Asymptotic estimates to a free boundary problem for the stationary Navier-Stokes equations

Transcription

1 Asymptotic estimates to a free boundary problem for the stationary Navier-Stokes equations Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades einer Doktorin der Naturwissenschaften genehmigte Dissertation vorgelegt von Dipl.-Math. Ute May aus Aachen Berichter: Prof. em. Dr. Josef Bemelmans Prof. Dr. Heiko von der Mosel Prof. Giovanni P. Galdi Tag der mündlichen Prüfung: 20. Dezember 2013 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar.

2 Abstract We analyse the decay of the flow behind a moving body B in a domain Ω h filled with a Newtonian incompressible fluid, where the body is partly immersed in the fluid. The only force acting in this setting is the body moving forward with constant velocity v. To neglect the influence of the bottom and walls of the domain, we assume that the container filled with fluid is infinitely deep and wide. The shape of the surface of the fluid and the position of the body are unknowns of the problem. Our aim is to find a solution to the free boundary value problem and to specify the asymptotic behaviour of the solution. The equations of motion in the fluid are given by the Navier-Stokes equations. We follow the usual approach to show existence of a stationary solution to a free boundary value problem by showing convergence to an iteration process: First we consider the problem on a fixed domain. We solve the Navier-Stokes equations and obtain the velocity u and the pressure p. This solution is then used to calculate the surface and the position of the body in a new domain. In the new domain we again solve the Navier-Stokes equations and so on. If we can show convergence to the sequence of solutions, we get a stationary solution to the free boundary problem. In the main part of the thesis we investigate the asymptotic structure of the static solution on a fixed domain. With the help of a reflection argument we show a representation formula for the velocity on the basis of a representation formula in an exterior domain. The result is exactly what we would expect: The behaviour of the solution at large distance from the body corresponds to the one in an exterior domain, but is perturbed by a term that depends on the surface of the fluid h. When we consider the Oseen equations as a linearisation we can also prove a wake region behind the body. It has the shape of a half paraboloid again perturbed by terms depending on h. As the asymptotic representation formula holds on Ω h we can also describe the asymptotic structure on the surface of the fluid. The last chapter deals with a two dimensional problem of a similar type. We show, that the decay behind a floating body in a two dimensional layer is exponentially fast. As we consider a layer instead of a half-space, we have a Poincaré-inequality, which directly gives uniqueness of the solution. Therefore we concentrate on the asymptotic structure of the solution in this chapter. i

3 Kurzfassung Wir analysieren das Abklingverhalten der Strömung hinter einem schwimmenden Körper B in einem Gebiet Ω h in einer inkompressiblen Newtonschen Flüssigkeit. Die einzige wirkende Kraft ist die Vorwärtsbewegung des Körpers mit konstanter Geschwindigkeit v. Um den Einfluss des Bodens und der Wände in dem Gebiet auszublenden, nehmen wir an, dass der Behälter mit der Flüssigkeit unendlich tief und breit ist. Die Oberflächenform der Flüssigkeit und die Position des Körpers sind Unbekannte des Problems. Unser Ziel ist es, eine Lösung des freien Randwertproblems zu finden und das asymptotische Verhalten der Lösung zu analysieren. Die Bewegungsgleichungen in der Flüssigkeit sind durch die Navier-Stokes-Gleichungen gegeben. Mit einem Iterationsverfahren zeigen wir die Existenz einer Lösung des freien Randwertproblems: Zunächst betrachten wir das Problem auf einem festen Gebiet. Wir lösen die Navier-Stokes-Gleichungen und erhalten die Geschwindigkeit u und den Druck p. Mithilfe dieser Lösung berechnen wir die Oberfläche und die Position des Körpers in einem neuen Gebiet. In dem neuen Gebiet lösen wir abermals die Navier-Stokes-Gleichungen. Wenn wir zeigen können, dass diese Folge der Lösungen konvergiert, erhalten wir eine stationäre Lösung des freien Randwertproblems. Um das asymptotische Verhalten der Lösung zu analysieren, zeigen wir mit Hilfe eines Spiegelungsargumentes eine Darstellungsformel für die Geschwindigkeit. Das Abklingverhalten der Lösung entspricht dem im Außenraum, abgesehen von einem Term, der von der Oberfläche der Flüssigkeit h abhängt. Wenn wir die Oseen- Gleichungen betrachten, können wir auch einen Nachlauf hinter dem Körper beweisen. Er hat die Form eines halben Paraboloids, gestört durch einen von h abhängigen Faktor. Da die asymptotische Darstellungsformel auf Ω h gilt, können wir auch die asymptotische Struktur der Geschwindigkeit auf der Oberfläche der Flüssigkeit beschreiben. Das letzte Kapitel beschäftigt sich mit einem ähnlichen zweidimensionalen Problem. Wir zeigen, dass die Geschwindigkeit hinter einem schwimmenden Körper in einer zweidimensionalen Schicht exponentiell schnell abklingt. Da wir eine Schicht anstelle einer Halbraumes betrachten, gilt die Poincaré-Ungleichung, die direkt die Eindeutigkeit der Lösung liefert. Daher untersuchen wir nur die asymptotische Struktur der Lösung. ii

4 Ich möchte mich herzlich bei Professor Bemelmans für die motivierende und hilfsbereite Betreuung der Arbeit bedanken. Niemand versteht es besser, Begeisterung für mathematische Problemstellungen zu wecken. A special thanks to Professor Galdi, who has also contributed significantly to the success of this work. Die Dissertation und insbesondere der Auslandsaufenthalt bei Professor Galdi wurde unterstützt von der Graduiertenförderung der RWTH Aachen. Auch hierfür möchte ich mich herzlich bedanken. Ein großes Dankeschön geht an meine Kollegen und meine Familie, die mir immer unterstützend zur Seite gestanden haben. Vielen Dank an Fabian und Julia Heimann. Fabian hat die diese Dissertation von der ersten bis zur letzten Zeile in allen Höhen und Tiefen begleitet, Julia sogar ihr ganzes bisheriges Leben lang. Vielen Dank für stundenlanges Zuhören, geduldiges Lesen und unerschütterlichen Optimismus. iii

5 Contents 1 Introduction Motivation Statement of the problem Outline of the thesis Notations 7 3 Existence on a fixed domain 11 4 Transformation to a reference domain 21 5 Regularity and Uniqueness 25 6 Asymptotic behaviour of the solution Decay of the velocity Decay of the surface function The free boundary value problem 51 8 The two dimensional problem in a layer 58 iv

6 1 Introduction 1.1 Motivation The behaviour of a flow in a Newtonian fluid, that is induced by a moving body, partly or completely immersed in the fluid, has been worked on for a long time. One of the first mathematicians and physicians concerned with this problem was Oseen, who already in 1937 found exact representations of solutions to the - in that time - so called generalized Stokes equations. The model problem always was and still is a body that is moving in an infinite container filled with fluid. Oseen thought about the influence of a bottom of the domain on the fluid flow. In his book [Ose27] he analyses the flow around a small ball that is moving next to a wall and he obtains an explicit representation of the solution. Later the model problem of the moving body in the whole space was analysed very elaborately by Finn and by Galdi, the decay behind the body was especially the topic of [Gal11] and [Fin65]. It turns out, that in three dimensions the velocity decays like x 1, which is the same asymptotic behaviour as shown by the fundamental solution to the Oseen equations. Finn calls the solutions to this model problem physically reasonable solutions, or in short PR-solutions. The velocity shows a different decay depending on the direction behind the moving body. Outside a paraboloidal region behind the body the velocity decays like x 2, inside the so called wake region the decay gets slower the more the direction approaches the horizontal axis. This behaviour is exactly what you can observe in reality and is therefore physically reasonable. The case of a layer like domain was worked on by Nazarov and Pileckas in [NP99a] and [NP99b]. Here we can see very clearly the influence of a restriction of the domain in one dimension, but they do not consider a moving body in the layer. Instead, the flow is induced by an external force. For example they prescribe the flow through the cross section of the layer. They find out, that the flow has a profile of the Poiseuille type and that the velocity has no component in the direction normal to the bottom and the top of the layer. Unfortunately their argument does not work any longer, if we assume, that the flow is induced by a moving body, because in this case the velocity in the normal direction does not vanish. 1

