Infinite energy solutions for Dissipative Euler equations in R^2
aa r X i v : . [ m a t h . A P ] J a n , INFINITE ENERGY SOLUTIONS FOR DISSIPATIVE EULER EQUATIONSIN R VLADIMIR CHEPYZHOV AND SERGEY ZELIK Abstract.
We study the Euler equations with the so-called Ekman damping in the whole2D space. The global well-posedness and dissipativity for the weak infinite energy solutionsof this problem in the uniformly local spaces is verified based on the further development ofthe weighted energy theory for the Navier-Stokes and Euler type problems. In addition, theexistence of weak locally compact global attractor is proved and some extra compactness of thisattractor is obtained.
Contents
1. Introduction 12. Preliminaries I: Weighted and uniformly local spaces 33. Preliminaries II: Estimating the pressure 64. Dissipative estimates for the velocity field 85. Uniqueness and enstrophy equality 116. The attractor 157. Appendix. The interpolation inequality 18References 191.
Introduction
We study the following dissipative Euler system in the whole plane x ∈ R :(1.1) ( ∂ t u + ( u, ∇ x ) u + αu + ∇ x p = g, div u = 0 , u (cid:12)(cid:12) t =0 = u , which differs from the classical Euler equations by the presence of the so-called Ekman dampingterm αu with α >
0. These equations describe, for instance, a 2-dimensional fluid moving ona rough surface and are used in geophysical models for large-scale processes in atmosphere andocean. The term αu parameterizes the main dissipation occurring in the planetary boundarylayer (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 tothe 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 domainshave 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 Mathematics Subject Classification.
Key words and phrases.
Euler equations, Ekman damping, infinite energy solutions, weighted energy estimates,unbounded domains.The second author would like to thank Thierry Gallay for the fruitful discussions.This work is partially supported by the Russian Foundation of Basic Researches (projects 14-01-00346 and15-01-03587) and by the grant 14-41-00044 of RSF. a finite time interval. Moreover, even the asymptotic smoothness as time tends to infinity ismuch 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 thesmooth norms of these solutions grow faster than exponential in time, so they are not helpfulfor the attractor theory. Thus, to the best of our knowledge, there is no way to obtain moreregularity of the attractor than the L ∞ (Ω) bounds for the vorticity even in the case of boundedunderlying domain Ω or periodic boundary conditions. On the other hand, the L ∞ -bounds forthe vorticity cannot be essentially relaxed if we want to have the uniqueness of a solution. Bythis reason, the weak attractors are normally used in order to describe the longtime behavior ofsolutions of the damped Euler equations. Some exception is the paper [8] where the so-calledtrajectory attractor is constructed for this system in a strong topology of W , (Ω) based on theenstrophy equality and the energy method.The situation becomes more complicated where the underlying domain becomes unbounded,say, Ω = R and we are interested in the infinite energy solutions. Indeed, although in this casewe have an immediate control of the L ∞ -norm of the vorticity from the maximum principle:(1.2) k ω ( t ) k L ∞ ≤ Ce − αt k ω (0) k L ∞ + 1 α k curl g k L ∞ , ω := curl u, 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 theanalogous results for Euler equations), so in order to get the dissipative bounds for the velocityfield, we need to use the energy type estimates. For the case of damped Navier-Stokes equationsthese estimates have been obtained in [36] for the case where the initial data u belong to theso-called uniformly local Sobolev spaces, see also [33, 34] for the analogous results for the caseof Navier-Stokes equations in cylindrical domains as well as [10, 11, 23] and references thereinfor 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 := (cid:26) u ∈ [ L b ( R )] , div u = 0 , curl u ∈ L ∞ ( R ) (cid:27) and L b ( R ) is the usual uniformly local space determined by the norm(1.4) k u k L b := sup x ∈ R k u k L ( B x ) < ∞ . Here and below, B Rx means the unit ball of radius R in R centered at x ∈ R . The norm inthe space H b is defined by the following natural formula:(1.5) k u k H b := k u k L b + k curl u k L ∞ , 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 ≥ Theorem 1.1.
Let the external forces g ∈ H b . Then, for every u ∈ H b , problem (1.1) possessesa unique weak solution u ( t ) and this solution satisfies the following dissipative estimate: (1.6) k u ( t ) k H b ≤ Q ( k u k H b ) e − βt + Q ( k g k H b ) , where the positive constant β and monotone increasing function Q are independent of t ≥ and u ∈ H b . The solution semigroup S ( t ) : H b → H b associated with this equation possessesa weak locally compact global attractor A in the phase space H b , see Definition 6.1. Moreover, ISSIPATIVE EULER EQUATIONS 3 this attractor is a compact set in [ W ,ploc ( R )] , for every p < ∞ and attracts bounded sets of H b in the topology of [ W ,ploc ( R )] . The paper is organized as follows.In Section 2, we recall the definitions and basic properties of the weighted and uniformlylocal Sobolev spaces, introduce special classes of weights and remind a number of elementaryinequalities which will be used throughout of the paper.In Section 3, we introduce (following [36]) a number of technical tools which allows us to treatthe pressure term in the proper weighted and uniformly local spaces as well as to exclude it fromthe 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 extendthe method initially suggested in [36] for the case of damped Navier-Stokes equations to moredifficult case of zero viscosity. Moreover, we also indicate here some improvements of the resultsconcerning 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 rateis restricted by the quintic polynomial in time. As indicated in Remark 4.4, using the approachdeveloped in this paper, we may replace the quintic polynomial by the cubic one:(1.7) k u ( t ) k L b ( R ) ≤ C ( t + 1) . Moreover, in the particular case g = 0 this estimate can be further improved:(1.8) k u ( t ) k L b ( R ) ≤ C ( t + 1) , see also the recent work [13] where the analogous linear growth estimate has been establishedfor 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 famousYudovich 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 Yudovichtype estimates in weighted L -spaces. Moreover, following [9], we establish here the so-calledweighted enstrophy equality which plays a crucial role in verifying the strong compactness ofthe attractor.Finally, the weak locally compact attractor A is constructed in Section 6. Moreover, usingthe 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 ,ploc ( R )] for any p < ∞ .Note also that, in contrast to the case of Navier-Stokes equations, in the case of Euler equa-tions, the L ∞ -estimate for the vorticity holds not only for R , but for more or less generalunbounded domains. This allows to extend the results of the paper to the case of unboundeddomains different from R . To this end, one just needs to modify formula (3.3) for pressure byincluding the proper boundary terms. We return to this problem somewhere else.2. Preliminaries I: Weighted and uniformly local spaces
In this section, we briefly discuss the definitions and basic properties of the weighted anduniformly local Sobolev spaces (see [23, 33, 35] for more detailed exposition). We start with theclass of admissible weight functions and associated weighted spaces.
