= {\left\{ \begin{array}{ll} \mu (s_*), &{} 0<s\le s_*,\\ \mu (s), &{} s>s_*, \end{array}\right. } \end{aligned}$$

we consider the further functional

$$\begin{aligned} \Phi (t)=-\frac{4}{\kappa }\int _0^\infty \mu _*(s)\langle \eta ^t(s),u(t)\rangle _1 \mathrm{d}s. \end{aligned}$$

Lemma 4.6

For any \(\varepsilon >0\), we have the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Phi + \Vert u\Vert ^2_1\le \frac{8\mu (s_*)}{\kappa ^2}\Pi + \frac{4}{\alpha \kappa \varepsilon } \Vert \eta \Vert _{\mathcal M}^2+ \alpha \varepsilon \Vert \partial _t u\Vert _1^2. \end{aligned}$$
(4.4)

Proof

We start from the identity

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Phi +\frac{4}{\kappa }\int _0^\infty \mu _*(s)\Vert u\Vert ^2_1 \mathrm{d}s= & {} -\frac{4}{\kappa }\int _0^\infty \mu _*(s)\langle T \eta (s), u\rangle _1 \mathrm{d}s\\&\quad -\,\frac{4}{\kappa }\int _0^\infty \mu _*(s) \langle \eta (s),\partial _ t u \rangle _1 \mathrm{d}s. \end{aligned}$$

According to the assumptions on \(\mu \) and the definition of \(\mu _*\), we get

$$\begin{aligned} \frac{4}{\kappa }\int _{0}^\infty \mu _*(s)\Vert u\Vert ^2_1 \mathrm{d}s \ge \frac{4}{\kappa }\int _{s_*}^\infty \mu _*(s)\Vert u\Vert ^2_1 \mathrm{d}s\ge 2 \Vert u\Vert ^2_1. \end{aligned}$$

Integrating by parts, and using (2.2), we find the controls

$$\begin{aligned} -\frac{4}{\kappa }\int _0^\infty \mu _*(s)\langle T \eta (s), u\rangle _1 \mathrm{d}s&= -\frac{4}{\kappa }\int _{s_*}^{\infty } \mu '(s) \langle \eta (s),u\rangle _1 \mathrm{d}s \\&\le \frac{4\sqrt{2\mu (s_*)}}{\kappa } \Vert u\Vert _1 \Pi ^{\frac{1}{2}} \\&\le \Vert u\Vert _1^2 + \frac{8\mu (s_*)}{\kappa ^2}\Pi , \end{aligned}$$

and

$$\begin{aligned} -\frac{4}{\kappa }\int _0^\infty \mu _*(s)\langle \eta (s),\partial _t u \rangle _1\mathrm{d}s&\le \frac{4}{\kappa }\int _0^\infty \mu _*(s) \Vert \eta (s)\Vert _1\Vert \partial _t u\Vert _1 \mathrm{d}s \\&\le \frac{4}{\kappa }\int _0^\infty \mu (s) \Vert \eta (s)\Vert _1\Vert \partial _t u\Vert _1 \mathrm{d}s\\&\le \alpha \varepsilon \Vert \partial _t u\Vert _1^2+ \frac{4}{\alpha \kappa \varepsilon } \Vert \eta \Vert _{\mathcal M}^2. \end{aligned}$$

Collecting the estimates above, we meet the thesis. \(\square \)

Finally, we define the functional

$$\begin{aligned} \Psi (t)=2\beta \Vert u(t)\Vert ^2_{-\vartheta }. \end{aligned}$$

Lemma 4.7

We have the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Psi +2\alpha \Vert \partial _t u\Vert ^2_1 \le C \Vert \eta \Vert _{{\mathcal M}}^2+ C \Vert f\Vert ^2+C\Vert u\Vert \Vert u\Vert _1^{3}. \end{aligned}$$
(4.5)

Proof

Differentiating in time we find

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Psi +4\Vert \partial _t u\Vert ^2 + 4\alpha \Vert \partial _t u\Vert ^2_1 = 4\langle f, \partial _t u\rangle -4\langle \eta , \partial _t u\rangle _{{\mathcal M}} -4b(u,u,\partial _t u). \end{aligned}$$

Since

$$\begin{aligned} 4\langle f, \partial _t u\rangle -4\langle \eta , \partial _t u\rangle _{{\mathcal M}}\le \alpha \Vert \partial _t u\Vert _1^2 + C \Vert \eta \Vert _{{\mathcal M}}^2+ C \Vert f\Vert ^2, \end{aligned}$$

while, by interpolation and the Young inequality,

$$\begin{aligned} -4b(u,u,\partial _t u)\le C\Vert u\Vert ^{\frac{1}{2}}\Vert u\Vert _1^{\frac{3}{2} } \Vert \partial _t u\Vert _1 \le \alpha \Vert \partial _t u\Vert _1^2 +C\Vert u\Vert \Vert u\Vert _1^{3}, \end{aligned}$$

we obtain (4.5). \(\square \)

4.3 Proof of Theorem 4.2

Denoting

$$\begin{aligned} \nu =\min \Big \lbrace \frac{ \alpha \kappa \delta }{32},1\Big \rbrace , \end{aligned}$$

we introduce the family of energy functionals depending on the parameter \(\varepsilon >0\)

$$\begin{aligned} \Lambda _\varepsilon (t)={\mathcal E}(t)+\nu \varepsilon \Phi (t)+ \varepsilon ^2\Psi (t). \end{aligned}$$

It is apparent that the control

$$\begin{aligned} \frac{1}{2}{\mathcal E}(t)\le \Lambda _\varepsilon (t)\le 2{\mathcal E}(t) \end{aligned}$$
(4.6)

holds for any \(\varepsilon \) small enough. Collecting (4.3)–(4.5) we deduce the family of differential inequalities

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda _\varepsilon +\nu \varepsilon \Vert u\Vert _1^2 +\Big (1- \frac{8\mu (s_*) \nu \varepsilon }{\kappa ^2} \Big )\Pi -\Big ( \frac{4\nu }{\alpha \kappa }+ C\varepsilon ^2\Big ) \Vert \eta \Vert _{{\mathcal M}}^2 +\alpha \varepsilon ^2\Vert \partial _t u\Vert _1^2 \le \Theta _\varepsilon , \end{aligned}$$

having set

$$\begin{aligned} \Theta _\varepsilon =-\beta \Vert u\Vert _{-\vartheta }^2 +\langle f,u\rangle +C\Vert f\Vert ^2+C \varepsilon ^2 \Vert u\Vert \Vert u\Vert _1^{3}. \end{aligned}$$

Since by (2.2)

$$\begin{aligned} \frac{\delta }{2}\Vert \eta \Vert ^2_{{\mathcal M}}\le \Pi , \end{aligned}$$

from the very definition of \(\nu \) we find

$$\begin{aligned} \Big (1- \frac{8\mu (s_*) \nu \varepsilon }{\kappa ^2} \Big )\Pi -\Big ( \frac{4\nu }{\alpha \kappa }+ C\varepsilon ^2\Big ) \Vert \eta \Vert _{{\mathcal M}}^2&\ge \frac{1}{2} \Pi -\Big ( \frac{4\nu }{\alpha \kappa } + C\varepsilon ^2\Big ) \Vert \eta \Vert _{{\mathcal M}}^2\\&\ge \Big ( \frac{\delta }{4}- \frac{4\nu }{\alpha \kappa } -C\varepsilon ^2\Big ) \Vert \eta \Vert _{{\mathcal M}}^2 \\&\ge \frac{\delta }{16} \Vert \eta \Vert _{{\mathcal M}}^2, \end{aligned}$$

provided that \(\varepsilon >0\) is sufficiently small. This gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda _\varepsilon + \nu \varepsilon \Vert u\Vert _1^2 +\frac{\delta }{16}\Vert \eta \Vert _{{\mathcal M}}^2+\alpha \varepsilon ^2\Vert \partial _t u\Vert _1^2 \le \Theta _\varepsilon . \end{aligned}$$