7 1 Introduction A paper of Hillairet and Wittwer [HW09] deals with the velocity normal to the bottom of a domain, that is unbounded in every other direction. They consider a body moving next to a wall, for example a bubble that rises up parallel to a wall. They prove, that the rate of decay in the normal direction is like x 3/2, which is, as expected, better than in an exterior domain. Aside of moving bodies we focus on another important topic in the analysis of fluid mechanics. The boundaries of the domains we consider will consist not only of the boundary of the moving body, but also of a free boundary. The domain in that case becomes also an unknown of the problem. There are various papers on free boundary value problems, especially in the most recent years by Galdi, by Bemelmans and by Kyed. Already in 1991 Gellrich analysed the decay of a fluid flow in a layer with a upper free boundary in her dissertation. Her work is by many aspects the basis for our results. In [Gel91] she determines the decay of a flow induced by an external force in an infinite layer. The layer has a fixed bottom and a free upper surface. She finds out, that the decay of the flow and the free surface are exponential. In contrast to our problem, there is no floating body and the external force is of compact support or decays exponentially fast. The second difference is the shape of the domain: In a layer we can use a Poincaréinequality, whereas in a half-space as in our case Poincaré s inequality does not hold. Nevertheless the techniques to deal with the free boundary problem are essentially the same for both problems. 1.2 Statement of the problem We consider two different problems to analyse the decay of a flow behind a moving body, one in three dimensions and another in two space dimensions. For the three dimensional case we consider a rigid body B floating on a Newtonian incompressible fluid. The body is assumed to be convex and have a C 2 -boundary. The only force acting in this setting is the body moving forward with constant velocity v in the direction of the negative x 1 -axis. In this context, we are not interested in how this movement is realized, we imagine for example a stick that is attached to the highest point of B, that pushes the body in a constant movement. The angle between the free surface and the body is assumed to be π 2. To neglect the influence of the bottom and walls of the domain, we assume, that the container filled with fluid is infinitely deep and wide. We can also allow for an external force f to be acting in the fluid. As this does not represent a natural phenomenon we consider the problem with a right hand side f 0 only for technical reasons in some cases. 2

8 1 Introduction v B Figure 1.1: Constantly moving body Our aim is to find a solution to the free boundary value problem and to specify the asymptotic behaviour of the solution. The equations of motion in the fluid are given by the Navier-Stokes equations ν u + (u )u + p = f in Ω h, where ν is the viscosity, u is the velocity vector-field, p is the pressure and f is the external force mentioned above. Ω h is the domain that is filled with the fluid. As we only consider incompressible fluids, u has to fulfil the condition u = 0. The boundary Ω h consists of the wetted part of B, which we call Γ B := Ω h B and of the free surface Γ h. On Γ B we assume no slip boundary conditions, on the free surface we have a kinematic boundary condition that ensures, that there is no flow through the free surface and a dynamic boundary condition, that is, in our case, realized by a tangential stress condition. We further assume, that the fluid is at rest at infinity. We assume a coordinate system attached to the body. Therefore we choose the origin to be in the center of B and get the following system of equations on the domain Ω h ν v + (v )v + p = f in Ω h v = 0 in Ω h v = 0 on Γ B v n = 0 on Γ h τ (i) T (v, p) n = 0 on Γ h for i = 1, 2, v v for x, where n is the outer normal on the free surface Γ h, τ i, i = 1, 2 are the tangential vectors and T (v, p) is the stress tensor T (v, p) = p1 + ν 2 ( v + ( v) T ). 3

9 1 Introduction In the free boundary value problem the domain is also an unknown of the problem, we need a further condition to calculate the free surface and the position of the body, that is given by a function h : R 2 R. x 3 h(x 1, x 2 ) (x 1, x 2 ) Figure 1.2: Function h, that describes the surface of the fluid The condition for h is given by where H(h) is the mean curvature n T (v, p) n + ρgh = 2κH(h) H(h) := h 1 + h 2 of h, ρ is the density of the fluid and g is the gravitational acceleration, κ stands for the capillarity constant of the fluid. As we have our coordinate system attached to the body and the domain is infinitely deep, the position of the body in the fluid is only characterized by the position of the contact line Σ between the surface of the fluid and the boundary of the body. We do not allow the body to change its orientation or to rotate. This restriction is already realized by the prescribed motion of the body. We can again imagine the body to be attached to a stick and just to let it move up or down, while the stick is moving forwards with the prescribed velocity v. The position of the contact line is calculated with a variational approach by Bemelmans, Galdi and Kyed ([BGK11]). They show that the solution of the variational problem satisfies the equation for the function h mentioned above. 4

10 1.3 Outline of the thesis 1 Introduction We follow the usual approach to show existence of a stationary solution to a free boundary value problem (see for example [Gel91]) by showing convergence to an iteration process: First we consider the problem on a fixed domain Ω h. For simplicity the boundary of the domain is assumed to be planar in the first step except for the floating body B. We solve the Navier-Stokes equations and obtain the velocity u and the pressure p on the domain Ω h. This solution is then used to calculate the surface and the position of the body in a new domain. The calculation of the surface is more complicated than in [Gel91] because of the floating body. The existence of a solution to the corresponding problem is established in [BGK11] with the help of a variational approach. In the new domain we again solve the Navier-Stokes equations and so on. If we can show convergence to the sequence of solutions, we get a stationary solution to the free boundary problem. We can of course analyse the properties of this stationary solution if we consider a solution on an arbitrary fixed domain. The dependence on the boundary of the domain is in our case a dependence on the surface of the fluid and the position of the body as the domain is unbounded in every other direction. We use the Banach fix point theorem to show convergence to the sequence of solutions and so we need to compare the velocities and the pressures on different domains and to different data. Therefore we transform the domain onto a reference domain, that is, in our case, the half-space R 3 where the floating body is transformed onto a ball of radius one. We chose this transform, because we have to make sure, that the contact angle between the surface of the fluid and the boundary of the body does not change. The velocity and pressure are transformed with the help of Piola s identity (see [Sim05]) and the transformed Navier-Stokes equations are calculated in [Bem87]. With the help of this mapping we can use a reflection argument to show regularity and uniqueness of the solution on every fixed domain. A similar reflection argument was established in [Jin05], the regularity and uniqueness arguments we can use after the reflection can be found in [Gal11]. In the next chapter we investigate the asymptotic structure of the static solution on a fixed domain. Again with the help of a reflection argument we show a representation formula for the velocity on the basis of a representation formula in an exterior domain (see[gal11]). The result is exactly what we would expect: The behaviour of the solution at large distance from the body corresponds to the one in an exterior domain, but is perturbed by a term that depends on the surface of the fluid h. When we consider the Oseen equations as a linearisation we can also prove a wake region behind the body. It has the shape of a half paraboloid again perturbed by terms depending on h. As the asymptotic representation formula holds on Ω h we can also 5

11 1 Introduction describe the asymptotic structure on the surface of the fluid. We can also estimate the asymptotic behaviour of the function h that describes the surface of the fluid with the help of an indirect proof. The last chapter deals with a two dimensional problem of a similar type. We show, that the decay behind a floating body in a two dimensional layer is exponentially fast. The technique here is different from the three dimensional case, we use an approach of Galdi ([Gal08]). As we consider a layer instead of a half-space, we have a Poincaré-inequality, which directly gives uniqueness of the solution. Therefore we concentrate on the asymptotic structure of the solution in this chapter. 6

12 2 Notations In what follows we denote the domain filled with fluid by Ω h := {x R 3 : x 3 < h(x 1, x 2 )} \ B, where B is the body floating on the fluid and h : R 2 R is the surface function. For simplicity we write x := (x 1, x 2 ), then we have h h(x ). The flat reference domain, that we need to show regularity and uniqueness of the solution, is S := R 3 \ B 1 (0). x 3 Γ h Γ B (x 1, x 2 ) Figure 2.1: The domain Ω h The boundaries of the domains are Γ B := Ω h B, Γ h := {x R 3 : x 3 = h(x 1, x 2 )} \ B and respectively. Γ 1 := B 1 (0) R 3, Γ 0 := R 2 \ B 1 (0) The mapping from Ω h onto S is denoted by φ : S Ω h whereas the inverse mapping ψ : Ω h S maps Ω h onto the reference domain. 7

13 2 Notations x 3 ψ x 3 Γ h Γ B (x 1, x 2 ) Γ 1 Γ 0 (x 1, x 2 ) Ω h S φ Figure 2.2: Ω h and the reference domain S We define the following function spaces: For k N or k = and Ω a domain like Ω h or S we define C k 0 (Ω) := {u C k (Ω) : supp u Ω} and Moreover we set with the norm C k 0 (Ω) := {u Ω : u C k 0 (R 3 )}. W 1,2 (Ω) := {u : Ω R 3 : u W 1,2 (Ω) < } ( u W 1,2 (Ω) := u 2 + u 2 dx Ω ) 1 2 W 1,2 (Ω) is a Hilbert space with scalar-product u, v W 1,2 (Ω) := u v + u v dx. Ω We define the quotient space with the semi-norm and the normed space D 1,2 (Ω) := {u : Ω R 3 : u D 1,2 (Ω) < } ( u D 1,2 (Ω) := u 2 dx Ω D 1,2 (Ω) := D 1,2 (Ω)/R that contains all equivalence classes [u] of functions in D 1,2 (Ω), that satisfy ) 1 2 [u] D1,2 (Ω) = 0 [u] 0. 8

14 2 Notations We denote by u a representative of the class [u]. D 1,2 (Ω) is a Hilbert space with the scalar-product u, v D 1,2 (Ω) := u v dx. As the fluid is incompressible we consider solenoidal functions. We define and in the same way C k 0,σ(Ω) = {u C k 0 (Ω) : u = 0 in Ω} W 1,2 σ (Ω) := {u : Ω R 3 : u W 1,2 (Ω) <, u = 0 in Ω} Ω and Dσ 1,2 (Ω) := {u : Ω R 3 : u D 1,2 (Ω) <, u = 0 in Ω}. We show, that the functions in Wσ 1,2 (Ω) and Dσ 1,2 (Ω) can be approximated by functions from C0,σ (Ω), at least for the domain Ω as it is specified above. In [Gal11, Theorem III.4.3.] the result is shown for a halfspace. To obtain the result for the domain Ω we have to check the conditions i)-iv) in [Gal11, p.198]. We treat the case Wσ 1,2 (Ω), the case Dσ 1,2 (Ω) going completely analogously. (i) It is clear, that Ω R and Ω R,2R as defined in [Gal11] are domains. (ii) We have to check, that problem [Gal11, (III.4.2)] has a solution with a constant c independent of R. This is the case, if we can apply [Gal11, Theorem III.3.1]. Ω and also Ω R and Ω 2R consist of a set of starshaped domains with Lipschitzboundary, as the only critical part of the boundary is the contact line Σ, where we have a right angle. Therefore [Gal11, Theorem III.3.1.] holds. (iii) As we have the Poincaré-inequality [Gal11, (II.5.5.)] in the domain Ω R,2R, the constant c 2 in [Gal11, (III.4.3)] is independent of R (iv) As Ω 2R is a compact domain with Lipschitz-boundary by [Gal11, Theorem III.4.1.]. All conditions are satisfied and we have W 1,2 σ (Ω 2R ) = C 0,σ (Ω 2R) W 1,2 (Ω 2R ) Wσ 1,2 (Ω) = C0,σ (Ω) W 1,2 (Ω) and D 1,2 σ (Ω) = C 0,σ (Ω) D 1,2 (Ω). 9

15 2 Notations We want to consider functions with homogeneous Dirichlet boundary data on Γ B and mixed boundary data on Γ h. We set C k σ,γ(ω) := {ϕ C k 0,σ(Ω) : ϕ = 0 on Γ B, ϕ n = 0 on Γ h } and we define the Sobolev spaces W 1,2 σ,γ (Ω) := C σ,γ (Ω) W 1,2 (Ω) and D 1,2 σ,γ (Ω) := C σ,γ (Ω) D 1,2 (Ω). ( ) The dual space D 1,2 σ,γ (Ω) of D 1,2 σ,γ (Ω) contains all linear functionals on D1,2 and is equipped with the norm We have ([Gal94a][Theorem 2.5, p.39]) l 1,2 := sup l(u). (D σ,γ (Ω)) u D 1,2 (Ω) =1 ( D 1,2 σ,γ (Ω) ) = D 1, 2 σ,γ (Ω). σ,γ (Ω) For the (weak) derivative of a function u in the direction x i we write i u, the whole gradient is u. As we are not interested in the explicit calculation of all constants, we use the generic constant c R, that may change during a calculation. In most cases we consider a coordinate system, that is attached to the moving body. In that case, we call the velocity v. If we shift the velocity to obtain a vector-field that vanishes at infinity, we designate the resulting vector-field as u := v v. 10

16 3 Existence on a fixed domain In this chapter we want to show, that there exists at least one weak solution to the Navier-Stokes equations in Ω := Ω h, where we assume, that Ω h is a given domain with a fixed height function h = h(x ). For simplicity we set v = (1, 0, 0) T. We start by defining a weak solution to the Navier-Stokes equations in the domain Ω. Definition 3.1. A weak solution to the Navier-Stokes equations with right hand side f D 1,2 0 in Ω is a function v D 1,2 σ,γ (Ω) with ν v ϕ dx + (v )vϕ dx = f ϕ dx Ω Ω Ω for all ϕ D 1,2 σ,γ (Ω) and lim v v dx = 0. r B r(0) For a weak solution u to the Navier-Stokes equations the boundary conditions v = 0 on Γ B v n = 0 on Γ h τ (i) T (v, p) n = 0 on Γ h for i = 1, 2, hold in a weak sense (see [Bem81, Definition 2.]). 11

