Universität Stuttgart Fachereich Mathematik Spatially nondecaying solutions of 2D Navier-Stokes equation in a strip Sergey Zelik Preprint 25/6
Universität Stuttgart Fachereich Mathematik Spatially nondecaying solutions of 2D Navier-Stokes equation in a strip Sergey Zelik Preprint 25/6
Fachereich Mathematik Fakultät Mathematik und Physik Universität Stuttgart Pfaffenwaldring 57 D-7 569 Stuttgart E-Mail: WWW: preprints@mathematik.uni-stuttgart.de http://www.mathematik/uni-stuttgart.de/preprints ISSN 1613-839 c Alle Rechte vorehalten. Nachdruck nur mit Genehmigung des Autors. L A T E X-Style: Winfried Geis, Thomas Merkle
5 1. Introduction It is well-known that the Navier-Stokes system { t u + (u, x )u = ν x u x p + g, (1.1) div u =, u =, u t= = u in a ounded 2D domain Ω R 2 is well-posed and generates a dissipative semigroup S(t) in the appropriate phase space (of square integrale divergent-free vector fields), see [6], [23], [24] and references therein. We also recall that these results are strongly ased on the so-called energy estimate. In order to otain this energy estimate one multiplies equation (1.1) y u, integrate over Ω and uses the fact that the nonlinear term disappears: (1.2) ((u, x ), u) := (u(x), x )u(x).u(x) dx, x Ω for every divergent-free vector field with Dirichlet oundary conditions. In contrast to that, the situation is essentially less understood when the domain Ω is unounded. Moreover, although there exists a highly developed theory of dissipative PDEs in unounded domains (mainly ased on the so-called weighted energy estimates, see [7]-[1], [18]-[19], [27]- [3] and references therein), up to the moment, it was very difficult to extend it to the concrete Navier-Stokes prolem in unounded domains, due to several principal ostacles. Indeed, in contrast to ounded domains, in the unounded ones the space of square integrale (divergent-free) vector fields is not a convenient phase space, since the assumption u L 2 (Ω) imposes too restrictive decay conditions on u(x) as x. So, under this choice of the phase space, many classical hydrodynamical ojects, like Poiseuille flows, Couette-Taylor flows, Kolmogorov flows etc. are automatically out of the consideration. Thus, following the general theory mentioned aove, it is reasonale to replace the assumption u L 2 (Ω) y more relevant one: u L 2 (Ω) where the uniformly local Soolev spaces W l,p (Ω) are defined via the following standard expression: W l,p (Ω) := {u D (Ω), u W l,p (Ω) := sup u W l,p (Ω Bx 1 x ) < }. Ω Here Bx 1 denotes the all of radius one of R 2 centered at x R 2 and W l,p means the classical Soolev space, see Section 1 for details. But here arises the main difficulty: how to otain a priori estimates for the solution u(t) in the uniformly local spaces? Indeed, since u(t) is not square integrale any more, we cannot multiply (1.1) simply y u and use identity (1.2) (the integrals do not have sense). So, following the general strategy, we need to multiply it y u where = (x) is an appropriate weight function. But in that case the nonlinear term does not vanish and produce the additional cuic term like u 3. We note that this cuic term is not sign-defined and the rest terms in the energy equality are at most quadratic with respect to u, so it was not clear how to control this cuic term in order to produce reasonale a priori estimate. Another ostacle is related with the fact that u is not divergent free, so the pressure p does not disappear in the weighted energy equality and one should e also ale to control the term ( p, u). Of course, this prolem is closely related with finding the reasonale extension of the Helmholtz projector (to divergent free vector fields) to uniformly local spaces. The aove mentioned difficulties stimulated the developing of the alternative methods to handle the Navier-Stokes equations in unounded domains. In particular, rather helpful is the so-called vorticity equation (1.3) t ω x ω + (u, x )ω = x2 g 1 x1 g 2 where ω := x2 u 1 x1 u 2. Indeed, if Ω does not contain oundary, e.g. Ω = R 2 or Ω = S 1 R where S 1 is a circle (like in the Kolmogorov prolem), the maximum principle applied to (1.3) allows to otain gloal a priori estimate for the vorticity ω which, together with the accurate analysis of the explicit formulae for the Helmholtz projectors, allow to otain the gloal in time a priori estimates for the solution u(t) and, thus, to prove the gloal solvaility of the Navier-Stokes equation in the uniformly local phase spaces, see [2] and [12]. Unfortunately, a priori estimate
6 for vorticity otained from the maximum principle grows linearly in time, so all of the further estimates will also growing in time (to the est of our knowledge, for the case Ω = R 2, it gives doule exponential ( e CeCt ) growth rate and polynomial ( t 3 ) growth rate for Ω = S 1 R). The other essential drawack is that this method seems to e non-applicale to the prolems with oundary, e.g. in the case where Ω is a cylindrical domain. Another attractive possiility to avoid direct weighted energy estimates is to use the ifurcation analysis. Indeed, in the situation where the asic steady state of the Navier-Stokes prolem is slightly aove the instaility threshold, the solutions remaining close to that steady state can e descried in terms of the so-called modulation equations which are essentially simpler than the initial Navier-Stokes prolem (usually it is Ginzurg-Landau or Swift-Hohenerg equations), see [1], [13]-[15], [17] and references therein. Since the well-posedness and dissipativity of these modulation equations is well-understood, the standard perturation methods allow sometimes to otain gloal in time estimates for solutions of the initial Navier-Stokes prolem starting from the small neighorhood of the asic steady state. In particular, the gloal existence and dissipativity of such solutions for the 3D Couette-Taylor flow is otained in [21] and almost gloal solvaility (on the exponentially long with respect to perturation parameter time interval) for the case of Poiseuille flow can e found in [22]. It is worth to emphasize that, in the case where the domain Ω R 2 possesses the Friedrichs inequality (1.4) u 2 L 2 (Ω) λ 1 x u 2 L 2 (Ω), u W 1,2 (Ω) with positive λ 1 and under the restrictive assumption that u is square integrale, all of the aove mentioned ostacles disappear and Navier-Stokes prolem (1.1) possesses a standard (unweighted) energy theory similar to the case of ounded domains, see [5], [24]. We also mention the survey paper [3] on existence of spatially decaying solutions of the Navier-Stokes prolem in various domains (not necessarily satisfying (.4)), see also [11] and [26]. The main aim of the present paper is to develop weighted energy theory for the 2D Navier- Stokes prolems in a strip Ω := R ( 1, 1), (x 1, x 2 ) Ω overcoming the ostacles mentioned aove. For simplicity, we will mainly consider the model Navier-Stokes prolem { t u + (u, x )u = x u x p + g, (1.5) div u =, u =, u t= = u with ν = 1 (the case of aritrary ν can e reduced to ν = 1 y the appropriate scaling, see the end of Section 8). Moreover, in order to make prolem (1.5) well posed, we need to add the average flux condition: (1.6) (Su 1 )(t, x 1 ) := 1/2 1 1 u 1 (t, x 1, x 2 ) dx 2 c, where c R is a given constant (assumption (1.3) can e considered as a kind of oundary conditions at x 1 = ± ). The main result of the paper is a comprehensive study of the Navier-Stokes prolem (1.5), (1.6) in the uniformly local spaces (i.e. requiring the solution u(t) e only ounded as x 1 ±, no decaying conditions are imposed). In particular, we prove the existence of a solution, its uniqueness and regularity, dissipativity and existence of a locally compact gloal attractors for the Navier- Stokes prolem (1.5), (1.6). We emphasize that, in contrast to the previous results on this topic, our phase space contains all of the Poiseuille flows and all known structures ifurcating from them. Moreover, our result allows to emed the 2D Navier-Stokes prolem in a strip into a general scheme of investigating dissipative PDEs in unounded domains mentioned aove, including the study of the dimension and Kolmogorov s entropy of attractors, topological entropies, spatial and temporal chaos, etc. We return to these questions in the forthcoming paper [31]. The paper is organized as follows. We recall in Sections 2 and 3 some asic facts on the theory of weighted spaces and the regularity of elliptic oundary value prolems in these spaces which will e systematically used throughout of the paper.
7 Section 4 is devoted to study the Helmholtz projector Π and the Stokes operator A := Π x in weighted and uniformly local Soolev spaces. The results of this section are similar to [4] and [5] (and are, factually, inspired y these papers). In Section 5, we study the auxiliary linear non-divergent free prolem (1.7) t v = x v + x q, Πv t=t =, div v = u, v = where (x) is the appropriate weight function and u(t) is a solution of the Navier-Stokes prolem. This auxiliary prolem is necessary in order to overcome the ostacle related with the appearance of the term containing pressure in the weighted energy equality. Roughly speaking, we will multiply equation (1.5) y the function u(t) v(t) where v solves (1.7). Then, since div(u v) = the pressure term disappears (and the derivative of our weights will e small, so the corrector v will e also small and do not produce any essential difficulties in its estimating, see Sections 5 and 6 for the details). We note that it is not clear how to overcome this ostacle in more simple way. Indeed, the most natural multiplication y Π(u) does not work since Π(u) has nonzero trace at the oundary which leads to additional uncontrollale oundary terms under the integration y parts in ( x u, Π(u)). Another possiility is to construct a new projector Q to divergent free vector fields which preserves the oundary conditions and multiply the equation y Q(u). This, however, leads to essential difficulties with the term ( t u, Q(u)) which should e a complete time derivative from something. We also note that the multiplication of the equation y the comination of t u and Π x u (as in [4] and [5]) is useless for us, since it works only if the unweighted L 2 -norm of x u is a priori known. In Section 6 we overcome the main ostacle to the weighted energy theory for Navier-Stokes equations the cuic term u 3 mentioned aove. In order to do so, we use the special weights (1.8) θ ε,x (x) := (1 + ε 2 x x 2 ) 1/2 with very small ε which factually depends on the solution u considered. Then, the careful analysis of the otained weighted energy inequality allows us to otain the gloally in time ounded a priori estimate of the L 2 -norm of u(t). Based on this a priori estimate, we then estalish the existence of such solution. In a fact, we first consider the case of zero flux c = (see Theorem 6.5) and, after that reduce the general case to that particular one using the trick with the auxiliary energy stale equilirium (see Theorem 6.6). The uniqueness of such solution is verified in Section 7 (see Theorem 7.1). Moreover, we also verify here the L 2 1,2 -W smoothing property for that solutions which is necessary for gloal attractors (see Theorem 7.4). Finally, in Section 8, we prove the dissipative estimate (=existence of an asoring all) for solutions of Navier-Stokes prolem in the uniformly local phase space (Theorem 8.1) and estalish the existence of a gloal attractor A. Moreover, using the scaling arguments, we otain the following estimate for the size of attractor in L 2 -norm in terms of the kinematic viscosity ν: (1.9) A L 2 (Ω) Cν 3 (c 3 ν + g 2 L 2 (Ω) + ν4 ) where the constant C is independent of ν, c and g. We recall that in ounded domains (in square integrale case), the est known estimate is the following one: (1.1) A L 2 (Ω) Cν 1 g L 2 (Ω). We see that, although estimate (1.9) is worse than (1.1), ut it remains polynomial as ν (with a reasonale degree 3). Thus, our method is not extremely rough and can e used in order to otain reasonale quantitative ounds for the solutions. To conclude, we mention that our method seems to e applicale to more general 2D domains satisfying (1.4) and even to 3D cylindrical domains (of course, up to the uniqueness prolem). We return to these topics somewhere else. Acknowledgement. This work is partially supported y Alexander von Humoldt foundation and y the CRDF grant RUM1-2654-MO-5. The author is also grateful to A.Afendikov and A.Mielke for stimulating discussions.
8 2. Functional spaces In this section, we riefly recall the definitions and asic properties of weight functions and weighted functional spaces which will e systematically used throughout of the paper (see also [9], [28] for more details). We start with the class of admissile weight functions. Definition 2.1. A function C loc (R n ) is a weight function of exponential growth rate µ > if the following inequalities hold: (2.1) (x + y) C (x)e µ y, (x) >, for all x, y R n. The following proposition collects the evident properties of that weights. Proposition 2.2. 1. Let e a weight function with exponential growth rate µ. Then, for every ε > µ, is a weight function of exponential growth rate ε (with the same constant C ). 2. Let and ψ e weight functions of exponential growth rate µ. Then the functions Ψ 1 = (x)ψ(x) and Ψ 2 = (x)/ψ(x) are weight functions of exponential growth rate 2µ with the constant C Ψi C C ψ. 3. Let e a weight function of exponential growth rate µ and let ψ C loc (R n ) satisfies (2.2) C 1 (x) ψ(x) C 2 (x), x R n. Then ψ is also a weight function of exponential growth rate µ and C ψ C 1 1 C 2C. 4. Let ε > and (x) e a weight function of exponential growth rate µ. Then the function ε (x) := (εx) is of exponential growth rate εµ and with C ε = C. All of the assertions of the proposition are simple corollaries of estimate (2.1). The natural example of such weights is the following one: (2.3) µ,x (x) := e µ x x, x R n, µ R. Oviously, they are of exponential growth rate µ and the constant C µ,x = 1 (independent of x R n ). However, these weights are nonsmooth at x = x. In order to overcome this drawack, it is natural to use the following equivalent weights: (2.4) ϕ µ,x (x) := e µ 1+ x x 2, x R n. Indeed, since x x 2 + 1 x + 1, then these weights satisfy (2.5) e µ µ,x (x) ϕ µ,x (x) e µ µ,x (x), x R n and, consequently, ϕ µ,x are also weight functions of exponential growth rate µ (with C ϕµ,x = e 2 µ ). Moreover, in contrast to (2.3) these weights are smooth and satisfy, for µ 1 the additional ovious inequality (2.6) D k x ϕ µ,x (x) C k µ ϕ µ,x (x), x R n where k N, D k x denotes a collection of all x-derivatives of order k and the constant C k is independent of x and µ. This inequality is crucial for otaining the regularity estimates in weighted spaces (see [9] [1], [27] [3] and Section 3 elow). Another important class of weight functions is the so-called polynomial ones: (2.7) θ m x (x) := (1 + x x 2 ) m/2, m R. It is not difficult to verify that these weights are of exponential growth rate µ for every µ > with the constant C θm,x depending on µ and m, ut independent of x Ω. We now introduce a class of weighted Soolev spaces in a regular unounded domain Ω associated with weights introduced aove. Since we factually need elow only the case where Ω := R ( 1, 1) is a strip which oviously have regular oundary, in order to avoid the technicalities, we do not formulate precise assumptions on the oundary (which can e found e.g. in [9] or [1]).