In light of (2.1) and (4.6), it is apparent that the latter inequality can be rewritten in the form

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda _\varepsilon +2\varkappa \varepsilon \Lambda _\varepsilon +\alpha \varepsilon ^2\Vert \partial _t u\Vert _1^2 \le \Theta _\varepsilon , \end{aligned}$$

for every \(\varepsilon >0\) small and a suitably fixed \(\varkappa >0\). We are left to estimate the term \(\Theta _\varepsilon \). To this end, let us put

$$\begin{aligned} p=\frac{4(1+\vartheta )}{1+2\vartheta },\quad q=\frac{3+4\vartheta }{1+2\vartheta },\quad r=\frac{\vartheta -\varrho }{1+\vartheta }. \end{aligned}$$

Observe that p, q, r comply with the hypotheses of Lemma 4.4. In particular, \(r<1\). Then, by  (4.6) and Lemma 4.5 with \(a=-\vartheta , b=-\varrho , c=1\) and \(\varpi =r\),

$$\begin{aligned} \langle f,u\rangle&\le \Vert f\Vert _{\varrho } \Vert u\Vert _1^{r}\Vert u\Vert _{-\vartheta }^{1-r}\\ {}&\le \frac{\beta }{2}\Vert u\Vert _{-\vartheta }^2+ C \Vert f\Vert _\varrho ^{\frac{2}{1+r}} \Vert u\Vert _1^{\frac{2r}{1+r}}\\&\le \frac{\beta }{2}\Vert u\Vert _{-\vartheta }^2 + \varkappa \varepsilon \Lambda _\varepsilon + \frac{C\Vert f\Vert _\varrho ^2}{\varepsilon ^r} \end{aligned}$$

By the same token, using now \(a=-\vartheta , b=0, c=1\) and \(\varpi =\vartheta /(1+\vartheta )\), and subsequently applying the Young inequality with exponents \(2(1+\vartheta )\) and p / 2,

$$\begin{aligned} C \varepsilon ^2 \Vert u\Vert \Vert u\Vert _1^3&\le C \varepsilon ^2 \Vert u\Vert _{-\vartheta }^{\frac{1}{1+\vartheta }} \Vert u\Vert _1^{\frac{3+4\vartheta }{1+\vartheta }}\\&\le C \varepsilon ^2 \Vert u\Vert _{-\vartheta }^{\frac{1}{1+\vartheta }} \Lambda _\varepsilon ^{\frac{2q}{p}}\\&\le \frac{\beta }{2} \Vert u\Vert _{-\vartheta }^2 + C \varepsilon ^{p}\Lambda _\varepsilon ^q. \end{aligned}$$

Summarizing (recall that \(V^\varrho \subset {\varvec{H}}\)), we obtain the estimate

$$\begin{aligned} \Theta _\varepsilon \le \varkappa \varepsilon \Lambda _\varepsilon +C \varepsilon ^{p}\Lambda _\varepsilon ^q+\frac{C \Vert f\Vert _\varrho ^2}{\varepsilon ^r}, \end{aligned}$$

and we end up with the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda _\varepsilon +\varkappa \varepsilon \Lambda _\varepsilon +\alpha \varepsilon ^2\Vert \partial _t u\Vert _1^2 \le C \varepsilon ^{p}\Lambda _\varepsilon ^q+\frac{C \Vert f\Vert _\varrho ^2}{\varepsilon ^r}. \end{aligned}$$
(4.7)

Using again (4.6), the desired conclusion follows from an application of Lemma 4.4. \(\square \)

Once one has Theorem 4.2, an integration of (4.7) for a fixed \(\varepsilon >0\) provides a useful estimate needed in the sequel.

Corollary 4.8

Let the assumptions of Theorem 4.2 hold. Then for every \(t\ge 0\)

$$\begin{aligned} \int _0^{t} \Vert \partial _t u(y)\Vert _1^2\mathrm{d}y\le (1+t){\mathcal Q}({\mathcal E}(0)). \end{aligned}$$
(4.8)

5 Exponential Decay of Solutions

In absence of a forcing term, Theorem 4.2 establishes the exponential decay of the energy.

Corollary 5.1

Let \(\beta >0\) be fixed, and let \(f\equiv 0\). Then, for any initial datum \(U_0\in {\mathcal H}\), the corresponding energy fulfills the decay estimate

$$\begin{aligned} {\mathcal E}(t)\le {\mathcal Q}({\mathcal E}(0))\mathrm{e}^{-\omega t}, \end{aligned}$$

where \(\omega >0\) is a universal constant depending only on the structural parameters of the problem.

Nonetheless, the conclusion above remains true also when \(\beta =0\), that is, when no extra damping of the form \(\beta A^{-\vartheta } u\) is present. The result, however, is proved in a different way, with an argument that cannot be exported to the case of a nonzero f.

Theorem 5.2

Let \(\beta =0\), and let \(f\equiv 0\). Then, for any initial datum \(U_0\in {\mathcal H}\), the corresponding energy fulfills the decay estimate

$$\begin{aligned} {\mathcal E}(t)\le {\mathcal Q}({\mathcal E}(0))\mathrm{e}^{-\omega t}, \end{aligned}$$

where \(\omega >0\) is a universal constant depending only on the structural parameters of the problem.

Proof

For an arbitrarily given \(R\ge 0\), let us consider an initial datum \(U_0\in {\mathbb B}_{{\mathcal H}}(R)\). Since \(f\equiv 0\), we readily see from (4.3) that

$$\begin{aligned} {\mathcal E}(t)\le {\mathcal E}(0). \end{aligned}$$
(5.1)

Setting

$$\begin{aligned} \nu = \min \Big \lbrace \frac{\alpha \kappa \delta }{32}, 1 \Big \rbrace , \end{aligned}$$

for \(\varepsilon >0\) small to be fixed later, we define this time

$$\begin{aligned} \Lambda (t)= {\mathcal E}(t)+ \nu \varepsilon \Phi (t), \end{aligned}$$

with \(\Phi \) as in the previous section. It is apparent that

$$\begin{aligned} \frac{1}{2} {\mathcal E}(t)\le \Lambda (t)\le 2{\mathcal E}(t). \end{aligned}$$

In addition, we have the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda + \nu \varepsilon \Vert u\Vert _1^2 +\Big (1- \frac{8\mu (s_*) \nu \varepsilon }{\kappa ^2} \Big )\Pi -\frac{4\nu }{\alpha \kappa } \Vert \eta \Vert _{{\mathcal M}}^2 \le \alpha \nu \varepsilon ^2 \Vert \partial _t u\Vert _1^2. \end{aligned}$$