Definition 2.1.
A positive function φ ( x ), x ∈ R , is a weight function of exponential growthrate µ ≥ φ ( x + y ) ≤ Ce µ | y | φ ( x ) , x, y ∈ R . V. CHEPYZHOV AND S. ZELIK
The associated weighted Lebesgue space L pφ ( R ), 1 ≤ p < ∞ , is defined as a subspace of functionsbelonging to L ploc ( R ) for which the following norm is finite:(2.2) k u k pL pφ := Z R φ ( x ) | u ( x ) | p dx < ∞ and the Sobolev space W l,pφ ( R ) is the subspace of distributions u ∈ D ′ ( R ) whose derivatives upto order l inclusively belong to L pφ ( R ) (this works for positive integer l only, for fractional andnegative l , the space W l,pφ 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(2.3) φ ( x ) := e − ε | x − x | or φ ( x ) := e − √ ε | x − x | , ε ∈ R , x ∈ R . Another class of admissible weights of exponential growth rate are the so-called polynomialweights and, in particular, the weight function(2.4) θ x ( x ) := 11 + | x − x | , x ∈ R , which will be essentially used throughout of the paper.Next, we define the so-called uniformly local Sobolev spaces. Definition 2.2.
The space L pb ( R ) is defined as the subspace of functions of L ploc ( R ) for whichthe following norm is finite:(2.5) k u k L pb := sup x ∈ R k u k L p ( B x ) < ∞ (here and below B Rx denotes the R -ball in R centered at x ). The spaces W l,pb ( R ) are definedas subspaces of distributions u ∈ D ′ ( R ) whose derivatives up to order l inclusively belong tothe space L pb ( R ).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 ≤ p < ∞ , l ∈ R and R > . Then, (2.6) C Z x ∈ R φ ( x ) k u k pW l,p ( B Rx ) dx ≤ k u k pW l,pφ ≤ C Z x ∈ R φ ( x ) k u k pW l,p ( B Rx ) dx , where the constants C i depend on R , l and p and the constants C and µ from (2.1) , but areindependent of u and of the concrete choice of the weight φ . For the proof of these estimates, see e.g., [11].Thus, the norms R x ∈ R φ ( x ) k u k pW l,p ( B Rx ) dx computed with different R ’s are equivalent.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 R x ∈ R φ dx < ∞ .Then, for every u ∈ W l,pb ( R ) and every κ ≥ , (2.7) k u k pW l,p ( B κx ) ≤ C Z y ∈ B κx k u k pW l,p ( B y ) dy ≤ C κ Z y ∈ R φ ( y − x ) k u k pW l,p ( B y ) dy and, in particular, fixing κ = 1 in (2.7) and taking the supremum with respect to x ∈ R , wehave (2.8) k u k W l,pb ≤ C sup x ∈ R k u k W l,pφ ( ·− x , ISSIPATIVE EULER EQUATIONS 5 where C is independent of u and the concrete choice of the weight φ . In addition, (2.9) k u k pW l,pφ ≤ C k φ k L k u k pW l,pb , 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 ( x ) which willallow us to handle the convolution operators in weighted spaces. Lemma 2.5.
Let θ x ( x ) be the weight defined via (2.4) . Then, the following estimate holds: (2.10) Z x ∈ R θ x ( x ) θ y ( x ) dx ≤ Cθ x ( y ) , where C is independent of x , y ∈ R . For the proof of this lemma see, e.g., [36].
Corollary 2.6.
Let θ x ( x ) be defined via (2.4) . Then, for every u ∈ L pθ x ( R ) , we have (2.11) k u k pL pθy ≤ C Z x ∈ R θ y ( x ) k u k pL pθx dx , where C is independent of y ∈ R . 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 inthe space W l,pb ( R ):(2.12) k u k W l,pb,R := sup x ∈ R k u k W l,p ( B Rx ) . Then, according to (2.7),(2.13) k u k W l,pb ≤ k u k W l,pb,R ≤ CR /p k u k W l,pb , where the constant C is independent of R ≥
1. We also introduce the scaled weight function(2.14) θ R,x ( x ) := 1 R + | x − x | = R − θ x /R ( x/R ) . Then, the scaled analogue of (2.7) reads(2.15) k u k pW l,p ( B κRx ) ≤ CR − Z y ∈ B κRx k u k pW l,p ( B Ry ) dy ≤ C κ R Z y ∈ R θ R,x ( y ) k u k pW l,p ( B Ry ) dy, where the constants C and C κ are independent of R and the scaled analogue of (2.10)(2.16) Z x ∈ R θ R,x ( x ) θ R,y ( x ) dx ≤ CR − θ R,x ( y ) , where C is independent of x , y ∈ R and R >
0. Moreover, multiplying inequality (2.15) by θ R,y ( x ), integrating over x and using (2.16), we see that(2.17) Z x ∈ R θ R,x ( x ) k u k pW l,p ( B κRx ) dx ≤ C κ Z x ∈ R θ R,x ( x ) k u k pW l,p ( B Rx ) dx, where C κ is independent of R . We also note that, analogously to (2.9) and using (2.13),(2.18) Z x ∈ R θ R,x ( x ) k u k W l, ( B Rx ) dx ≤ CR − k u k W l, b,R ≤ C R k u k W l, b , where the constants C and C are independent of R ≫ V. CHEPYZHOV AND S. ZELIK 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 thegeneral vector fields belonging to L b ( R ), so the standard (for the bounded domains) approachdoes 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 thedivergence from both sides of (1.1) and assuming that the external forces g are divergence free:(3.1) div g = 0 , we have(3.2) − ∆ x p = div(( u, ∇ x ) u ) = X i,j =1 ∂ x i ∂ x j ( u i u j ) . Thus, formally, p can be expressed through u by the following singular integral operator:(3.3) p ( y ) := Z R X ij K ij ( x − y ) u i ( x ) u j ( x ) dx, K ij ( x ) := 12 π | x | δ ij − x i x j | x | which we present in the form(3.4) p = K w := K ∗ w, w := u ⊗ u, K ∗ w = X ij K ij ∗ w ij . It is well-known that the convolution operator K is well-defined as a bounded linear operatorfrom w ∈ [ L q ( R )] to p ∈ L q ( R ), 1 < q < ∞ , but it is not true neither for q = ∞ nor for theuniformly local space L qb ( R ). However, as the following lemma shows, the gradient of p (whichis sufficient in order to define a solution of (1.1)) is well-defined in uniformly local spaces andhas natural regularity properties. Lemma 3.1.
