Geissert, M.

Hieber, M.

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 11M. Geissert; M. HieberLp-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacleIn: Marek Fila and Karol Mikula and Pavol Quittner (eds.): Proceedings of Equadiff 11, International Conference onDifferential Equations. Czecho-Slovak series, Bratislava, July 25-29, 2005, [Part 1] Contributions of plenary speakersand minisymposia organizers. Comenius University Press, Bratislava, 2007. Presented in electronic form on theInternet. pp. 69--76.Persistent URL: http://dml.cz/dmlcz/700394Terms of use:© Comenius University, 2007Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronic delivery and stamped with digitalsignature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czActa Math. Univ. ComenianaeVol. LXXVI, 1(2007), pp. 69–76Proceedings of Equadiff 1169Lp-THEORY OF THE NAVIER-STOKES FLOW IN THEEXTERIOR OF A MOVING OR ROTATING OBSTACLEM. GEISSERT and M. HIEBERAbstract. In this paper we describe two recent approaches for the Lp-theory ofthe Navier-Stokes flow in the exterior of a moving or rotating obstacle.1. IntroductionConsider a compact set O ⊂ Rn, the obstacle, with boundary Γ := ∂O of classC1,1. Set Ω := Rn\O. For t > 0 and a real n × n-matrix M we setΩ(t) := {y(t) = etMx, x ∈ Ω} and Γ(t) := {y(t) = etMx, x ∈ Γ}.Then the motion past the moving obstacle O is governed by the equations ofNavier-Stokes given by∂tw − ∆w + w · ∇w + ∇q = 0, in Ω(t) × R+,∇ · w = 0, in Ω(t) × R+,w(y, t) = My, on Γ(t) × R+,w(y, 0) = w0(y), in Ω.(1)Here w = w(y, t) and q(y, t) denote the velocity and the pressure of the fluid,respectively. The boundary condition on Γ(t) is the usual no-slip boundary condi-tion. Quite a few articles recently dealt with the equation above, see [2], [3], [4],[5], [6], [8], [10], [11], [15], [16].In this paper, we describe two approaches to the above equations for the Lp-setting where 1 < p < ∞. The basic idea for both approaches is to transferthe problem given on a domain Ω(t) depending on t to a fixed domain. Thefirst transformation described in the following Section 2 yields additional termsin the equations which are of Ornstein-Uhlenbeck type. We shortly describe thetechniques used in [15] and [12] in order to construct a local mild solution of (1).In contrast to the first transformation, the second one, inspired by [17] and [6],allows to invoke maximal Lp-estimates for the classical Stokes operator in exteriordomains and like this we obtain a unique strong solution to (1). This approach isdescribed in section 3.Received December 1, 2005.2000 Mathematics Subject Classification. Primary 35Q30, 76D03.Key words and phrases. Navier-Stokes, rotating obstacle, mild and strong solutions.Supported by the DFG-Graduiertenkolleg 853.70 M. GEISSERT and M. HIEBER2. Mild solutionsIn this section we construct mild solutions to the Navier-Stokes problem (1). Todo this we first transform the equations (1) to a fixed domain. Let Ω, Ω(t) andΓ(t) be as in the introduction and suppose that M is unitary. Then by the changeof variables x = e−tMy and by setting v(x, t) = e−tMw(etMx, t) and p(x, t) =q(etMx, t) we obtain the following set of equations defined on the fixed domain Ω:∂tv − ∆v + v · ∇v − Mx · ∇v + Mv + ∇p = 0, in Ω × R+,∇ · v = 0, in Ω × R+,v(x, t) = Mx, on Γ × R+,v(x, 0) = w0(x), in Ω.