17 3 Existence on a fixed domain We have the following existence theorem: ( ) Theorem 3.2. Let f D 1,2 0 (Ω), h(x ) = O x 3 2, i h = O( x 5 2 ) for i {1; 2}. Then there exists at least one weak solution to the Navier-Stokes equations ν v + (v )v + p = f in Ω v = 0 in Ω v = 0 on Γ B v n = 0 on Γ h τ (i) T (v, p) n = 0 on Γ h for i = 1, 2, v (1, 0, 0) T for x. Proof. (I) In the first step we change the data of the problem to obtain homogeneous boundary conditions on every part of the boundary, including v (0, 0, 0) for x. As in [Gel93, Lemma 3.1.1] we therefore want to find a flux-carrier-function g D 1,2 (Ω) with the following properties: g = 0 in Ω g = 0 on Γ B g n = 0 on Γ h g (1, 0, 0) T for x. In order to fulfil g (1, 0, 0) T we redefine g on the domain S which has a flat surface for large x. We define the transform F : S Ω (x 1, x 2, x 3) (x 1, x 2, x 3 + h(x 1, x 2)χ(x 1, x 2)), where χ(x 1, x 2 ) is a cut-off function with { χ(x 1, x 0 for x < δ 2) = 1 for x > 2δ for some δ > 0. We have F (x 1, x 2, x 3) = hχ + 1 χh 2 hχ + 2 χh 1 12

18 3 Existence on a fixed domain 2δ δ δ 2δ χ = 0 χ = 1 Figure 3.1: Cut-off function χ and det F = 1. Then ζ : Ω R 3 is defined by (see [Sim05]) ζ(x 1, x 2, x 3 ) := F (F 1 (x 1, x 2, x 3 ))ζ (F 1 (x 1, x 2, x 3 )) [ det F (F 1 (x 1, x 2, x 3 )) ] 1, where ζ : S R 3 is a vector field on the (for large x ) flat domain S. To make sure ζ is solenoidal it is now sufficient to make ζ solenoidal. We set 0 q (x 1, x 2, x 3) := x 3 ρ( x ), 0 where ρ is a cut-off function with ρ( x ) = { 0 for x < 3δ 1 for x > 4δ and ρ( x ) + x ζ (x 1, x 2, x 3) := q (x 1, x 2, x 3 3ρ( x ) 3) = 0. x 3 1ρ( x ) For ζ there holds ζ = 0 in S ζ = 0 on F 1 (Γ B ) ζ n = 0 on F 1 (Γ h ) ζ (1, 0, 0) T for x. 13

19 3 Existence on a fixed domain 4δ 3δ 2δ δ δ 2δ χ = 0 χ = 1 3δ 4δ ρ = 0 ρ = 1 Figure 3.2: Cut-off function ρ As h(x ) 0 for x and lim ζ = lim x x ζ ζ 1 hχ + 1 χh 2 hχ + 2 χh = 0 + lim 0, x 0 1 hχ + 1 χh the function ζ goes to (1, 0, 0) for x, whenever x < δ or x. We have to make sure, that ζ (1, 0, 0) T for x also for x = const > δ. Therefore we set g := ζ + ξ, where 0 χ h 3 η ξ = χ h η = 0 0 ( 1 hχ + 1 χh)η, with a cut-off function η(x 3 ) = { 0 for x3 < 5δ 1 for x 3 > 6δ. Then g = ρ + x 3 3 ρ + χ h 3 η 0 ( 1 hχ + 1 χh)(ρ η) + x 3 3 ρ x 3 1 ρ is solenoidal and fulfils the required Dirichlet boundary conditions. 14

20 3 Existence on a fixed domain 4δ 3δ 2δ δ δ 2δ 3δ χ = 0 χ = 1 4δ ρ = 0 ρ = 1 5δ η = 0 6δ η = 1 Figure 3.3: Cut-off function η In addition Γ h τ (i) T (g, p) n dσ = 0 because of Green s formula and the properties of g (see for example [Bem81, (2.7)]). Due to the structures of the cut-off functions, g shows the following behaviour for large x : 1 g h 3 η 0 ( 1 hχ + h 1 χ)(1 η). For the gradient of g we have: i g 1 (x) = i ρ + δ i3 3 ρ + x 3 i 3 ρ + i χh 3 η + χ i h 3 η + χh i 3 η i g 2 (x) = 0 i g 3 (x) = ( i 1 hχ + 1 h i χ + i 1 χh + 1 χ i h)(ρ η) + ( 1 hχ + 1 χh)( i ρ i η) δ i3 3 ρ + x 3 i 3 ρ δ i3 1 ρ x 3 i 1 ρ. For large x this leads to i g 1 (x) i h 3 η + h i 3 η i g 2 (x) = 0 i g 3 (x) ( i 1 hχ + 1 h i χ + 1 χ i h + h i 3 χ)(ρ η) 1 hχ i η and we obtain ( ) g = O x 3 2 for x because of the the asymptotic behaviour of h and g = 0 for x 3 < 7δ. 15

21 3 Existence on a fixed domain (II) The existence of a weak solution can now be proved exactly as in [Gal94b, Theorem IX.4.1] with the help of the Galerkin method. The solution is of the form v = u + g, where g is the function constructed above and u is a weak solution to ν u + (u )u + p + (g )u + (u )g = f + ν g (g )g in Ω u = 0 in Ω u = 0 on Γ B u n = 0 on Γ h τ (i) T (u, p) n = 0 on Γ h for i = 1, 2, u 0 for x. We can choose a sequence {ψ k } k N C Γ,σ(Ω) such that like in ([Gal94a][Lemma VII.2.1.]) for {ψ k } k N C 0,σ (Ω) 1. the linear hull of {ψ k } is dense in D 1,2 Γ,σ (Ω), 2. (ψ k, ψ j ) = δ kj k, j N, 3. given ϕ CΓ,σ (Ω) and ɛ > 0 there is a m = m(ɛ) N and γ 1,..., γ m R such that m ϕ γ i ψ i C 1 < ɛ. i=1 We then have to find a sequence of approximating solutions (u m ) m=1 form m u m = ξ km ψ k such that k=1 of the ν( u m, ψ k ) + ((u m )u m, ψ k ) + ((u m )g, ψ k ) + ((g )u m, ψ k ) (3.1) = [f, ψ k ] ν( g, ψ k ) ((g )g, ψ k ), holds for k = 1,..., m. We want to use [Gal94b, Lemma VIII.3.2] to find a solution for this system. We multiply the equation by ξ km and sum over k from 1 to m. Then we have to find estimates for the terms ((u m )g, u m ), ( g, u m ) and ((g )g, u m ), since with [Gal94b, Lemma VIII.2.1] ((g )u m, u m ) = ((u m )u m, u m ) = 0. 16

22 3 Existence on a fixed domain To estimate ((u m )g, u m ) we can use the fact that g 0 in the same way as h 0 for x and g = 0 for x 3 < 7δ. We have ((u m )g, u m ) = (u m )g u m ) dx Ω u m 2 g dx Ω c u m 2 L 6 (Ω) g L 3/2 (Ω {x 3 < 7δ}) c u m 2 L 2 (Ω) g L 3/2 (Ω {x 3 < 7δ}). To use [Gal94b, Lemma VIII.3.2] we have to make sure, that But as c g L 3/2 (Ω {x 3 < 7δ}) < ν. c g L 3/2 (Ω {x 3 < 7δ}) = c c c Ω {x 3 < 7δ} Ω {x 3 < 7δ} 1 g 3/2 dx x 9/4 dx r 5/4 dr δ c δ < ν, we can choose δ small enough to obtain the desired relation. The integrals ( g, u m ) and ((g )g, u m ) can be estimated exactly as in the proof of [Gal94b, Theorem IX.4.1]: which is bounded because g L 2 (Ω) ( g, u m ) g L 2 (Ω) u m L 2 (Ω), Ω {x 3 > 7δ} 7δ c R 2 \B ɛ(0) 1 h(x ) 2 dx x 3 dx r 2 dr c <. We still have to estimate the integral Ω g g u m dx. We obtain (( ((g ) g, u m ) g (1, 0, 0) T ) ) g u m dx + ( (1, 0, 0)T ) g u m dx. Ω The first integral can be estimated with the Hoelder inequality (( g (1, 0, 0) T ) ) g u m dx c (( g (1, 0, 0) T ) ) g 6 L 5 u m (Ω) L 6 (Ω) Ω Ω c ( g (1, 0, 0) T ) L 3 (Ω) g L 2 (Ω) u m L 2 (Ω). 17