The operator w → ∇ x p , where p is defined via (3.4) can be extended by continuity(in L qloc ) in a unique way to the bounded operator from [ W s,qb ( R )] to [ W s − ,qb ( R )] , < q < ∞ and s ∈ R . 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(3.5) ∇ x P ( u ⊗ u ) : [ W , qb ( R )] → [ L qb ( R )] , < q < ∞ . Here the operator ∇ x P ( u ⊗ u ) is considered as a nonlinear (quadratic) operator u → ∇ x P ( u ⊗ u ).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:(3.6) H b := { u ∈ [ L b ( R )] , div u = 0 , curl u ∈ L ∞ ( R ) } . Here and below curl u := ∂ x u − ∂ x u 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(3.7) u ∈ L ∞ ( R + , H b )and the equation is satisfied in the sense of distributions with ∇ x p = ∇ x P ( u ⊗ u ) defined inLemma 3.1. Note that, according to the interpolation and embedding theorems,(3.8) u ∈ L ∞ ( R + , W ,qb ( R )) ⊂ L ∞ ( R + × R ) , < q < ∞ , see e.g., [20]. Thus, u ⊗ u ∈ L ∞ ( R + , W ,qb ( R )), so, due to the previous lemma, the pressure term ∇ x p ∈ L ∞ ([0 , T ] , L qb ( R )) and the equation (1.1) can be understood as equality in this space. ISSIPATIVE EULER EQUATIONS 7
Moreover, from equation (1.1), we then conclude that ∂ t u ∈ L ∞ ( R + , L qb ( R )) and, therefore, u ∈ C ([0 , T ] , L qb (Ω)). Thus, the initial data is also well-defined. Remark 3.3.
We emphasize once more that only the gradient of pressure ∇ p is well-definedas an element of L ∞ ([0 , T ] , L qb ), but the pressure itself may be unbounded as | x | → ∞ . To bemore precise, the operator ∇ P defined above satisfies(3.9) − div ∇ P ( w ) = X ij ∂ x i ∂ x j w ij , curl ∇ P ( w ) = 0in the sense of distributions. These relations can be justified by approximating w by finitefunctions and passing to the limit analogously to the proof of Lemma 3.1 given in [36]. Therefore,there is a function p ∈ L ∞ ([0 , T ] , W ,qloc ( R )) such that ∇ p = ∇ P ( w ) , see [30], but this function may grow as | x | → ∞ (in a fact, one can only guarantee that p ∈ BM O ( R ), but the functions with bounded mean oscillation may grow as | x | → ∞ , say, as apolynomial of log | x | , see [20]). Thus, in general p / ∈ L ∞ ( R ).Note also that the choice of ∇ p = ∇ P ( u ⊗ u ) is not unique. However, if p and p bothsatisfy (3.2) (for the same velocity field u ), then the difference p − p solves ∆( p − p ) = 0 (inthe sense of distributions) and, consequently is a harmonic function. Moreover, every harmonicfunction with bounded gradient is linear, so, if we want the velocity field u to be in the properuniformly local space, the most general choice of the pressure is(3.10) ∇ p = ∇ P ( u ⊗ u ) + ~C ( t ) , where the constant vector ~C ( t ) depends only on time (and is independent of x ) and ∇ P isdefined 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, butthis does not lead to more general theory since everything can be reduced to the case of ~C ≡ g by g − ~C ( t ).We conclude this preliminary section by reminding the key estimate which allows us handlethe pressure term in weighted energy estimates, see [36] for more details. To this end, weintroduce for every x ∈ R and R > ϕ R,x which satisfies(3.11) ϕ R,x ( x ) ≡ , for x ∈ B Rx , ϕ R,x ( x ) ≡ , for x / ∈ B Rx , and(3.12) |∇ x ϕ R,x ( x ) | ≤ CR − ϕ / R,x ( x ) , where C is independent of R (obviously such family of cut-off functions exist). Then, thefollowing result holds. Lemma 3.4.
Let the exponents < p, q < ∞ , p + q = 1 , w ∈ [ L pb ( R )] and v ∈ [ W ,q ( R )] bedivergence free. Then the following estimate holds: (3.13) | ( ∇ x P ( w ) , ϕ R,x v ) | ≤ C Z R θ R,x ( x ) k w k L p ( B Rx ) dx · k ϕ / R,x v k L q , where C is independent of R and x and θ R,x ( x ) is defined by (2.14) . For the proof of this lemma, see [36].
V. CHEPYZHOV AND S. ZELIK Dissipative estimates for the velocity field
Our aim here is to prove the following dissipative estimate for the solutions of (1.1) in thephase space H b . We start with recalling the L ∞ -estimate for the vorticity ω = curl u whichsatisfies the following scalar transport equation:(4.1) ∂ t ω + αω + ( u, ∇ x ) ω = curl g, ω (cid:12)(cid:12) t =0 = ω := curl u . Lemma 4.1.
Let u be a weak solution of the Euler problem (1.1) . Then, the vorticity ω satisfiesthe following estimate: (4.2) k ω ( t ) k L ∞ ≤ C k w (0) k L ∞ e − αt + k curl g k L ∞ α , where the constant C is independent of t and u . Indeed, the desired estimate (4.2) is an immediate corollary of the maximum principle appliedto the transport equation (4.1). The validity of the maximum principle can be easily justifiedusing the fact that the weak solution of the damped Euler equation is unique (which will beproved in the next section) and approximating the solution u by the smooth ones by the vanishingviscosity method. Thus, we only need to estimate the L b -norm of u . To do that, we will extendthe approach developed in [36] to the case of Euler equations. Theorem 4.2.