(2)Note that the coefficient of the convection term Mx · ∇u is unbounded, whichimplies that this term cannot be treated as a perturbation of the Stokes operator.This problem was first considered by Hishida in L2σ(Ω) for Ω ⊂ R3 and Mx =ω × x with ω = (0, 0, 1)T in [15] and [16]. The Lp-theory was developed by Heckand the authors in [12] even for general M .We will construct mild solutions for w0 ∈ Lpσ(Ω), p ≥ n, to the problem (2)with Kato’s iteration (see [18]).The starting point is the linear problem∂tu − ∆u − Mx · ∇u + Mu + b · ∇u + u · ∇b + ∇p = 0, in Ω × R+,∇ · u = 0, in Ω × R+,u = 0, on Γ × R+,u(x, 0) = w0(x), in Ω,(3)where b ∈ C∞c (Ω). The additional term b · ∇u + u · ∇b simplifies the treatment ofthe Navier-Stokes problem (see (11) below). We will first show that the solutionof (3) is governed by a C0-semigroup on Lpσ(Ω). More precisely, let LΩ,b be definedbyLΩ,bu := PΩLbuD(LΩ,b) := {u ∈ W 2,p(Ω) ∩ W 1,p0 (Ω) ∩ Lpσ(Ω) : Mx · ∇u ∈ Lp(Ω)},where Lbu := ∆u + Mx · ∇u−Mu + b · ∇u + u · ∇b. Then the following theoremis proved in [12].Theorem 2.1. Let 1 < p < ∞ and let Ω ⊂ Rn be an exterior domain withC1,1-boundary. Assume that tr M = 0 and b ∈ C∞c (Ω). Then the operator LΩ,bgenerates a C0-semigroup TΩ,b on Lpσ(Ω).Sketch of the proof. The proof is devided into several steps. First it is shownthat LΩ,b is the generator of an C0-semigroup TΩ,b on L2σ(Ω). Then a-priori Lp-estimates for TΩ,b are proved. Once we have shown this we can easily define aconsistent family of semigroups TΩ,b on Lpσ(Ω) for 1 < p < ∞. In the last step thegenerator of TΩ,b on Lpσ(Ω) is identified to be LΩ,b.Lp-THEORY OF THE NAVIER-STOKES FLOW 71We start by showing that LΩ,b is the generator of a C0-semigroup on L2σ(Ω).Choose R > 0 such that supp b ∪ Ωc ⊂ BR(0) = {x ∈ Rn : |x| < R}. We then setD = Ω ∩ BR+5(0),K1 = {x ∈ Ω : R < |x| < R + 3},K2 = {x ∈ Ω : R + 2 < |x| < R + 5}.Denote by Bi for i ∈ {1, 2} Bogovskĭı’s operator (see [1], [9, Chapter III.3], [13])associated to the domain Ki and choose cut-off functions ϕ, η ∈ C∞(Rn) suchthat 0 ≤ ϕ, η ≤ 1 andϕ(x) ={0, |x| ≤ R + 1,1, |x| ≥ R + 2, and η(x) ={1, |x| ≤ R + 3,0, |x| ≥ R + 4.For f ∈ Lpσ(Ω) we denote by fR the extension of f by 0 to all of Rn. Then,since C∞c,σ(Ω) is dense in Lpσ(Ω), fR ∈ Lpσ(Rn). Furthermore, we set fD =ηf − B2((∇η)f). Since∫K2(∇η)f = 0 it follows from [9, Chapter III.3] thatfD ∈ Lpσ(D).By the perturbation theorem for analytic semigroups there exists ω1 ≥ 0 suchthat for λ > ω1 there exist functions uDλ and pDλ satisfying the equations(λ − Lb)uDλ + ∇pDλ = fD, in D × R+,∇ · uDλ = 0, in D × R+,uDλ = 0, on ∂D × R+.