Infinite energy solutions for Dissipative Euler equations in R^2

We study the Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier-Stokes and Euler type problems. In addition, the existence of weak locally compact global attractor is proved and some extra compactness of this attractor is obtained.


Introduction
We study the following dissipative Euler system in the whole plane x ∈ R 2 : (1.1) ∂ t u + (u, ∇ x )u + αu + ∇ x p = g, div u = 0, u t=0 = u 0 , which differs from the classical Euler equations by the presence of the so-called Ekman damping term αu with α > 0. These equations describe, for instance, a 2-dimensional fluid moving on a rough surface and are used in geophysical models for large-scale processes in atmosphere and ocean. The term αu parameterizes the main dissipation occurring in the planetary boundary layer (see, e.g., [25]; see also [6] for the alternative source of damped Euler equations). The mathematical features of these and related equations are studied in a number of papers (see, for instance, [3,5,7,15,17,16]) including the analytic properties (which are very similar to the classical Euler equations without dissipative term, see [4,21,22,30] and references therein), stability analysis, vanishing viscosity limit, etc.
The attractors for damped Euler equations (1.1) in the case of bounded underlying domains have been studied in [15,5,16,17,7,8]. Remind that, in contrast to the Navier-Stokes case, the damped Euler equations remain hyperbolic and we do not have any smoothing property on a finite time interval. Moreover, even the asymptotic smoothness as time tends to infinity is much more delicate here. Indeed, similar to the classical Euler equations, following to Yudovich, see [31], we have the global existence of smooth solutions, but the best possible estimate for the smooth norms of these solutions grow faster than exponential in time, so they are not helpful for the attractor theory. Thus, to the best of our knowledge, there is no way to obtain more regularity of the attractor than the L ∞ (Ω) bounds for the vorticity even in the case of bounded underlying domain Ω or periodic boundary conditions. On the other hand, the L ∞ -bounds for the vorticity cannot be essentially relaxed if we want to have the uniqueness of a solution. By this reason, the weak attractors are normally used in order to describe the longtime behavior of solutions of the damped Euler equations. Some exception is the paper [8] where the so-called trajectory attractor is constructed for this system in a strong topology of W 1,2 (Ω) based on the enstrophy equality and the energy method.
The situation becomes more complicated where the underlying domain becomes unbounded, say, Ω = R 2 and we are interested in the infinite energy solutions. Indeed, although in this case we have an immediate control of the L ∞ -norm of the vorticity from the maximum principle: this gives only growing in time (faster than exponentially) estimates for the velocity u, see [14,28] even in a more simple case of damped Navier-Stokes equations (see also [29,19] for the analogous results for Euler equations), so in order to get the dissipative bounds for the velocity field, we need to use the energy type estimates. For the case of damped Navier-Stokes equations these estimates have been obtained in [36] for the case where the initial data u 0 belong to the so-called uniformly local Sobolev spaces, see also [33,34] for the analogous results for the case of Navier-Stokes equations in cylindrical domains as well as [10,11,23] and references therein for general theory of dissipative PDEs in unbounded domains. The aim of the present paper is to build up the attractor theory for the damped Euler equation (1.1) in uniformly local spaces extending the results of [8] and [36]. We assume that g ∈ H b , where (1.3) H b := u ∈ [L 2 b (R 2 )] 2 , div u = 0, curl u ∈ L ∞ (R 2 ) and L 2 b (R 2 ) is the usual uniformly local space determined by the norm (1.4) u L 2 b := sup Here and below, B R x 0 means the unit ball of radius R in R 2 centered at x 0 ∈ R 2 . The norm in the space H b is defined by the following natural formula: see Section 2 for more details. By definition, function u = u(t, x) is a weak solution of (1.1) if u(t) ∈ H b for t ≥ 0 and satisfies equation (1.1) in the sense of distributions, see Definition 3.2 below.
The main result of the paper is the following theorem.