Repeating the calculations of the former proof, we arrive at the inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda +2\varkappa \varepsilon \Lambda \le \alpha \varepsilon ^2 \Vert \partial _t u\Vert _1^2, \end{aligned}$$
(5.2)

for some \(\varkappa >0\). In order to bound the right-hand side, we multiply the first equation of (3.2) by \(2\varepsilon ^2 \partial _t u\), to get

$$\begin{aligned} 2\alpha \varepsilon ^2 \Vert \partial _t u\Vert ^2_1=-2\varepsilon ^2 \Vert \partial _t u\Vert ^2 -2\varepsilon ^2\langle \eta , \partial _t u\rangle _{{\mathcal M}} -2\varepsilon ^2 b(u,u,\partial _t u). \end{aligned}$$

Invoking the Poincaré inequality (2.1), standard computations together with (5.1) yield

$$\begin{aligned} \alpha \varepsilon ^2 \Vert \partial _t u\Vert _1^2 \le C \varepsilon ^2 \Vert u\Vert _1^4+C \varepsilon ^2\Vert \eta \Vert ^2_{\mathcal M}\le C({\mathcal E}(0)+1)\varepsilon ^2\Lambda . \end{aligned}$$

Accordingly, we end up with

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda +\big [2\varkappa -C({\mathcal E}(0)+1)\varepsilon \big ]\varepsilon \Lambda \le 0. \end{aligned}$$

At this point, we choose \(\varepsilon \) small enough such that \(C({\mathcal E}(0)+1)\varepsilon \le \varkappa \), and an application of the Gronwall lemma entails

$$\begin{aligned} {\mathcal E}(t)\le {\mathcal Q}({\mathcal E}(0))\mathrm{e}^{-\omega t}, \end{aligned}$$
(5.3)

with \(\omega =\varkappa \varepsilon \). This is the desired inequality, except that the exponential rate \(\omega \) depends on \({\mathcal E}(0)\). To complete the argument, we use a rather standard trick of semigroup theory. Due to (5.3), there is a time \(t_0\ge 0\), depending on \({\mathcal E}(0)\), such that \({\mathcal E}(t_0)\le 1\). Hence, for \(t\ge t_0\) we can repeat the argument above, obtaining

$$\begin{aligned} {\mathcal E}(t)\le {\mathcal Q}(1)\mathrm{e}^{-\omega (t-t_0)}, \quad \forall \, t\ge t_0, \end{aligned}$$
(5.4)

where now \(\omega >0\) is independent of \({\mathcal E}(0)\). Collecting (5.3), where \(\omega =\omega ({\mathcal E}(0))\), and (5.4), we reach the desired conclusion. \(\square \)

6 Regular Exponentially Attracting Sets

It is well known that dynamical systems generated by equation with memory do not regularize in finite time, due to the intrinsic hyperbolicity of the memory component. In particular, this prevents the existence of absorbing sets having higher regularity than the initial data. Nonetheless, one can still hope that trajectories originating from bounded sets are exponentially attracted by more regular bounded sets.

Definition 6.1

A bounded set \(\mathcal {B}_\star \) is said to be exponentially attracting for S(t) in \({\mathcal H}\) if there exists \(\omega >0\) such that

$$\begin{aligned} \text {dist}_{{\mathcal H}}(S(t)\mathcal {B},\mathcal {B}_\star )\le {\mathcal Q}(\Vert \mathcal {B}\Vert _{{\mathcal H}})\mathrm{e}^{-\omega t} \end{aligned}$$

for every bounded subset \(\mathcal {B}\subset {\mathcal H}\).

Here, with standard notation,

$$\begin{aligned} \text {dist}_{{\mathcal H}}(\mathcal {B}_1,\mathcal {B}_2)=\sup _{b_1\in \mathcal {B}_1}\inf _{b_2\in \mathcal {B}_2} \Vert b_1-b_2\Vert _{\mathcal H}\end{aligned}$$

is the Hausdorff semidistance in \({\mathcal H}\) between two (nonempty) sets \(\mathcal {B}_1\) and \(\mathcal {B}_2\).

Proposition 6.2

There exists \(R_\star >0\) such that the ball

$$\begin{aligned} \mathcal {B}_\star ={\mathbb B}_{{\mathcal H}^1}(R_\star ) \end{aligned}$$

is exponentially attracting for S(t).

It is enough showing that the ball \(\mathcal {B}_\star \) exponentially attracts the absorbing set \(\mathcal {B}_0\) found in Sect. 4. To this end, for every initial data \(U_0=(u_0,\eta _0) \in \mathcal {B}_0\), in the same spirit of Temam (1997) we decompose the solution \(S(t)U_0\) into the sum

$$\begin{aligned} S(t)U_0= L(t)U_0+ K(t)U_0, \end{aligned}$$

where the maps

$$\begin{aligned} L(t)U_0=(v(t),\xi ^t)\quad \text {and}\quad K(t)U_0=(w(t),\zeta ^t) \end{aligned}$$

solve the problems

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( v + \alpha Av\right) + B(u,v) + \displaystyle \int _0^\infty \mu (s) A \xi (s)\mathrm{d}s =0, &{} \\ \partial _t \xi = T \xi + v,&{}\\ {} (v(0),\xi ^0)=(u_0,\eta _0), \end{array}\right. } \end{aligned}$$
(6.1)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( w + \alpha Aw\right) + B(u,w) + \displaystyle \int _0^\infty \mu (s) A \zeta (s)\mathrm{d}s + \beta A^{-\vartheta } w =\tilde{f}, &{} \\ \partial _t \zeta = T \zeta + w,&{}\\ (w(0),\zeta ^0)=(0,0), \end{array}\right. } \end{aligned}$$
(6.2)

with

$$\begin{aligned} \tilde{f}(t)=f-\beta A^{-\vartheta } v(t). \end{aligned}$$

Existence and uniqueness of these problems are rather standard issues, since (6.1) and (6.2) are linear with a globally Lipschitz perturbation. Following a standard procedure, we will show that system (6.1) is exponentially stable, whereas the solutions to (6.2) are uniformly bounded in the more regular space \({\mathcal H}_1\).

In what follows, the generic positive constant C may depend on \(\Vert f\Vert \) as well as on the radius \(R_0\) of the absorbing set \(\mathcal {B}_0\). In particular, by Theorem 4.2 we know that

$$\begin{aligned} \Vert S(t)U_0\Vert _{{\mathcal H}}\le C. \end{aligned}$$
(6.3)

The proof of Proposition 6.2 is an immediate consequence of the following two lemmas.

Lemma 6.3

For every initial datum \(U_0\in \mathcal {B}_0\), we have the estimate

$$\begin{aligned} \Vert L(t)U_0\Vert _{{\mathcal H}}\le C \mathrm{e}^{-\omega t}, \end{aligned}$$

where \(\omega >0\) is a universal constant.