(4)Moreover, by [14, Lemma 3.3 and Prop. 3.4], there exists ω2 ≥ 0 such that forλ > ω2 there exists a function uRλ satisfying(λ − L0)uRλ = fR, in Rn × R+,∇ · uRλ = 0, in Rn × R+.(5)For λ > max{ω1, ω2} we now define the operator Uλ : Lpσ(Ω) → Lpσ(Ω) byUλf = ϕuRλ + (1 − ϕ)uDλ + B1(∇ϕ(uRλ − uDλ )),(6)where uRλ and uDλ are the functions given above, depending of course on f . Bydefinition, we haveUλf ∈ {v ∈ W 2,p(Ω) ∩ W 1,p0 (Ω) ∩ Lpσ(Ω) : Mx · ∇v ∈ Lpσ(Ω)}.(7)Setting Pλf = (1 − ϕ)pDλ , we verify that (Uλf, Pλf) satisfies(λ − Lb)Uλf + ∇Pλf = f + Tλf, in Ω × R+,∇ · Uλf = 0, in Ω × R+,Uλf = 0, on ∂Ω × R+,where Tλ is given byTλf = −2(∇ϕ)∇(uRλ − uDλ ) − (∆ϕ + Mx · (∇ϕ))(uRλ − uDλ ) + (∇ϕ)pDλ+ (λ − ∆ − Mx · ∇ + M)B1((∇ϕ)(uRλ − uDλ )).72 M. GEISSERT and M. HIEBERIt follows from [12, Lemma 4.4] that for α ∈ (0, 12p′ ), where 1p + 1p′ = 1, thereexists a strongly continuous function H : (0,∞) → L(Lpσ(Ω)) satisfying‖H(t)‖L(Lpσ(Ω)) ≤ Ctα−1eω̃t, t > 0(8)for some ω̃ ≥ 0 and C > 0 such that λ → PΩTλ is the Laplace Transform of H.We thus easily calculate‖PΩTλ‖L(Lpσ(Ω)) ≤ Cλ−α, λ > ω.Therefore, Rλ := Uλ∑∞j=0(PΩTλ)j exists for λ large enough and (λ−Lb)Rλf = ffor f ∈ L2σ(Ω). Since LΩ,b is dissipative in L2σ(Ω), LΩ,b generates a C0-semigroupTΩ,b on L2σ(Ω). Moreover, we have the representationTΩ,b(t)f =∞∑n=0Tn(t)f, f ∈ L2σ(Ω),(9)where Tn(t) :=∫ t0Tn−1(t − s)H(s) ds for n ∈ N andT0(t) = ϕTR(t)fR + (1−ϕ)TD,b(t)fD + B1((∇ϕ)(TR(t)fR − TD,b(t)fD)), t ≥ 0.Here TR denotes the semigroup on Lpσ(Rn) generated by LRn,0 and TD,b denotesthe semigroup on Lpσ(D) generated by LD,b. Note that λ → Uλ is the LaplaceTransform of T0. Since the right hand side of the representation (9) is well definedand exponentially bounded in Lpσ(Ω) by [12, Lemma 4.6], we can define a familyof consistent semigroups TΩ,b on Lp(Ω) for 1 < p < ∞. Finally, the generator ofTΩ,b on Lp(Ω) is LΩ,b which can be proved by using duality arguments (cf. [12,Theorem 4.1]). Remark 2.2. (a) The semigroup TΩ,b is not expected to be analytic since, by[16, Proposition 3.7], the semigroup TR3 in R3 is not analytic.(b) As the cut-off function ϕ is used for the localization argument similarly to[15] the purpose of η is to ensure that fD ∈ Lpσ(Ω). This is essential to est-ablish a decay property in λ for the pressure PDλ (cf. [12, Lemma 3.5]) and Tλ.(c) The crucial point for a-priori Lp-estimates for TΩ,b on L2σ(Ω) is the existenceof H satisfying (8).Since Lp-Lq smoothing estimates for TR and TD,b follow from [14, Lemma 3.3and Prop. 3.4] and [12, Prop. 3.2], the representation of the semigroup TΩ,b givenby (9) and estimates for sums of convolutions of this type (cf. [12, Lemma 4.6])yield the following proposition.Proposition 2.3. Let 1 < p < q < ∞ and let Ω ⊂ Rn be an exterior domainwith C1,1-boundary. Assume that trM = 0 and b ∈ C∞c (Ω). Then there existconstants C > 0, ω ≥ 0 such that for f ∈ Lpσ(Ω)(a) ‖TΩ,b(t)f‖Lqσ(Ω) ≤ Ct−n2(1p− 1q)eωt‖f‖Lpσ(Ω), t > 0,(b) ‖∇TΩ,b(t)f‖Lp(Ω) ≤ Ct− 12 eωt‖f‖Lpσ(Ω), t > 0.Moreover, for f ∈ Lpσ(Ω)‖tn2(1p− 1q)TΩ,b(t)f‖Lqσ(Ω) + ‖t12∇TΩ,b(t)f‖Lp(Ω) → 0, for t → 0.Lp-THEORY