Attainability of Time-Periodic flow of a Viscous Liquid Past an Oscillating Body
aa r X i v : . [ m a t h . A P ] J a n Attainability of Time-Periodic flow of aViscous Liquid Past an Oscillating Body
Giovanni P. Galdi ∗ and Toshiaki Hishida † January 22, 2020
Abstract
A body B is started from rest by translational motion in an other-wise quiescent Navier-Stokes liquid filling the whole space. We show,for small data, that if after some time B reaches a spinless oscilla-tory motion of period T , the liquid will eventually execute also a timeperiodic motion with same period T . This problem is a suitable gen-eralization of the famous Finn’s starting problem for steady-states, tothe case of time-periodic motions. Consider a rigid body, B , at rest and completely immersed in a quiescentNavier-Stokes liquid filling the whole three-dimensional space, Ω, outside B .Next, suppose that, at time t = 0 (say), B is smoothly set in translationalmotion (no spin) and that after the time t = 1 (say), its velocity η = η ( t )coincides with a periodic function, ξ = ξ ( t ), of period T whose average overthe time interval [0 , T ] vanishes. In the particular case where both η and ξ are parallel to a given direction, the above means that B is brought fromrest to a regime where it oscillates between two fixed configurations. In thegeneral case, B is taken from rest to a (spinless) motion where its center ofmass moves periodically along a given closed curve.On physical grounds, it is expected that, under the given assumptions,the liquid will eventually reach a time-periodic flow of period T , at least if themagnitude of η and (possibly) some of its derivatives is not “too large.” Thisspecific circumstance is often referred to as attainability property of the flow. ∗ Partially supported by NSF grant DMS-1614011 † Partially supported by the Grant-in-Aid for Scientific Research 18K03363 from JSPS
1n this regard, it is worth mentioning a famous problem of attainability,the so called “Finn’s starting problem” [2] where B accelerates (withoutspinning) from rest to a given constant translational velocity. In such a case,the terminal flow of the liquid is expected to be steady-state. Finn’s problemwas eventually and affirmatively solved by Galdi, Heywood and Shibata [5]and, with more general assumptions, very recently by Hishida and Maremonti[10].In analogy with these results, the main objective of this paper will be toshow that, under the given hypothesis on the motion of B , the liquid indeedattains a corresponding time-periodic flow of period T .We shall next give a rigorous mathematical formulation of the problem.Let us begin to observe that the translational velocity η ( t ) can be written as η ( t ) = h ( t ) ξ ( t ) , where we assume ξ ( t + T ) = ξ ( t ) ( t ∈ R ) , Z T ξ ( t ) dt = 0 ,ξ ∈ W , (0 , T ) = W , (0 , T ; R ) , (1.1)and h ∈ C ( R ; [0 , , h ( t ) = 0 ( t ≤ , h ( t ) = 1 ( t ≥ . (1.2)The governing equations of the liquid flow, driven by the translational veloc-ity η of the body, are thus given by ∂ t u + u · ∇ u = ∆ u + η ( t ) · ∇ u − ∇ p u , div u = 0 ) in Ω × (0 , ∞ ) ,u | ∂ Ω = η ( t ) ,u → | x | → ∞ ,u ( · ,
0) = 0 , (1.3)where u = u ( x, t ) and p u = p u ( x, t ) are, respectively, the velocity vectorfield and pressure field of the liquid, and Ω (the exterior of the body in R ) is assumed to have a sufficiently smooth boundary ∂ Ω. Likewise, if thetranslational velocity of B is the time-periodic function ξ , it is reasonable toexpect that, the corresponding velocity field of the liquid v = v ( x, t ) is time-2eriodic of period T ( T -periodic) as well, and obeys the following equations ∂ t v + v · ∇ v = ∆ v + ξ ( t ) · ∇ v − ∇ p v , div v = 0 ) in Ω × R / T Z ,v | ∂ Ω = ξ ( t ) ,v → | x | → ∞ , (1.4)where p v denotes the pressure associated with v .In [4] the first author showed existence, uniqueness and regularity of a T -periodic solution ( v ( t ) , p v ( t )) to (1.4) for all “small” ξ ( t ) satisfying (1.1).Furthermore, he provided a detailed analysis of the asymptotic representationof v ( t ) at spatial infinity, by showing that the leading term of v ( t ) is givenby a distinctive steady-state velocity field U ( x ) that decays at large spatialdistances like | x | − . Therefore, in general, v ( t ) L (Ω), for all t ∈ R .Let u = hv + w, (1.5)where h is the function given in (1.2). Then, from (1.3), we deduce that the“perturbation” w ( t ) should obey, together with the corresponding pressure p w = p u − hp v , the following system of equations ∂ t w + w · ∇ w + h ( t )( v · ∇ w + w · ∇ v )= ∆ w + η ( t ) · ∇ w − ∇ p w + f, div w = 0 in Ω × (0 , ∞ ) ,w | ∂ Ω = 0 ,w → | x | → ∞ ,w ( · ,
0) = 0 , (1.6)with the forcing term ( h ′ := dh/dt ) f := − h ′ v + ( h − h )( v − ξ ) · ∇ v . (1.7)The desired attainability property consists then in showing that the solution w ( t ) to (1.6) (exists, is unique and) tends to 0 as t → ∞ in a suitable norm.In this respect, some comments are in order. Since u (0) = 0, one wouldexpect that the solution u ( t ) to (1.3) has finite energy, namely u ( t ) ∈ L (Ω)for all t ≥
0. Moreover, as noticed earlier on, v ( t ) is, in general, not in L .Consequently, in view of (1.5), w ( t ) need not be in L (Ω), as also suggestedby the fact that f is not in L (Ω). This implies that “energy-based methods”might not be an appropriate tool to analyze the asymptotic behavior of w ( t ),and one has thus to resort to the more general L q -theory. This difficulty is3nalogous to that encountered in Finn’s starting problem, which was in factsolved in [5] thanks to the asymptotic properties of the Oseen semigroup in L q -spaces, proved for the first time in [13].However, in comparison with [5], our problem presents the following twofurther complications. (i) The velocity field v ( t ), t ∈ R , possesses weakersummability properties at large spatial distances than its steady-state coun-terpart considered in [5]. This is due to the fact that ξ ( t ) has zero average, see(1.1), so that, unlike [5], the motion of B produces no wake structure in theflow. (ii) The non-autonomous character of the principal linear part, wherethe drift term η ( t ) · ∇ w cannot be seen as a perturbation to the main (Stokes)operator, for all sufficiently large times. In order to overcome the difficulty in(i) we adapt to the case at hand the duality method developed by Yamazaki[18] that allows us to handle the additional linear terms h ( t )( v · ∇ w + w · ∇ v )in (1.6), in spite of the “poor” summability of v at large distances. As far asthe other difficulty, we shall employ the theory recently developed in [7, 8]by the second author, which provides L q - L r decay estimates of the evolutionoperator, { T ( t, s ) } t ≥ s ≥ , generated by the non-autonomous Oseen operator − P [∆ + η ( t ) · ∇ ] –with P Helmholtz projection on the space of L q -vectorfields– entirely analogous to those available in the autonomous case for Stokesand Oseen semigroups [11], [13], [14].By suitably combining the above arguments and using the results in [4],in the present paper we are able to show, in particular, the decay to 0 of w ( t ), as t → ∞ , in appropriate L q -spaces; see Theorem 2.1. Moreover, bydeveloping an idea of Koba [12], we shall also show the decay of w ( t ) in L ∞ -norm (see (2.7) below). However, our proof –based on the L ∞ -estimateof the composite operator T ( t, s ) P div given in Proposition 3.3– turns out tobe simpler and more direct than that given in [12].The plan of the paper is as follows. In the next section we shall state themain results, collected in Theorem 2.1. In Section 3 we present some resultsfrom [4] and [7, 8] and deduce some relevant consequences. The final Section4 is devoted the proof of Theorem 2.1. Notation. C ∞ ,σ (Ω) is the subclass of vector functions u in C ∞ (Ω) withdiv u = 0. By L q (Ω), 1 ≤ q ≤ ∞ , W m,q (Ω) , m ≥ , ( W ,q ≡ L q ), we denoteusual Lebesgue and Sobolev classes of vector functions, with correspondingnorms k . k q and k . k m,q . Also, L qσ (Ω) denotes the completion of C ∞ ,σ (Ω) in L q (Ω), and P : L q L qσ the associated Helmholtz projection ([3], [15],[16]). For 1 < p < ∞ and 1 ≤ q ≤ ∞ , let L p,q (Ω) denote the Lorentzspace with norm k . k p,q ; see [1] for details about this space. Since P definesa bounded operator on L p,q (Ω), we set L p,qσ (Ω) = P [ L p,q (Ω)]. Moreover, D m, (Ω) stands for the space of (equivalence classes of) functions u ∈ L loc (Ω)4uch that P | k | = m k D k u k < ∞ . Obviously, the latter defines a seminorm in D m, . Let B be a function space of spatial variable endowed with seminorm k · k B . For r = [1 , ∞ ], T > L r ( B ) is the class of functions u : (0 , T ) → B such that k u k L r ( B ) ≡ (cid:0)Z T k u ( t ) k rB (cid:1) r < ∞ , if r ∈ [1 , ∞ ) ;ess sup t ∈ [0 , T ] k u ( t ) k B < ∞ , if r = ∞ . Likewise, we put W m,r ( B ) = n u ∈ L r ( B ) : ∂ kt u ∈ L r ( B ) , k = 1 , . . . , m o . By use of the evolution operator T ( t, s ) mentioned in the introductory sec-tion, problem (1.6) is transformed into the integral equation w ( t ) = w ( t ) − Z t T ( t, s ) P div ( F w )( s ) ds (2.1)with w ( t ) = Z t T ( t, s ) P f ( s ) ds, (2.2) F w = F v w = w ⊗ w + h ( w ⊗ v + v ⊗ w ) . (2.3)The main result reads Theorem 2.1.
Suppose (1.1) and (1.2) hold and let | h ′ | := sup t ≥ | h ′ ( t ) | .For every ε ∈ (0 , ) , there is a constant δ = δ ( ε ) such that if k ξ k W , (0 , T ) ≤ δ | h ′ | (2.4) then problem (2.1) admits a unique solution w ∈ C w ∗ ((0 , ∞ ); L , ∞ σ (Ω)) withthe following properties:1. The equation (2.1) is satisfied in L , ∞ σ (Ω) .2. The initial condition: lim t → k w ( t ) k , ∞ = 0 . (2.5)5 . There is a constant C > such that k w ( t ) k , ∞ ≤ C k ξ k W , (0 , T ) (2.6) for all t ≥ .4. Attainability: k w ( t ) k q = (cid:26) O (cid:0) t − / / q (cid:1) , q ∈ (3 , q ) ,O (cid:0) t − / ε (cid:1) , q ∈ ( q , ∞ ] , k w ( t ) k q , ∞ = O ( t − / ε ) , (2.7) as t → ∞ , where q = 3 / ε . Remark 2.1.
The unique existence of the evolution operator T ( t, s ) or, inother words, the well-posedness of the initial boundary value problem for thelinearized system, was successfully proved by Hansel and Rhandi [6] evenin the case when the body B rotates. The key point of their argument ishow to overcome difficulties due to the rotational term; in fact, the Tanabe–Sobolevskii theory [17] of parabolic evolution operators does not work in thissituation. Remark 2.2.
We apply the theory in [17] to the non-autonomous Oseenoperator without rotation. Thus, in such a case, the regularity propertiesof T ( t, s ) basically coincide with those of analytic semigroups for the au-tonomous case. As a consequence, one could show that the solution w ( t ) inTheorem 2.1 becomes “strong” provided only h ′ ( t ) , in addition to satisfying (1.2) , is H¨older continuous. We will not give details of such a claim, sinceour main objective is to show the attainability property. Remark 2.3.
We observe that our approach furnishes, in particular, alsothe stability of the time-periodic solution v ( t ) . In fact, this property canbe established by studying an integral equation of the type (2.1) obtained bysetting formally h ( t ) ≡ (which implies that the term f in (1.7) vanishesidentically) and replacing the function w ( t ) with f w ( t ) = T ( t, w (0) , where w (0) is the initial perturbation. One can slightly modify the proof of Theo-rem 2.1 to show that the asymptotic decay property of w ( t ) stated in (2.7) continues to hold, provided, in addition to (2.4) , that w (0) ∈ L , ∞ σ (Ω) withsufficiently small norm. Preparatory Results
Let us begin to recall the following result concerning the existence, uniquenessand asymptotic spatial behavior of solutions to (1.4).