Let the above assumptions hold. Then the Euler equation (1.1) possesses at leastone weak solution which satisfies the following dissipative estimate: (4.3) k u ( t ) k H b ≤ Q ( k u k H b ) e − βt + Q ( k g k H b ) , where β > and Q is a monotone function.Proof. For simplicity, we first derive the desired estimate (4.3) in the non-dissipative case β = 0and then indicate the changes to be made in order to verify the dissipation. The existence of asolution can be obtained after that in a standard way using e.g., the vanishing viscosity methodwhich we leave to the reader.We will systematically use the weight functions(4.4) θ R,x ( x ) := 1 R + | x − x | and the family of cut-off functions ϕ R,x ( x ) which equal to one if x ∈ B Rx and zero outside of B Rx such that(4.5) |∇ x ϕ R,x ( x ) | ≤ CR − ϕ R,x ( x ) / . introduced in Sections 2 and 3.Then, multiplying equation (1.1) by uϕ R,x (where R is sufficiently large number and x ∈ R )and following [36], after the integration over x , estimation of the nonlinear term as follows: | (( u, ∇ x u ) , ϕ R,x u ) | = 12 | ( u. ∇ x ϕ R,x , | u | ) | ≤ CR − k u k L ( B Rx ) and standard transformations, we get(4.6) ddt k u ( t ) k L ϕR,x + α k u k L ϕR,x ≤ C k g k L ϕR,x + CR − k u k L ( B Rx ) + 2 | ( ∇ x P ( u ⊗ u ) , ϕ R,x u ) | . Here the constant C is independent of R . To estimate the term containing pressure, we useLemma 3.4 with q = 3 and p = 3 /
2. Then, due to (3.13) together with the H¨older and Young
ISSIPATIVE EULER EQUATIONS 9 inequalities,(4.7) | ( ∇ x P ( u ⊗ u ) , ϕ R,x u ) | ≤ C Z R θ R,x ( x ) k u ⊗ u k L / ( B Rx ) dx · k u k L ( B Rx ) ≤≤ C Z R θ R,x ( x ) k u k L ( B Rx ) dx · k u k L ( B Rx ) ≤≤ C (cid:18)Z R θ R,x ( x ) dx (cid:19) / (cid:18)Z R θ R,x ( x ) k u k L ( B Rx ) dx (cid:19) / · k u k L ( B Rx ) ≤≤ CR − / (cid:18)Z R θ R,x ( x ) k u k L ( B Rx ) dx (cid:19) / · k u k L ( B Rx ) ≤≤ C Z R θ R,x ( x ) k u k L ( B Rx ) dx + CR − k u k L ( B Rx ) , where all constants are independent of R ≫
1. Thus, (4.6) now reads(4.8) ddt k u ( t ) k L ϕR,x + α k u k L ϕR,x ≤ C k g k L ϕR,x ++ CR − k u k L ( B Rx ) + C Z R θ R,x ( x ) k u k L ( B Rx ) dx. We introduce(4.9) Z R,y ( u ) := Z x ∈ R θ R,y ( x ) k u k L ϕR,x dx . Then, using (2.17), we have(4.10) C Z y ∈ R θ R,y ( y ) k u k L ( B Ry ) dy ≤ Z R,y ( u ) ≤ C Z y ∈ R θ R,y ( y ) k u k L ( B Ry ) dy, where C i are independent of R . Multiplying now equation (4.8) on θ R,y ( x ), integrating over x ∈ R and using (2.16), we see that, for sufficiently large R ,(4.11) ddt Z R,x ( u ( t )) + 2 βZ R,x ( u ( t )) ≤ CZ R,x ( g ) + CR − Z x ∈ R θ R,x ( x ) k u k L ( B Rx ) 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 bitmore delicate than in [36] since now we do not have the control of the H -norm. We use theinterpolation inequality proved in Appendix(4.12) k u k L ( B R ) ≤ C (cid:18) R k u k L ( B R ) + k u k / L ( B R ) k ω k / L ∞ ( B R ) (cid:19) . which holds for every R > u ∈ H b . Using this inequality, we have(4.13) Z x ∈ R θ R,x ( x ) k u k L ( B Rx ) dx ≤≤ C Z x ∈ R θ R,x ( x ) (cid:18) R k u k L ( B Rx ) + k u k / L ( B Rx ) k ω k / L ∞ ( B Rx ) (cid:19) dx ≤≤ C Z x ∈ R θ R,x ( x ) k u k L ( B Rx ) (cid:18) R k u k L ( B Rx ) + k u k / L ( B Rx ) k ω k / L ∞ ( B Rx ) (cid:19) dx ≤≤ C (cid:18) R − k u k L b, R + k u k / L b, R k ω k / L ∞ (cid:19) Z x ∈ R θ R,x ( x ) k u k L ( B Rx ) dx ≤≤ CR / ( k ω k L ∞ + R − k u k L b,R ) Z R,x ( u ) . Inserting this estimate into (4.11), we finally get(4.14) ddt Z
R,x ( u ( t )) + βZ R,x ( u ( t ))++ (cid:16) β − CR − / ( k ω ( t ) k L ∞ + R − k u ( t ) k L b,R ) (cid:17) Z R,x ( u ( t )) ≤ CZ R,x ( g ) . That is the complete analogue of estimate (5.18) of [36], so arguing exactly as in the proof ofestimate (5.23) there, we end up with the estimate(4.15) k u ( t ) k L b ≤ C ( k u k H b + k g k H b + 1) . For the convenience of the reader, we give below a schematic derivation of (4.15) from the keyestimate (4.14). The details can be found in [36]. Indeed, under the additional assumption thatthe constant R = R ( u , g ) satisfies(4.16) KR − / ( R − k u k L b,R + k ω k L ∞ ) ≤ β, t ≥ , the Gronwall estimate applied to (4.14) gives(4.17) Z R,x ( t ) ≤ Z R,x ( u ) e − βt + CZ R,x ( g ) ≤ CR ( k u k L b + k g k L b )and, therefore, taking into the account (4.2) and (2.15), we have the desired control(4.18) R − k u ( t ) k L b,R + k w ( t ) k L ∞ ≤ C ( k u k H b + k g k H b + 1) . Thus, to finish the proof of (4.15), we only need to fix the parameter R in such way that ourextra assumption (4.16) is satisfied. Inserting the obtained estimate (4.18) into the left-handside of (4.16), we see that it will be formally satisfied if(4.19) R − / = βKC ( k u k L b + k curl u k L ∞ + 1 + k g k L b + k curl g k L ∞ ) − and this estimate together with (4.18) gives the desired estimate (4.15). Of course, the abovearguments 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] (seealso [26] for the analogous estimate in the case of the Cahn-Hilliard equation in R ). Indeed, asshown there, if we replace (4.19) by(4.20) R ( t ) − / = βKC (cid:0) k u k H b e − γt + 1 + k g k H b (cid:1) − , where γ > k u ( t ) k L b ≤ k u k L b,R ≤ C k u k H b e − γt + C (1 + k g k H b ) . Thus, the desired estimate (4.3) for the L b -norm of the velocity field is obtained. Since thecontrol of the L ∞ -norm of the vorticity has been already obtained in (4.2), the theorem isproved. (cid:3) Remark 4.3.