23 3 Existence on a fixed domain For the second term we use the fact that we differentiate only in the direction of x 1. For large x we have 1 g 1 (x) 1 h 3 η 1 g 2 (x) = 0 1 g 3 (x) ( 1 1 hχ + 1 h 1 χ + 1 χ 1 h)(1 η) and the support of 1 g is bounded in x 3 -direction. Therefore ( (1, 0, 0)T ) g = 1 g c h = O ( x 5/2) and we have the following estimate: ( (1, 0, 0)T ) g u m dx c 1g L 6/5 (Ω {x 3 < 7δ}) u m L 6 (Ω) Ω c x ( 2) dx um L 2 (Ω) Ω {x 3 < 7δ} c x 3 dx u m L 2 (Ω) Ω {x 3 < 7δ} c u m L 2 (Ω). With [Gal94b, Lemma VIII.3.2] we have a solution ξ to (3.1) for all m. Again from (3.1) we find (ν + c) u m L 2 (Ω) f D 1,2 (Ω) + ν g L 2 (Ω) + (g (1, 0, 0) T ) L 3 (Ω) g L 2 (Ω) + 1 g L 6/5 (Ω) and so u m c for all m. We can now select a subsequence that is weakly convergent to u D 1,2 Γ,σ (Ω) and with [Gal11, Exercise II.5.8] strongly convergent in L 2 (Ω R ). With [Gal11, Theorem II.3.1] it follows that u L 2 (Ω) c. We want to show that u obeys ν( u, ψ k ) + ((u )u, ψ k ) + ((u )g, ψ k ) + ((g )u, ψ k ) = [f, ψ k ] ν( g, ψ k ) ((g )g, ψ k ). Clearly ( u m, ψ k ) ( u, ψ k ) for m, ((u m )g, ψ k ) ((u )g, ψ k ) for m 18

24 3 Existence on a fixed domain and (((g )u m ), ψ k ) (((g )u), ψ k ) for m, because of the weak convergence in D 1,2 σ,γ (Ω) and the strong convergence in L2 on Ω k, where we set Ω k = supp(ψ k ). Due to the fact, that supp(ψ k ) = Ω k is bounded we can use the same proof as in [Gal11, Theorem IX.3.1] to show that ((u m )u m, ψ k ) ((u )u, ψ k ) for m. It follows that u D 1,2 σ,γ (Ω) is a weak solution to the Navier-Stokes equations in Ω. There holds the estimate (ν + c) u L 2 (Ω) f D 1,2 (Ω) + ν g L 2 (Ω) + (g (1, 0, 0) T ) L 3 (Ω) g L 2 (Ω) + 1 g L 6/5 (Ω) (3.2) Remark 3.3. We use [Gal11, Theorem III.5.3] to show existence and uniqueness of the pressure p L 6 (Ω) corresponding to the solution v Dσ 1,2 (Ω) to the Navier- Stokes problem ν u + (u )u + 1 u + p = 0 in Ω u = 0 in Ω u 0 for x u = v on Γ B u n = v n on Γ h τ (i) T (u, p) n = 0 on Γ h for i = 1, 2. We define F(ψ) := ν( u, ψ) + (u ψ, u) + (u, 1 ψ). Then F defines a bounded linear functional on D 1,6/5 0 (Ω h ), because for ψ D 1,6/5 0 (Ω) we have ( u, ψ) u L 2 (Ω) ψ L 6/5 (Ω) (u ψ, u) u 2 L 6 (Ω) ψ L 6/5 (Ω) (u, 1 ψ) u L 6 (Ω) ψ L 6/5 (Ω). 19

25 3 Existence on a fixed domain The domain Ω satisfies the cone condition, and so by [Gal11, Remark III.3.14] the problem [Gal11, (III.3.65)] is solvable in Ω h. With [Gal11, Theorem III.5.3] there follows the existence of a uniquely determined p L 6 (Ω h ) such that F(ψ) = p ψ for all ψ D 1,6/5 0 (Ω). Ω 20

26 4 Transformation to a reference domain In this chapter we define a mapping from the plane reference domain S to Ω h and the inverse mapping from Ω h to S. We will need these mappings for the proof of regularity and uniqueness for the solution of the Navier-Stokes equations. It is important, that the part of Ω h with Dirichlet boundary conditions is mapped on the appropriate part of S and likewise for the mixed boundary conditions. In addition we chose the mappings in a way that we can maintain the right angle on the contact line between the surface of the Newtonian fluid and the boundary of the body. As the body has a C 2 -boundary and the surface of the fluid is a graph over R 2 \ B of class C 2, the mapping can be chosen to have no singularities. As we are more interested in the behaviour of the fluid at large distances from the body, than on the effect of the body shape on the fluid flow, we define the mapping piecewisely for small and for large values of x. In our calculations we can then easily estimate the mapping for large x. x 3 ψ x 3 Γ h Γ B (x 1, x 2 ) Γ 1 Γ 0 (x 1, x 2 ) φ Figure 4.1: Ω h and the reference domain S Close to the body the mapping φ : S Ω h is defined by first mapping the unit ball B 1 (0) onto B and then mapping the plane {x 3 = 0} \ B onto the free boundary of the fluid. We have r r B (ϕ, θ) φ (r, ϕ, θ) := ϕ θ ( π γ(r r B(ϕ, θ), ϕ) ) 21

27 4 Transformation to a reference domain in spherical coordinates, where γ(r, ϕ) is the angle between the (x 1, x 2 )-plane and a ray with horizontal angle ϕ that cuts the surface of the fluid at length r, and r B (ϕ, θ) is the distance of B to the origin at the horizontal and vertical angles ϕ and θ. In Cartesian coordinates we use the variables x S and x Ω h. Between the function h(x 1, x 2 ), that describes the surface of the fluid and the angle γ we have the following relation: ( ) h( x1, x 2 ) γ(r, ϕ) = arcsin r where x 1 = r cos(γ(r, ϕ)) cos(ϕ), x 2 = r cos(γ(r, ϕ)) sin(ϕ). The planar domain R 2 {0} \ B 1 (0) is mapped onto the free part of the boundary, whereas the boundary of the unit sphere is mapped onto the boundary of the moving body B. For large x there exists a smooth continuation of φ to the following mapping, which we denote also by φ where η is a cut-off function with for some δ > 0. φ( x 1, x 2, x 3 ) = ( x 1, x 2, x 3 + h( x 1, x 2 ) η( x 3 )), η = { 1 for x3 > δ 0 for x 3 < 2δ We consider the behaviour of the mapping for large x whenever we have to estimate the velocity and pressure on the whole exterior domain E or on S. In these cases we multiply with a constant to clarify that we have different, but bounded integrals on a large ball around the moving body, that can be estimated by the integrals over the rest of the domain. For the Jacobian matrix of φ we get φ = h η 2 h η 1 + h η and hence det φ = 1 + h η. 22

28 4 Transformation to a reference domain We can choose η small enough to ensure, that h η 1 and so we have det φ 0. The second derivatives of φ read as follows: ij φ k = 0 for k = 1, 2 ij φ 3 = i j h η for i, j = 1, 2 3i φ 3 = i h η for i = 1, 2 33 φ 3 = h η. As det φ 0 for all x S there exists the inverse mapping ψ : Ω h S and we obtain: ψ(x 1, x 2, x 3 ) = (x 1, x 2, x 3 h(x 1, x 2 )η(x 3 )), with a suitable cut-off function η, ψ = 0 1 0, 1 hη 2 hη 1 + hη det ψ = 1 + hη, ij ψ k = 0 for k = 1, 2 ij ψ 3 = i j h for i, j = 1, 2. 3i ψ 3 = i hη for i = 1, 2 33 ψ 3 = hη. The terms occurring in the Navier-Stokes equations are transformed in the following way (see [Sim05] and [Bem87]): The velocity u : Ω R 3 is transformed to for the pressure p : Ω R we have ũ( x) := [det ψ(φ( x))] 1 ψ(φ( x))u(φ( x)), p( x) := p(φ( x)) and the external force f : Ω R 3 on the right hand side changes into f( x) := f(φ( x)). Then the transformed Navier-Stokes equations read as follows ([Bem87, (25)-(28)]): Lũ + A p + N(ũ, ũ) = Ã f ũ = 0 23

29 4 Transformation to a reference domain in the flat domain S := R 3 \ B 1 (0), with Lũ := v ( (A ũ) + B ũ + Cũ) N(ũ, u) := a 1 ũ u + Bũũ. The coefficients in L and N depend on the mappings φ and ψ and its derivatives: a := (det ψ) 1, (4.1) ( ) A := ψ( ψ) T ψi ψ n j = = (a ij ) n x k x i,j=1, (4.2) k i,j=1 A := a A = a (a ij ) n ij=1 ( ), (4.3) n ψi à := a ψ = a, (4.4) x j i,j=1 ( ( ) B := (b ikl ) n i,k,l=1 = ank δ il 2 ψ k 2a ψ i ψ k x n x s x s x r x s x s As ( B := ( b ikl ) n i,k,l=1 = ψi φ m x s x k x m ( C := (c i j) n ψi 2 i,j=1 = a x s x r x r for x and for x we obtain ( a 1 φ s x l ( a 1 φ s x j )) n )) n φ( x) = x + O(h), i φ( x) = 1 + O(h), i j φ( x) = O( h) ψ(x) = x + O(h), i ψ(x) = 1 + O(h), i j ψ(x) = O(h) ( a 1 φ )) n r, x l i,k,l=1 (4.5), (4.6) i,k,l=1. (4.7) i,j=1 a = 1 + O(h), A, A, à = 1 + O(h), A, B, B, C = O(h). 24

30 5 Regularity and Uniqueness To show regularity and uniqueness of the solution to the Navier-Stokes equations in the domain Ω h we use the following approach: First we map the domain Ω h on a flat domain S, where the moving body is mapped onto a ball of radius 1. We transform the terms in the Navier-Stokes equations with the help of Piola s identity to get a solenoidal solution to the transformed Navier-Stokes equations on the domain S. We want to use a reflection argument to prove, that the solution coincides on S with the solution to the transformed Navier-Stokes equations on the exterior domain S S = R 3 \ B 1 (0) =: E. This is the case if we can show uniqueness of the solution to the transformed Navier-Stokes equations in the domain E. Therefore we establish a L 3 -estimate for the solution of the transformed equations in the exterior domain based on the same theorem for the Navier-Stokes equations (see [Gal11, Lemma X.6.1]) and with the help of this estimate we can show uniqueness of the solution to the transformed equations in E. transf ormation ref lection x 3 x 3 x 3 Γ h Ω h (x 1, x 2 ) S Γ 0 (x 1, x 2 ) E (x 1, x 2 ) Figure 5.1: The domains Ω h, S and E Theorem 5.1. Let (v, p) D 1,2 (E) L 6 (E) be a weak solution to the transformed Navier-Stokes equations in E with a right hand side f D 1,2 0 (E). Then u := v v L p (E) D 1,s (E) for all p > 2, s > 4 3 and p Lr (E) D 1,t (E) for r > 3 2, t > 1. Proof. If we can show, that u is a solution to the Navier-Stokes equations in E with a right hand side of class L q (E) for all q > 1, we can use [Gal11, Lemma X.6.1] and [Gal11, Theorem X.6.4] to show the desired summability properties of u and p. 25

31 5 Regularity and Uniqueness Therefore we even set q = 1 in [Gal11, Lemma X.6.1] to obtain u L 2q 2 q (E) from [Gal11, (X.6.5)] for all q > 1. The map φ : S Ω is regular enough to assume that a := det φ x we can write u as a solution to the Navier-Stokes equations 0. Therefore ν u + (u )u + p =A 1 Ãf + (1 A 1 A) p + (1 A 1 a 1 )u u A 1 Bu + νa 1 A u + νa 1 B u + A 1 Cu, with A 1 = 1 + O(h) a 1 = 1 + O(h) and a = 1 + O(h), A, A, Ã = 1 + O(h), A, B, B, C = O(h). We have to estimate the right hand side of this equation in L q (E) for q = 1. The Oseen term produced by shifting the solution is here put into the right hand side term f := 1 v + f with a possible external force f L 2 (Ω h ) of compact support. We have 1 A 1 A = 1 A 1 Aa = 1 a1 and therefore ( (1 A 1 A) p ) dx E ( (1 A 1 A)p ) dx E { x 3 <2δ} + (1 A 1 A)p n dσ B h 6 p L 5 (E { x3 <2δ}) L 6 (E { x 3 <2δ}) + (1 A 1 A)p n dσ. The first integral is bounded because B h L s (E { x 3 <2δ}) < for all s > 1. We now that (u, p) is a solution to the Navier-Stokes equations in E. And so (u, p) solves the Navier-Stokes equations in a bounded domain M E with B M, for 26

32 5 Regularity and Uniqueness example M := B 3 (0) with boundary data u M W 1 1/2,2 ( M). We know from [Gal11, Theorem IX.5.1] that p W 1,2 (M). Then p has a trace in L 2 ( (B) and as 1 A 1 A is also bounded on the boundary of B, there holds (1 A 1 A)p n dσ <. B For the other terms on the right hand side of the Navier-Stokes equations we obtain ( ( E E ( (1 A 1 a 1 )u u ) q dx ) 1/q u L 6 (E) u L 2 (E) 1 A 1 a 1 L 3q 3 2q (E) ( E c h L 3 2q 3q L (E { x 3 <2δ}) u 6 (E) u L 2 (E) ( ) ) q 1/q A 1 Bu u dx A 1 B L 3 q 3q (E) u 2 L 3 (E) h L 3q 3 q (E { x 3 <2δ}) u 2 L 6 (E) (( A 1 A + A 1 B ) u ) q dx ) 1/q A 1 A + A 1 B L 2q 2 q (E) u L 2 (E) ( E c h L 2q 2 q (E { x 3 <2δ}) u L 2 (E) ( A 1 Cu ) q dx ) 1/q u L 6 (E) A 1 C L 6q 6 q (E) c h L 6 q 6q u L (E { x 3 <2δ}) 6 (E). As we get which is finite if ( h = O x 3/2) and h = O ( x 5/2) h s 1 L s 1 (E { x 3 <2δ}) = (E { x 3 <2δ}) (E { x 3 <2δ}) h s 1 dx x ( 3/2)s 1 dx 2δ r ( 3/2)s1+1 dr 1 2δ [ r ] ( 3/2)s 1+2 ( 3/2)s < 0 s 1 > Therefore the L q -Norm of h and h is bounded if q is large enough. We obtain h L s 1 (E { x3 <2δ}) < for all s 1 >