OF THE NAVIER-STOKES FLOW 73In order to construct a mild solution to (2) choose ζ ∈ C∞c (Rn) with 0 ≤ ζ ≤ 1and ζ = 1 near Γ. Further let K ⊂ Rn be a domain such that supp∇ζ ⊂ K. Wethen define b : Rn → Rn byb(x) := ζMx − BK((∇ζ)Mx),(10)where BK is Bogovskĭı’s operator associated to the domain K. Then div b = 0and b(x) = Mx on Γ. Setting u := v − b, it follows that u satisfies∂tu − Lbu + ∇p = F in Ω × (0, T ),∇ · u = 0 in Ω × (0, T ),u = 0 on Γ × (0, T ),u(x, 0) = u0(x) − b(x), in Ω,(11)with ∇ · (u0 − b) = 0 in Ω and F = −∆b − Mx · ∇b + Mb + b · ∇b, provided usatisfies (2). Applying the Helmholtz projection PΩ to (11), we may rewrite (11)as an evolution equation in Lpσ(Ω):u′ − LΩ,bu + PΩ(u · ∇u) = PΩF, 0 < t < T,u(0) = u0 − b.(12)Note that we need the compatibility condition u0(x) · n = Mx · n on ∂Ω toobtain u0 − b ∈ Lpσ(Ω). In the following, given 0 < T < ∞, we call a functionu ∈ C([0, T );Lpσ(Ω)) a mild solution of (12) if u satisfies the integral equation for0 < t < Tu(t) = TΩ,b(t)(u0 − b) −t∫0TΩ,b(t − s)PΩ(u · ∇u)(s) ds +t∫0TΩ,b(t − s)PΩF (s) ds.Then the main result of [12] is the following theorem.Theorem 2.4. Let n ≥ 2, n ≤ p ≤ q < ∞ and let Ω ⊂ Rn be an exteriordomain with C1,1-boundary. Assume that trM = 0 and b ∈ C∞c (Ω) and u0 − b ∈Lpσ(Ω). Then there exist T0 > 0 and a unique mild solution u of (12) such thatt → tn2 ( 1p− 1q )u(t) ∈ C ([0, T0] ;Lqσ(Ω)) ,t → tn2 ( 1p− 1q )+ 12∇u(t) ∈ C ([0, T0] ;Lq(Ω)) .3. Strong solutionsIn this section we construct strong solutions to problem (1) for Ω ⊂ Rn, n ≥ 2 andtrM = 0. The main difference to the method presented in the previous sectionis another change of variables. Indeed, we construct a change of variables whichcoincides with a simple rotation in a neighborhood of the rotating body but itequals to the identity operator far away from the rotating body. More precisely,74 M. GEISSERT and M. HIEBERlet X(·, t) : Rn → Rn denote the time dependent vector field satisfying∂X∂t(y, t) = −b(X(y, t)), y ∈ Rn, t > 0,X(y, 0) = y, y ∈ Rn,where b is as in (10). Similarly to [6, Lemma 3.2], the vector field X(·, t) is aC∞-diffeomorphism form Ω onto Ω(t) and X ∈ C∞([0,∞) × Rn). Let us denotethe inverse of X(·, t) by Y (·, t). Then, Y ∈ C∞([0,∞)×Rn). Moreover, it can beshown that for any T > 0 and |α| + k > 0 there exists Ck,α,T > 0 such thatsupy∈Rn,0≤t≤T∣∣∣∣ ∂k∂tk ∂α∂yα X(y, t)∣∣∣∣ + supx∈Rn,0≤t≤T∣∣∣∣ ∂k∂tk ∂α∂xα Y (x, t)∣∣∣∣ ≤ Ck,α,T0 .(13)Settingv(x, t) = JX(Y (x, t), t)w(Y (x, t), t), x ∈ Ω, t ≥ 0,where JX denotes the Jacobian of X(·, t) andp(x, t) = q(Y (x, t), t), x ∈ Ω, t ≥ 0,similarly to [6, Prop. 3.5] and [17], we obtain the following set of equations whichare equivalent to (1).∂tv − Lv + Mv + N v + Gp = 0, in Ω × R+,∇ · v = 0, in Ω × R+,v(x, t) = Mx, on Γ × R+,v(x, 0) = w0(x), in Ω.