Remind that the analogue of the dissipative estimate (4.3) for the case of dampedNavier-Stokes equation(4.22) ∂ t u + ( u, ∇ x ) u + αu + ∇ x p = ν ∆ x u + g, div u = 0has been previously obtained in [36]. However, the proof given there used essentially the viscousterm ν ∆ x u and the obtained estimate was not uniform with respect to ν →
0. In contrastto this, based on the new version of the interpolation inequality, see (4.12), we have checkedthat the above estimate holds for the limit case ν = 0. Moreover, as not difficult to see, thedissipative estimate (4.3) is now uniform with respect to ν → ISSIPATIVE EULER EQUATIONS 11
Remark 4.4.
The method described above works in the case of classical Euler equations (whichcorresponds to α = 0) as well. However, in this case we cannot expect any dissipative estimatesand the analogue of (4.3) will be growing in time:(4.23) k u ( t ) k H b ≤ C (1 + k g k H b + k u k H b ) ( t + 1) . The proof of this estimate repeats word by word the one given in [36] for the case of dampedNavier-Stokes equations (if we use the new version of the key interpolation inequality). Inaddition, the following estimate stated in [36] remains true:(4.24) 1( t + 1) k u ( t ) k L b, ( t +1)4 ≤ C ( t + 1) , where the constant C depends on u and g , but is independent of t . As elementary exampleswith g = const , u = tg show, in contrast to (4.23), estimate (4.24) on the mean value of theenergy with respect to the expanding balls of radii R ( t ) = ( t + 1) is sharp.Moreover, in the important particular case g = 0 this estimate can be essentially improvedarguing exactly as in [36]:(4.25) 1( t + 1) k u ( t ) k L b, ( t +1)2 ≤ C, so the L b -norm of the velocity field in this case can grow at most as a quadratic polynomial intime. The usage of the following L ∞ analogue of the interpolation inequality (4.12):(4.26) k u k L ∞ ( B R ) ≤ C (cid:18) k curl u k / L ∞ ( B R ) k u k / L ( B R ) + 1 R k u k L ( B R ) (cid:19) 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) and using that(4.27) k curl u ( t ) k L ∞ ≤ C ( k curl u k L ∞ + t k curl g k L ∞ )(which is the analogue of (4.2) for the case of α = 0) together with estimate (4.24), we see that(4.28) k u ( t ) k L b ≤ k u ( t ) k L ∞ ≤ C ( t + 1) , where C depends on g and u but is independent of t . Finally, in the particular case where g = 0, taking R = ( t + 1) and using estimate (4.25), we see that(4.29) k u ( t ) k L b ≤ C ( t + 1) , where C depends on u , but is independent of t . Actually, we do not know whether or not the k u ( t ) k L b norm can grow as t → ∞ . However, it has been recently established in [12] that in thecase of damped Navier-Stokes equation in an infinite cylinder (with the periodicity assumptionwith respect to one variable, say, x ), the corresponding solution remains bounded as t → ∞ .We also mention that estimate (4.29) has been recently obtained in [13] based on a slightlydifferent representation of the non-linearity and pressure term in Navier-Stokes equation whichallows to avoid the usage of rather delicate interpolation inequality (4.26).5. 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 thefollowing lemma.
Lemma 5.1.
Let the vector field u ∈ H b . Then, the following estimate holds (5.1) k u k W ,pb ≤ Cp k u k H b , where the constant C is independent of p > and u ∈ H b . Proof.
Estimate (5.1) follows from the following analogous estimate in the case of boundeddomains established by Yudovich.
Proposition 5.2.
Let Ω ⊂ R be a smooth bounded domain. Then, for any vector field v ∈ [ W ,p (Ω)] such that v.n (cid:12)(cid:12) ∂ Ω = 0 and any < p < ∞ , the following estimate holds: (5.2) k v k W ,p (Ω) ≤ C ( p + 1 p − (cid:0) k div v k L p (Ω) + k curl v k L p (Ω) (cid:1) , where the constant C is independent of p and v . Indeed, due to Leray-Helmholtz decomposition, the vector field v can be expressed in termsof inverse Laplacians as follows:(5.3) v = −∇ x ( − ∆ x ) − N div v − ∇ ⊥ x ( − ∆ x ) − D curl v, where ( − ∆ x ) − D and ( − ∆ x ) − N are inverse Laplacians with Dirichlet and Neumann boundaryconditions respectively and ∇ ⊥ x := ( − ∂ x , ∂ x ) is the orthogonal complement to the gradient.Thus, to verify (5.2), it is enough to know that(5.4) k Av k W ,p (Ω) ≤ C ( p + 1 p − k v k L p (Ω) , for the case A = ( − ∆ x ) − D and A = ( − ∆ x ) − N (of course, for the case of Neumann boundaryconditions we need the extra zero mean assumption). The proof of estimate (5.4) can be foundin [32]. Thus, estimate (5.2) is verified and we may return to the proof of the desired estimate(5.1). Let ϕ x be the smooth cut-off function which equals one identically if x ∈ B x and zerooutside of the ball B x . Then applying (5.2) to the function v := ϕ x u and Ω = B x , we get(5.5) k u k W ,p ( B x ) ≤ Cp ( k curl u k L p ( B x ) + k u k L ∞ ( B x ) ) ≤ Cp ( k curl u k L ∞ ( B x ) + k u k L ( B x ) ) , where we have used the obvious estimate k u k L ∞ ( B x ) ≤ C ( k curl u k L ∞ ( B x ) + k u k L ( B x ) ) and thefact that p > p = 1. Taking the supremum over x ∈ R ,we obtain the desired estimate (5.1) and finish the proof of the lemma. (cid:3) The main result of the section is the following theorem.
Theorem 5.3.