Proof

Let \(\Lambda \) be as in the proof of Theorem 5.2, with \((v,\xi )\) in place of \((u,\eta )\). It is apparent that

$$\begin{aligned} \frac{1}{4} \Vert L(t)U_0\Vert _{{\mathcal H}}^2 \le \Lambda (t)\le \Vert L(t)U_0\Vert _{{\mathcal H}}^2. \end{aligned}$$

Repeating the same reasonings, making use of (6.3) to handle the term \(b(u,v,\partial _t v)\), we find the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda +2\omega \Lambda \le 0 \end{aligned}$$

for some \(\omega >0\), and the Gronwall lemma completes the argument. \(\square \)

Lemma 6.4

For every initial datum \(U_0\in \mathcal {B}_0\), we have the estimate

$$\begin{aligned} \Vert K(t)U_0\Vert _{{\mathcal H}^1}\le C. \end{aligned}$$

Proof

We first note that, by virtue of (6.3) and the previous Lemma 6.3,

$$\begin{aligned} \Vert K(t)U_0\Vert _{{\mathcal H}}\le C \end{aligned}$$
(6.4)

and

$$\begin{aligned} \Vert \tilde{f}(t)\Vert \le C. \end{aligned}$$

We introduce the analogous functionals of Sect. 4 for the variable \((w,\zeta )\) in higher order spaces. Namely,

$$\begin{aligned} {\mathcal E}_1(t)= \frac{1}{2} \Vert K(t)U_0\Vert _{{\mathcal H}^1}^2, \end{aligned}$$

and

$$\begin{aligned} \Pi _1(t)= -\frac{1}{2} \int _0^\infty \mu '(s)\Vert \zeta ^t(s)\Vert ^2_{{\mathcal M}^1}\mathrm{d}s. \end{aligned}$$

In particular, we have the identity

$$\begin{aligned} \Pi _1=-\langle T A^{\frac{1}{2}}\zeta , A^{\frac{1}{2}}\zeta \rangle _{{\mathcal M}} =-\langle T \zeta ,\zeta \rangle _{{\mathcal M}^1}. \end{aligned}$$

Hence, testing the first equation of (6.2) by Aw in \({\varvec{H}}\) and the second one by \(\zeta \) in \({\mathcal M}^1\), we obtain

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} {\mathcal E}_1 +b(u,w,Aw)+ \beta \Vert w\Vert _{1-\vartheta }^2+ \Pi _1=(\tilde{f},Aw). \end{aligned}$$
(6.5)

In a similar fashion, we define

$$\begin{aligned} \Phi _1(t)&=-\frac{6}{\kappa }\int _0^\infty \mu _*(s)\langle \zeta ^t(s),w(t)\rangle _2 \mathrm{d}s,\\ {} \Psi _1(t)&=\beta \Vert w(t)\Vert ^2_{1-\vartheta }. \end{aligned}$$

Arguing as in Lemma 4.6, we deduce the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Phi _1+2\Vert w\Vert ^2_2\le \frac{18\mu (s_*)}{\kappa ^2} \Pi _1+ \frac{9}{\alpha \kappa \varepsilon } \Vert \zeta \Vert _{{\mathcal M}^1}^2 +\alpha \varepsilon \Vert \partial _t w\Vert _2^2, \end{aligned}$$
(6.6)

while

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Psi _1 +2\Vert \partial _t w\Vert _1^2 + 2\alpha \Vert \partial _t w\Vert _2^2= 2\langle \tilde{f}, A \partial _t w\rangle -2\langle \zeta , \partial _t w\rangle _{{\mathcal M}^1}-2b(u,w,A\partial _t w). \end{aligned}$$

Exploiting (6.3),

$$\begin{aligned} -2b(u,w,A \partial _t w)\le C \Vert w\Vert _2\Vert \partial _t w\Vert _2, \end{aligned}$$

and we readily get

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Psi _1+ \alpha \Vert \partial _t w\Vert _2^2 \le C \Vert \zeta \Vert _{{\mathcal M}^1}^2+ C \Vert w\Vert _2^2+C. \end{aligned}$$
(6.7)

At this point, setting

$$\begin{aligned} \nu =\min \Big \lbrace \frac{\alpha \kappa \delta }{72}, 1\Big \rbrace , \end{aligned}$$

for some \(\varepsilon >0\) small to be determined later, we define the functional

$$\begin{aligned} \Lambda _1 (t)= {\mathcal E}_1(t)+\nu \varepsilon \Phi _1(t)+ \varepsilon ^2 \Psi _1(t), \end{aligned}$$

which fulfills the controls

$$\begin{aligned} \frac{1}{2} {\mathcal E}_1(t) \le \Lambda _1(t)\le 2{\mathcal E}_1(t). \end{aligned}$$

Collecting (6.5)–(6.7), we find the differential inequality

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \Lambda _1+\nu \varepsilon \Vert w\Vert _2^2+ \Big (1- \frac{18\mu (s_*) \nu \varepsilon }{\kappa ^2} \Big ) \Pi _1-\Big (\frac{9\nu }{\alpha \kappa } + C \varepsilon ^2\Big ) \Vert \zeta \Vert _{{\mathcal M}^1}^2\le \Theta _1, \end{aligned}$$

where

$$\begin{aligned} \Theta _1=-\nu \varepsilon \Vert w\Vert _2^2+C\varepsilon ^2\Vert w\Vert _2^2+(\tilde{f},Aw) -b(u,w,Aw)+ C. \end{aligned}$$

On account of (6.3)–(6.4),

$$\begin{aligned} (\tilde{f},Aw)-b(u,w,Aw)&\le \Vert \tilde{f}\Vert \Vert w\Vert _2+\Vert u\Vert _{L^6} \Vert \nabla w\Vert _{L^3} \Vert Aw\Vert \\&\le \frac{\nu \varepsilon }{4} \Vert w\Vert _2^2+ C \Vert w\Vert _1^{\frac{1}{2}}\Vert w \Vert _2^{\frac{3}{2}}+C\\&\le \frac{\nu \varepsilon }{2} \Vert w\Vert _2^2+ C. \end{aligned}$$

It is then apparent that (for \(\varepsilon >0\) small)

$$\begin{aligned} \Theta _1\le C. \end{aligned}$$

Note that the constants C above depend on \(\varepsilon \), which however will be eventually fixed. Indeed, once \(\varepsilon \) is chosen suitably small, and recasting almost word by word the proof of Theorem 4.2, we end up with

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \Lambda _1+ \omega \Lambda _1 \le C, \end{aligned}$$

for some \(\omega >0\). Since \(\Lambda (0)=0\), a final application of the Gronwall lemma will do. \(\square \)

For a semigroup S(t) satisfying the continuity property \(S(t)\in {\mathcal C}({\mathcal H},{\mathcal H})\) for every \(t\ge 0\), as in our case, having a compact exponentially attracting set is a sufficient condition in order for the global attractor to exist (see e.g. Temam 1997). By definition, this is the (unique) compact set \(\mathfrak {A}\subset {\mathcal H}\) which is at the same time

  • fully invariant, i.e. \(S(t)\mathfrak {A}=\mathfrak {A}\) for every \(t\ge 0\);

  • attracting for the semigroup, i.e

    $$\begin{aligned} \lim _{t\rightarrow \infty }\big [\text {dist}_{{\mathcal H}}(S(t)\mathcal {B},\mathfrak {A})\big ]=0 \end{aligned}$$

    for every bounded subset \(\mathcal {B}\subset {\mathcal H}\).

Unfortunately, our attracting set \({\mathcal B}_\star \), although closed and bounded in \({\mathcal H}^1\), is not compact in \({\mathcal H}\). Indeed, even if the embedding \({\varvec{V}}\subset {\varvec{H}}\) is compact, the same cannot be said for the embedding \({\mathcal M}^1\subset {\mathcal M}\) (see Pata and Zucchi 2001 for a counterexample to compactness). Accordingly, the embedding \({\mathcal H}^1\subset {\mathcal H}\) is in general not compact as well. Nevertheless, there is a general argument devised in Pata and Zucchi (2001) that allows to recover the sought compactness with a little effort, producing a compact set \({\mathcal B}_\star '\subset {\mathcal B}_\star \) which is still exponentially attracting. In turn, this entails the existence of the global attractor \(\mathfrak {A}\). We do not enter into more details, since in the next sections we will prove the existence of an exponential attractor. As a byproduct, this will yield the existence of \(\mathfrak {A}\), along with the finiteness of its fractal dimension.