Theorem 1.1. Let the external forces g ∈ H b . Then, for every u 0 ∈ H b , problem (1.1) possesses a unique weak solution u(t) and this solution satisfies the following dissipative estimate: where the positive constant β and monotone increasing function Q are independent of t ≥ 0 and u 0 ∈ H b . The solution semigroup S(t) : H b → H b associated with this equation possesses a weak locally compact global attractor A in the phase space H b , see Definition 6.1. Moreover, this attractor is a compact set in [W 1,p loc (R 2 )] 2 , for every p < ∞ and attracts bounded sets of H b in the topology of [W 1,p loc (R 2 )] 2 . The paper is organized as follows. In Section 2, we recall the definitions and basic properties of the weighted and uniformly local Sobolev spaces, introduce special classes of weights and remind a number of elementary inequalities which will be used throughout of the paper.
In Section 3, we introduce (following [36]) a number of technical tools which allows us to treat the pressure term in the proper weighted and uniformly local spaces as well as to exclude it from the various weighted energy estimates. Section 4 is devoted to the derivation of the basic dissipative estimate (1.6). In this section, based on the new version of the interpolation inequality (proved in the Appendix), we extend the method initially suggested in [36] for the case of damped Navier-Stokes equations to more difficult case of zero viscosity. Moreover, we also indicate here some improvements of the results concerning the classical Navier-Stokes and Euler equations (which corresponds to the case of α = 0). In this case, the solution u(t) may grow as t → ∞ and as proved in [36] the growth rate is restricted by the quintic polynomial in time. As indicated in Remark 4.4, using the approach developed in this paper, we may replace the quintic polynomial by the cubic one: Moreover, in the particular case g = 0 this estimate can be further improved: , see also the recent work [13] where the analogous linear growth estimate has been established for the infinite energy solutions of the Navier-Stokes equations.
The uniqueness of the weak solution of (1.1) is verified in Section 5 by adapting the famous Yudovich proof to the case of weighted and uniformly local spaces. Note that, in contrast to [29] and [19], our approach does not use the so-called Serfati identity and is based on the Yudovich type estimates in weighted L 2 -spaces. Moreover, following [9], we establish here the so-called weighted enstrophy equality which plays a crucial role in verifying the strong compactness of the attractor.
Finally, the weak locally compact attractor A is constructed in Section 6. Moreover, using the above mentioned weighted enstrophy equality and the energy method (analogously to [8]), we prove the compactness of this weak attractor in the strong topology of the space [W 1,p loc (R 2 )] 2 for any p < ∞.
Note also that, in contrast to the case of Navier-Stokes equations, in the case of Euler equations, the L ∞ -estimate for the vorticity holds not only for R 2 , but for more or less general unbounded domains. This allows to extend the results of the paper to the case of unbounded domains different from R 2 . To this end, one just needs to modify formula (3.3) for pressure by including the proper boundary terms. We return to this problem somewhere else.

Preliminaries I: Weighted and uniformly local spaces
In this section, we briefly discuss the definitions and basic properties of the weighted and uniformly local Sobolev spaces (see [23,33,35] for more detailed exposition). We start with the class of admissible weight functions and associated weighted spaces.
The associated weighted Lebesgue space L p φ (R 2 ), 1 ≤ p < ∞, is defined as a subspace of functions belonging to L p loc (R 2 ) for which the following norm is finite: is the subspace of distributions u ∈ D ′ (R 2 ) whose derivatives up to order l inclusively belong to L p φ (R 2 ) (this works for positive integer l only, for fractional and negative l, the space W l,p φ is defined using the interpolation and duality arguments, see [11,33] for more details).
The typical examples of weight functions of exponential growth rate are Another class of admissible weights of exponential growth rate are the so-called polynomial weights and, in particular, the weight function which will be essentially used throughout of the paper.
Next, we define the so-called uniformly local Sobolev spaces.
is defined as the subspace of functions of L p loc (R 2 ) for which the following norm is finite: (here and below B R x 0 denotes the R-ball in R 2 centered at x 0 ). The spaces W l,p b (R 2 ) are defined as subspaces of distributions u ∈ D ′ (R 2 ) whose derivatives up to order l inclusively belong to the space L p b (R 2 ). The next proposition gives the useful equivalent norms in the weighted Sobolev spaces Proposition 2.3. Let φ be the weight function of exponential growth rate and let 1 ≤ p < ∞, l ∈ R and R > 0. Then, where the constants C i depend on R, l and p and the constants C and µ from (2.1), but are independent of u and of the concrete choice of the weight φ.