Let u ( t ) and u ( t ) be two weak solutions of the damped Euler equation (1.1) .Then, the following estimate holds: (5.6) k u ( t ) − u ( t ) k L b ≤ Ke k u (0) − u (0) k L b K ! e − Lt , where the positive constants K and L depend on the H b -norms of u (0) and u (0) , but areindependent of t . In particular, the weak solution of the damped Euler solution is unique.Proof. Let u and u be two solutions of (1.1) and w = u − u . Then, this function solves(5.7) ∂ t w + ( w, ∇ x ) u + ( u , ∇ x ) w + αw + ∇ x p = 0 . Multiplying (5.7) by wϕ R,x , where ϕ R,x is the same as in (4.5) and R > x ∈ R arearbitrary, after the straightforward calculations, we have(5.8) 12 ddt k w k L ϕR,x + α k w k L ϕR,x ≤ C ( |∇ x u | + |∇ x u | , w ϕ R,x )++ CR − ( | u | , w ) L ( B Rx ) + | ( ∇ x P ( u ⊗ u − u ⊗ u ) , wϕ R,x ) | . ISSIPATIVE EULER EQUATIONS 13
To estimate the term with pressure, we use (3.13) with p = 2 which gives(5.9) | ( ∇ x P ( u ⊗ u − u ⊗ u ) , wϕ R,x ) | ≤≤ C k w k L ( B Rx ) Z y ∈ R θ R,x ( y ) k u ⊗ u − u ⊗ u k L ( B Ry ) dy. Using now that k u i k L ∞ ≤ C , i = 1 ,
2, where the constant C depends on the H b -norms of theinitial 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 k u k L ( B Rx ) ≤ C R Z y ∈ R θ R,x ( y ) k u k L ( B Ry ) dy, we end up with(5.10) 12 ddt k w k L ϕR,x + α k w k L ϕR,x ≤ C R Z y ∈ R θ R,x ( y ) k w k L ( B Ry ) dy ++ C ( |∇ x u | + |∇ x u | , w ϕ R,x ) . Thus, we only need to estimate the most complicated last term in the RHS of (5.10). To thisend, we will essentially use (5.1) and the fact that k u i k H b ≤ C . Then, due to the interpolationinequality k w k L p/ ( p − ≤ C k w k θL ∞ k w k − θL , θ = 1 p which holds for any p >
2, we end up with(5.11) | ( |∇ x u | + |∇ x u | , w ϕ R,x ) | ≤ C R ( k u k W ,pb + k u k W ,pb ) k w k L p/ ( p − ( B Rx ) ≤≤ Cp k w k /pL ∞ k w k p − /pL ( B Rx ) ≤ Cp k w k p − /pL ( B Rx ) Let us take here p = ln K k w k L B Rx ! , where K is large enough to guarantee that p >
2. Such K = K ( k u (0) k H b , k u (0) k H b ) exists since u and u are globally bounded in the L b -norm and,consequently, k w k L ( B Rx ) ≤ k u k L b, R + k u k L b,R ≤ Q R ( k u (0) k H b + k u (0) k H b )for some monotone increasing function Q R . Then, we get | ( |∇ x u | + |∇ x u | , w ϕ R,x ) | ≤ C k w k L ( B Rx ) ln K k w k L ( B Rx ) and (5.10) reads(5.12) ddt k w k L ϕR,x ≤ C R Z y ∈ R θ R,x ( y ) k w k L ( B Ry ) dy + C k w k L ( B Rx ) ln K k w k L ( B Rx ) . We now use that the function z → z log Kz is concave. Then, due to Jensen inequality(5.13) Z x ∈ R θ R,y ( x ) k w k L ( B Rx ) ln K k w k L ( B Rx ) dx ≤≤ Z x ∈ R θ R,y ( x ) k w k L ( B Rx ) dx ln K R x ∈ R θ R,y ( x ) dx R x ∈ R θ R,y ( x ) k w k L ( B Rx ) dx . Using also that the function z → z ln Kz is monotone increasing if z ≤ Ke − , together with(4.10) and (2.17), we get Z x ∈ R θ R,y ( x ) k w k L ( B Rx ) ln K k w k L ( B Rx ) dx ≤ CZ R,y ( w ) ln K Z R,y ( w )for some new constant K >
0. Multiplying now (5.12) by θ R,y ( x ), y ∈ R , integrating over x ∈ R and (2.10), we finally arrive at(5.14) ddt Z R,y ( w ) ≤ LZ R,y ( w ) + CZ R,y ( w ) ln KZ R,y ( w )for some new constants L and K depending on the H b -norms of the initial data. Integratingthis inequality, we have(5.15) Z R,y ( w ( t )) ≤ K (cid:18) Z R,y ( w (0)) K (cid:19) e − Lt e − e − Lt . Fixing, say, R = 1 in this estimate and taking the supremum over y ∈ R , we end up with thedesired estimate (5.6) and finish the proof of the theorem. (cid:3) We conclude this section by reminding the so-called weighted enstrophy equality which will beused in the next section in order to verify the convergence to the attractor in a strong topology.Indeed, multiplying formally equation (4.1) by ωφ ε,x where φ ε,x ( x ) := e − ε | x − x | , integratingover x ∈ R and using that div u = 0, we arrive at(5.16) 12 ddt k ω k L φε,x + α k ω k L φε,x − ( u. ∇ x φ ε,x , | ω | ) = (curl g, ωφ ε,x ) . However, these arguments require justification since for weak solutions ω ∈ L ∞ ([0 , T ] × R ) onlyand the term (( u, ∇ x ) ω, ωφ ε,x ) is not rigorously defined. We overcome this difficulty using themollification operators and arguing as in [9]. Theorem 5.4.
Let u be a weak solution of the damped Euler problem (1.1) and let ε > and x ∈ R be arbitrary. Then, the function t → k ω ( t ) k L φε,x is absolutely continuous and theequality (5.16) holds for almost all t .Proof. Indeed, let S µ v := ρ µ ∗ v where ρ µ ( x ) = µ − φ ( xµ − ), µ > ρ is a standard molli-fication kernel. Then, applying S µ to both sides of equation (4.1) and denoting ω µ := S µ ω , wehave(5.17) ∂ t ω µ + αω µ + ( u, ∇ x ) ω µ = curl g µ + R µ , where R µ := (( u, ∇ x ) ω )) ∗ ρ µ − ( u, ∇ x )( ω ∗ ρ µ ). Using the fact that u ( t ) ∈ W ,pb ( R ) for all p < ∞ and arguing exactly as in [9], we see that R µ is uniformly with respect to µ → L ∞ ([0 , T ] , L pb ( R )) and(5.18) R µ → L ([0 , T ] , L ploc ( R )) , ISSIPATIVE EULER EQUATIONS 15 see [9], Lemma II.1, page 516. Multiplying (5.17) by ω µ φ ε,x (which is now allowed since ω µ issmooth in x ) after the integration over x and t , we get that for every t ≥ s ≥ (cid:18) k ω µ ( t ) k L φε,x − k ω µ ( s ) k L φε,x (cid:19) = Z ts ((curl g µ , ω µ ( τ ) φ ε,x )++( R µ ( τ ) , ω µ ( τ ) φ ε,x ) + ( u ( τ ) . ∇ x φ ε,x , | ω µ ( τ ) | ) − α k ω µ ( τ ) k L φε,x (cid:19) dτ. Passing to the limit µ → ω ∈ L ∞ ( R + × R ) ∩ C w ([0 , T ] , L φ ε,x ( R )) , we end up with the integral equality(5.21) 12 (cid:18) k ω ( t ) k L φε,x − k ω ( s ) k L φε,x (cid:19) = Z ts ((curl g, ω ( τ ) φ ε,x )++( u ( τ ) . ∇ x φ ε,x , | ω ( τ ) | ) − α k ω ( τ ) k L φε,x (cid:19) dτ which is equivalent to (5.16) and finish the proof of the theorem. (cid:3) The attractor
The aim of this section is to verify the existence of the attractor for the damped Euler equationin 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 :(6.1) S ( t ) : H b → H b , S ( t ) u := u ( t ) , t ≥ , where u ( t ) is a unique solution of (1.1) with the initial data u ∈ H b . Moreover, according toestimate (4.3) it is dissipative in the space H b , i.e., the following estimate holds:(6.2) k S ( t ) u k H b ≤ Q ( k u k H b ) e − βt + Q ( k g k H b )for some positive β and monotone increasing Q which are independent of t and u and, accordingto estimates (5.6) and (5.15), the maps S ( t ) are locally H¨older continuous in the space L b ( R )and as well as in the space L θ R,x ( R ).As usual, in the case of unbounded domains and infinite energy solutions, see [23] for moredetails, we cannot expect the existence of a global attractor in the uniform topology of H b , butonly in the local topology of(6.3) H loc := { u ∈ [ L loc ( R )] , div u = 0 , ω ∈ L ∞ loc ( R ) } . However, we do not know whether or not the above defined semigroup S ( t ) is asymptoticallycompact in the strong topology of H b , so we have to use the weak star topology in H loc (whichwe will further denote by H w ∗ loc ) in order to define the convergence to the global attractor. Werecall that a sequence u n ∈ H loc converges weakly star in H loc to some function u ∈ H loc iff forany ball B Rx the restrictions u n (cid:12)(cid:12) B Rx converge weakly to u (cid:12)(cid:12) B Rx in L ( B Rx ) and the restrictions ω n (cid:12)(cid:12) B Rx converge weakly star to ω (cid:12)(cid:12) B Rx in the space L ∞ ( B Rx ). 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 followingversion of a global attractor. Definition 6.1.
Let S ( t ) : H b → H b be a semigroup. Then, a set A ⊂ H b is a weak locallycompact attractor for this semigroup iff:1) The set A is bounded and closed in H b and is compact in the topology of H w ∗ loc ;2) It is strictly invariant: S ( t ) A = A for all t ≥
3) It attracts bounded (in the topology of H b ) sets in the topology of H w ∗ loc , i.e., for everybounded set B ⊂ H b and every neighbourhood O ( A ) of the attractor A in the topology of H w ∗ loc ,there exists T = T ( B, O ) such that(6.4) S ( t ) B ⊂ O ( A )for all t ≥ T .The main result of the section is the following theorem. Theorem 6.2.
Let the above assumptions hold. Then the solution semigroup S ( t ) : H b → H b associated with the damped Euler equation (1.1) possesses a weak locally compact attractor A in H b which is generated by all bounded solutions of the equation: (6.5) A = K (cid:12)(cid:12) t =0 , where K ⊂ L ∞ ( R , H b ) is the set of all weak solutions u ( t ) of equation (1.1) which are definedfor all t ∈ R and are bounded in H b .Proof. Indeed, according to the dissipative estimate (6.2), the ball(6.6) B R := { u ∈ H b , k u k H b ≤ R } is an absorbing ball for the semigroup S ( t ) if R is large enough. This ball is metrizable andcompact in the weak star topology of H w ∗ loc . Thus, the considered semigroup possesses an ab-sorbing ball B R which is bounded in H b and is compact in H w ∗ loc . Moreover, using the fact thatthe semigroup is H¨older continuous on B R (due to estimate (5.15)) together with the compact-ness of the embedding H loc ⊂ L loc , it is straightforward to show that, for every fixed t ≥ S ( t ) are continuous on B R in the topology of H w ∗ loc . Thus, all assumptions of theabstract 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 thistheorem. So, Theorem 6.2 is proved. (cid:3) Corollary 6.3.
Let the above assumptions hold. Then, for every ε > and every p < ∞ , theweak locally compact attractor A is compact in [ W − ε,ploc ( R )] and attracts the images of boundedsets in H b in the strong topology of W − ε,ploc ( R ) , i.e., for every bounded set B ⊂ H b and every R > and x ∈ R (6.7) lim t →∞ dist W − ε,p ( B Rx ) (cid:16) ( S ( t ) B ) (cid:12)(cid:12) B Rx , A (cid:12)(cid:12) B Rx (cid:17) = 0 , where dist V ( X, Y ) is a non-symmetric Hausdorff distance between sets X and Y of a metricspace 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 − ε,ploc ( R ).