33 5 Regularity and Uniqueness and in the same way h L s 2 (E { x3 <2δ}) < for all s 2 > 1. We still have to check if the exponents in the upper estimates are large enough for q = 1: 2q 2 q = q 3 2q = q 3 q = q 6 q = = = = = 5 6 1, where we can use the estimate for h in the last line. We can now apply Lemma X.6.1 in [Gal11] to obtain that u L p (E) D 1,s (E) for all p > 2, s > 4 3 and p Lr (E) D 1,t (E) for r > 3 2, t > 1. With the help of the L 3 -estimate that we obtain with this Lemma, we are now able to show uniqueness of the solution to the transformed Navier-Stokes equations in the exterior domain E. Lemma 5.2. Let (v, p) D 1,2 (E) L 6 (E) be a weak solution to the transformed Navier-Stokes equations in E. Then v is unique. Proof. The uniqueness of v follows at once, if we apply [Gal11, Theorem X.3.1]. The uniqueness in the exterior domain is necessary for the reflection argument that we want to use. We are now able to show, that a solution to the transformed Navier- Stokes equations on S corresponds on S to a solution of the Navier-Stokes equations on E, where the right hand side is symmetric to the x -plane and consists of the terms that occur in the transformation. Lemma 5.3. If Ω is symmetric in x 3 -direction, and for f D 1,2 0 (Ω) there holds f i (x 1, x 2, x 3 ) = f i (x 1, x 2, x 3 ), for i = 1, 2 and f 3 (x 1, x 2, x 3 ) = f 3 (x 1, x 2, x 3 ), 28

34 5 Regularity and Uniqueness then for the unique solution v Dσ 1,2 (Ω) to the Navier-Stokes equations in Ω to this right hand side there also holds the symmetry property v i (x 1, x 2, x 3 ) = v i (x 1, x 2, x 3 ), for i = 1, 2 and v 3 (x 1, x 2, x 3 ) = v 3 (x 1, x 2, x 3 ). Proof. We fix one point x in Ω and check, if v i (x 1, x 2, x 3 ) = v i (x 1, x 2, x 3 ) =: u i (x 1, x 2, x 3 ) for i = 1, 2, v 3 (x 1, x 2, x 3 ) = v 3 (x 1, x 2, x 3 ) =: u 3 (x 1, x 2, x 3 ). We define p(x 1, x 2, x 3 ) := π(x 1, x 2, x 3 ), then we have: ν v 1 (x 1, x 2, x 3 ) + v 1 (x 1, x 2, x 3 ) 1 v 1 (x 1, x 2, x 3 ) + v 2 (x 1, x 2, x 3 ) 2 v 1 (x 1, x 2, x 3 ) + v 3 (x 1, x 2, x 3 ) 3 v 1 (x 1, x 2, x 3 ) + 1 p(x 1, x 2, x 3 ) = ν v 1 (x 1, x 2, x 3 ) + v 1 (x 1, x 2, x 3 ) 1 v 1 (x 1, x 2, x 3 ) + v 2 (x 1, x 2, x 3 ) 2 v 1 (x 1, x 2, x 3 ) + v 3 (x 1, x 2, x 3 ) 3 v 1 (x 1, x 2, x 3 ) + 1 p(x 1, x 2, x 3 ) = ν u 1 (x 1, x 2, x 3 ) + u 1 (x 1, x 2, x 3 ) 1 u 1 (x 1, x 2, x 3 ) + u 2 (x 1, x 2, x 3 ) 2 u 1 (x 1, x 2, x 3 ) u 3 (x 1, x 2, x 3 )( 1) 3 u 1 (x 1, x 2, x 3 ) + 1 π(x 1, x 2, x 3 ). 29

35 5 Regularity and Uniqueness The same holds for the second equation. For the third equation we obtain ν v 3 (x 1, x 2, x 3 ) + v 1 (x 1, x 2, x 3 ) 1 v 3 (x 1, x 2, x 3 ) + v 2 (x 1, x 2, x 3 ) 2 v 3 (x 1, x 2, x 3 ) + v 3 (x 1, x 2, x 3 ) 3 v 3 (x 1, x 2, x 3 ) + 3 p(x 1, x 2, x 3 ) = ν v 3 (x 1, x 2, x 3 ) v 1 (x 1, x 2, x 3 ) 1 v 3 (x 1, x 2, x 3 ) v 2 (x 1, x 2, x 3 ) 2 v 3 (x 1, x 2, x 3 ) v 3 (x 1, x 2, x 3 ) 3 v 3 (x 1, x 2, x 3 ) 1 p(x 1, x 2, x 3 ) = ν u 3 (x 1, x 2, x 3 ) + u 1 (x 1, x 2, x 3 ) 1 u 3 (x 1, x 2, x 3 ) + u 2 (x 1, x 2, x 3 ) 2 u 3 (x 1, x 2, x 3 ) + u 3 (x 1, x 2, x 3 )( 1) 3 ( 1)u 3 (x 1, x 2, x 3 ) + 3 π(x 1, x 2, x 3 ). and because of the uniqueness of the solution we obtain v i (x 1, x 2, x 3 ) = v i (x 1, x 2, x 3 ) for i = 1, 2, v 3 (x 1, x 2, x 3 ) = v 3 (x 1, x 2, x 3 ). Corollary 5.4. Let (v, p) D 1,2 (E) L 6 (E) be a weak solution to the transformed Navier-Stokes equations in E. Then (v, p) corresponds on S to the solution of the transformed Navier-Stokes equations with mixed boundary conditions on S. Proof. We have to show, that v and p satisfy the boundary conditions required for the free boundary value problem in S, i.e. v = 0 on Γ B v n = 0 on Γ 0 τ (i) T (v, p) n = 0 on Γ 0 for i = 1, 2 v (1, 0, 0) T for x. Clearly the first and the last condition are fulfilled because of the corresponding conditions on E. The other two conditions reduce to simpler expressions caused by the shape of the domain S. As the outer normal on Γ 0 is just the vertical unit vector (0, 0, 1) we obtain the homogeneous Dirichlet boundary condition v n = 0 on Γ 0 v 3 = 0 on Γ 0. 30

36 5 Regularity and Uniqueness If we multiply the stress tensor on Γ 0 with the outer normal n we just get τ (i) T (v, p) n = (δ ij p + i v j + j v i )δ j3 = δ i3 p + i v v i. The tangential vectors are (1, 0, 0) and (0, 1, 0) and so we get τ (i) T (v, p) n = i v v i for i = 1, 2. We now have to take into account, that from the Dirichlet condition v 3 0 on Γ 0 and so especially the derivatives in tangential directions vanish on the flat boundary. In the end we have 3 v i = 0 on Γ 0 for i = 1, 2. We have to show that these conditions hold for the solution to the Navier-Stokes equations in E. As the domain is symmetric in x 3 -direction we can use Lemma 5.3 to obtain On Γ 0, where x 3 = 0, this leads to v 3 (x, x 3 ) = v 3 (x, x 3 ) v i (x, x 3 ) = v i (x, x 3 ). v 3 (x, 0) = 0 and 3 v i (x, x 3 ) = 3 v i (x, x 3 ) = 0. Now we can establish regularity of the solution to the transformed Navier-Stokes equations in E to conclude the same regularity properties for the solution in S and thus on the non-flat domain Ω h. Lemma 5.5. Let (v, p) D 1,2 (E) L 2 loc (E) be a weak solution to the transformed Navier-Stokes equations in E. Then v is regular. Corollary 5.6. Let (v, p) D 1,2 (S) L 2 loc (S) be a weak solution to the transformed Navier-Stokes equations in S. Then v is unique and regular up to the boundary Γ h. Proof. Regularity of the weak solution up to the boundary follows with the same reflection argument as above and [Gal11, Theorem X.1.1], as the boundary Γ h is a part of the interior of E. Corollary 5.7. Let (v, p) D 1,2 (Ω h ) L 2 loc (Ω h) be a weak solution to the Navier- Stokes equations in Ω h. Then v is unique and regular. 31