For the proof of these estimates, see e.g., [11]. Thus, the norms The next Proposition gives relations between the weighted and uniformly local norms.
Proposition 2.4. Let φ be the weight of exponential growth rate such that x∈R 2 φ dx < ∞. Then, for every u ∈ W l,p b (R 2 ) and every κ ≥ 1, and, in particular, fixing κ = 1 in (2.7) and taking the supremum with respect to x 0 ∈ R 2 , we have where C is independent of u and the concrete choice of the weight φ. In addition, where C is also independent of u and the concrete choice of φ.
For the proof of these results, see e.g., [33,35]. The next lemma gives a simple, but important estimate for the weights θ x 0 (x) which will allow us to handle the convolution operators in weighted spaces.
Lemma 2.5. Let θ x 0 (x) be the weight defined via (2.4). Then, the following estimate holds: For the proof of this lemma see, e.g., [36].
The proof of this corollary can also be found in [36]. We conclude this section by introducing some weights and norms depending on a big parameter R which will be crucial for what follows. First, we introduce the following equivalent norm in the space W l,p b (R 2 ): Then, according to (2.7), where the constant C is independent of R ≥ 1. We also introduce the scaled weight function Then, the scaled analogue of (2.7) reads where the constants C and C κ are independent of R and the scaled analogue of (2.10) where C is independent of x 0 , y 0 ∈ R 2 and R > 0. Moreover, multiplying inequality (2.15) by θ R,y 0 (x 0 ), integrating over x 0 and using (2.16), we see that We also note that, analogously to (2.9) and using (2.13), where the constants C and C 1 are independent of R ≫ 1.

Preliminaries II: Estimating the pressure
In this section, we introduce the key estimates which allow us to work with the pressure term ∇p in the uniformly local spaces. Note that the Helmholtz decomposition does not work for the general vector fields belonging to L 2 b (R 2 ), so the standard (for the bounded domains) approach does not work at least directly and we need to proceed in a bit more accurate way.
As usual, we assume that (1.1) is satisfied in the sense of distributions. Then, taking the divergence from both sides of (1.1) and assuming that the external forces g are divergence free: we have Thus, formally, p can be expressed through u by the following singular integral operator: which we present in the form It is well-known that the convolution operator K is well-defined as a bounded linear operator . However, as the following lemma shows, the gradient of p (which is sufficient in order to define a solution of (1.1)) is well-defined in uniformly local spaces and has natural regularity properties.
For the proof of this lemma see [36]. We will denote the operator obtained in the lemma by ∇ x P and the corresponding term in the Euler equation will be denoted by ∇ x P (u ⊗ u). Then, in particular We are now ready to give the definition of a weak solution of problem (1.1).
Definition 3.2. Let the external forces g ∈ H b , where the space H b is defined as follows: Here and below curl u := ∂ x 2 u 1 − ∂ x 1 u 2 and the norm in this space is given by (1.5).
A vector field u(t, x) is a weak solution of the damped Euler problem (1.1) if and the equation is satisfied in the sense of distributions with ∇ x p = ∇ x P (u ⊗ u) defined in Lemma 3.1. Note that, according to the interpolation and embedding theorems, [20]. Thus, u⊗u ∈ L ∞ (R + , W 1,q b (R 2 )), so, due to the previous lemma, the pressure term ) and the equation (1.1) can be understood as equality in this space.

Remark 3.3.
We emphasize once more that only the gradient of pressure ∇p is well-defined as an element of L ∞ ([0, T ], L q b ), but the pressure itself may be unbounded as |x| → ∞.
To be more precise, the operator ∇P defined above satisfies in the sense of distributions. These relations can be justified by approximating w by finite functions and passing to the limit analogously to the proof of Lemma 3.1 given in [36]. Therefore, see [30], but this function may grow as |x| → ∞ (in a fact, one can only guarantee that p ∈ BM O(R 2 ), but the functions with bounded mean oscillation may grow as |x| → ∞, say, as a polynomial of log |x|, see [20]). Thus, in general p / ∈ L ∞ (R 2 ). Note also that the choice of ∇p = ∇P (u ⊗ u) is not unique. However, if p 1 and p 2 both satisfy (3.2) (for the same velocity field u), then the difference p 1 − p 2 solves ∆(p 1 − p 2 ) = 0 (in the sense of distributions) and, consequently is a harmonic function. Moreover, every harmonic function with bounded gradient is linear, so, if we want the velocity field u to be in the proper uniformly local space, the most general choice of the pressure is where the constant vector C(t) depends only on time (and is independent of x) and ∇P is defined in Lemma 3.1. In the present paper, we consider only the choice C(t) ≡ 0. In a fact, the vector C(t) should be treated as one more external data and can be chosen arbitrarily, but this does not lead to more general theory since everything can be reduced to the case of C ≡ 0 by replacing the external force g by g − C(t).
We conclude this preliminary section by reminding the key estimate which allows us handle the pressure term in weighted energy estimates, see [36] for more details. To this end, we introduce for every x 0 ∈ R 2 and R > 1 the cut-off function ϕ R,x 0 which satisfies where C is independent of R (obviously such family of cut-off functions exist). Then, the following result holds. 4 and v ∈ [W 1,q (R 2 )] 2 be divergence free. Then the following estimate holds: where C is independent of R and x 0 and θ R,x 0 (x) is defined by (2.14).
For the proof of this lemma, see [36].

Dissipative estimates for the velocity field
Our aim here is to prove the following dissipative estimate for the solutions of (1.1) in the phase space H b . We start with recalling the L ∞ -estimate for the vorticity ω = curl u which satisfies the following scalar transport equation: Lemma 4.1. Let u be a weak solution of the Euler problem (1.1). Then, the vorticity ω satisfies the following estimate: where the constant C is independent of t and u.
Indeed, the desired estimate (4.2) is an immediate corollary of the maximum principle applied to the transport equation (4.1). The validity of the maximum principle can be easily justified using the fact that the weak solution of the damped Euler equation is unique (which will be proved in the next section) and approximating the solution u by the smooth ones by the vanishing viscosity method. Thus, we only need to estimate the L 2 b -norm of u. To do that, we will extend the approach developed in [36] to the case of Euler equations.
where β > 0 and Q is a monotone function.
Proof. For simplicity, we first derive the desired estimate (4.3) in the non-dissipative case β = 0 and then indicate the changes to be made in order to verify the dissipation. The existence of a solution can be obtained after that in a standard way using e.g., the vanishing viscosity method which we leave to the reader. We will systematically use the weight functions and the family of cut-off functions ϕ R,x 0 (x) which equal to one if x ∈ B R x 0 and zero outside of B 2R x 0 such that introduced in Sections 2 and 3. Then, multiplying equation (1.1) by uϕ R,x 0 (where R is sufficiently large number and x 0 ∈ R 2 ) and following [36], after the integration over x, estimation of the nonlinear term as follows: and standard transformations, we get Here the constant C is independent of R. To estimate the term containing pressure, we use Lemma 3.4 with q = 3 and p = 3/2. Then, due to (3.13) together with the Hölder and Young inequalities, x 0 ) , where all constants are independent of R ≫ 1. Thus, (4.6) now reads Then, using (2.17), we have (4.10) Multiplying now equation (4.8) on θ R,y 0 (x 0 ), integrating over x 0 ∈ R 2 and using (2.16), we see that, for sufficiently large R, x ) dx, where the positive constants C and β are independent of R.