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 semigroupassociated with the damped Euler equation (1.1) is compact in W ,ploc ( R ) for any p < ∞ andattracts 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 , loc ( R ) or, which is the same, the asymptotic compactness of the associated vorticity ω in L loc ( R ). To verify it, following [8], we will use the so-called energy method, see also [2, 24].Let u n ∈ B R be a sequence of the initial data and t n → ∞ be a sequence of times. We need toverify that the sequence S ( t n ) u n is precompact in W , loc ( R ).Let u n ( t ), t ≥ − t n , be the solutions of the following damped Euler problems:(6.8) ∂ t u n + ( u n , ∇ x ) u n + ∇ x p n + αu n = g, u n (cid:12)(cid:12) t = − t n = u n . ISSIPATIVE EULER EQUATIONS 17 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 (cid:12)(cid:12) t =0 = curl u n . To verify the desired asymptotic compactness (and to finish the proof of the theorem), we onlyneed to verify that the sequence ω n (0) is precompact in L loc ( R ). We first note that, due to thedissipative estimate (4.3), the sequence u n is uniformly bounded in H b :(6.10) k u n k L ∞ ( R , H b ) + k ∂ t u n k L ∞ ( R ,L qb ( R )) , q < ∞ , where the control over the norm of ∂ t u n is obtained from equation (6.8) analogously to Definition3.2 (to simplify the notations, we extend u n and ∂ t u n by zero for t ≤ t n ). Thus, without loss ofgenerality, 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 qloc ( R )) . Then, due to the compactness arguments,(6.12) u n → u strongly in C loc ( R × R )and, in particular,(6.13) ω n → ω weakly star in L ∞ loc ( R × R ) and ω n → ω strongly in C loc ( R , W − , loc ( R ))Passing to the limit n → ∞ in a straightforward way in equations (6.8), we see that the limitfunction 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 loc ( R )and using that ω n (0) is uniformly bounded in L b ( R ) the last convergence implies that(6.15) ω n (0) → ω (0) weakly in L φ ε,x ( R )for all ε > x ∈ R .At the second step, we will show that the convergence in (6.15) is actually strong which willcomplete the proof of the theorem. To this end, it is enough to prove that(6.16) k ω n (0) k L φε,x → k ω (0) k L φε,x for some ε > x ∈ R . To this end, we will use the enstrophy equality (5.16) for equations(6.9) which we rewrite in the following form:(6.17) k ω n (0) k L φε,x ++ Z − t n e αs Z x ∈ R ( α − φ ε,x ( x ) − u n ( x, s ) . ∇ x φ ε,x ( x )) φ ε,x ( x ) | ω n ( s, x ) | dx ds == k ω n ( − t n ) k L φε,x e − αt n + Z − t n e αs (curl g, ω n ( s ) φ ε,x ) ds. Remind that(6.18) |∇ x φ ε,x ( x ) | ≤ Cεφ ε,x ( x )and therefore we may fix ε > α − φ ε,x ( x ) − u n ( x, s ) . ∇ x φ ε,x ( x ) ≥ for all ( s, x ) ∈ R − × R . Then, using the classical result on the weak lower semicontinuity of con-vex functionals, see e.g., [18], together with the strong convergence (6.12) and weak convergence(6.13), we conclude that Z s ∈ R − Z x ∈ R F ( s, x, u ( s, x ) , ω ( s, x )) ds dx ≤ lim inf n →∞ Z − t n Z x ∈ R 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 ( x ) − u. ∇ x φ ε,x ( x )) φ ε,x ( x ) | ω | . Passing now to the limit n → ∞ in (6.17), we arrive at(6.21) lim sup n →∞ k ω n (0) k L φε,x + Z −∞ Z x ∈ R F ( s, x, u ( s, x ) , ω ( s, x )) dx ds ≤≤ Z −∞ e αs (curl g, ω ( s ) φ ε,x ) ds. On the other hand, according to the enstrophy equality for the limit functions u and ω ,(6.22) k ω (0) k L φε,x + Z −∞ Z x ∈ R F ( s, x, u ( s, x ) , ω ( s, x )) dx ds = Z −∞ e αs (curl g, ω ( s ) φ ε,x ) ds. Thus,(6.23) lim sup n →∞ k ω n (0) k L φε,x ≤ k ω (0) k L φε,x ≤ lim inf n →∞ k ω n (0) k L φε,x . Therefore, the convergence (6.16) is verified and the theorem is proved. (cid:3) Appendix. The interpolation inequality
The aim of this Appendix is to verify the following interpolation inequality.
Lemma 7.1.
Let u ∈ L b ( R ) be the divergent free vector field such that ω := curl u ∈ L ∞ ( R ) .Then, the following inequality holds: (7.1) k u k L ( B Rx ) ≤ C (cid:18) R k u k L ( B Rx ) + k u k / L ( B Rx ) k ω k / L ∞ ( B Rx ) (cid:19) . where R > , x ∈ R are arbitrary and the constant C is independent of R , x , and u . To verify this inequality, we use the following result proved in [36].
Proposition 7.2.
Let the vector field u ∈ [ W , ( B Rx )] be such that div u, curl u ∈ L ∞ ( B Rx ) .Then, (7.2) k u k L ( B Rx ) ≤ C k u k / L ( B Rx ) (cid:16) k curl u k L ∞ ( B Rx ) + k div u k L ∞ ( B Rx ) (cid:17) / , where the constant C is independent of R and x . Moreover, for any < p < ∞ , (7.3) k u k L ∞ ( B Rx ) ≤ C k u k θL ( B Rx ) (cid:16) k curl u k L p ( B Rx ) + k div u k L p ( B Rx ) (cid:17) − θ , where θ = − p − , C may depend on p , but is independent of R and x ∈ R .Proof of the lemma. We first note that (7.1) is homogeneous, so (scaling x → Rx if necessary) itis enough to prove it for R = 1 and x = 0 only. Let now ϕ ∈ C ∞ ( B / ) be the cut-off functionsuch that ϕ ( x ) ≡ x ∈ B . Then applying inequality (7.2) with R = 2 to the vector field ϕu ,we get(7.4) k u k L ( B ) ≤ k ϕu k L ( B / ) ≤ C k u k / L ( B ) k ω k / L ∞ ( B ) + C k u k / L ( B ) k u k / L ∞ ( B / ) . ISSIPATIVE EULER EQUATIONS 19
Applying now inequality (7.3) with p = 4 and θ = 1 / ϕ u , where ϕ ∈ C ∞ ( B / ) is a new cut-off function which equals to one if x ∈ B / , we get(7.5) k u k / L ∞ ( B / ) ≤ C k u k / L ( B ) k ω k / L ( B ) + k u k / L ( B ) k u k / L ( B / ) . In the first term, we replace the L -norm of ω by its L ∞ -norm, insert the obtained result to(7.4) and use the Young inequality, this gives(7.6) k u k L ( B ) ≤ C k u k / L ( B ) k ω k / L ∞ ( B ) + C k u k / L ( B ) k ω k / L ∞ ( B ) ++ C k u k / L ( B ) k u k / L ( B / ) ≤ C k u k L ( B ) + C k u k / L ( B ) k ω k / L ∞ ( B ) ++ C k u k / L ( B ) k u k / L ( B / ) . Thus, we only need to estimate the L -norm in the RHS of (7.6). To this end, we introduceone more cut-off function ϕ ∈ C ∞ ( B ) such that ϕ ( x ) = 1 for x ∈ B / and use the followingLadyzhenskaya type inequality for vector fields v ∈ W , ( B ):(7.7) k v k L ( B ) ≤ C k v k L ( B ) (cid:16) k div v k L ( B ) + k curl v k L ( B ) (cid:17) . Applying this inequality to the vector field ϕ u and estimating again the L -norm of the vorticityby its L ∞ -norm, we have(7.8) k u k / L ( B / ) ≤ C k u k / L ( B ) k ω k / L ∞ ( B ) + C k u k / L ( B ) . Inserting this estimate to the RHS of (7.6) and using Young inequality again, we derive thedesired estimate (7.1). (cid:3)
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