9 Definition 2.3. Let Ω e a regular domain and let e a weight function of exponential growth rate. Then, for every 1 p, we set (2.8) L p (Ω) := {u Lp loc (Ω), u p := (x) p u(x) p dx < } L p and (2.9) L p, (Ω) := {u Lp loc (Ω), u L p, Ω := sup ((x ) u L p (Ω Bx 1 x )) < }. Ω Here and elow Bx r denotes an r-all of R n centered at x and we write L p instead of Lp,1. Moreover, for every l N, we define the weighted Soolev spaces W l,p l,p (Ω) and W, (Ω) as spaces of distriutions whose derivatives up to order l elong to L p (Ω) and Lp, (Ω) respectively. Furthermore, the weighted Soolev spaces W l,p l,p () and W, () on the oundary can e defined analogously only the integral over Ω (resp. supremum in (2.9)) in (2.8) should e naturally replaced y the integral (resp. supremum) over the oundary, see [9], [1]. Remark 2.4. In the sequel, we will also use the functions u(t) with values in the weighted Soolev spaces defined aove. In slight ause the notations, we denote y L p l,p (R, W ) the space, generated y the following norm: (2.1) u L p l,p (R,W ) := sup sup u L p ([T,T +1],W l,p (Ω Bx 1 x Ω T R )). The following proposition collects some useful facts on the spaces introduced efore. Proposition 2.5. Let Ω e a regular domain and e a weight of exponential growth rate µ. Then, 1) For every r > and every u L p (Ω), 1 p <, ( 1/p (2.11) Cr 1 u L p (Ω) p (x ) u p L p (Ω Bx r x Ω ) ) dx C r u L p (Ω) where the constant C r depends on r, µ and on the constant C from (2.1), ut is independent of and of the concrete choice of the weight. 2) For every α > µ, every q [1, ] and every u L 1 (Ω), we have ( ( q ) 1/q (2.12) (x ) q e α x x u(x) dx) dx C α u L 1 x Ω x Ω (Ω) where the constant C α depends on α, µ and on the constant C, ut is independent of u and of the concrete choice of and q. 3) For every α > µ and every u L p, (Ω), we have (2.13) Cα 1 u p sup {(x L p, (Ω) ) p e αp x x u(x) p dx} C α u L p, (Ω) x Ω x Ω where the constant C α depends on α, µ and on the constant C, ut is independent of u and of the concrete choice of. The proof of that estimates is given in [9] (see also [1], [26]). Remark 2.6. As we will see elow, estimate (2.11) allows to reduce the proofs of emedding and interpolation theorems for weighted Soolev spaces to the classical unweighted case in a ounded domain. Estimates (2.12) and (2.13) allow, in turns, to otain the elliptic regularity in weighted spaces with aritrary weights of exponential growth rate if the analogous result for the special weights e α x x (or which is the same, for the equivalent smooth weights (2.4)) is known, see Section 3. Moreover, these estimates allow to control the dependence of the constants in emedding, interpolation and regularity theorems on the concrete choice of the weights which is crucial for our study of the nondecaying solutions of NS equations.
1 We need now to introduce also the weighted Soolev spaces with fractional derivatives. To this end, we first recall that in the unweighted case the space W l+s,p (Ω) for s (, 1) and l Z + is usually defined via (2.14) u p W l+s,p (Ω) := u p W l,p (Ω) + x Ω y Ω D l xu(x) D l xu(y) p x y n+sp dx dy and, for negative l, the space W l,p (Ω) is defined as a conjugate space to W l,q (Ω) where 1/p+1/q = 1, see [16], [25]. Then, estimate (2.11) justifies the following definition. Definition 2.7. Let Ω e a regular domain and e a weight function of exponential growth rate. For every 1 < p and every l R, we define the space W l,p (Ω) as a suspace of distriutions for which the following norm is finite: (2.15) u p := (x W l,p ) p u p (Ω) W l,p (Ω Bx r ) dx x Ω where r is some positive numer (it is not difficult to verify that, this space is independent of r). Analogously the norm in W l,p, is defined via (2.16) u p := sup W l,p, (Ω) x Ω {(x ) p u p W l,p (Ω B r x ) }, for simplicity, we fix elow r = 1 in definitions (2.15) and (2.16) of the weighted norms. Indeed, according to (2.11), we see that, for l Z + the spaces thus defined coincide with the spaces from Definition 2.1. Moreover, it is not difficult to verify, using the explicit formula (2.14) that in the unweighted case = 1, the norm (2.15) is equivalent to (2.14). The following proposition descries the weighted negative Soolev spaces in terms of conjugate spaces. Proposition 2.8. Let Ω e a regular domain and let e a weight function of exponential growth rate µ. Then, for every l >, and every 1 < p, q < with 1/p + 1/q = 1, (2.17) W l,p (Ω) = [W l,q, (Ω)] 1 where W l,q, (Ω) denotes the closure of C l,q (Ω) in the W -norm and means the conjugate space (with respect to the standard inner product in L 2 (Ω)). Moreover, (2.18) C 1 u W l,p (Ω) u [W C l,q 2 u, 1 (Ω)] W l,p where the constants C 1 and C 2 depend on µ, l, p and C, ut are independent of the concrete choice of u and C. Proof. In order to avoid the technicalities, we give elow the proof of (2.18) only for the case of a cylindrical domain Ω := R ω where ω is a smooth ounded domain of R n 1 (only that case will e used in the sequel) although the slightly modified proof works for a general regular domain. In that particular case, we can restrict ourselves to consider only one dimensional weights C loc (R). Indeed, since ω is ounded, (2.1) implies that (2.19) C 1 (s, ξ ) (s, ξ) C 2 (s, ξ ), s R, ξ ω where ξ ω is some fixed point and, consequently, the weights (s, ξ) is equivalent to ξ (s) := (s, ξ ). Moreover, it is more convenient to use, instead of alls Bx r the finite cylinders Ω s := (s, s + 1) ω, i.e. to define the norm in W l,p (Ω) via (2.2) u p = (s) p u p W l,p (Ω) W l,p (Ω ds s) s R (since the norms (2.15) are equivalent for different r and ω is ounded then (2.15) and (2.2) are also equivalent). (Ω)