Thus, we only need to estimate the integral in the RHS of (4.11) which, however, a bit more delicate than in [36] since now we do not have the control of the H 1 -norm. We use the interpolation inequality proved in Appendix which holds for every R > 0 and every u ∈ H b . Using this inequality, we have Inserting this estimate into (4.11), we finally get (4.14) d dt Z R,x 0 (u(t)) + βZ R,x 0 (u(t))+ That is the complete analogue of estimate (5.18) of [36], so arguing exactly as in the proof of estimate (5.23) there, we end up with the estimate For the convenience of the reader, we give below a schematic derivation of (4.15) from the key estimate (4.14). The details can be found in [36]. Indeed, under the additional assumption that the constant R = R(u 0 , g) satisfies the Gronwall estimate applied to (4.14) gives and, therefore, taking into the account (4.2) and (2.15), we have the desired control Thus, to finish the proof of (4.15), we only need to fix the parameter R in such way that our extra assumption (4.16) is satisfied. Inserting the obtained estimate (4.18) into the left-hand side of (4.16), we see that it will be formally satisfied if and this estimate together with (4.18) gives the desired estimate (4.15). Of course, the above arguments are formal but they can be made rigorous exactly as in [36]. Remind that estimate is still not dissipative in time. In order to obtain its dissipative analogue, we just need to take R = R(t) depending on time and argue exactly as in Section 6 of [36] (see also [26] for the analogous estimate in the case of the Cahn-Hilliard equation in R 3 ). Indeed, as shown there, if we replace (4.19) by (4.20) where γ > 0 is small enough then, arguing exactly as above we obtain that Thus, the desired estimate (4.3) for the L 2 b -norm of the velocity field is obtained. Since the control of the L ∞ -norm of the vorticity has been already obtained in (4.2), the theorem is proved.  ∂ t u + (u, ∇ x )u + αu + ∇ x p = ν∆ x u + g, div u = 0 has been previously obtained in [36]. However, the proof given there used essentially the viscous term ν∆ x u and the obtained estimate was not uniform with respect to ν → 0. In contrast to this, based on the new version of the interpolation inequality, see (4.12), we have checked that the above estimate holds for the limit case ν = 0. Moreover, as not difficult to see, the dissipative estimate (4.3) is now uniform with respect to ν → 0.
Remark 4.4. The method described above works in the case of classical Euler equations (which corresponds to α = 0) as well. However, in this case we cannot expect any dissipative estimates and the analogue of (4.3) will be growing in time: The proof of this estimate repeats word by word the one given in [36] for the case of damped Navier-Stokes equations (if we use the new version of the key interpolation inequality). In addition, the following estimate stated in [36] remains true: where the constant C depends on u 0 and g, but is independent of t. As elementary examples with g = const, u = tg show, in contrast to (4.23), estimate (4.24) on the mean value of the energy with respect to the expanding balls of radii R(t) = (t + 1) 4 is sharp.
Moreover, in the important particular case g = 0 this estimate can be essentially improved arguing exactly as in [36]: b -norm of the velocity field in this case can grow at most as a quadratic polynomial in time. The usage of the following L ∞ analogue of the interpolation inequality (4.12): which can be proved analogously to (4.12) allows us to improve essentially the inequality (4.23). Indeed, applying (4.26) for the vector field u(t), fixing R = (t + 1) 4 and using that (which is the analogue of (4.2) for the case of α = 0) together with estimate (4.24), we see that where C depends on g and u 0 but is independent of t. Finally, in the particular case where g = 0, taking R = (t + 1) 2 and using estimate (4.25), we see that where C depends on u 0 , but is independent of t. Actually, we do not know whether or not the u(t) L 2 b norm can grow as t → ∞. However, it has been recently established in [12] that in the case of damped Navier-Stokes equation in an infinite cylinder (with the periodicity assumption with respect to one variable, say, x 1 ), the corresponding solution remains bounded as t → ∞. We also mention that estimate (4.29) has been recently obtained in [13] based on a slightly different representation of the non-linearity and pressure term in Navier-Stokes equation which allows to avoid the usage of rather delicate interpolation inequality (4.26).

Uniqueness and enstrophy equality
The aim of this section is to adapt the Yudovich proof of uniqueness for the Euler equations (see [31]) to the case of uniformly local spaces. The key technical thing for this proof is the following lemma.
Lemma 5.1. Let the vector field u ∈ H b . Then, the following estimate holds where the constant C is independent of p > 2 and u ∈ H b .
Proof. Estimate (5.1) follows from the following analogous estimate in the case of bounded domains established by Yudovich.

Proposition 5.2.
Let Ω ⊂ R 2 be a smooth bounded domain. Then, for any vector field v ∈ [W 1,p (Ω)] 2 such that v.n ∂Ω = 0 and any 1 < p < ∞, the following estimate holds: where the constant C is independent of p and v.
Indeed, due to Leray-Helmholtz decomposition, the vector field v can be expressed in terms of inverse Laplacians as follows: Laplacians with Dirichlet and Neumann boundary conditions respectively and ∇ ⊥ x := (−∂ x 2 , ∂ x 1 ) is the orthogonal complement to the gradient. Thus, to verify (5.2), it is enough to know that for the case of Neumann boundary conditions we need the extra zero mean assumption). The proof of estimate (5.4) can be found in [32]. Thus, estimate (5.2) is verified and we may return to the proof of the desired estimate (5.1). Let ϕ x 0 be the smooth cut-off function which equals one identically if x ∈ B 1 x 0 and zero outside of the ball B 2 x 0 . Then applying (5.2) to the function v := ϕ x 0 u and Ω = B 2 x 0 , we get x 0 ) ) and the fact that p > 2 is separated from the singularity at p = 1. Taking the supremum over x 0 ∈ R 2 , we obtain the desired estimate (5.1) and finish the proof of the lemma.
The main result of the section is the following theorem. Theorem 5.3. Let u 1 (t) and u 2 (t) be two weak solutions of the damped Euler equation (1.1). Then, the following estimate holds: where the positive constants K and L depend on the H b -norms of u 1 (0) and u 2 (0), but are independent of t. In particular, the weak solution of the damped Euler solution is unique.
Proof. Let u 1 and u 2 be two solutions of (1.1) and w = u 1 − u 2 . Then, this function solves Multiplying (5.7) by wϕ R,x 0 , where ϕ R,x 0 is the same as in (4.5) and R > 1 and x 0 ∈ R 2 are arbitrary, after the straightforward calculations, we have To estimate the term with pressure, we use (3.13) with p = 2 which gives Using now that u i L ∞ ≤ C, i = 1, 2, where the constant C depends on the H b -norms of the initial data (thanks to the dissipative estimate (4.3) and the obvious embedding H b ⊂ L ∞ ), together with the Cauchy-Schwartz inequality and the straightforward inequality we end up with Thus, we only need to estimate the most complicated last term in the RHS of (5.10). To this end, we will essentially use (5.1) and the fact that u i H b ≤ C. Then, due to the interpolation inequality which holds for any p > 2, we end up with Let us take here p = ln , where K is large enough to guarantee that p > 2. Such exists since u 1 and u 2 are globally bounded in the L 2 b -norm and, consequently, for some monotone increasing function Q R . Then, we get x 0 )   and (5.10) reads We now use that the function z → z log K z is concave. Then, due to Jensen inequality (5.13) Using also that the function z → z ln K z is monotone increasing if z ≤ Ke −1 , together with (4.10) and (2.17), we get for some new constant K 1 > 0. Multiplying now (5.12) by θ R,y (x 0 ), y ∈ R 2 , integrating over x 0 ∈ R 2 and (2.10), we finally arrive at for some new constants L and K depending on the H b -norms of the initial data. Integrating this inequality, we have Fixing, say, R = 1 in this estimate and taking the supremum over y ∈ R 2 , we end up with the desired estimate (5.6) and finish the proof of the theorem.
Passing to the limit µ → 0 in this equality and using (5.18) together with the standard convergence properties of the mollification operators as well as that we end up with the integral equality φε,x 0 dτ which is equivalent to (5.16) and finish the proof of the theorem.