37 5 Regularity and Uniqueness Proof. If v and p are a solution to the Navier-Stokes equations in Ω h, then and ṽ( x) := [det ψ(φ( x))] 1 ψ(φ( x))v(φ( x)), p( x) := p(φ( x)) are a solution to the transformed Navier-Stokes equations in S, where φ and ψ are the transform of the domain and its inverse. We re-transform the functions to obtain and v(x) = [det φ(ψ(x))] 1 φ(ψ(x))ṽ(ψ(x)) p(x) := p(ψ(x)). As ṽ and p are unique and regular, the same holds for v and p as long as h is regular enough. But this follows immediately because h depends continuously on v and p (see [BGK11]). 32

38 6 Asymptotic behaviour of the solution In this chapter we show, that the asymptotic behaviour of the solution to the free boundary value problem coincides with the asymptotic behaviour of the solution in an exterior domain in the way that the shape of the wake region behind the floating body is the same as in the exterior domain, only perturbed by a function that depends on the surface of the domain. Especially we want to show, that the (two dimensional) wake on the surface of the fluid has the shape of a parabola, perturbed only by the deviation of the graph of h from a plane. First we give uniform estimates for u, u in Ω h and afterwards we consider the asymptotic behaviour inside and outside the wake region behind the body. In the second part of the chapter we prove a uniform estimate for the function h, that describes the surface of the fluid. As we use the estimate for the velocity to analyse the asymptotic behaviour of h, we can show a better estimate for the decay of the function h before, than behind the moving body. 6.1 Decay of the velocity We start by proving an asymptotic representation formula for the solution to the Navier-Stokes equations. In the following we use the notation x S and x Ω h. Note that for large values of x we have x = x. The following representation formula directly gives us a uniform estimate for u, but we can also use it to prove a better rate of decay outside the wake region. Lemma 6.1. Let v D 1,2 σ,γ (Ω h) be a solution to the free boundary value problem for the Navier-Stokes equations, with a surface function h(x). For h there holds h x 3 2 for large x, h x 5 2. Then we have the following asymptotic representation for u := v v as x : u k (x) = [det ψ(x)] k φ j (ψ(x)) ( ) M i E ij (ψ(x)) + R E ij (ψ(x) ỹ)ũ l (ỹ) l ũ i (ỹ) dỹ + s j (x), E 33

39 6 Asymptotic behaviour of the solution where ψ(x) = O( x + h(x)), φ( x) = O( x + h( x)) M i = R T il (ũ, p)n l + Rδ 1l ũ i (ỹ)n l dσỹ, B ũ = O( u ), p = O( p ) ( s j (x) = O ψ(x) 3/2). Proof. Let φ : S Ω h be the mapping from the flat domain S onto Ω h and ψ : Ω h S the inverse mapping. We choose the transform in a way, that φ = 0 for x 3 < 2δ, for example for large x with φ( x) = ( x 1, x 2, x 3 + h( x )η( x 3 )) η( x 3 ) := { 1 for x3 > δ 0 for x 3 < 2δ for some δ > 0. For the construction and properties of this mapping see Chapter 4. Then ũ( x) := [det ψ(φ( x))] 1 ψ(φ( x))u(φ( x)) is a solution to the Navier-Stokes equations on S with the right hand side F = A 1 Ã f + (1 A 1 A) p + (1 A 1 a 1 )ũ ũ A 1 Bũ + νa 1 A ũ + νa 1 B ũ + A 1 Cũ where a, A, A, Ã, B, B, C are defined in (4.1)-(4.7) and supp (F ) { x 3 2δ}. We extend this F symmetrically on E := R 3 \ B, where B := B 1 (0). Then we solve the Navier-Stokes equations in E with the new right hand side. Because of the uniqueness and the symmetry of the solution we find that it corresponds to ũ on S. We therefore have the following representation formula for x E and therefore especially for x S (see [Gal11, (X.5.32)]) ũ j ( x) = R E ij ( x ỹ)f i (ỹ) dỹ + R E ij ( x ỹ)ũ l (ỹ) l ũ i (ỹ) dỹ E E + [ũ i (ỹ)t il (w j, e j )( x ỹ) E ij ( x ỹ)t il (ũ, p)(ỹ) B Rũ i (ỹ)e ij ( x ỹ)δ 1l ] n l dσỹ where R is the Reynolds number, E = E ij ( x ỹ) is the Oseen fundamental solution (see [Gal11, VII.3.1]) and w j, e j are the columns of E and the corresponding pressure respectively. 34

40 6 Asymptotic behaviour of the solution Note that the representation formula also holds on the free boundary of the domain, which makes sense, because we have already shown that the solution is regular up to the boundary. With x = φ( x) we have in Ω h : [ u k (x) = [det ψ(x)] k φ j (ψ(x)) R E ij (ψ(x) ỹ)f i (ỹ) dỹ E + R E ij (ψ(x) ỹ)ũ l (ỹ) l ũ i (ỹ) dỹ E + [ũ i (ỹ)t il (w j, e j )(ψ(x) ỹ) E ij (ψ(x) ỹ)t il (ũ, p)(ỹ) B ] Rũ i (ỹ)e ij (ψ(x) ỹ)δ 1l ] n l dσỹ. We want to use the same method as in [Gal11, Theorem V.3.2] to show the asymptotic representation formula. By adding and subtracting the term ( ) E ij (ψ(x)) [T il (ũ, p)(ỹ) + Rũ i (ỹ)δ 1l ] n l dσỹ B and estimating the resulting integrals we reduce the asymptotic behaviour of u to the asymptotic structure of the Oseen fundamental solution perturbed by functions depending on the surface h of the fluid. We obtain [ u k (x) = [det ψ(x)] k φ j (ψ(x)) R E ij (ψ(x) ỹ)f i (ỹ) dỹ E + R E ij (ψ(x) ỹ)ũ l (ỹ) l ũ i (ỹ) dỹ E + ũ i (ỹ)t il (w j, e j )(ψ(x) ỹ)n l dσỹ B E ij (ψ(x)) [T il (ũ, p)(ỹ) + Rũ i (ỹ)δ 1l ] n l dσỹ B (E ij (ψ(x) ỹ) E ij (ψ(x))) (T il (ũ, p)(ỹ) + Rũ i (ỹ)δ 1l ) n l dσỹ B We consider the integrals separately. First we want to estimate the integral E E ij(ψ(x) ỹ)f i (ỹ) dỹ. Unfortunately the support of F is not bounded, but as F can be estimated for large ỹ by the surface function h we can divide the domain of integration such that in one part of the domain the asymptotic structure of the Oseen fundamental solution and in the ]. 35