11 We first verify the right inequality of (2.18). To this end, we introduce a partition of unity {ψ y } y R C (R) such that 1. supp ψ y (y, y + 1), (2.21) 2. y R ψ y(s) dy 1, 3. Ds k ψ y (s) C k, where the constant C k is independent of s R (oviously such partition of unity exists and can e chosen in a smooth way with respect to y R). Let now u [W l,q, (Ω)] e a functional over W l,q 1, (Ω) and let v e an aritrary test function 1 from that space. Then, using (2.21) and Hölder inequality, we have (2.22) u, v u, ψ y v dy u W l,p (Ω y) ψ y v W l,q (Ω y) dy y R y R C (y) v W l,p (Ω y) (y) 1 v W l,q (Ω y) dy C u W l,p (Ω) v W l,q 1 (Ω) y R which, together with the definition of the norm in a conjugate space gives the right-hand side of inequality (2.18). Let us now verify the left-hand side of that inequality. Indeed, let u W l,p (Ω). We fix a family of functions v y W l,q (Ω y), such that (2.23) u, v y = u W l,p (Ω y) v y W l,q (Ω y) and normalize these functions as follows: (2.24) v y W l,q (Ω y) = (y) p u p 1 W l,p (Ω. y) Since the spaces W l,q (Ω y ) are uniformly convex, these family are uniquely defined and, moreover, continuous with respect to y R. Let us define also the function v(x) as follows (2.25) v(x) := v y (x) dy. y R We claim that v W l,q, (Ω). Indeed, since v 1 y W l,q (Ω y), it can e naturally continued y zero to the function v y W l,q (Ω) with supp v y Ω y. Thus, the integral (2.25) is well posed and defines a function v W l,q loc (Ω) vanishing at the oundary. So, we only need to estimate the W l,q 1 (Ω)-norm of it. Using now that v y W l,q (Ω s) = if s y 1, we have (2.26) v W l,q (Ω s) s y 1 C(s) p v y W l,q (Ω y) dy = s y 1 s y 1 u p 1 W l,p (Ω y) dy C 1(s) p (y) p u p 1 W l,p (Ω y) dy y R e α s y u p 1 W l,p (Ω y) dy where the constant α > 2pµ/q can e aritrary (here we have implicitly used (2.1) in order to estimate (y) via (s)). Taking the q-th power from the oth sides of that relation, applying the Hölder inequality and using that q(p 1) = p, we arrive at (s) q v q W l,q (Ω C(s)p s) e αq s y /2 u p W l,p (Ω dy. y) Integrating this relation over s R and using (2.12), we finally infer y R (2.27) v q W l,q 1 (Ω) C 2 u p W l,p (Ω).
12 We are now ready to finish the proof of the proposition. Indeed, due to (2.23) (2.25), we have u, v = u W l,p (Ω y) v y W l,q (Ω y) dy = u p W l,p (Ω) y Ω and, consequently, due to (2.27), (2.28) u [W l,q, 1 (Ω)] u, v C u p(1 1/q) v W l,q 1 (Ω) W l,p (Ω). Since p(1 1/q) = 1, then (2.28) implies the left-hand side of inequality (2.18). Proposition 2.8 is proven. Remark 2.9. Proposition 2.8 shows, in particular, that in the case = 1, the spaces W l,p (Ω) introduced in Definition 2.7, coincide with the standard Soolev spaces for any l R. Moreover, arguing analogously to the proof of Proposition 2.8, one can verify the interpolation representation of the weighted spaces W l+α,p (Ω) with fractional derivatives (l Z, α (, 1)) (2.29) W l+α,p (Ω) = ( W l,p ) l+1,p (Ω), W (Ω) in a complete analogy with the unweighted case, see e.g. [25]. We now recall also the emedding and trace theorems for the weighted functional spaces. Proposition 2.1. Let Ω e a regular domain and e a weight function of exponential growth rate µ. Then 1) For every 1 < p 1 p 2 < and every l 2 l 1 satisfying (2.3) 1 p 2 l 2 n 1 p 1 l 1 n, there is a continuous emedding W l1,p1 (Ω) W l2,p2 (Ω) and the norm of the emedding operator depends on l i, p i, µ and C, ut is independent of the concrete form of the weight function. If the inequality (2.3) is strict, then we can take also p 2 =. 2) For every m Z +, 1 < p < and l > m + 1/p the trace operator Π m (2.31) Π m Ω u := (u, n u,, m n u ) (where n u denotes the normal derivative of the function u at the oundary ) maps W l,p (Ω) to m k= W l k 1/p,p () and there exists the associated extension operator [Π m ] 1 (right inverse to Π m ) and the norms of that operators depend on l, m, p, µ and C, ut are independent of the concrete choice of the weight. Furthermore, the aove results hold also for the family of spaces W l,p, (Ω). Proof. As in the proof of Proposition 2.8, we restrict ourselves to consider only the case of a cylindrical domain Ω := R ω, one dimensional weights and the equivalent norms (2.2). Moreover, we will consider elow only the case of spaces W l,p l,p (the spaces W, can e considered analogously). Indeed, let u W l1,p1 (Ω). Then, according to the classical Soolev emedding theorem (see [25]), we have (2.32) u W l 2,p 2 (Ωs) C u W l 1,p 1 (Ωs) where the constant C is independent of s. Taking the power p 2 from the oth sides of that inequality, we transform it to the following form (for simplicity, we consider only the case p 2 < ) ( ) p2/p1 u p2 W l 2,p 2 (Ω s) Cp2 u p2 W l 1,p 1 (Ω C s) 1 e αp1 s y u p1 W l 1,p 1 (Ω dy y) s R where α > µ is aritrary and the constant C 1 is independent of u. Multiplying this relation y (s) p2 integrating y s R and using inequality (2.12), we infer u p2 W l 2,p 2 (Ω) C 2 u p2 W l 1,p 1 (Ω) α,p
13 which proves the first part of the proposition. Let us verify the second assertion of the proposition. Indeed, the existence and oundedness of the trace operator Π m can e verified ased on the analogous property for domains Ω s exactly as efore (so we rest it to the reader). Thus, we only need to construct the extension operator [Π m ] 1. Indeed, let U := {u k } m k= m k= W l k 1/p,p () e aritrary. Using now the partition of unity (2.21), we construct the family U s := ψ s U = {ψ s u k } m k=. Then, since all of that functions vanish at the origins of the cylinder Ω s, there exists an extension operator [Π m s ] 1 for ounded domain Ω s which maps U s to W l,p (Ω s ) and its norm is independent of U and s, see [25]. The required extension operator [Π m ] 1 can e now constructed as follows: (2.33) [Π m ] 1 U := [Π m s ] 1 U s ds. s R Indeed, the fact that this operator is well defined and the required uniform (with respect to ) estimate for its norm as the map from m k= W l k 1/p,p () to W l,p (Ω) can e verified exactly as estimate (2.27) for the function (2.25) from the proof of Proposition 2.8. Proposition 2.1 is proven. Our next task is formulate some trace theorems for classes of less smooth functions which are closely related with the theory of NS equations. To this end, we need the following definition. Definition 2.11. Let Ω e a regular domain of R n, e a weight function of exponential growth rate µ and 1 < p <. Let us define the space E p (Ω) of vector-valued functions u := (u1,, u n ) [D(Ω)] n y the following norm: (2.34) u p E p (Ω) := u p [L p (Ω)]n + div u p L p (Ω). The spaces E p, (Ω) are defined analogously. Moreover, for every sufficiently smooth vector-valued function u := (u 1,, u n ), we denote y l n u := ( u, n) the normal component of that function at the oundary. Proposition 2.12. Let Ω e a regular domain and e a weight function of exponential growth rate µ. Then the operator