7 Exponential Attractors

Definition 7.1

A compact set \(\mathfrak {E}\subset {\mathcal H}\) is an exponential attractor for S(t) if

  • \(\mathfrak {E}\) is positively invariant, i.e. \(S(t)\mathfrak {E}\subset \mathfrak {E}\) for every \(t\ge 0\);

  • \(\mathfrak {E}\) is exponentially attracting for the semigroup;

  • \(\mathfrak {E}\) has finite fractal dimension in \({\mathcal H}\).

Recall that the fractal dimension of \(\mathfrak {E}\) in \({\mathcal H}\) is defined as

$$\begin{aligned} \text {dim}_{{\mathcal H}}(\mathfrak {E})=\limsup _{\varepsilon \rightarrow 0} \frac{\ln N(\varepsilon )}{\ln \frac{1}{\varepsilon }}, \end{aligned}$$

where \(N(\varepsilon )\) is the smallest number of \(\varepsilon \)-balls of \({\mathcal H}\) necessary to cover \(\mathfrak {E}\).

The main result of the paper reads as follows.

Theorem 7.2

The dynamical system S(t) on \({\mathcal H}\) possesses an exponential attractor \(\mathfrak {E}\), which is bounded in \({\mathcal H}^1\).

As a consequence of the existence of a compact attracting set, S(t) possesses the global attractor \(\mathfrak {A}\), which is the smallest among the compact attracting sets (hence contained in the exponential attractor \(\mathfrak {E}\)).

Corollary 7.3

The dynamical system S(t) on \({\mathcal H}\) possesses the global attractor \(\mathfrak {A}\). Moreover, \(\mathfrak {A}\) has finite fractal dimension in \({\mathcal H}\) and is a bounded in \({\mathcal H}^1\).

The proof of Theorem 7.2, carried out in the next section, is based on an abstract result from Danese et al. (2015) (see Theorem 5.1 therein), that we report here below as a lemma, in a version specifically tailored to fit our particular problem. To this end, we will make use of the projections \(\mathbb {P}_1\) and \(\mathbb {P}_2\) of \({\mathcal H}\) onto its components \({\varvec{V}}\) and \({\mathcal M}\), namely,

$$\begin{aligned} \mathbb {P}_1 (u,\eta ) = u\quad \text {and}\quad \mathbb {P}_2(u,\eta ) = \eta . \end{aligned}$$

Lemma 7.4

Let the following assumptions hold.

(i) :

There exists \(R_\star > 0\) such that the ball \(\mathcal {B}_\star ={\mathbb B}_{{\mathcal H}^1} (R_\star )\) is exponentially attracting.

(ii) :

For every \(R \ge 0\) and every \(\theta > 0\) sufficiently large,

$$\begin{aligned} \int _\theta ^{2\theta }\Vert \partial _t u(t)\Vert _{1}^2 \mathrm{d}t\le {\mathcal Q}(R+\theta ), \end{aligned}$$

for all \(u(t) = \mathbb {P}_1 S(t)U_0\) with \(U_0 \in {\mathbb B}_{{\mathcal H}^1} (R)\).

(iii) :

There exists \(R_1> 0\) with the following property: for any given \(R\ge 0\), there exists a nonnegative function \(\psi \) vanishing at infinity such that

$$\begin{aligned} \Vert S(t)U_0\Vert _{{\mathcal H}^1}\le \psi (t) + R_1, \end{aligned}$$

for all \(U_0 \in {\mathbb B}_{{\mathcal H}^1}(R)\).

(iv) :

For every fixed \(R\ge 0\), the semigroup S(t) admits a decomposition of the form

$$\begin{aligned} S(t)=L(t)+K(t) \end{aligned}$$

satisfying for all initial data \(U_{0i}\in {\mathbb B}_{{\mathcal H}^1}(R)\)

$$\begin{aligned} \Vert L(t)U_{01}-L(t)U_{02}\Vert _{{\mathcal H}}&\le \psi (t)\Vert U_{01}-U_{02}\Vert _{{\mathcal H}},\\ \Vert K(t)U_{01}-K(t)U_{02}\Vert _{{\mathcal H}^1}&\le {\mathcal Q}(t)\Vert U_{01}-U_{02}\Vert _{{\mathcal H}}. \end{aligned}$$

Here, both \({\mathcal Q}\) and the nonnegative function \(\psi \) vanishing at infinity depend on R. Moreover, the function

$$\begin{aligned} \bar{\zeta }^t = \mathbb {P}_2 K(t)U_{01}-\mathbb {P}_2 K(t)U_{02} \end{aligned}$$

fulfills the Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\bar{\zeta }^t= T\bar{\zeta }^t+\bar{w}(t),\\ \bar{\zeta }^0=0, \end{array}\right. } \end{aligned}$$

for some \(\bar{w}\) satisfying the estimate

$$\begin{aligned} \Vert \bar{w}(t)\Vert _{1} \le {\mathcal Q}(t)\Vert U_{01}-U_{02}\Vert _{{\mathcal H}}. \end{aligned}$$

Then S(t) possesses an exponential attractor \(\mathfrak {E}\) contained in the ball \({\mathbb B}_{{\mathcal H}^1}(R_1)\).

Remark 7.5

Actually, in the abstract result from Danese et al. (2015) a further assumption is needed, involving a certain operator that in our case is just the identity (and the assumption is trivially satisfied).

8 Proof of Theorem 7.2

The proof amounts to verifying the four points of the above Lemma 7.4. Indeed, (i) is the content of Proposition 6.2, while (ii) is an immediate consequence of the continuous embedding \({\mathcal H}^1\subset {\mathcal H}\) and Corollary 4.8. Accordingly, we are left to show the validity of (iii) and (iv). In what follows, the generic positive constant C may depend on \(\Vert f\Vert \) and on the radius \(R_0\) of the absorbing set \(\mathcal {B}_0\).

\(\bullet \) Verifying (iii) Given \(R\ge 0\), let us consider the ball \({\mathbb B}_{{\mathcal H}^1}(R)\). We easily infer from the continuous embedding \({\mathcal H}^1\subset {\mathcal H}\) that

$$\begin{aligned} {\mathbb B}_{{\mathcal H}^1}(R)\subset {\mathbb B}_{{\mathcal H}}({\mathcal Q}(R)). \end{aligned}$$

Therefore, on account of Theorem 4.2, there exists \(t_\mathrm{e}=t_\mathrm{e}(R)\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert S(t){\mathbb B}_{{\mathcal H}^1}(R)\Vert \le {\mathcal Q}(R), &{} \forall \, t\le t_\mathrm{e},\\ \Vert S(t){\mathbb B}_{{\mathcal H}^1}(R)\Vert \le R_0, &{} \forall \, t\ge t_\mathrm{e}. \end{array}\right. } \end{aligned}$$
(8.1)

Taking an arbitrary \(U_0 \in {\mathbb B}_{{\mathcal H}^1}(R)\), we define the higher-order energy functional

$$\begin{aligned} {\mathcal E}_1(t)= \frac{1}{2} \Vert S(t)U_0\Vert _{{\mathcal H}^1}^2, \end{aligned}$$

and the nonnegative functional

$$\begin{aligned} \Pi _1(t)= -\frac{1}{2} \int _0^\infty \mu '(s)\Vert \eta ^t(s)\Vert ^2_{{\mathcal M}^1}\mathrm{d}s. \end{aligned}$$