The attractor
The aim of this section is to verify the existence of the attractor for the damped Euler equation in the uniformly local spaces. We first remain that according to Theorems 4.2 and 5.3, equation (1.1) generates a solution semigroup S(t) in the phase space H b : where u(t) is a unique solution of (1.1) with the initial data u 0 ∈ H b . Moreover, according to estimate (4.3) it is dissipative in the space H b , i.e., the following estimate holds: for some positive β and monotone increasing Q which are independent of t and u 0 and, according to estimates (5.6) and (5.15), the maps S(t) are locally Hölder continuous in the space L 2 b (R 2 ) and as well as in the space L 2 θ R,x 0 (R 2 ). As usual, in the case of unbounded domains and infinite energy solutions, see [23] for more details, we cannot expect the existence of a global attractor in the uniform topology of H b , but only in the local topology of (6.3) }. However, we do not know whether or not the above defined semigroup S(t) is asymptotically compact in the strong topology of H b , so we have to use the weak star topology in H loc (which we will further denote by H w * loc ) in order to define the convergence to the global attractor. We recall that a sequence u n ∈ H loc converges weakly star in H loc to some function u ∈ H loc iff for any ball B R x 0 the restrictions u n B R in the space L ∞ (B R x 0 ). Remind also that any closed ball in H b is metrizable and is compact in the topology of H w * loc , see [27]. Thus, we will use the following version of a global attractor. Proof. Indeed, according to the dissipative estimate (6.2), the ball (6.6) is an absorbing ball for the semigroup S(t) if R is large enough. This ball is metrizable and compact in the weak star topology of H w * loc . Thus, the considered semigroup possesses an absorbing ball B R which is bounded in H b and is compact in H w * loc . Moreover, using the fact that the semigroup is Hölder continuous on B R (due to estimate (5.15)) together with the compactness of the embedding H loc ⊂ L 2 loc , it is straightforward to show that, for every fixed t ≥ 0, the operators S(t) are continuous on B R in the topology of H w * loc . Thus, all assumptions of the abstract attractor existence theorem (see e.g., [1]) are satisfied and the existence of the attractor A is proved. Formula (6.5) for the attractor's structure is also an immediate corollary of this theorem. So, Theorem 6.2 is proved. Corollary 6.3. Let the above assumptions hold. Then, for every ε > 0 and every p < ∞, the weak locally compact attractor A is compact in [W 1−ε,p loc (R 2 )] 2 and attracts the images of bounded sets in H b in the strong topology of W 1−ε,p loc (R 2 ), i.e., for every bounded set B ⊂ H b and every R > 0 and x 0 ∈ R 2 (6.7) lim where dist V (X, Y ) is a non-symmetric Hausdorff distance between sets X and Y of a metric space V .
Indeed, the convergence (6.7) is an immediate corollary of the definition of the attractor A and the compactness of the embedding H loc ⊂ W 1−ε,p loc (R 2 ). We conclude this section by establishing, analogously to [8], that we may take ε = 0 in (6.7). Theorem 6.4. Let the above assumptions hold. Then the attractor A of the solution semigroup associated with the damped Euler equation (1.1) is compact in W 1,p loc (R 2 ) for any p < ∞ and attracts the images of bounded sets in H b in the topology of this space.
Proof. Indeed, due to the interpolation, it is sufficient to verify the asymptotic compactness of S(t) in W 1,2 loc (R 2 ) or, which is the same, the asymptotic compactness of the associated vorticity ω in L 2 loc (R 2 ). To verify it, following [8], we will use the so-called energy method, see also [2,24]. Let u n 0 ∈ B R be a sequence of the initial data and t n → ∞ be a sequence of times. We need to verify that the sequence S(t n )u n 0 is precompact in W 1,2 loc (R 2 ). Let u n (t), t ≥ −t n , be the solutions of the following damped Euler problems: ∂ t u n + (u n , ∇ x )u n + ∇ x p n + αu n = g, u n t=−tn = u n 0 . and the associated vorticities ω n = curl u n solve (6.9) ∂ t ω n + (u n , ∇ x )ω n + αω n = curl g, ω n t=0 = curl u n 0 .