l n : E p 1/p,p (Ω) W () is well-defined and (2.35) l n u 1/p,p W () C u E p (Ω) where the constant C depends on µ and C, ut is independent of the concrete choice of the weight function. Moreover, the analogous result holds also for the spaces E p, (Ω) Proof. As efore, we verify estimate (2.35) only for the cylindrical domains. Indeed, let u and v s e smooth functions in Ω s. Then, due to Green s formula (2.36) (l n u, v) s := (div u, v) Ωs (u, x v) Ωs. As usual, we see that the right-hand side of (2.36) is well-defined for all u E p (Ω s ) and v W 1,q (Ω s ) where 1/p + 1/q = 1. Moreover, due to the classical trace theorems, there exists an extension operator [Π s ] 1 : W 1 1/q,q ( s ) W 1,q (Ω s ) whose norm is, oviously independent of s. Thus, (2.36) shows that the functional l n u is well-defined and satisfies (2.37) l n u W 1/p,p ( s) = l n u [W 1 1/q,q ( s)] C u E p (Ω s). Multiplying this relation y (s) p and integrating over s R, we deduce (2.35) and finish the proof of the proposition. Here we have implicitly used that l n u W 1/p,p ((s,s+1) ω) l n u W 1/p,p ( s). The estimate for E p, (Ω) can e otained analogously using the supremum instead of integral over s R.
14 As we have already mentioned, estimates of Proposition 2.5 allow to reduce the proofs of elliptic regularity in aritrary weighted spaces to the particular case of special weights (2.4). The following evident proposition will e useful in order to reduce the case of that special weights to the classical unweighted case = 1. Proposition 2.13. Let Ω e a regular domain and let T µ,x e a multiplication operator y the weight ϕ µ,x (x) (i.e. (T µ,x u)(x) := ϕ µ,x (x)u(x)). Then, for every l R and 1 p, this operator realizes an isomorphism etween the spaces Wϕ l,p µ,x (Ω) and W l,p (Ω). Moreover, (2.38) C 1 u W l,p ϕµ,x (Ω) T µ,x u W l,p (Ω) C u W l,p ϕµ,x (Ω) where the constant C depends on l, p and µ, ut is independent of u and x R n. Indeed, this estimate is an immediate corollary of inequalities (2.6) and Definition 2.7 of the corresponding weighted spaces. We now formulate the weighted analogue of one standard interpolation inequality which is crucial for the theory of 2D Navier-Stokes equation. Proposition 2.14. Let Ω := R (, 1) e a strip ((x 1, x 2 ) Ω) and let i, i = 1, 2 e weight functions of the exponential growth rate µ. Then, the following interpolation inequality holds: (2.39) u L 4 ( 1 2 ) 1/2 (Ω) C u 1/2 L 2 (Ω) u 1/2 1 W 1,2 (Ω) 2 where the constant C depends on C i and µ, ut is independent of the concrete choice of weights i. Moreover, the analogous estimate holds for the spaces W l,2, (Ω) as well. Proof. Indeed, due to the interpolation inequality, we have (2.4) u 4 L 4 (Ω s) u 2 L 2 (Ω s) u 2 W 1,2 (Ω s) where the constant C is independent of s, see e.g. [16]. We transform this inequality as follows: (2.41) u 4 L 4 (Ω C ( 2 s) u L 2 (Ω s) u W 1,2 (Ω s)) ( 2 C 1 e α s y u L 2 (Ω y Ω y+1) u W 1,2 (Ω y Ω y+1) dy). s R Multiplying this relation y 1 (s) 2 2 (s) 2 and using estimate (2.12) and Hölder inequality, we infer (2.42) u 4 L 4 (Ω) ( 1 2 ) 1/2 ( 2 C 2 1 (s) 2 (s) u L2 (Ω s Ω s+1) u W 1,2 (Ω s Ω s+1) ds) s R C 2 1 (s) 2 u 2 L 2 (Ω ds s Ω s+1) 2 (s) 2 u 2 W 1,2 (Ω ds s Ω s+1) s R s R C 3 1 (s) 2 u 2 L 2 (Ω ds s) 2 (s) 2 u 2 W 1,2 (Ω ds s) s R which implies (2.4). The case of spaces W l,2, can e considered analogously. Proposition 2.14 is proven. Remark 2.15. Rem1.5 The proof of Proposition 2.14 shows a general way of proving the weighted analogue of various interpolation inequalities. The most important for us here is the fact that the constants in that inequalities will depend only on the exponential growth rate µ and on the constants C and will e independent of the concrete choice of the weights. We conclude y formulating some useful result on the weighted and local topologies on ounded sets of W l,p (Ω). s R
15 Proposition 2.16. Let Ω e a ounded domain l R and p [1, ] let also B e a ounded suset of W l,p (Ω). Then, for every weight function of exponential growth rate µ satisfying (2.43) L p (R n ) <, the set B elongs to W l,p (Ω) and the topology, generated on B y this emedding is independent of the weight and coincides with the local topology on B generated y emedding to W l,p loc (Ω). Proof. Indeed, due to (2.43), we have u p = p (x W l,p ) u p (Ω) W l,p (Ω Bx 1 ) dx p L p (R n ) u p W l,p (Ω) x Ω which shows that W l,p (Ω) W l,p (Ω). Let us now the sequence u n u in W l,p loc (Ω). This means that, for every x Ω and every R R +, (2.44) lim n u n u W l,p (Ω B R x ) =. Let also u n, u B and e an integrale (in the sense of (2.43)) weight. Then, since the set B is assumed to e ounded in W l,p (Ω), (2.45) lim R u n W l,p (Ω\BR ) = uniformly with respect to n N. Assertions (2.44) and (2.45) imply in a standard way that u n u in W l,p l,p l,p (Ω). Since the emedding W (Ω) Wloc (Ω) is ovious, then Proposition 1.8 is proven. 3. Elliptic regularity in weighted spaces In this Section, we recall some standard elliptic regularity results in weighted Soolev spaces which are necessary to deals with the Navier-Stokes equations in unounded domains. For simplicity, we restrict ourselves to consider only the case of a strip Ω := R ( 1, 1) (x := (x 1, x 2 ) Ω) although some of the results of this section remain true for general regular domains, see [9]-[1], [27]-[3] for details. We start with the weighted regularity estimate for the Laplacian with Dirichlet oundary conditions. Proposition 3.1. Let us consider the following Dirichlet prolem in a strip Ω: (3.1) x u = h, u =. Then, for every 1 < p < and l = 1,, 1, there exists positive µ = µ (p) such that, for every weight function with sufficiently small exponential growth rate µ (µ µ ) and every h W l,p l+2,p (Ω), equation (2.1) possesses a unique solution u W (Ω) and the following estimate holds: (3.2) u W l+2,p (Ω) C h W l,p where the constant C depends on C, ut is independent of the concrete choice of the weight. Moreover, the analogous estimate holds also for the spaces W l,p, (Ω). Proof. We restrict ourselves to verify a priori estimate (3.2) only (the existence and uniqueness of a solution can e then verified in a standard way, see e.g. [9], [1]). As we have already mentioned, due to estimates (2.12) and (2.13), it is sufficient to verify estimate (3.2) only for the special class of weights ϕ µ,x (x) introduced in (2.4). Indeed, if we have estimate (3.2) for such weights with the constant C independent of x, then we oviously have the following estimate: (3.3) u p W l+2,p (Ω s) C µ u p W l+2,p ϕµ,s (Ω) C 1 h p W l,p ϕµ,s (Ω) (Ω) C 2 y R e pµ s y h p W l,p (Ω y) dy