We recall the identity

$$\begin{aligned} \Pi _1=-\langle T \eta ,\eta \rangle _{{\mathcal M}^1}. \end{aligned}$$

Testing the first equation of (3.2) by Au in \({\varvec{H}}\) and the second one by \(\eta \) in \({\mathcal M}^1\), we obtain the differential equality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} {\mathcal E}_1 + \beta \Vert u\Vert _{1-\vartheta }^2+ \Pi _1(\eta )= (f,Au)-b(u,u,Au). \end{aligned}$$
(8.2)

For any \(t\le t_\mathrm{e}\), according to (8.1) and exploiting the standard Sobolev embeddings, we have

$$\begin{aligned} (f,Au)-b(u,u,Au)&\le \Vert f\Vert \Vert u\Vert _2 +\Vert u\Vert _{L^6} \Vert \nabla u\Vert _{L^3}\Vert u\Vert _2\\&\le C\Vert u\Vert _2 + C \Vert u\Vert _1 \Vert u\Vert _2^2\\&\le {\mathcal Q}(R)\Vert u\Vert _2^2+C, \end{aligned}$$

which in turn gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} {\mathcal E}_1 \le {\mathcal Q}(R){\mathcal E}_1+ C. \end{aligned}$$

The Gronwall lemma entails

$$\begin{aligned} {\mathcal E}_1(t)\le {\mathcal Q}(R)\mathrm{e}^{{\mathcal Q}(R)t_\mathrm{e}}={\mathcal Q}(R), \quad \forall \, t\le t_\mathrm{e}. \end{aligned}$$
(8.3)

In order to show the existence of an absorbing set for the semigroup S(t) on \({\mathcal H}^1\), and similarly to the proof of Lemma 6.4, we define some further functionals. For any \(t\ge t_\mathrm{e}\), let

$$\begin{aligned} \Phi _1(t)=-\frac{6}{\kappa }\int _0^\infty \mu _*(s)\langle \eta ^t(s),u(t)\rangle _2 \mathrm{d}s, \end{aligned}$$

and

$$\begin{aligned} \Psi _1(t)=\beta \Vert u(t)\Vert ^2_{1-\vartheta }. \end{aligned}$$

Arguing as in Lemma 6.4,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Phi _1+ 2\Vert u\Vert ^2_2 \le \frac{18\mu (s_*)}{\kappa ^2}\Pi _1 +\frac{9}{\alpha \kappa \varepsilon } \Vert \eta \Vert _{{\mathcal M}^1}^2 +\alpha \varepsilon \Vert \partial _t u\Vert _2^2, \end{aligned}$$
(8.4)

whereas a differentiation in time yields

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Psi _1 +2\Vert \partial _t u\Vert _1^2 +2\alpha \Vert \partial _t u\Vert _2^2 = 2\langle f, A\partial _t u\rangle -2\langle \eta ,\partial _t u\rangle _{{\mathcal M}^1} -2b(u,u,A\partial _t u). \end{aligned}$$

Making use of (8.1),

$$\begin{aligned} -2b(u,u,A\partial _t u)\le C \Vert u\Vert _2\Vert \partial _t u\Vert _2, \end{aligned}$$

and we end up with

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Psi _1 +\alpha \Vert \partial _t u\Vert _2^2 \le C\Vert u\Vert _2^2 + C \Vert \eta \Vert _{{\mathcal M}^1}^2+C. \end{aligned}$$
(8.5)

Denoting

$$\begin{aligned} \nu =\min \Big \lbrace \frac{\alpha \kappa \delta }{72}, 1\Big \rbrace , \end{aligned}$$

we define

$$\begin{aligned} \Lambda _1(t)={\mathcal E}_1(t)+\nu \varepsilon \Phi _1(t)+ \varepsilon ^2 \Psi _1(t), \end{aligned}$$

depending on \(\varepsilon >0\) to be determined later. As customary, we have the controls

$$\begin{aligned} \frac{1}{2} {\mathcal E}_1(t) \le \Lambda _1(t)\le 2{\mathcal E}_1(t), \end{aligned}$$
(8.6)

provided \(\varepsilon \) is small enough. Moreover, adding (8.2), (8.4) and (8.5), we are led to the differential inequality

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \Lambda _1+\nu \varepsilon \Vert u\Vert _2^2 +\Big (1-\frac{18\mu (s_*)\nu \varepsilon }{\kappa ^2}\Big ) \Pi _1 -\Big (\frac{9\nu }{\alpha \kappa }+C\varepsilon ^2\Big )\Vert \eta \Vert _{{\mathcal M}^1}^2\le \Theta _1, \end{aligned}$$

having set

$$\begin{aligned} \Theta _1=-\nu \varepsilon \Vert u\Vert _2^2+(f,Au)-b(u,u,Au)+ C\varepsilon ^2\Vert u\Vert ^2_2+C. \end{aligned}$$

Since

$$\begin{aligned} \frac{\delta }{2}\Vert \eta \Vert _{{\mathcal M}^1}^2\le \Pi _1, \end{aligned}$$

we deduce that

$$\begin{aligned} \Big (1-\frac{18\mu (s_*)\nu \varepsilon }{\kappa ^2}\Big ) \Pi _1 -\Big (\frac{9\nu }{\alpha \kappa }+C\varepsilon ^2\Big )\Vert \eta \Vert _{{\mathcal M}^1}^2&\ge \frac{1}{2}\Pi _1-\Big (\frac{9\nu }{\alpha \kappa }+C\varepsilon ^2\Big )\Vert \eta \Vert _{{\mathcal M}^1}^2 \\&\ge \Big (\frac{\delta }{4}- \frac{\delta }{8}+C\varepsilon ^2\Big )\Vert \eta \Vert _{{\mathcal M}^1}^2\\&\ge \frac{\delta }{16}\Vert \eta \Vert _{{\mathcal M}^1}^2, \end{aligned}$$

provided that \(\varepsilon >0\) is sufficiently small. Accordingly,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\Lambda _1+ \varkappa \varepsilon \Lambda _1\le \Theta _1, \end{aligned}$$

for some \(\varkappa >0\). Regarding the right-hand side \(\Theta _1\), by interpolation and (8.1), we have the estimate

$$\begin{aligned} (f,Au)-b(u,u,Au)&\le \Vert f\Vert \Vert u\Vert _2 + \Vert u\Vert _{L^6}\Vert \nabla u\Vert _{L^3}\Vert u\Vert _2 \\&\le C\Vert u\Vert _2 + C\Vert u\Vert _2^{\frac{3}{2}}\\ {}&\le \varepsilon ^2 \Vert u\Vert _2^2+ C. \end{aligned}$$

Then, up to choosing \(\varepsilon \) small enough, we find

$$\begin{aligned} \Theta \le C, \end{aligned}$$

which in turn entails

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \Lambda _1+ \varkappa \varepsilon \Lambda _1 \le C. \end{aligned}$$

The Gronwall lemma on the time-interval \((t_\mathrm{e},t)\) together with (8.6) yield

$$\begin{aligned} {\mathcal E}_1(t)\le {\mathcal Q}(R)\mathrm{e}^{-\omega (t-t_\mathrm{e})}+ C, \quad \forall \, t\ge t_\mathrm{e}, \end{aligned}$$
(8.7)

with \(\omega =\varkappa \varepsilon \). Collecting (8.3) and (8.7), we readily get

$$\begin{aligned} {\mathcal E}_1(t)\le {\mathcal Q}(R)\mathrm{e}^{-\omega t}+ C,\quad \forall \, t\ge 0, \end{aligned}$$

implying the desired conclusion. \(\square \)

\(\bullet \) Verifying (iv) Given any initial datum \(U_0=(u_0,\eta _0)\), we consider this time the trivial splitting

$$\begin{aligned} S(t)U_0=L(t)U_0+K(t)U_0 \end{aligned}$$

where the maps

$$\begin{aligned} L(t)U_0=(v(t),\xi ^t)\quad \text {and}\quad K(t)U_0=(w(t),\zeta ^t) \end{aligned}$$

solve the problems

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( v + \alpha Av\right) +\displaystyle \int _0^\infty \mu (s) A \xi (s)\mathrm{d}s=0, &{} \\ \partial _t \xi = T \xi + v,&{}\\ (v(0),\xi ^0)=(u_0,\eta _0), \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( w + \alpha Aw\right) +\displaystyle \int _0^\infty \mu (s) A \zeta (s)\mathrm{d}s =f-B(u,u)-\beta A^{-\vartheta }u, &{} \\ \partial _t \zeta = T \zeta + w,&{}\\ (w(0),\zeta ^0)=(0,0). \end{array}\right. } \end{aligned}$$

Note that L(t) is a strongly continuous linear semigroup on \({\mathcal H}\). Besides, L(t) is exponentially stable. This can be easily seen by recasting the proof of Theorem 5.2.

Let now \(R\ge 0\) be fixed, and let \(U_{01},U_{02} \in {\mathbb B}_{{\mathcal H}^1}(R)\). Along this proof, the generic positive constant C is allowed to depend on R. Then, we decompose the difference

$$\begin{aligned} (\bar{u}(t),\bar{\eta }^t)=S(t)U_{01}-S(t)U_{02} \end{aligned}$$

into the sum

$$\begin{aligned} (\bar{u}(t),\bar{\eta }^t)=(\bar{v}(t),\bar{\xi }^t)+(\bar{w}(t),\bar{\zeta }^t), \end{aligned}$$

where

$$\begin{aligned} (\bar{v}(t),\bar{\xi }^t)=L(t)U_{01}-L(t)U_{02}, \quad \text {and} \quad (\bar{w}(t),\bar{\zeta }^t)= K(t)U_{01}-K(t)U_{02}. \end{aligned}$$

We first note that, on account of \(\mathbf (iii) \),

$$\begin{aligned} \Vert S(t)U_{0i}\Vert _{{\mathcal H}^1}\le C. \end{aligned}$$
(8.8)

Besides, the exponential stability of L(t) implies the existence of a universal constant \(\omega >0\) such that

$$\begin{aligned} \Vert L(t)U_{01}-L(t)U_{02}\Vert _{{\mathcal H}}\le C\mathrm{e}^{-\omega t} \Vert U_{01}-U_{02}\Vert _{{\mathcal H}}. \end{aligned}$$
(8.9)

We are left to prove the desired estimate for the difference \((\bar{w},\bar{\zeta })\), solution to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \left( \bar{w} + \alpha A\bar{w}\right) +\displaystyle \int _0^\infty \mu (s) A \bar{\zeta }(s)\mathrm{d}s =-B(\bar{u},u_1)-B(u_2,\bar{u})-\beta A^{-\vartheta }\bar{u}, &{} \\ \partial _t \bar{\zeta } = T \bar{\zeta } + \bar{w},&{}\\ (\bar{w}(0),\bar{\zeta }^0)=(0,0). \end{array}\right. } \end{aligned}$$

To this end, introducing the higher-order energy

$$\begin{aligned} {\mathcal E}_1(t)=\frac{1}{2} \Vert K(t)U_{01}-K(t)U_{02}\Vert _{{\mathcal H}^1}^2, \end{aligned}$$

we find the identity

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{\mathcal E}_1 -\frac{1}{2} \int _0^\infty \mu '(s)\Vert \bar{\zeta }^t(s)\Vert ^2_{{\mathcal M}^1}\mathrm{d}s =-b(\bar{u},u_1,A\bar{w})-b(u_2,\bar{u},A\bar{w})-\beta \langle A^{-\vartheta }\bar{u}, A\bar{w}\rangle . \end{aligned}$$

Owing to (8.8), and appealing to the embedding \({\varvec{W}}\subset [L^\infty (\Omega )]^3\), we have the controls

$$\begin{aligned} -b(\bar{u},u_1,A\bar{w})&\le C \Vert u_1\Vert _2 \Vert \bar{u}\Vert _1 \Vert \bar{w}\Vert _2\\&\le C \Vert \bar{v}\Vert _1 \Vert \bar{w}\Vert _2+ C \Vert \bar{w}\Vert _2^2\\&\le C\Vert \bar{w}\Vert _2^2+C \Vert \bar{v}\Vert _1^2,\\ -b(u_2,\bar{u},A\bar{w})&\le \Vert u_2\Vert _{L^\infty } \Vert \bar{u}\Vert _1 \Vert \bar{w}\Vert _2\\&\le C\Vert \bar{v}\Vert _1\Vert \bar{w}\Vert _2 +C\Vert \bar{w}\Vert _2^2\\&\le C\Vert \bar{w}\Vert _2^2+C \Vert \bar{v}\Vert _1^2, \end{aligned}$$

and

$$\begin{aligned} -\beta \langle A^{-\vartheta }\bar{u},A\bar{w}\rangle \le -\beta \langle A^{-\vartheta }\bar{v},A\bar{w}\rangle \le C\Vert \bar{w}\Vert _2^2 +C \Vert \bar{v}\Vert _1^2. \end{aligned}$$

Therefore, we arrive at

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{\mathcal E}_1\le C{\mathcal E}_1 + C \Vert \bar{v}\Vert _1^2. \end{aligned}$$

Recalling that \((\bar{w}(0),\bar{\zeta }^0)=(0,0)\) and exploiting (8.9), an application of the Gronwall lemma provides the sought inequality

$$\begin{aligned} {\mathcal E}_1(t)\le C\int _0^t \mathrm{e}^{C(t-y)} \Vert \bar{v}(y)\Vert _1^2 \mathrm{d}y \le C \mathrm{e}^{Ct} \Vert U_{01}-U_{02}\Vert ^2. \end{aligned}$$

In particular, we learn that

$$\begin{aligned} \Vert \bar{w}(t)\Vert _{1} \le C \mathrm{e}^{Ct}\Vert U_{01}-U_{02}\Vert _{{\mathcal H}}, \end{aligned}$$

which is exactly the last point of \(\mathbf (iv) \) to be verified. \(\square \)

9 Further Developments

In this final section, we discuss some open issues that might be the object of future investigations.

I. Exploiting the techniques of this work, it is actually possible to study generalized versions of (3.1), such as

$$\begin{aligned} \partial _t\left( u + \alpha Au\right) +\int _0^\infty g(s) A u(t-s)\mathrm{d}s + \beta h(u) + B(u,u)=f. \end{aligned}$$

for some suitable nonlinearity h(u), e.g. the Brinkman-Forchheimer correction term (see e.g Straughan 2008)

$$\begin{aligned} h(u)=a|u|^p u+bu, \quad p>0, \end{aligned}$$

where \(a>0\) and \(b\in \mathbb {R}\).

II. A further interesting problem is concerned with the analysis of nonautonomous NSV equations in presence of singularly oscillating external forces, depending on \(\varepsilon >0\), of the form

$$\begin{aligned} f^\varepsilon (t)=f_0(t)+\varepsilon ^{-\rho } f_1\Big (\frac{t}{\varepsilon }\Big ),\quad \rho >0, \end{aligned}$$

along with the formal limit obtained as \(\varepsilon \rightarrow 0\), corresponding to

$$\begin{aligned} f^0(t)=f_0(t). \end{aligned}$$

The existence of global attractors depending on \(\varepsilon >0\) and their stability as \(\varepsilon \rightarrow 0\) for NSV equations without memory has been addressed in Qin et al. (2012) (see also Chepyzhov et al. 2009; Chepyzhov and Vishik 2007 for Navier–Stokes models). In connection with memory equations, the techniques to handle singularly oscillating forces have been introduced in the novel paper (Chepyzhov et al. 2017).

III. In place of (3.1) we may consider the family of equations

$$\begin{aligned} \partial _t\left( u + \alpha Au\right) +\int _0^\infty g_\varepsilon (s) A u(t-s)\mathrm{d}s + \beta u + B(u,u)=f. \end{aligned}$$
(9.1)

where

$$\begin{aligned} g_\varepsilon (s)=\frac{1}{\varepsilon }g\Big (\frac{s}{\varepsilon }\Big ),\quad \varepsilon \in (0,1], \end{aligned}$$

is a rescaling of the original kernel g. Such a rescaling has been firstly introduced in Conti et al. (2006). We assume without loss of generality that

$$\begin{aligned} \int _0^\infty g(s)\mathrm{d}s=1. \end{aligned}$$

In the formal limit \(\varepsilon \rightarrow 0\) we have the distributional convergence

$$\begin{aligned} g_\varepsilon \rightarrow \delta _0, \end{aligned}$$

\(\delta _0\) being the Dirac mass at \(0^+\). Accordingly, (9.1) collapses into the NSV equation with Ekman damping

$$\begin{aligned} \partial _t \left( u - \alpha \Delta u\right) - \Delta u + \beta u + B(u,u)=f. \end{aligned}$$
(9.2)

Writing as before \(\mu =-g'\), and defining

$$\begin{aligned} \mu _\varepsilon (s)=\frac{1}{\varepsilon ^2}\mu \Big (\frac{s}{\varepsilon }\Big ), \end{aligned}$$

both (9.1) and (9.2) generate dynamical systems \(S_\varepsilon (t)\), acting on their respective phase spaces

$$\begin{aligned} {\mathcal H}_\varepsilon ={\varvec{V}}\times {\mathcal M}_\varepsilon , \end{aligned}$$

where

$$\begin{aligned} {\mathcal M}_\varepsilon = {\left\{ \begin{array}{ll} L^2_{\mu _\varepsilon }(\mathbb {R}^+;{\varvec{V}})&{}\text { if }\varepsilon >0,\\ \{0\}&{}\text { if }\varepsilon =0. \end{array}\right. } \end{aligned}$$

Then, for all \(\varepsilon \in [0,1]\), the semigroups \(S_\varepsilon (t)\) possess exponential attractors \(\mathfrak {E}_\varepsilon \) on \({\mathcal H}_\varepsilon \). Besides, by means of the general theory developed in Gatti et al. (2010), one can show that the family \(\{\mathfrak {E}_\varepsilon \}\) fulfills the following properties:

  • The exponential attraction rate of \(\mathfrak {E}_\varepsilon \) is uniform with respect to \(\varepsilon \); namely, there exist \({\mathcal Q}\) and \(\omega >0\), both independent of \(\varepsilon \), such that

    $$\begin{aligned} \text {dist}_{{\mathcal H}_\varepsilon }(S_\varepsilon (t)\mathbb {B}_{{\mathcal H}_\varepsilon }(R),\mathfrak {E}_\varepsilon ) \le {\mathcal Q}(R)\mathrm{e}^{-\omega t},\quad \forall \,R\ge 0. \end{aligned}$$

  • The (finite) fractal dimension \(\text {dim}_{{\mathcal H}_\varepsilon }(\mathfrak {E}_\varepsilon )\) is uniformly bounded with respect to \(\varepsilon \).

  • The family \(\{\mathfrak {E}_\varepsilon \}\) is (Hölder) continuous at \(\varepsilon =0\); namely, there exist constants \(C\ge 0\) and \(\alpha \in (0,1)\) such that

    $$\begin{aligned} \text {dist}^\mathrm{sym}_{{\mathcal H}_{\varepsilon }}(\mathfrak {E}_\varepsilon ,\mathfrak {E}_0)\le C \varepsilon ^\alpha , \end{aligned}$$

    where

    $$\begin{aligned} \text {dist}^\mathrm{sym}_{{\mathcal H}_{\varepsilon }}(\mathfrak {E}_\varepsilon ,\mathfrak {E}_0)= \max \big \{\text {dist}_{{\mathcal H}_{\varepsilon }}(\mathfrak {E}_\varepsilon ,\mathfrak {E}_0), \text {dist}_{{\mathcal H}_{\varepsilon }}(\mathfrak {E}_0,\mathfrak {E}_\varepsilon )\big \} \end{aligned}$$

    is the symmetric Hausdorff distance in \({\mathcal H}_\varepsilon \).

A similar project has been carried out in Gal and Tachim-Medjo (2013) for the equation

$$\begin{aligned} \partial _t\left( u + \alpha Au\right) +\nu Au+(1-\nu )\int _0^\infty g_\varepsilon (s) A u(t-s)\mathrm{d}s + \beta u + B(u,u)=f \end{aligned}$$

where \(\nu \in (0,1)\) is a fixed parameter. However, in this case, the presence of the instantaneous kinematic viscosity \(\nu A u\) renders the problem easier.

IV. An enhanced version of the previous analysis would be letting \(\beta >0\) in (9.1) to be a free parameter as well, and then considering the double limit

$$\begin{aligned} \varepsilon \rightarrow 0\quad \text {and}\quad \beta \rightarrow 0. \end{aligned}$$

In this situation, we have a family of two-parameter semigroups \(S_{\varepsilon ,\beta }(t)\) acting on \({\mathcal H}_\varepsilon \), the limiting case \(S_{0,0}(t)\) corresponding to the NSV equation

$$\begin{aligned} \partial _t \left( u - \alpha \Delta u\right) - \Delta u + B(u,u)=f. \end{aligned}$$

For every fixed \(\varepsilon >0\) and \(\beta >0\), the semigroup \(S_{\varepsilon ,\beta }(t)\) possesses an exponential attractor \(\mathfrak {E}_{\varepsilon ,\beta }\), and the same is true for the limiting semigroup \(S_{0,0}(t)\) (see Coti Zelati and Gal 2015). Again, the task is proving the convergence

$$\begin{aligned} \mathfrak {E}_{\varepsilon ,\beta }\rightarrow \mathfrak {E}_{0,0} \end{aligned}$$

as \(\varepsilon \rightarrow 0\) and \(\beta \rightarrow 0\), in the sense of the symmetric Hausdorff distance. Since we are not able to provide uniform estimates when \(\varepsilon >0\) and \(\beta =0\) (except in the case when \(f\equiv 0\)), we expect to obtain the desired result under an additional constraint on the double limit, of the kind \(\beta \ge F(\varepsilon )\), for a suitable positive function F vanishing at zero.