Proposition 3.1 ([4]) . Let ξ satisfy (1.1) . Then, there exists a constant ε > such that if D := k ξ k W , (0 , T ) < ε , (3.1) problem (1.4) has one and only one time-periodic solution ( v, p v ) of period T in the class v ∈ W , ( D , ) ∩ W , ( D , ) ∩ W , ∞ ( W , ) ∩ L ∞ ( D , ) ,p v ∈ L ∞ ( W , ) ∩ W , ( D , ) , with all corresponding norms of ( v, p v ) bounded from above by D . Moreover,there exists a constant C > such that this solution obeys the followingestimates (1 + | x | ) | v ( x, t ) | + (1 + | x | ) {|∇ v ( x, t ) | + | p v ( x, t ) |} + (1 + | x | ) {|∇ v ( x, t ) | + |∇ p v ( x, t ) |} ≤ C D , (3.2) for all ( x, t ) ∈ Ω × R / T Z . Remark 3.1.
The constant δ in (2.4) of Theorem 2.1 must be taken smallerthan ε in (3.1) . The next result regards the large time behavior of the evolution operator T ( t, s ) and its adjoint T ( t, s ) ∗ . These properties, among others, have beenestablished in [7, 8]. Proposition 3.2 ([7, 8]) . Let m ∈ (0 , ∞ ) and assume sup t ≥ | η ( t ) | + sup t>s ≥ | η ( t ) − η ( s ) | t − s ≤ m. (3.3)
1. Let < q < ∞ and q ≤ r ≤ ∞ . Then, there is a constant C = C ( m, q, r, Ω) > such that k T ( t, s ) f k r ≤ C ( t − s ) − (3 /q − /r ) / k f k q (3.4) for all t > s ≥ , f ∈ L qσ (Ω) and that k T ( t, s ) f k r, ∞ ≤ C ( t − s ) − (3 /q − /r ) / k f k q, ∞ (3.5) for all t > s ≥ and f ∈ L q, ∞ σ (Ω) . . Let < q ≤ r ≤ . Then there is a constant C = C ( m, q, r, Ω) > such that k∇ T ( t, s ) ∗ g k r ≤ C ( t − s ) − (3 /q − /r ) / − / k g k q (3.6) for all t > s ≥ , g ∈ L qσ (Ω) and that k∇ T ( t, s ) ∗ g k r, ≤ C ( t − s ) − (3 /q − /r ) / − / k g k q, (3.7) for all t > s ≥ and g ∈ L q, σ (Ω) . If in particular /q − /r = 1 / aswell as < q
In [7, 8] the assumption on η is made in terms of the H¨olderseminorm, that is controlled by the left-hand side of (3.3) , which is, in turn,controlled by D ; see (3.1) . Estimate (3.5) with r < ∞ immediately followsfrom (3.4) by interpolation. The proof of L q, ∞ - L ∞ estimate, that is, (3.5) with r = ∞ , is not given in [7, 8], but it can be easily proved by use ofthe semigroup property, following the lines of the proof of (3.9) – (3.10) belowwith r = ∞ . The remaining three bounds (3.6) – (3.8) are shown in [8]. How-ever, we emphasize that (3.7) with r = 3 does not follow directly from (3.6) by interpolation. The idea of deducing (3.8) from (3.7) is, in fact, due toYamazaki [18]. We next prove an important consequence of the previous proposition.
Proposition 3.3.
Let m ∈ (0 , ∞ ) and assume (3.3) . The following proper-ties hold.1. Let / ≤ q < ∞ and q ≤ r ≤ ∞ . Then there is a constant C = C ( m, q, r, Ω) > such that the composite operator T ( t, s ) P div extendsto a bounded operator from L q (Ω) × to L rσ (Ω) , r < ∞ , and to L ∞ (Ω) subject to estimate k T ( t, s ) P div F k r ≤ C ( t − s ) − (3 /q − /r ) / − / k F k q (3.9) for all t > s ≥ and F ∈ L q (Ω) × . . Let / < q < r ≤ ∞ . Then there is a constant C = C ( m, q, r, Ω) > such that the composite operator T ( t, s ) P div extends to a boundedoperator from L q, ∞ (Ω) × to L rσ (Ω) , r < ∞ , and to L ∞ (Ω) subject toestimate k T ( t, s ) P div F k r ≤ C ( t − s ) − (3 /q − /r ) / − / k F k q, ∞ (3.10) for all t > s ≥ and F ∈ L q, ∞ (Ω) × .Proof. By density, it suffices to show (3.9) for F ∈ C ∞ (Ω) × . We firstconsider the case 3 / ≤ q ≤ r < ∞ , so that 1 < r ′ ≤ q ′ ≤
3. By (3.6) wehave |h T ( t, s ) P div F , ϕ i| = |h F, ∇ T ( t, s ) ∗ ϕ i|≤ k F k q k∇ T ( t, s ) ∗ ϕ k q ′ ≤ C ( t − s ) − (3 /q − /r ) / − / k F k q k ϕ k r ′ for all t > s ≥ ϕ ∈ L r ′ σ (Ω), which leads to (3.9) with r < ∞ . Thiscombined with (3.4) ( r = ∞ ) implies that k T ( t, s ) P div F k ∞ ≤ C ( t − s ) − / q k T (( t + s ) / , s ) P div F k q ≤ C ( t − s ) − / q − / k F k q yielding (3.9) with r = ∞ .Let 3 / < q ≤ r < ∞ , then (3.9) implies k T ( t, s ) P div F k r, ∞ ≤ C ( t − s ) − (3 /q − /r ) / − / k F k q, ∞ (3.11)for all t > s ≥ F ∈ L q, ∞ (Ω) × . Since k u k r ≤ C k u k − θr , ∞ k u k θr , ∞ (3.12)where 1 /r = (1 − θ ) /r + θ/r as well as 0 < θ < < r < r < r ≤ ∞ ,we obtain (3.10) from (3.11) as long as 3 / < q < r < ∞ . This combinedwith (3.4) ( r = ∞ ) leads to (3.10) when 3 / < q < r = ∞ . The proof iscomplete. Following Yamazaki [18], we consider the following weak form of (2.1): h w ( t ) , ϕ i = h w ( t ) , ϕ i + Z t h ( F w )( s ) , ∇ T ( t, s ) ∗ ϕ i ds ∀ ϕ ∈ C ∞ ,σ (Ω) . (4.1)9or q ∈ [3 , ∞ ), let us introduce the space X q = { w ∈ C w ∗ ((0 , ∞ ); L , ∞ σ (Ω) ∩ L q, ∞ σ (Ω)); [ w ] + [ w ] q < ∞ , lim t → k w ( t ) k , ∞ = 0 } , where [ w ] q := sup t> t / − / q k w ( t ) k q, ∞ . (4.2)Clearly, X q becomes a Banach space when endowed with norm [ w ] + [ w ] q .Under the smallness condition (3.1), the solution v obtained in Proposi-tion 3.1 and the force f defined by (1.7) fulfill v ( t ) , f ( t ) ∈ L , ∞ (Ω) ∩ L ∞ (Ω)with sup t ≥ ( k v ( t ) k , ∞ + k v ( t ) k ∞ ) ≤ C D , sup t ≥ ( k f ( t ) k , ∞ + k f ( t ) k ∞ ) ≤ C ( | h ′ | + D ) D , (4.3)which immediately follows from (3.2). This, combined with (3.5), implies thefollowing lemma. Lemma 4.1.
Suppose (1.1) , (1.2) and (3.1) . Then the function w defined by (2.2) belongs to X q for every q ∈ [3 , ∞ ) . Moreover, we have w ( t ) ∈ L ∞ (Ω) for each t > . Finally, for every r ∈ [3 , ∞ ] , there is a constant c r > suchthat k w ( t ) k r, ∞ ≤ c r ( | h ′ | + D ) D (1 + t ) − / / r (4.4) for all t > , with D given in (3.1) .Proof. Let 0 ≤ t < t + τ , then we have w ( t + τ ) − w ( t )= Z t { T ( t + τ, s ) − T ( t, s ) } P f ( s ) ds + Z t + τt T ( t + τ, s ) P f ( s ) ds =: I + II.
By (3.5) and (4.3), we know that k T ( t, s ) P f ( s ) k q, ∞ ≤ C ( | h ′ | + D ) D =: C with some constant C = C ( q ) > t, s ) for every q ∈ [3 , ∞ ).From the Lebesgue convergence theorem we infer I → τ →
0, whereasit follows at once II ≤ C τ . For the other case 0 < t/ < t + τ < t , we have w ( t + τ ) − w ( t ) = Z t + τ { T ( t + τ, s ) − T ( t, s ) } P f ( s ) ds − Z tt + τ T ( t, s ) P f ( s ) ds τ → w ( t ) is even strongly continuous up to t = 0 with values in L q, ∞ (Ω) aswell as k w ( t ) k , ∞ → t → L r, ∞ (Ω) with r ∈ [3 , ∞ ], we consider only the one involving k w ( t ) k ∞ , since the other onesare obtained similarly. Since f ( t ) = 0 for t ≥
1, we use (3.5) to find k w ( t ) k ∞ ≤ C Z ( t − s ) − / k P f ( s ) k , ∞ ds ≤ Ct − / ( | h ′ | + D ) D for t ≥
2, while we have k w ( t ) k ∞ ≤ Ct / ( | h ′ | + D ) D for t <
2. We thus obtain the desired estimate.Let us begin to prove the uniqueness property. In fact, the solution ob-tained in Theorem 2.1 is unique in the sense of the following lemma, providedwe choose the constant δ in (2.4) smaller than the constant δ defined below. Lemma 4.2.
There is a constant δ > such that if D ≤ δ , then the solutionto (4.1) is unique in the ball { w ∈ X ; [ w ] ≤ δ } .Proof. Let both w, e w ∈ X satisfy (4.1). By duality L , σ (Ω) ∗ = L / , ∞ σ (Ω)together with the weak-H¨older inequality, we have |h w ( t ) − e w ( t ) , ϕ i| ≤ C ([ w ] + [ e w ] + [ v ] )[ w − e w ] Z t k∇ T ( t, s ) ∗ ϕ k , ds for all ϕ ∈ C ∞ ,σ (Ω). We employ (3.8) and (4.3) to obtain[ w − e w ] ≤ c ∗ ([ w ] + [ e w ] + D )[ w − e w ] by duality, which yields the assertion by taking δ = 1 / c ∗ .Given ε ∈ (0 , ), we set q = 3 / ε ∈ (6 , ∞ ) and intend to find a solution w ∈ X q to (4.1) provided D is small enough. Given w ∈ X q and t >
0, wedefine (Ψ w )( t ) by h (Ψ w )( t ) , ϕ i = Z t h ( F w )( s ) , ∇ T ( t, s ) ∗ ϕ i ds ∀ ϕ ∈ C ∞ ,σ (Ω) . We then find [Ψ w ] ≤ C ([ w ] + [ v ] )[ w ] , [Ψ w ] q ≤ C ([ w ] + [ v ] )[ w ] q . (4.5)11he former is deduced along the same lines as in Lemma 4.2, while the latteris verified by splitting the integral as Z t/ + Z tt/ ! s − / / q k∇ T ( t, s ) ∗ ϕ k r, ds =: I + II (4.6)where r ∈ (3 / ,
2) is determined by the condition 1 /r = 2 / − /q . In fact,in view of (3.7), we get I ≤ C Z t/ s − / / q ( t − s ) − ds k ϕ k q ′ , that leads to the desired estimate, where 1 /q ′ + 1 /q = 1. Also, employing(3.8), we show II ≤ Ct − / / q Z tt/ k∇ T ( t, s ) ∗ ϕ k r, ds. By the same token we can show[Ψ w − Ψ e w ] r ≤ C ([ w ] + [ e w ] + [ v ] )[ w − e w ] r r ∈ { , q } , (4.7)for all w, e w ∈ X q .The above computations are exactly the same as in [9, Section 8]. How-ever, because in our case the equation is non-autonomous, the argument toshow the continuity with respect to time is different from the one adoptedby Yamazaki [18, Section 3] in which the strong continuity is deduced for t >
0. Here, we show merely the weak* continuity for t >
0, while we stillhave strong convergence to 0 at the initial time, namely, k (Ψ w )( t ) k , ∞ ≤ C ([ w ] + [ v ] ) sup
2) be the same exponent as in (4.6). By using the backwardsemigroup property we have I ≤ C ([ w ] + [ v ] )[ w ] q Z t s − / / q k∇ T ( t, s ) ∗ { T ( t + τ, t ) ∗ ϕ − ϕ }k r, ds ≤ C ([ w ] + [ v ] )[ w ] q k T ( t + τ, t ) ∗ ϕ − ϕ k / , which goes to zero as τ → ϕ ∈ C ∞ ,σ (Ω). Concerning the other part,we have II ≤ C ([ w ] + [ v ] )[ w ] q Z t + τt s − / / q k∇ T ( t + τ, s ) ∗ ϕ k r, ds ≤ C ([ w ] + [ v ] )[ w ] q t − / / q τ / − / q k ϕ k / , for all ϕ ∈ C ∞ ,σ (Ω), which implies the strong convergence with values in L , ∞ σ (Ω) also of this part. Summing up, by density argument, we can statethat the left-hand side of (4.8) goes to zero as τ → ϕ ∈ L / , σ (Ω)and also for all ϕ ∈ L q ′ , σ (Ω) in view of (4.5). The case 0 < t/ < t + τ < t is similarly discussed with h (Ψ w )( t + τ ) − (Ψ w )( t ) , ϕ i = Z t + τ h ( F w )( s ) , ∇{ T ( t + τ, s ) ∗ − T ( t, s ) ∗ } ϕ i ds − Z tt + τ h ( F w )( s ) , ∇ T ( t, s ) ∗ ϕ i ds to conclude that Ψ w is weak* continuous with values in L , ∞ σ (Ω) and in L q , ∞ σ (Ω).By these results, we can then conclude that w + Ψ w ∈ X q , for every w ∈ X q . Assume now D ≤
1. By taking into account (4.3), (4.4), (4.5) and(4.7), one can easily show the existence of a fixed point w ∈ X q of the map w w + Ψ w in a closed ball of X q with radius 2( c + c q )( | h ′ | +1) D , provided ( | h ′ | +1) D issmall enough, where the smallness depends on ε (recall that q = 3 / ε > X . From theinterpolation inequality (3.12), the solution w ( t ) satisfies (2.7) for q ∈ (3 , q ).For the solution w ( t ) constructed above, it follows from (3.10) and (4.3)that the second term on the right-hand side of (2.1) is Bochner integrablewith values in L (Ω); in fact, Z t k T ( t, s ) P div ( F w )( s ) k ds ≤ C ([ w ] + [ v ] )[ w ] q t >
0. The latter, in conjunction with Lemma 4.1, shows that the weakform (4.1) leads, in fact, to the the conclusion that the integral equation(2.1) is meaningful in L , ∞ σ (Ω). Actually, from the computations that weshall perform in the next paragraph, it turns out that the second term of theright-hand side of (2.1) is also Bochner integrable in L ∞ (Ω).It remains to show (2.7) for the other case q ∈ ( q , ∞ ], q = 3 / ε . Tothis end, on the account of the interpolation inequality (3.12), it is enoughto prove the decay of w ( t ) in the L ∞ -norm. The argument that follows isessentially due to Koba [12], but, unlike [12], we shall not use a duality pro-cedure; rather, we will directly apply the L q, ∞ - L ∞ estimate of the compositeoperator T ( t, s ) P div proved in Proposition 3.3. As a consequence, the proofis considerably shortened and more direct. By means of (3.10), it is easilyseen that Z t k T ( t, s ) P div ( w ⊗ w )( s ) k ∞ ds ≤ C [ w ] q t − / for all t >
0, where the summability of the integral is ensured since q > t >
2. We split the other part of the integral of (2.1) into two parts (cid:18)Z t − + Z tt − (cid:19) k T ( t, s ) P div [ h ( w ⊗ v + v ⊗ w )]( s ) k ∞ ds =: I + II.
We utilize (3.10) again to find that I ≤ Z t − ( t − s ) − − ε k v ( s ) k , ∞ k w ( s ) k q , ∞ ds = Z t/ + Z t − t/ =: I + I with I ≤ C D [ w ] q t − / , I ≤ C D [ w ] q t − / ε , and that II ≤ Z tt − ( t − s ) − (3 /r +3 /q ) / − / k v ( s ) k r, ∞ k w ( s ) k q , ∞ ds ≤ C D [ w ] q t − / ε where r ∈ (3 , ∞ ) is chosen in such a way that 1 /r + 1 /q < /
3, see (4.3).The proof is complete.
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