16 where the constant C 2 is also independent of s R. Multiplying now estimate (3.3) y (s) p (where is a weight function with exponential growth rate µ < µ ), integrating over s R and using estimate (2.12), we infer the required estimate (2.2). Analogously, estimate (3.2) for the spaces W l,p, can e otained y multiplication (3.3) y (s)p, taking the supremum over s R and using estimate (2.13). Thus, it only remains to verify (3.2) for the special weights ϕ µ,s with a sufficiently small positive µ and every s R. In turns, due to Proposition 2.13 and estimates (2.6), the case of special weights ϕ µ,s can e easily reduced to the unweighted case 1. Indeed, the function u Wϕ l+2,p µ (Ω) solves (3.3) if and only if the function v := ϕ µ,su W l+2,p (Ω) solves the following pertured version of prolem (3.2): (3.4) x v = ϕ µ,sh + [ϕ µ,sϕ µ,s 2(ϕ µ,s )2 ϕ 2 µ,s ]v+ We recall that, due to (2.6), (3.5) h µ (v) W l,p (Ω) Cµ v W l+2,p (Ω) + 2 µ,s µ,s x1 v := T µ,sh + h µ (v), v =. where the constant C is independent of s and µ. Thus, if estimate (3.2) for 1 is known, then applying it to equation (3.4) and using (3.5), we infer v W l+2,p (Ω) C( T µ,sh W l,p (Ω) + µ v W l+2,p (Ω)) with the constant C independent of µ and s. Fixing now µ to e small enough that Cµ < 1/2, we deduce from the last estimate that (3.6) v W l+2,p (Ω) 2C T µ,sh W l,p (Ω) which together with Proposition 2.13 imply estimate (3.2) for special weights ϕ µ,s. Thus, we have reduced the verifying of the regularity estimate (3.2) in weighted spaces to the unweighted case 1. It only remains to note that (3.2) with 1 is a classical L p -regularity estimate for the solutions of the Laplace operator, see e.g. [16], [25]. Proposition 3.1 is proven. Remark 3.2. Surely, regularity estimate (3.2) holds not only for l = 1,, 1, ut we will need it in the sequel only for that values of l. We also note that estimate (3.2) holds for the unweighted space since the spectrum of the Laplacian in a strip with Dirichlet oundary conditions is strictly negative. The next proposition gives the elliptic regularity for the ilaplace operator in a strip Ω. Proposition 3.3. Let Ω e a strip and let us consider the following oundary value prolem in Ω: { 2 (3.7) xu = h, u = h, n u = h 1. Then, for every 1 < p < and l =, 1, 2, there exists µ = µ (p) such that, for every weight function of a sufficiently small exponential growth rate µ (µ µ ) and every (h, h, h 1 ) W l 2,p (Ω) W l+2 1/p,p () W l+1 1/p,p () prolem (3.7) has a unique solution u W l+2,p (Ω) and the following estimate holds: (3.8) u W l+2,p (Ω) C( h W l 2,p (Ω) + h l+2 1/p,p W () + h 1 l+1 1/p,p W () ) where the constant C depends on C, ut is independent of the concrete choice of weight function. Moreover, the analogous result holds for the spaces W l,p, as well. Proof. We first note that, due to the emedding (trace) theorem for weighted spaces formulated in Proposition 2.1, we can assume without loss of generality that h = h 1 =. Moreover, arguing as in the proof of Proposition 3.1, we can reduce the derivation of estimate (3.8) to the unweighted case 1. After that it only remains to note that the spectrum of the ilaplacian 2 in a
17 strip Ω with homogeneous Dirichlet oundary conditions u = n u = is strictly negative. Thus, for the unweighted case (3.8) is just a classical L p -regularity result for the 4th order elliptic operator 2 x, see [25]. Proposition 3.3 is proven. We are now going to consider the Newmann-type oundary value prolems for the Laplacian in a strip Ω. The main difficulty here is the fact that, in contrast to the Dirichlet prolems considered aove, the Newmann prolem for the Laplacian has an essential spectrum at λ =, which makes the situation much more delicate. We however start with the regularized Newmann-type prolem where the spectrum remains strictly negative. Proposition 3.4. Let Ω e a strip and let us consider the following oundary value prolem in Ω: (3.9) x u u =, n u = h, Then, for every 1 < p < and l =, 1, 2, there exists µ = µ (p) such that, for every weight function of sufficiently small exponential growth rate µ (µ µ ) and every h W l 1/p,p () prolem (3.9) has a unique solution u W l+1,p (Ω) and the following estimate holds: (3.1) u W l+1,p (Ω) C h W l 1/p,p where the constant C depends on C, ut is independent of the concrete choice of weight function. Moreover, the analogous result holds for the spaces W l,p, as well. Proof. Indeed, in the case l = 1, 2 estimate (3.1) can e verified exactly as in Propositions 3.1 and 3.3 (y reducing to the homogeneous and unweighted case), so we rest it to the reader. In the case l = the situation is slightly more delicate since we do not formulate the extension theorem for the space W 1/p,p (Ω) in Proposition 2.1 and, consequently, we need to work with the nonhomogeneous oundary value prolem. Nevertheless, the reduction to the unweighted case ased on introducing the function v := ϕ µ,su works in this case as well. Indeed, this function oviously satisfies () (3.11) x v v = h µ (v), n v := T µ,sh and (3.12) h µ (v) L p (Ω) Cµ v W 1,p (Ω) Thus, we can split the solution v of (3.11) as follows: v = v 1 +v 2 where v 1 solves the homogeneous prolem (3.13) x v 1 v 1 = h µ (v), n v 1 = and the remainder v 2 solves the analogue of (3.9) with h replaced y T µ,sh. We see also that the right-hand side of (3.11) elongs to L p (Ω) and, consequently, due to the classical L p -regularity, we have (3.14) v 1 W 2,p (Ω) C h µ (v) L p (Ω) C 1 µ v W 1,p (Ω). If we assume now that estimate (3.1) for the unweighted case = 1 and l = is known, then, due to (3.14), we infer v W 1,p (Ω) v 1 W 1,p (Ω) + v 2 W 1,p (Ω) C T µ,sh W 1/p.p () + Cµ v W 1,p (Ω) which implies the estimate (3.15) v W 1,p (Ω) 2C T µ,sh W 1/p,p () if µ is small. Thus, the case of general weight naturally reduces to the case of 1 for l = as well. It remains to recall that, for 1, estimate (3.1) is a classical L p -regularity result for the Laplacian, see [25]. Proposition 3.4 is proven.
18 In order to treat the case of Newmann prolem without the regularizing term u, we need to introduce the following averaging operator with respect to the variale x 2 ((x 1, x 2 ) R ( 1, 1) := Ω): (3.16) (Su)(x 1 ) := 1 2 1 1 u(x 1, s) ds. The next proposition gives the solvaility of the Newmann prolem for some natural closed suspace of the the space of external forces h. Proposition 3.5. Let Ω e a strip and let us consider the following oundary value prolem in Ω: (3.17) x u = h, n u =. Then, for every 1 < p < and l =, 1, 2, there exists µ = µ (p) such that, for every weight function of a sufficiently small exponential growth rate µ (µ µ ) and every h W l,p (Ω) satisfying Sh, prolem (3.17) has a unique solution u W l+2,p (Ω), Su and the following estimate holds: (3.18) u W l+2,p (Ω) C h W l,p where the constant C depends on C, ut is independent of the concrete choice of weight function. Moreover, the analogous result holds for the spaces W l,p, as well. Proof. We first note that the operator S commutes with the multiplication operator T µ and with the x 1 -derivatives x1. Thus, arguing exactly as efore, we can reduce the proof of (3.18) to the unweighted case 1. So, we will prove elow (3.18) for the case 1 only. To this end, we first consider the case p = 2. In that case we can multiply equation (3.17) y u and otain, after the integration y parts that (3.19) x u 2 L 2 (Ω) h L 2 (Ω) u L2 (Ω) Since we have assumed additionally that Su then, we have the Friedrichs inequality (3.2) u W 1,2 (Ω) C x u L 2 (Ω) which together with (3.19) implies that (3.21) u W 1,2 (Ω) C h L 2 (Ω). In order to prove estimate (3.18) for p = 2 and 1, we now use the following standard interior regularity estimate: (3.22) u 2 W l+2,2 (Ω s) C( u 2 W 1,2 (Ω + s 1 Ω s Ω s+1) h 2 W l,2 (Ω ) s) C 1 e α s y ( u 2 W 1,2 (Ω + y) h 2 W l,2 (Ω ) dy. y) Integrating this estimate over s R and using (2.12) and (3.21), we infer the unweighted estimate (3.18) for p = 2. Thus, due to the trick with the multiplication operator T µ,s, estimate (3.18) is verified for p = 2 and all weights with sufficiently small exponential growth rate. Moreover, we have also the analogue of estimate (3.18) with p = 2 for the spaces W l,p, (Ω). Let us now consider the case p 2. We first consider the case p > 2 and will prove estimate (3.18) for the spaces W l,p (Ω). Indeed, since W l,p (Ω) W l,2 (Ω), then we already have the estimate (3.23) u W 1,2 (Ω) C h L 2 (Ω) C 1 h L p (Ω). Using now the interior regularity estimate y Ω u W l+2,p (Ω s) C( u W 1,2 (Ω s 1 Ω s Ω s+1) + h W l,p (Ω s)) (Ω) C 1 sup{e α s y ( u W 1,2 (Ω y) + h W l,p (Ω y))}, y R
19 taking a supremum over s R from the oth parts of that inequality and using (2.3) and (3.23), we finally infer (3.24) u W l+2,p (Ω) C h W l,p (Ω). Let now 1 < p < 2. Then, we split the solution u of (3.17) as follows: u = u 1 + u 2 where u 1 solves prolem (3.25) x u 1 u 1 = h, n u 1 = and the remainder u 2 solves (3.26) x u 2 = u 1, n u 2 =. We first note that, due to the L p -regularity (see Proposition 3.4), for equation (3.25), we have (3.27) u 1 W l+2,p (Ω) C h W l,p (Ω). Moreover, applying the operator S to oth sides of equation (3.25) and using that Sh, we have (3.28) (Su 1 ) Su 1 and, consequently, Su 1. Furthermore, due to the emedding theorem (see Proposition 2.1), we have (3.29) u 1 W l,2 (Ω) C u 1 W l+2,p (Ω), for every 1 < p < 2. Thus, we can apply estimate (3.23) for equation (3.26) which together with (3.27) gives estimate (3.24) for 1 < p < 2 as well. Thus, estimate (3.24) is verified for all 1 < p <. Then, due to the aove descried trick with the multiplication operator T µ,s, we deduce estimate (3.18) for the spaces W l+2,p, (Ω) for all weight functions of sufficiently small exponential growth rate. So, it only remains to otain it for the spaces W l,p (Ω). To this end, we note that (3.18) for the spaces W l,p,ϕ µ (Ω) implies, in particular, that,s (3.3) u p W l+2,p (Ω C sup s) {e µp s y h p W y R l,p (Ω } y) C 1 y Ω e µp s y /2 h p W l,p (Ω) dy. Multiplying (3.3) y (s) p, integrating over s R and using (2.12), we deduce finally estimate (3.18) and finish the proof of Proposition 3.5. Remark 3.6. As we see from the proof of Proposition 3.5, the weighted regularity estimates can e deduced not only from the unweighted estimates in W l,p (Ω), ut also from its analogues in the spaces W l,p (Ω). The last scale of spaces is sometimes (e.g., in the proof of Proposition 3.5) more convenient, since, in contrast to spaces L p (Ω), the spaces L p (Ω) have usual (for ounded domains) emedding properties (L p1 (Ω) Lp2 (Ω), for p 1 p 2 ). We now note that assumption Sh in Proposition 3.5 is essential for the weighted estimate (3.18). Indeed, in general case Sh, for the quantity Su = (Su)(x 1 ) we have the following equation: (3.31) (Su)(x 1 ) = (Sh)(x 1 ), x 1 R whose solution Su, oviously, does not possess any weighted regularity estimates for general h. Fortunately, for prolems arising in the weighted regularity theory for the Helmholtz operator, the function Sh has a special structure which allows to take one primitive of it remaining in weighted Soolev classes. To e more precise, the following proposition holds. Proposition 3.7. Let Ω e a strip and let us consider the following Newmann oundary value prolem in Ω: (3.32) x u =, n u = l n g
2 where g [L p (Ω)] 2 is a divergent free vector field (3.33) div g. Then, for every 1 < p < and l =, 1, 2, there exists µ = µ (p) such that, for every weight function of a sufficiently small exponential growth rate µ (µ µ ) and every g W l,p (Ω) satisfying (3.33), prolem (3.32) has a unique solution (up to adding a constant) satisfying x u W l,p (Ω), and (3.34) (Su)(x 1 ) = (Sg 1 )(x 1 ), x 1 R and the following estimate holds: (3.35) x u W l,p (Ω) C g W l,p (Ω) where the constant C depends on C, ut is independent of the concrete choice of weight function. Moreover, the analogous result holds for the spaces W l,p, as well. Proof. For simplicity, we deduce elow only a priori estimate (3.35). The existence and uniqueness of a solution can e verified in a standard way (see also [4]). We first define an auxiliary function v as a solution of the following prolem: (3.36) x v v =, n v = l n g. Then, due to Propositions 3.4 and 2.12, we have (3.37) v W l+1,p (Ω) C l ng l 1/p,p W () C 2 g W l,p Moreover, applying the x 2 -averaging operator S to equation (3.36), we have (3.38) (Sv)(x 1 ) (Sv)(x 1 ) = 1/2(g 2 (x 1, 1) g 2 (x 1, 1)), x 1 R. Furthermore, since the vector field g is divergence free, we have and, consequently, (Ω). 1/2(g 2 (x 1, 1) g 2 (x 1, 1)) = (S[ x2 g 2 ])(x 1 ) = (Sg 1 )(x 1 ) (3.39) (Sv)(x 1 ) (Sv)(x 1 ) = (Sg 1 )(x 1 ). Let us consider now the remainder w := u v which oviously satisfies the following equation: (3.4) x w = v, n w =. Then, according to Proposition 3.5, the function w := w Sw satisfies the following estimate: (3.41) w W l+1,p (Ω) C v W l,p (Ω) C 1 g W l,p So, it only remains to consider the equation for Sw, i.e. which together with (3.39) gives (Sw)(x 1 ) = (Sv)(x 1 ) (3.42) (Su)(x 1 ) = (Sg)(x 1 ). This relation shows that we can indeed to take one primitive and satisfy condition (3.34). It only remains to note that the function (Su)(x 1 ) is independent of x 2 and, consequently, (3.43) x u = x ū + ((Su), ). Thus, estimates (3.37), (3.41) together with the ovious fact that (Ω). (3.44) Sg W l,p (R) C g W l,p implies (3.35) and finishes the proof of Proposition 3.7. (Ω)
21 4. Weighted spaces and the Helmholtz projector In this Section, we discuss the weighted analogue of the classical Helmholtz decomposition of the space [L 2 (Ω)] 2 to divergent free and gradient vector fields which is necessary for excluding the pressure from Navier- Stokes equations. To this end, we first need to define the corresponding spaces of divergent free vector fields. Definition 4.1. Let Ω e a strip. Then, for every l, 1 < p < and every weight function of exponential growth rate, we define the following space of divergent free vector fields: (4.1) H l,p l,p (Ω) := {v [W (Ω)]2, div v, l n v =, Sv 1 } which is considered as a closed suspace of W l,p (Ω) and endowed y the norm induced y this emedding. Here the normal component l n v of the trace on the oundary is well-defined due to Proposition 2.12 and the x 2 -averaging operator S is defined y (3.16). The spaces H l,p, (Ω) can e defined analogously. Moreover, for simplicity, we will write elow H p (Ω) and Hp, (Ω) instead of H,p (Ω) and H,p, (Ω) respectively. We also define the space V p (Ω) as follows: and the analogous space V p, (Ω). V p (Ω) := {v H1,p (Ω), v = } The following natural proposition clarifies the additional conditions l n v = and Sv 1 in formula (4.1). Proposition 4.2. Let Ω e a strip and e a weight function of exponential growth rate µ and 1 < p <. Then the space H p (Ω) coincides with the closure of all divergent free vector fields v [D(Ω)] 2 in the topology of [L p (Ω)]2 : (4.2) H p (Ω) = [ v [D(Ω)] 2, div v = ] [L p (Ω)]2 where [ ] V denotes the closure in the topology of the space V. Proof. Indeed, let v e a divergent free vector field from [D(Ω)] 2. Then, oviously, l n v =. Moreover, integrating the relation x1 v 1 = x2 v 2, we infer that Sv 1 const = (since v 1 has a finite support). Since all these properties preserve under the closure (see Proposition 2.12), then the right-hand side of (4.2) is a suset of the left one. Thus, it only remains to approximate every function from u H p (Ω) y divergent free vector fields elonging to [D(Ω)] 2. In order to do so, it is natural to use the stream function Φ of a divergent free vector field u: (4.3) u 1 = x2 Φ, u 2 = x1 Φ which can e defined y the following natural formula: (4.4) Φ(x 1, x 2 ) := Indeed, oviously, Φ L p (Ω) and x2 1 u 1 (x 1, θ) dθ. (4.5) Φ Lp (Ω) C u 1 L p (Ω). Moreover, x2 Φ = u 1 and x1 Φ(x 1, x 2 ) = x2 1 x2 x1 u 1 (x 1, s) ds = x2 u 2 (x 1, s) ds = u 2 (x 1, x 2 ) 1 (here we have implicitly used that div u and u 2 (x 1, 1) u 2 (x 1, 1) = ). Thus, the function Φ satisfies indeed relations (4.3) and, consequently, we Φ W 1,p (Ω) and (4.6) Φ W l,p (Ω) C u L p (Ω).