To verify the desired asymptotic compactness (and to finish the proof of the theorem), we only need to verify that the sequence ω n (0) is precompact in L 2 loc (R 2 ). We first note that, due to the dissipative estimate (4.3), the sequence u n is uniformly bounded in H b : , q < ∞, where the control over the norm of ∂ t u n is obtained from equation (6.8) analogously to Definition 3.2 (to simplify the notations, we extend u n and ∂ t u n by zero for t ≤ t n ). Thus, without loss of generality, we may assume that (6.11) u n → u weakly star in L ∞ loc (R, H loc ), ∂ t u n → ∂ t u weakly star in L ∞ loc (R, L q loc (R 2 )). Then, due to the compactness arguments, and, in particular, (6.13) ω n → ω weakly star in L ∞ loc (R × R 2 ) and ω n → ω strongly in C loc (R, W −1,2 loc (R 2 )) Passing to the limit n → ∞ in a straightforward way in equations (6.8), we see that the limit function u(t), t ∈ R, solves the damped Euler equation (1.1) and, therefore, u ∈ K. Moreover, from (6.13), we conclude that (6.14) ω n (0) → ω(0) weakly in L 2 loc (R 2 ) and using that ω n (0) is uniformly bounded in L 2 b (R 2 ) the last convergence implies that for all ε > 0 and x 0 ∈ R 2 . At the second step, we will show that the convergence in (6.15) is actually strong which will complete the proof of the theorem. To this end, it is enough to prove that for some ε > 0 and x 0 ∈ R 2 . To this end, we will use the enstrophy equality (5.16) for equations (6.9) which we rewrite in the following form: Remind that and therefore we may fix ε > 0 being small enough that for all (s, x) ∈ R − × R 2 . Then, using the classical result on the weak lower semicontinuity of convex functionals, see e.g., [18], together with the strong convergence (6.12) and weak convergence (6.13), we conclude that s∈R − x∈R 2 F (s, x, u(s, x), ω(s, x)) ds dx ≤ lim inf n→∞ 0 −tn x∈R 2 F (s, x, u n (s, x), ω n (s, x)) ds dx, where (6.20) F (s, x, u, ω) := e αs (α − φ ε,x 0 (x) −1 u.∇ x φ ε,x 0 (x))φ ε,x 0 (x)|ω| 2 .
Passing now to the limit n → ∞ in (6.17), we arrive at On the other hand, according to the enstrophy equality for the limit functions u and ω, Therefore, the convergence (6.16) is verified and the theorem is proved.

Appendix. The interpolation inequality
The aim of this Appendix is to verify the following interpolation inequality.
Lemma 7.1. Let u ∈ L 2 b (R 2 ) be the divergent free vector field such that ω := curl u ∈ L ∞ (R 2 ). Then, the following inequality holds: x 0 ) . where R > 0, x 0 ∈ R 2 are arbitrary and the constant C is independent of R, x 0 , and u.
To verify this inequality, we use the following result proved in [36].
Proposition 7.2. Let the vector field u ∈ [W 1,2 0 (B 2R x 0 )] 2 be such that div u, curl u ∈ L ∞ (B 2R x 0 ). Then, where the constant C is independent of R and x 0 . Moreover, for any 2 < p < ∞, where θ = 1 2 − 1 2(p−1) , C may depend on p, but is independent of R and x 0 ∈ R 2 . Proof of the lemma. We first note that (7.1) is homogeneous, so (scaling x → Rx if necessary) it is enough to prove it for R = 1 and x 0 = 0 only. Let now ϕ ∈ C ∞ 0 (B 4/3 0 ) be the cut-off function such that ϕ(x) ≡ 1 if x ∈ B 1 0 . Then applying inequality (7.2) with R = 2 to the vector field ϕu, we get . Applying now inequality (7.3) with p = 4 and θ = 1/3 to the vector field ϕ 1 u, where ϕ 1 ∈ C ∞ 0 (B . In the first term, we replace the L 4 -norm of ω by its L ∞ -norm, insert the obtained result to (7.4) and use the Young inequality, this gives . Thus, we only need to estimate the L 4 -norm in the RHS of (7.6). To this end, we introduce one more cut-off function ϕ 2 ∈ C ∞ 0 (B 2 0 ) such that ϕ 2 (x) = 1 for x ∈ B 5/3 0 and use the following Ladyzhenskaya type inequality for vector fields v ∈ W 1,2 0 (B 2 0 ): . Applying this inequality to the vector field ϕ 2 u and estimating again the L 2 -norm of the vorticity by its L ∞ -norm, we have . Inserting this estimate to the RHS of (7.6) and using Young inequality again, we derive the desired estimate (7.1).