(14)Here(Lv)i =n∑j,k=1∂∂xj(gjk∂vi∂xk)+ 2n∑j,k,l=1gklΓijk∂vj∂xl+n∑j,k,l=1(∂∂xk(gklΓijl) +n∑m=1gklΓmjlΓikm)vj ,(N v)i =n∑j=1vj∂vi∂xj+n∑j,k=1Γijkvjvk,(Mv)i =n∑j=1∂Xj∂t∂vi∂xj+n∑j,k=1(Γijk∂Xk∂t+∂Xi∂xk∂2Yk∂xj∂t)vj ,(Gp)i =n∑j=1gij∂p∂xjwithgij =n∑k=1∂Xi∂yk∂Xj∂yk, gij =n∑k=1∂Yk∂xi∂Yk∂xjandΓkij =12n∑l=1gkl(∂gil∂xj+∂gjl∂xi+∂gij∂xl).Lp-THEORY OF THE NAVIER-STOKES FLOW 75The obvious advantage of this approach is that we do not have to deal with anunbounded drift term since all coefficients appearing in L, N , M and G are smoothand bounded on finite time intervals by (13). However, we have to consider a non-autonomous problem. Setting u = v − b, we obtain the following problem withhomogeneous boundary conditions which is equivalent to (14).∂tu − Lu + Mu + Nu + Bu + Gp = Fb, in Ω × R+,∇ · u = 0 in Ω × R+,u = 0, on Γ × R+,u(x, 0) = w0(x) − b(x), in Ω.(15)Here,(Bu)i =n∑j=1(uj∂bi∂xj+ bj∂ui∂xj)+ 2n∑j,k=1Γijkujbk, Fb = Lb −Mb −N b.Since gij is smooth and gij(·, 0) = δij by definition, it follows from (13) that‖gij(·, t) − δij‖L∞(Ω) → 0, t → 0.(16)In other words, L is a small perturbation of ∆ and G is a small perturbation of ∇for small times t. This motivates to write (15) in the following form.∂tu − ∆u + ∇p = F (u, p), in Ω × R+,∇ · u = 0, in Ω × R+,u = 0, on Γ × R+,u(x, 0) = w0(x) − b(x), in Ω,(17)where F (u, p) := (L − ∆)u − Mu − Nu + (∇ − G)p − Bu + Fb. We will usemaximal Lp-regularity of the Stokes operator and a fixed point theorem to showthe existence of a unique strong solution (u, p) of (15). More precisely, letXp,qT := W1,p(0, T ;Lq(Ω)) ∩ Lp(0, T ;D(Aq)) × Lp(0, T ; Ŵ 1,p(Ω)),where D(Aq) := W 2,q(Ω)∩W 1,q0 (Ω)∩Lqσ(Ω) is the domain of the Stokes operator.Then, by maximal Lp-regularity of the Stokes operator, Hölder’s inequality andSobolev’s embedding theorems Φ : Xp,qT → Xp,qT , Φ((ũ, p̃)) := (u, p) where (u, p) isthe unique solution of∂tu − ∆u + ∇p = F (ũ, p̃), in Ω × (0, T )∇ · u = 0, in Ω × (0, T ),u = 0, on Γ × (0, T ),u(x, 0) = w0(x) − b(x), in Ω,is well-defined for 1 < p, q < ∞ with n2q + 1p < 32 and T > 0. Here, the restrictionon p and q comes from the nonlinear term N .Finally, let Xp,qT,δ := {(u, p) ∈ Xp,qT : ‖(u, p)−(û, p̂)‖Xp,qT ≤ δ, u(0) = w0−b} with(û, p̂) = Φ(Φ(0, 0)). Then by (16), Hölder’s inequality and Sobolev’s embeddingtheorems, it can be shown that for small enough δ > 0 and T > 0, Ψ|Xp,qT,δ is acontraction.We summarize our considerations in the next theorem which is proved in [7].Note that the cases n = 2, 3 and p = q = 2 were already proved in [6].76 M. GEISSERT and M. HIEBERTheorem 3.1. Let 1 < p, q < ∞ such that n2q + 1p < 32 and let Ω ⊂ Rn be anexterior domain with C1,1-boundary. Assume that tr M = 0 and that w0 − b ∈(Lqσ(Ω),D(Aq))1− 1p ,p. Then there exist T > 0 and a unique solution (u, p) ∈ Xp,qTof problem (15).References1. Bogovskĭı M. E., Solution of the first boundary value problem for an equation of continuityof an incompressible medium. Dokl. Akad. Nauk SSSR 248 (1979), 1037–1040.2. Banin A. Mahalov A. and Nicolaenko B., Global regularity of 3D rotating Navier-Stokesequations for resonant domains. Indiana Univ. Math. J. 48 (1999), 1133–1176.3. , 3D Navier-Stokes and Euler equations with initial data characterized by uniformlylarge vorticity. Indiana Univ. Math. J. 50 (2001), 1–35.4. Borchers W., Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungeninkompressibler viskoser Flüssigkeiten, Habilitationschrift Universität Paderborn, 1992.5. Chen Z. and Miyakawa T., Decay properties of weak solutions to a perturbed Navier-Stokessystem in Rn. Adv. Math. Sci. Appl. 7 (1997), 741–770.6. Cumsille P. and Tucsnak M., Strong solutions for the Navier-Stokes flow in the exterior ofa rotation obstacle, Preprint, l’Institut Élie Cartan, 2004.7. Dintelmann E., Geissert M. and Hieber M., Strong solutions to the Navier-Stokes equationsin the exterior of a moving or rotating obstacle, in preparation.8. Farwig R., Hishida T. and Müller D., Lq-theory of a singular ‘winding’ integral operatorarising from fluid dynamics, Pacific J. Math. 215(2) (2004), 297–312. TU Darmstadt, 2003.9. Galdi G. P., An Introduction to the Mathematical Theory of the Navier-Stokes equations.Vol. I, Springer Tracts in Natural Philosophy, Vol. 38, Springer 1998.10. , Steady flow of a Navier-Stokes fluid around a rotating obstacle, J. Elasticity 71(1–3) (2003), 1–31.11. Galdi G. P., and Silvestre A. L., Strong Solutions to the Navier-Stokes Equations Around aRotating Obstacle, Arch. Ration. Mech. Anal. 1763(3) (2005), 331–350.12. Geissert M., Heck H. and Hieber M., Lp-theory of the Navier-Strokes flow in the exteriorof a moving or rotating obstacle, J. Reine Angew. Math., to appear.13. , On the equation div u = f and the Bogovskĭı Operator, in: G. Sweers (ed.), Func-tional Analysis and PDE, Birkhäuser, to appear.14. Hieber M. and Sawada O., The Navier-Stokes equations in Rn with linearly growing initialdata. Arch. Rational Mech. Anal., 175(2) (2005), 269–285.15. Hishida T., An existence theorem for the Navier-Stokes flow in the exterior of a rotatingobstacle. Arch. Rat. Mech. Anal., 150 (1999), 307–348.16. , The Stokes operator with rotation effect in exterior domains. Analysis, 19 (1999),51–67.17. Inoue A. and Wakimoto M., On existence of solutions of the Navier-Stokes equation in atime dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24(2) (1977), 303–319.18. Kato T., Strong Lp-solutions of Navier-Stokes equations in Rn with applications to weaksolutions. Math. Z., 187 (1984), 471–480.M. Geissert, Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7,D-64289 Darmstadt, Germany,e-mail : geissert@mathematik.tu-darmstadt.deM. Hieber, Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, D-64289 Darmstadt, Germany,e-mail : hieber@mathematik.tu-darmstadt.de

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$L^p$-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle

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$L^p$ -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle

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LpL^pLp-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle

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$L^p$ -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle