A Weak Solvability of the Navier Stokes Equation with Navier s Boundary Condition around a Ball Striking the Wall

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1 Preprint, Institute of Mathematics, AS CR, Prague INSTITUTE of MATHEMATICS Academy of Sciences Czech Republic A Weak Solvability of the Navier Stokes Equation with Navier s Boundary Condition around a Ball Striking the Wall Jiří Neustupa and Patrick Penel Abstract We assume that B t is a closed ball in R 3 + := {x,x 2,x 3 ) R 3 ; x 3 > }, striking the wall = the x,x 2 plane) at time t c,t ). The speed of the ball at the instant of the collision need not be zero. Although a weak solution to the Navier Stokes equation with Dirichlet s no slip boundary condition in R 3 + B t ),T ) does not exist if the speed of the stroke is non zero, we prove that such a solution may exist if Dirichlet s boundary condition is replaced by Navier s slip boundary condition. Motivation, introduction and notation The existence of a weak solution to the Navier Stokes equation in a fixed domain Ω R 3 on a given time interval,t ) belongs to fundamental results of the qualitative theory of the Navier Stokes equation. See e.g. J. Leray 934 [7], E. Hopf 952 [5], O. A. Ladyzhenskaya 969 [6], J. L. Lions 969 [8], R. Temam 977 [26] or G. P. Galdi 2 [].) Of all results on the existence of the weak solution in domains with given moving boundaries, we cite the papers by H. Fuita and N. Sauer 97 [7] the boundary of a variable domain Ω t consists of a finite number of moving simple closed surfaces of the class C 3, the distance of any two of these surfaces is never less than d > ) and J. Neustupa 27 [9] Ω t has an arbitrary shape and smoothness, the assumptions on Ω t involve simulation of collisions of bodies moving in a fluid). Jiří Neustupa Mathematical Institute of the Czech Academy of Sciences, Žitná 25, 5 67 Praha, Czech Republic, neustupa@math.cas.cz Patrick Penel Université du Sud Toulon Var, Département de Mathématique & Laboratoire Systèmes Navals Complexes, B.P. 232, La Garde Cedex, France, penel@univ-tln.fr

2 2 Jiří Neustupa and Patrick Penel There exists a series of other works dealing with flows in time varying domains that concern the motion of one or more bodies in a fluid. The fluid and the bodies are studied as an interconnected system so that the position of the bodies in the fluid is not apriori known. The weak solvability of such a problem, provided the bodies do not touch each other or they do not strike the boundary, was proved by B. Desardins and M. J. Esteban 999 [3], 2 [4], K. H. Hoffmann V. N. Starovoitov 999 [3] the 2D case), C. Conca, J. San Martín and M. Tucsnak 2 [2] and M. D. Gunzburger, H. C. Lee, G. Seregin 2 [2]. The analogous result, without the assumption on the lack of collisions, was proved by J. San Martín, V. N. Starovoitov and M. Tucsnak 22 [2] the 2D case), K. H. Hoffmann, V. N. Starovoitov 2 [4] the motion of a small ball in a fluid filling a large ball ) and E. Feireisl 23 [6] in a 3D bounded domain, the author uses the contact condition that once two bodies touch one another, they remain stuck together forever). All the mentioned authors consider the homogeneous Dirichlet boundary condition for velocity on the boundary of Ω t. The motion of the so called self propelled bodies which produce certain velocity profile on their surface), together with the motion of the fluid around them, was studied except others by G. P. Galdi, see the survey paper []. None of the mentioned papers provides the existence of a weak solution to the Navier Stokes equation at the geometrical configuration when the fluid fills a domain Ω t around a solid ball striking a wall with a finite non zero speed. Moreover, it follows from results of V. N. Starovoitov 23 [2] that the weak solution with the no slip Dirichlet boundary condition in such a situation cannot exist. Paper [9], where the no slip boundary condition is also considered, provides the weak solution only if the ball strikes the wall with the speed that tends to zero as time approaches the instant of the collision. With a non zero speed, the body must have another shape than the ball, see [9].) This state motivated us to study the Navier Stokes equation in the described domain Ω t with boundary conditions that enable the fluid to slip on the boundary. We assume that the motion of the ball is given. We use Navier s boundary condition and we prove the global in time existence of a weak solution under the restriction that the speed of the ball is sufficiently small at times close to the instant of the collision see Theorem. The considered case of a ball moving in a fluid and striking perpendicularly the wall represents a sample example. We actually prepare a generalization concerning flow around moving bodies of various shapes which may collide one with another. Nevertheless, the basic techniques is developed in the present paper. It is based on the construction of Rothe approximations. A series of steps require a different approach than in the case of homogeneous Dirichlet s boundary condition. For instance, Sobolev s imbedding inequalities cannot be used in a standard fashion because the constants in these inequalities now depend on time. Other difficulties appear in the part where we treat the limit transition in the nonlinear term and we therefore need an information on a strong convergence of a sequence of approximations in an appropriate norm. The argument based on the Lions Aubin lemma cannot be used in a usual way see Section 6.)

3 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 3 The time variable domain Ω t. We suppose that,t ) is a bounded time interval and t c,t ). We denote by R 3 + the half space {x = x,x 2,x 3 ) R 3 ; x 3 > }, by R + 3 the closure of R 3 + and by R 3 + the boundary of R 3 + = the x, x 2 plane). Further, we denote by B t the closed ball in R + 3 with radius R and center S t =,, δ t + R). We suppose that δ t the distance of the ball B t from R 3 +) is a continuous function of t for t [,T ] such that δ t c = and i) δ t is decreasing on [,t c ] and increasing on [t c,t ], ii) δ t the derivative of δ t ) is bounded on the intervals [,t c ) and t c,t ], iii) δ t the second derivative of δ t ) is integrable on,t ). We put Ω t := R 3 + B t. The boundary of Ω t is denoted by Γ t. Ω t represents the space filled by the fluid and B t represents a solid ball which moves in the fluid and strikes the fixed wall R 3 + at time t = t c. We assume, for simplicity, that ball B t does not rotate and all its particles have only the translational velocity. Thus, the velocity of the material points on the boundary Γ t of Ω t is V t x) := {,, δ t ) for t t c and x B t, for t t c and x R 3 +. Notation of norms and function spaces..,.) 2;Ω t is the scalar product and. 2;Ω t is the norm in L 2 Ω t ) or in L 2 Ω t ) 3 or in L 2 Ω t ) 9, respectively. The meaning of.,.) 2;Γ t and. 2;Γ t is analogous.. q;ω t is the norm in L q Ω t ) or in L q Ω t ) 3 or in L q Ω t ) 9, respectively. Cσ Ω t ) is the space of infinitely differentiable divergence free vector functions in Ω t with a compact support in Ω t and zero normal component on Γ t. Wσ,2Ω t ) is the closure of Cσ Ω t ) in W,2 Ω t ) 3. C,σ Ω t ) is a subspace of Cσ Ω t ), containing functions with a compact support in Ω t. Lσ q Ω t ) is the closure of C,σ Ω t ) in L q Ω t ) 3 for q < + ). If t,t ) {t c } then Wσ,2Ω t ) L q Ω t ) 3 for 2 q 6. Using the characterization of Lσ q Ω t ) see [8, p. ]), we can verify that Wσ,2Ω t ) Lσ q Ω t ). The initial boundary value problem. Put Q,T ) := { x,t); < t < T, x Ω t} and Γ,T ) := { x,t); < t < T, x Γ t}. Our aim is to prove the existence of a weak solution of the problem t v + v v + p = ν v + f in Q,T ), ) divv = in Q,T ), 2) v n = V t n in Γ,T ), 3) [T d v) n] τ + Kv V t ) = in Γ,T ), 4) v = v in Ω {}. 5)

4 4 Jiří Neustupa and Patrick Penel The equations ), 2) describe the motion of a viscous incompressible fluid in domain Ω t. The symbols v, p, ν, f, n and T d v) successively denote the velocity of the fluid, the pressure, the kinematic coefficient of viscosity, the specific external body force, the outer normal vector on the boundary of Ω t and the dynamic stress tensor associated with the flow v. The density of the fluid is supposed to be one. The subscript τ denotes the tangential component to Γ t. Since the considered fluid is Newtonian, the dynamic stress tensor has the form T d v) = 2ν v) s where v) s is the symmetrized gradient of v. Condition 3) expresses the impermeability of Γ t. Condition 4) is due to H. Navier, who proposed in 824 that the tangential component of the stress acting on the boundary should be proportional to the velocity of the fluid relative with respect to the material boundary). We suppose in accordance with physical arguments) that K. Introduction of function a t. In order to transform the inhomogeneous boundary condition 3) to the homogeneous one, we look for the solution v in the form v = a t + u where a t is considered to be a known function satisfying the condition a t n = V t n a.e. in Γ,T ) 6) and u is a new unknown function. The construction of an appropriate function a t is presented in Section 2. We shall see that function a t can be defined a.e. in R 3 + [,T ] so that it is divergence free and, in addition to the condition 6), it also satisfies the series of estimates a t 2 2;Ω t c δ t ) 2 ln + R ) δ t, 7) t a t, φ ) 2;Ω t c 2 δ t ) 2 φ 2;Ω t + c 3 δ t φ 2;Ω t, 8) a t a t, φ ) 2;Ω t c 4 δ t ) 2 φ 2;Ω t, 9) a t δ 5;Ω t c t 5 δ t, ) ) / φ a t, φ ) 2;Ω t c 6 δ t φ 2 2;Ω t, ) K a t V t, φ ) 2;Γ t 6 ν φ 2 2;Ω t + c 7 2) for t t c, all φ Wσ,2Ω t ), with constants c c 7 which are independent of φ and t. Obviously, the right hand side of 7) is integrable on,t ) with any power α and the right hand side of ) is integrable on,t ) with any power α [,). Using the continuous imbedding L 6 Ω t ) W,2 Ω t ), we can also derive the estimate φ a t, φ ) 2;Ω a t t 2;Ω t φ /2 2;Ω t φ 3/2 6;Ω t c 8 at) φ 2 2;Ω t + 6 ν φ 2 2;Ω t 3)

5 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 5 where at) := a t 2;Ω t + a t 4 2;Ω t. Inequality 3) holds at times t t c when domain Ω t has the cone property and W,2 Ω t ) 3 is therefore continuously imbedded into L 6 Ω t ) 3. Constant c 8 depends on ν and it also generally depends on t through the cone parameters appearing in the definition of the cone property of Ω t, see e.g. [, p. 3]. However, if we use 3) only at times t such that t t c > κ then c 8, although dependent on κ, can be considered to be independent of t. The value of κ will be fixed by condition iv) in Theorem. We shall also see in Section 2 that the initial value problem d dt Xt; t,x ) = at Xt; t,x ) ), Xt ; t,x ) = x 4) has a unique solution Xt; t,x ), defined for a.a. t,t ), all t [,T ] and all x R 3 +. The mapping x Xt; t,x ) is a transformation of Ω t l t onto Ω t l t where l t and l t are certain sets of measure zero), whose Jacobian equals one due to the incompressibility of the flow a t. This mapping can be used in order to transform volume integrals on Ω t to volume integrals on Ω t. 2 A formal study of the initial boundary value problem ) 5) and the main theorem A formal derivation of the weak formulation. The weak formulation of the problem ) 5) can be formally derived from the classical formulation if we multiply equation ) by an appropriate test function φ, integrate in Q,T ) and use all the conditions 2) 5). Thus, assume that φ is an infinitely differentiable divergence free vector function in R 3 + [,T ] that has a compact support in R 3 + [,T ) and satisfies the condition φ n = on Γ,T ). Assume that v is a sufficiently smooth solution of ) 5) of the form v = a t + u where a t has all the properties named in the last paragraph of Section and u Lσ 2 Ω t ) for a.a. t,t ). The product { t v+v t )v} φ equals the sum of { t v+a t )v} φ and u v φ. The integral of the first term can be treated as follows: { t vx,t) + a t x) vx,t) } φx,t) dxdt + v Ω t Ω x ) φx,) dx d = Ω dt v Xt;,x ),t ) φ Xt;,x ),t ) dx dt + v Ω x ) φx,) dx { = t φx,t) + a t x) φx,t) } vx,t) dxdt. 5) Ω t The integral of u v φ in Ω t can be transformed to the negative integral of u φ v by means of the integration by parts. Further, we have ν v φ dx = Ωt ν at φ dx + Ω t Γ t ν u n φ ds ν u : φ dx Ωt

6 6 Jiří Neustupa and Patrick Penel = ν [ 2n u) s n u ] [ φ ds + ν at φ ν u : φ ] dx Γ t Ω t = n [2ν v) Γ t s 2ν a t [ ) s ] φ ds + ν at φ 2ν u) s : φ ] dx Ω t = Kv Vt ) φ ds 2ν v) Γ t Ω t s : φ dx. 6) We have used the identities n 2ν v) Γ t s φ ds = n u φ ds = Γ t ν at φ dx = Ω t [n T Γ t d v)] τ φ ds = Kv Vt ) φ ds, Γ t u)t : φ dx, Ω t 2ν n at ) s φ ds 2ν at ) s : φ dx, Γ t Ω t the first of whose follows from 4). The integral of p φ on Ω t equals zero because the subspace of gradients of scalar functions is orthogonal to L 2 σ Ω t ) in L 2 Ω t ) 3. Thus, using 5) and 6), we obtain the integral identity { v t φ v φ v + 2ν v) s : φ } dxdt + Kv Vt ) φ dsdt Ω t Γ t T = f φ dxdt + v Ω t Ω φ.,) dx. Replacing v by the sum a t + u, we arrive at the definition: Definition the weak solution of ) 5)). Suppose that u Lσ 2 Ω ) and f L 2,T ; L 2 Ω t ) 3 ). The function v a t + u is called a weak solution of the problem ) 5) if u L 2,T ; Wσ,2Ω t )) L,T ; Lσ 2 Ω t )) satisfies { a t + u) t φ a t + u) φ a t + u) + 2ν [ a t + u)] s : φ } dxdt Ω t + Kat + u V t ) φ dsdt = f φ dxdt + + u Γ t Ω t Ω [a ] φ.,)dx 7) for all divergence free vector functions φ C R+ 3 [,T ) ), that satisfy the condition φ n = on Γ,T ). The readers can verify that this definition enables us the backward calculation, i.e. to show that if the weak solution v is sufficiently smooth and all other input data are also sufficiently smooth then there exists a pressure p so that the pair v, p is a classical solution of ) 5). We shall refer to the problem defined above as to the weak problem 7). A formal derivation of the energy inequality. The energy inequality is a fundamental apriori estimate of a solution of the problem ) 5). An analogous estimate can be rigorously derived for appropriate approximations of the solution. However,

7 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 7 in order to abstract from technical details connected with the approximations and to explain how we use the boundary conditions and apply estimates 7) 3), we include the formal derivation of the energy inequality already in this section. Lemma. Suppose that iv) there exists κ > so that c 6 δ t < 4 ν for t c κ < t < t c + κ. Recall that c 6 is the constant from inequality.).) Then there exist non negative integrable functions ω and G on,t ) such that if v, p) a t + u, p) is a smooth solution of the initial boundary value problem ) 5) and t,t ) then t t u.,t) 2 2;Ω t + ν u.,s) 2 2;Ω s ds + 2K u.,s) 2 2;Γ s ds u 2 2;Ω + t ωs) u.,s) 2 2;Ω s ds + Gt). 8) Proof. Assume that t t c, multiply equation ) where v = a t + u) by u and integrate in Ω t. We obtain { [ t a t + u) + a t a t + u)] u + u a t u ν v u } dx = f u dx. 9) Ω t Ωt Now we estimate or rewrite the terms in 9): Following 6), we have ν + ν Ω v u dx = K t Ω t u 2 dx + 2ν u 2 ds + K at V t ) u ds + ν Γ t Γ t Ω t at ) s : u dx. n u u ds Γ t Using the identity u n) u = valid a.e. on Γ t ) and the negative semi definiteness of the tensor n a.e. on Γ t following from the special geometry of Ω t ), we observe that Γ t n u u ds. Therefore, using 2), we get n u u ds + K at V t ) u ds Γ t Γ t 6 ν u 2 2;Ω t c 7, ν v u dx K u 2 Ω t 2;Γ t Ω ν u 2 dx 6ν at 2 dx c 7. 2) t Ω t Due to 8) and 9), we obtain with c 9 = [8c c2 4 )/ν] esssup δ t ) 4 ) [ ta t + a t a t ] u dx c Ω t 3 δ t u 2 2;Ω t + 4 c 3 δ t + 6 ν u 2 2;Ω t + c 9. Using the transformation x y = Xt + h; t,x) of Ω t l t onto Ω t+h l t+h, we can rewrite the next integral as follows:

8 8 Jiří Neustupa and Patrick Penel [ t u + a t u ] [ ] d ) u dx = u Xϑ; t,x),ϑ 2 dx Ω t Ω t dϑ 2 ϑ=t [ u ) = lim Xt + h; t,x),t + h 2 ) ) ] u Xt; t,x),t 2 dx h 2h Ω [ t ] = lim h 2h uy,t + h) 2 dy ux,t) 2 dx = d Ω t+h Ω t dt 2 u 2 dx. Ω t Now, due to the inequalities ) and 3) and denoting by χ the characteristic function of the interval t c κ,t c + κ ), we can estimate u at u dx c Ω t 6 χ t) δ t u 2 2;Ω t + 6 ν u 2 2;Ω t + c 8 at) u 2 2;Ω t. Finally, by means of condition iv) of the smallness of δ t on the interval t c κ,t c + κ ), the term c 6 χ t) δ t u 2 2;Ω t can also be absorbed by 4 6 ν u 2 2;Ω t see 2)). Substituting now all previous estimates or identities to 9), using inequality 7) and denoting ωt) = 2c 3 δ t + 2c 8 at) +, we obtain d dt u 2 2;Ω t + ν u 2 2;Ω t + 2K u 2 2;Γ t f 2 2;Ω t + ωt) u 2 2;Ω t + 2 c 3 δ t + 6c δ t ) 2 ln + R ) δ t + c 7 + c 9. To complete the proof, we integrate this inequality on the time interval,t). Our main theorem, whose proof is given in Sections 4 6, reads: Theorem. Suppose that function δ t satisfies conditions i) iii) and also the condition of smallness iv). Then the weak problem 7) has a solution. 3 Construction of the auxiliary function a t and its properties The purpose of this section is to define a divergence free function a t in R 3 + [,T ] which has the properties named and used in Section : identity 6), essentially a t n =,, δ t ) n in B t for t t c, and inequalities 7) 3). Except for the Cartesian coordinates x, x 2, x 3, we shall also use the cylindrical coordinates r, ϕ and x 3. Thus, r 2 = x 2 + x2 2. The lower half of the surface of Bt coincides with the graph of the function x 3 = g t r) := δ t + R R 2 r 2 ; r R. Domains Ω t r ), Ω ext and Ω int. Let us fix r := 3 4 R. The crucial sub domain of Ω t, where the collision occurs, is see Fig. )

9 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 9 Ω t r ) := { x = r,ϕ,x 3 ) Ω t ; r < r, x 3 < g t r) }. 2) We also denote by Ω ext the set of points x = r,ϕ,x 3 ) R 3 + such that either r > 2 2 R or x 3 > max{δ ;δ T }. The complementary set Ω int is defined as R 3 + Ω ext. x 3 Fig. : Ω t Ω ext Ω int R S t B t R R2 r 2 x Ω t r ) δ t 3 = g t r) r = r O,,) x, x 2 An auxiliary function b t. Suppose that t,t ) {t c }. We define β t = βr,β t ϕ,β t 3, t ) := rx ) 3 2g t r), δ t, b t = b t r,b t ϕ,b t 3 ) := curlβt = r 2g t r),, x 3 r rg t r) 2g t r) 2 + x ) 3 δ t g t r) 22) in the cylinder r < r, < x 3 < δ t + R. The derivative of g t r) with respect to r is r g t r) = r/ R 2 r 2. The function b t is divergence free and it satisfies the conditions of impermeability, b t n = b t 3 = for x 3 =, and for x 3 = gt r), b t n = r 2g t r),, r rg t r) ) 2g t r) + δ t r g t r),, ) =,, δ t ) n [g t r)] 2 + Thus, b t n = V t n on the lower and upper parts of the boundary of Ω t r ). Two auxiliary cut off functions. We shall use two cut off functions: infinitely differentiable cut off function of one variable such that η s) := for s < R, 2 for 2 R < s, [,] for R s 2 2 R η is an

10 Jiří Neustupa and Patrick Penel and η2 t is an infinitely differentiable cut off function in R + 3 whose support is a subset of {x = r,ϕ,x 3 ); r r, x 3 δ t +R} and η2 t x) = for r < 2 R and x 3 < gt r). Let e ϕ denote the unit vector in the direction of ϕ. Then curl [ 2 r δ t ] e ϕ =,, δ t ). Furthermore, curl [ η x S t ) 2 r δ t ] e ϕ coincides with,, δ t ) in B t and it equals zero if x S t > 2 2 R. Definition of function a t. We put a t x) := curl [η2 t x)βt x) + [ η2 t x)] η x S t ) 2 r δ ] t e ϕ. 23) Then, in the important regions, b t x) for x Ω t r ) with r := 2 R = 2 3 r, a t x) =,, δ t ) for x B t + := { x R 3 +; x S t R, x 3 > R + δ t}, for x Ω ext. Obviously, a t is divergence free and satisfies the identity 6). It can be proved all the estimates 7) 3) named in Section : Since a t is smooth outside the critical region Ω t r ), where the collision of the ball B t with the x,x 2 plane occurs, and a t = in Ω ext, we can focus only on the behavior of a t in Ω t r ), where a t = b t. Using the explicit form of b t, given by 22), one can show that b t indeed satisfies the same estimates as 7) 3). We only have to consider the norms or scalar products in Ω t r ) instead of Ω t on the left hand sides of 7) ). Similarly, b t satisfies an estimate analogous to 2) with Γ t Ω t r ) instead of Γ t. We verify only two of the estimates in the rest of this section. An estimate of φ b t, φ ) 2;Ω t r ) with φ W,2 σ Ω t ). We consider this estimate to be crucial, because although domain Ω t r ) is time dependent, it provides an estimate with constant C independent of t. Let us begin with the integral of r b t r)φ 2 r, where we can easily check that r b t r = r r δ t /2g t r) ) C δ t /g t r). We put φ r r,x 3 ) := 2π φ r r,ϕ,x 3 )dϕ. Since the flow φ is incompressible and it also satisfies the condition of impermeability φ n = on Γ t, we have g t r) φ r dx 3 = g t r) 2π φ r dϕ dx 3 = φ nds = divφ dx = 24) Ω t r) Ω t r) where < r r ). This implies that to each r,r ) there exists x 3 r) between and g t r) such that φ r r,x 3 r)) =. Using also the inequality r 2 2Rg t r) and applying Poincaré s inequality see e.g. [5, R. Dautray and J. L. Lions, p. 27]) to the integral 2π φr 2 dϕ, we obtain Ω t r ) r b t r)φr 2 dx C r δ t r dr g t r) g t r) dx 3 2π ) φr 2 dϕ

11 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall C δ r t = C δ r t C δ r t + C δ r t C δ r t + C δ t r dr g t r) r dr g t r) r 3 dr g t r) r r dr g t r) g t r) g t r) g t r) g t r) g t r) 2π dx 3 4π ϕ φ r ) 2 dϕ + 2π dx 3 2π dx 3 2π [ x3 ϕ φ r ) 2 dϕ +C δ t r r 2 [ ϕ φ r r,ϕ,x 3 ) ] 2 dϕ 2 y φ r r,y)dy] dx 3 x 3 r) [ 2π r dr g t r) 2π [ r dr dx 3 ϕ r 2 φ r r,ϕ,x 3 ) ] 2 dϕ g t g t r) r)r dr x3 φ r r,x 3 ) 2 dx 3 C δ t ] 2 ) φ r dϕ g t r) Ω t r ) φ r 2 dx 3 φ r 2 dx. The generic constant C is always independent of t. The integrals of 3 b t 3 )φ2 3 and r b t 3 )φ r φ 3 can be treated similarly. Here we can use the identity φ 3 r,ϕ,) =.) Thus, we finally estimate the modulus of φ b t, φ ) 2;Ω t r ) by C δ t φ 2 2;Ω t r ). An estimate of the surface integral of b t V t ) φ. We estimate the product b t V t ) φ on the lower part Γ tr ) := {x = r,ϕ,x 3 ) Γ t ; r < r, x 3 = } and on the upper part Γ tr ) := {x = r,ϕ,x 3 ) Γ t ; r < r, x 3 = g t r)} of Γ t Ω t r ). Using the explicit forms of b t V t on Γ tr ) and Γ tr ) and the identity φ 3 = r g t r)φ r on Γ tr ) following from the condition φ n = ), we get b t V t ) φ ds + b t V t ) φ ds Γ tr ) Γ tr ) = δ t r 2π [ r 2 2 g t r) φ rr,ϕ,) + r2 g t + [ r g t r) ] ) 2 φ r r,ϕ,g t r) )] dϕ dr r) = δ t r [ r 2 2 g t r) φ r r,) + r2 g t + [ r g t r) ] ) 2 φ r r,g t r) )] dr r) r C r 2 ) g t σ φ r r,σ) dσ dr r) x 3 r) r + C r 2 g t + [ r g t r) ] ) g 2 t ) r) σ φ r r,σ) dσ dr r) x 3 r) r g t r) r ε x3 φ r r,x 3 ) ) 2 +Cε) dx 3 dr. The generic constant again C does not depend on t. Choosing sufficiently small ε >, we obtain an inequality that further enables us to arrive at 2). The initial value problem 4). Suppose that t [,T ] and x Ω t l t where l t is the open line segment in Ω t with the end points,,) and,,δ t )). Then

12 2 Jiří Neustupa and Patrick Penel the initial value problem 4) has a unique solution Xt; t,x ) defined for t [,T ] by the Carathéodory theorem. The traectory of the solution stays in Ω t l t where l t is defined by analogy with l t ) due to the condition a t n = V t n satisfied by function a t on Γ t. The mapping x Xt; t,x ) is a regular mapping of Ω t l t onto Ω t l t, whose Jacobian equals one. Note that if x Ω ext then Xt; t,x ) = x independently of t and t because a t x,t) = for all t T. 4 The time discretized boundary value problems The time discretization. Let n N and k {; ;...; n}. We put h := T /n, t k := kh, Ω k := Ω t k and Γ k := Γ t k. We can assume without loss of generality that the critical time t c of the collision differs from all the time instants t k. The stationary boundary value problems. We put U := u. We successively solve, for k =,...,n, a sequence of these stationary boundary value problems: given U k Lσ 2 Ω k ) and f k L 2 Ω k ) 3, we look for U k, P k such that U k x) U k Xtk ; t k,x) ) + hu k x) {[ a] k x) + U k x) } + h P k x) = ν h { Div[ a] k x) + U k x) } + A k x) + hf k x) in Ω k, 25) divu k x) = in Ω k, 26) U k n = in Γ k, 27) [ Td ) k n ] τ + K a k + U k V k ) = in Γ k, 28) The meaning of the functions A k, [ a] k, f k, a k, V k and T d ) k is explained below: A k x) := a t k x) + a t k Xtk ; t k,x) ) tk = [ a] k x) := h tk t k a t x) dt, f k x) := h t k tk d dt at Xt; t k,x) ) dt, t k fx,t) dt for x Ω k and T d ) k := 2ν { [ a] k + U k }s on Γ k. Denoting by e 3 the unit vector,,), we define for x Γ s and t, s [,T ] such that s t) { x + δ t δ s) e Yt; s,x) := 3 if x B s, 29) x if x the x,x 2 plane. The mapping x Yt; s,x) represents the shift of the material point x on the boundary of the flow field in the time interval [s,t]. Now we denote for x Γ k a k x) := h tk t k a t Yt; t k,x) ) dt, V k x) := h tk t k V t Yt; t k,x) ) dt.

13 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 3 Note that the term a t u, which appears in equation ) if we write v in the form v = a t + u, is now related to the difference at the beginning of 25): U k x) U k Xtk ; t k,x) ) = tk t k U k Xt; tk,x) ) a t Xt; t k,x) ) dt. The weak formulation of the BV problem 25) 28). We can get rid of pressure P k in the classical formulation 25) 28) if we formally multiply equation 25) by a test function Φ k from Wσ,2Ω k ). Furthermore, we integrate by parts in the viscous term and we use the boundary conditions 27) and 28) in the same way as the conditions 3) and 4) were used in 6). Thus, we arrive at the weak formulation: we look for U k Wσ,2Ω k ) such that { Uk x) U k Xtk ; t k,x) ) + hu k x) {[ a] k x) + U k x) }} Φ k x) dx Ω k + 2ν h { [ a] k x) + U k x) } s : Φ k x) dx Ω k + Kh [ a k x) + U k x) V k x) ] Φ k x) ds Γ k = hf k x) Φ k x) dx + A k x) Φ k x) dx 3) Ω k Ω k for all Φ k Wσ,2Ω k ). The solvability of this nonlinear elliptic problem can be proved by standard methods, particularly of theory of the steady Navier Stokes equation. We refer e.g. to the book [9] by G. P. Galdi for the corresponding techniques. The coerciveness of an associated quadratic form follows from the next estimates. Apriori estimates of solutions of the BV problem 3). Using Φ k = U k in 3), we obtain: 2 U k 2 2;Ω + Uk x) U k 2 k Xtk ; t k x) ) 2 dx + ν h U k ) s : U k dx Ω k Ω k + Kh U k 2 ds Γ k 2 U k 2 2;Ω + k Ω h f k U k dx + A k U k dx k Ω k + Ω h U k [ a] k U k dx + ) ν h [ a]k s : U k dx + K a k V k ) U k ds. k Ω k Γ k The integral of U k ) s : U k can be estimated from below by 2 U k 2 2;Ω by k means of the integration by parts, the identity n U k ) U k = valid on Γ k ) and the negative semi definiteness of n on Γ k. The integrals on the right hand side can be treated by analogy with the procedure explained in Section, which now leads us to a discrete variant of the energy inequality 8):

14 4 Jiří Neustupa and Patrick Penel U 2 2;Ω + k= + 2Kh k= Ω k Uk x) U k Xtk ; t k,x) ) 2 dx + ν h U k 2 2;Γ k U 2 2;Ω + k= k= ω k U k 2 2;Ω k + U k 2 2;Ω k k= g k 3) for =,...,n, where ω k, g k are certain positive numbers, depending on the same quantities as functions ω and G in 8), and satisfying the estimates n k= ω k c and n k= g k c 2 with appropriate constants c and c 2 independent of n). 5 The non stationary approximations, their estimates and weak convergence We define for t k < t t k where k =,...,n) u n x,t) := { Uk x) if x Ω k, if x R 3 + Ω k, U n x,t) := { Uk x) if x Ω k, O if x R 3 + Ω k, u n x,t) := u n Yt k ; t,x),t ) = U k Ytk ; t,x) ) if x Γ t. Estimates of the sequences {u n }, {U n } and {u n }. Inequality 3) implies that there exist c 3 h) > and c 4 h) > such that both c 3 h) and c 4 h) tend to zero as h + and [ c3 h) ] u n.,t) 2 2;R 3 u 2 2;Ω + + t t + ν U n.,s) 2 2;R 3 + t ds + 2K u n.,s) 2 2;Γ s ds λ n s) u n.,s) 2 2;R 3 + ds + c 2 + c 4 h) 32) where λ n s) := ω k for t k < s t k. Applying Gronwall s lemma, we deduce that there exists c 5 > depending on c, c 2 and u 2;Ω ) such that for all n N so large that c 3 h) 2 and for all t,t ), we have u n.,t) 2;R 3 + c 5. 33) Using this estimate in 32), we observe that there exist c 6 and c 7 independent of n and such that T T U n.,s) 2 2;R 3 ds c + 6, u n.,s) 2 2;Γ s ds c 7. 34) Inequalities 33) and 34) conversely yield:

15 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 5 U k 2;Ωk c 5 k =,...,n) and h n k= U k 2 2;Ω k c 6. 35) Weak convergence of selected subsequences. Estimates 33) and 34) imply that there exist subsequences of {u n }, {U n } and {u n } we shall denote them again by {u n }, {U n } and {u n } in order not to complicate the notation) and functions u L,T ; L 2 R 3 +) 3 ), U L 2,T ; L 2 R 3 +) 9 ) and u L 2 Γ,T ) ) 3 such that u n u weakly in L,T ; L 2 R 3 +) 3 ) for n +, 36) U n U weakly in L 2,T ; L 2 R 3 +) 9 ) for n +, 37) u n u weakly in L 2 Γ,T ) ) 3 for n + 38) with the following relations between u, U and u : Lemma 2. a) U = u in the sense of distributions in Q,T ), b) u L 2,T ; W,2 σ Ω t )), c) u = u on Γ,T ) here u denotes the trace of function u Q,T ) on Γ,T ) ). The proof can be made by standard techniques. 6 The limit function u: a solution of the weak problem 7) Suppose that φ is a fixed infinitely differentiable divergence free vector function in R + 3 [,T ] with a compact support in R + 3 [,T ), such that φ n = on Γ [,T ]. Using the relation between u n and the solutions of the steady weak problem 3), one can verify that u n with U n standing for u n and u standing for the trace on Γ,T ) ) satisfies the non steady weak problem 7), up to a correction which tends to zero as n +. The intermediate step is to use 3) with Φ k = φ.,t k ).) Applying 36) 38), we can pass to the limit as n + in all the linear terms. Thus, the limit of the nonlinear term the integral of u n U n φ) also exists. So we obtain: { [ t φ + a t φ] a t + u) u φ a t + 2ν [ a t + u)] s : φ } dxdt Ω t + lim n + un U n φ dxdt + K [at + u V t ] φ dsdt Ω t Γ t T = f φ dxdt + + u Ω t Ω a ) φ.,) dx. 39) Comparing 39) with 7), we observe that in order to verify that u is a solution of the weak problem 7), it is sufficient to show that there exists a subsequence of

16 6 Jiří Neustupa and Patrick Penel {u n } we shall denote it again by {u n }) such that lim n + un U n φ dxdt = Ω t u u φ dxdt. 4) Ωt This limit procedure is not standard because of the variability of domain Ω t and the choice of the test function φ, which generally has only the normal component equal to zero on Γ,T ). We explain it in greater detail in the next six paragraphs. Cutting off function φ. Let ε > be given. Then, due to 33) and 34), there exists κ > so small that tc +κ un U n [ ] tc φ dxdt c Ω t 8 ess sup u n +κ.,t) 2;Ω t U n.,t) 2;Ω t dx <t<t t c κ t c κ c 8 c 5 2κ c 6 < ε 4) for all n N sufficiently large. Here c 8 is the maximum of φ on R 3 + [,T ].) Let η 3 be an infinitely differentiable cut off function of variable t defined on the interval [,T ], with values in [,], such that η 3 t) := { for t [,tc κ ] [t c + κ,t ], for t [ t c 2 κ,t c + 2 κ ], The function φ x,t) := η 3 t)φx,t) equals zero for t c 2 κ t t c + 2 κ and tc +κ un U n φ φ ) dxdt < ε Ω t t c κ due to 4). Since ε can be chosen arbitrarily small, it is sufficient to prove 4) with function φ instead of φ. Approximation of function φ. Since each of the domains Ω k for k =,...,n) has the cone property because all the time instants t k differ from t c ), inequalities 35) and the Sobolev imbedding theorem imply that U k L 6 Ω k ) 3. This means that u n.,t) L 6 R 3 +) 3 for all t,t ). Moreover, if we restrict ourselves to times t Iκ ), where Iκ ) := [,t c 2 κ ) tc + 2 κ,t ], then the cone parameters in the definition of the cone property of domain Ω t can be chosen to be independent of t. Hence the constants in the imbedding inequalities also become independent of t and we obtain the uniform estimate u n.,t) 6;R 3 + C u n.,t) 2;R U n.,t) 2;R 3 + ) for all t Iκ ). From this information and from 34), we can deduce that the product u n U n belongs to L 2 Iκ ); L R 3 +) 3) L Iκ ); L 3/2 R 3 +) 3). By interpolation, we obtain the inclusion u n U n L r Iκ ); L s R 3 +) 3) for r, s such that 2/r + 3/s = 4. Particularly, u n U n L 5/4 Iκ ); L 5/4 R 3 +) 3).

17 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 7 Function φ can be approximated by infinitely differentiable divergence free vector functions that have a compact support in Q [,T ) with an arbitrary accuracy in the norm of the space L 5 Iκ ); L 5 Ω t ) 3). Hence, given ε 2 >, there exists such a vector function φ which satisfies un U n φ dx un U n φ dx < ε Ω t Ω t 2 for all n N sufficiently large. Since ε 2 can be chosen to be arbitrarily small, we can prove 4) only with the function φ instead of φ respectively instead of φ ). Partition of function φ. Let m N. We denote τ = T /m for =,...,m). There exist m + infinitely differentiable functions θ,..., θ m on [,T ] with their values in the interval [,] such that suppθ I := [τ,τ ), suppθ I := τ,τ + ) for =,...,m ), suppθ m I m := τ m,τ m ] and m = θ t) = := θ φ for =,,...,m). The functions φ where Q I = { x,t) for t T. Now we put φ are divergence free, they have compact supports in Q I R 3 [,T ]; t I, x Ω t} ) and m = φ = φ in Q [,T ]. Denote by K be the orthogonal proection of suppφ into R 3. If m is large enough then the distance between K and Γ t is greater than one half of the distance between suppφ and Γ [,T ] for all t I. Thus, there exists a bounded open set Ω in R 3 with the boundary of the class C, such that K Ω Ω Ω t for all t I. So, we conclude that in order to prove 4), it is sufficient to treat 4) separately with φ = φ for =,,...,m) and to show that lim u n u n φ n + I Ω dxdt = u u φ I Ω dxdt. 42) The local Helmholtz decomposition of function u n. We denote by Pσ the Helmholtz proection in L 2 Ω )3. Put wn := P σ un. The function I Pσ )un has the form ϕn for an appropriate scalar function ϕ n. 42) can now be written as [ lim w n + I Ω n wn φ + wn 2 ϕn φ + ϕn wn φ + ϕn 2 ϕn φ ] dxdt = I Ω u )u φ dxdt. 43) Since ϕn 2 ϕn = 2 ϕn 2) and φ.,t) Lσ 2 Ω ), the integral of ϕ n 2 ϕn φ on Ω equals zero. The convergence 36) and 37), the coincidence of U n with u n on Ω I and the boundedness of operator Pσ in L2 Ω )3 and in W,2 Ω )3 imply that

18 8 Jiří Neustupa and Patrick Penel wn w = Pσ u, and ϕn ϕ = I Pσ )u for n + 44) weakly in L 2 I ; W,2 Ω )3 ) and weakly in L I ; L 2 σ Ω )). Strong convergence of a subsequence of {wn }. We are going to show that there exists a subsequence of {wn } that tends to w strongly in L 2 I ; Lσ 2 Ω )) as n +. We shall therefore use the next lemma, see J. L. Lions [8, Theorem 5.2]. Lemma 3. Let < γ < 2 and let H, H and H be Hilbert spaces such that H H H. Let H γ R; H, H ) denote the Banach space { w L 2 R; H ); ϑ γ ŵϑ) L 2 R; H ) } with the norm w γ;r := w 2 L 2 R;H ) + ϑ γ ŵϑ) 2 L 2 R;H )) /2. Here ŵϑ) is the Fourier transform of wt).) Let H γ a,b; H, H ) further denote the Banach space of restrictions of functions from H γ R; H, H ) onto the interval a,b), with the norm w γ;a,b) := inf z γ;r where the infimum is taken over all z H γ R; H, H ) such that z = w a.e. in a,b). Then H γ,t ; H, H ) L 2 a,b; H). Consider {;...; m} fixed. We shall use Lemma 3 with a,b) = I, H = Wσ,2Ω ), H = L2 σ Ω ) and H = W,2,σ Ω ). Here W,2,σ Ω ) denotes the dual to W,2,σ Ω ) where W,2,σ Ω ) is the closure of C,σ Ω ) in W,2 Ω )3. The space W,2,σ Ω the trace on Ω ) can be characterized as the space of functions from W,2 σ equal to zero.) We claim that {wn Ω ) that have } is bounded in the space H γ I ; H, H ). The boundedness of {wn } in L2 I ; H ) follows from 33), 34), from the coincidence of U n with u n on Ω I and from the boundedness of operator Pσ in L2 Ω )3 and in W,2 Ω )3. Thus, we only need to verify that { ϑ γ ŵn } is bounded in the space L 2 I ; H ), i.e. in L 2 I ; W,2,σ Ω )). Let z n by zero of wn from the time interval I onto R. Then ẑ nϑ) = + e 2π itϑ w nt)dt = be an extension tk e 2π itϑ Pσ U k dt 45) k Λ n t k where Λ n is the set of such indices k {;...;n} that [R 3 t k,t k )] suppφ /. Λ n has the form Λ n = {l;l +;...;q} where l q n. Calculating the integrals in 45), we obtain ẑ nϑ) = = 2π iϑ q k=l [ e 2π it ϑ k e 2π it k ϑ ] Pσ 2π iϑ U k [ e 2π it l P σ U l e 2π it q P σ U q ] + 2π iϑ q k=l+ e 2π it k ϑ [Pσ U k P σ U k ].

19 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 9 Since Ω Ω s for all s I, we also have Ω Ω k for all k Λ n if n is large enough). If ϑ then, using 45) and 35), we can estimate the norm of ϑ γ ẑn ϑ) in W,2,σ Ω ) as follows: ϑ γ ẑnϑ),2;ω CΩ ) ϑ γ q h U k 2:Ω CΩ ) ϑ γ. 46) k=l If ϑ > then we must proceed more subtly: ϑ γ ẑ nϑ),2;ω ϑ γ 2π + ϑ γ q 2π Pσ U k P σ U k,2;ω k=l+ CΩ ) ϑ γ ) U l 2:Ω + U q 2:Ω + ϑ γ 2π q sup k=l+ ψ k ψ k,2;ω P σ U l,2;ω + Pσ U ) q,2;ω Ω U k U k ) ψ k dx 47) where the supremum is taken over all ψ k W,2,σ Ω ) such that ψ k,2;ω >. The sum in 47) can be estimated by S + S 2 where S = S 2 = q sup k=l+ ψ k ψ k,2;ω q sup k=l+ ψ k ψ k,2;ω Ω Ω [ Uk x) U k Xtk ; t k,x) )] ψ k x)dx, [ Uk x) U k Xtk ; t k,x) )] ψ k x)dx. The function ψ k, extended by zero to R 3 + Ω, belongs to W,2 σ Ω k ). Hence the integral of [ U k x) U k Xtk ; t k,x) )] ψ k x) on Ω equals the integral of the same function in Ω k and it can be therefore expressed by means of 3). Thus, S can be estimated: S q k=l+ sup ψ k ψ k,2;ω Ω h U k x) [ a] k x) ψ k x) dx k h U k x) U k x) ψ k x) dx h ν { [ a] k x) + U k x) } s : ψ k x) dx Ω k Ω k K [ a k x) + U k x) V k x) ] ψ k x) ds Γ k + hf k x) ψ k x) dx + A k x) ψ k x) dx. Ω k Ω k The surface integral on Γ k equals zero because the function ψ k is zero on Γ k. The right hand side can be estimated by CΩ ) by means of 7), 35), standard inequalities based on the Sobolev imbedding theorem applied in Ω ) and the Hölder in-

20 2 Jiří Neustupa and Patrick Penel equality. Let us show the procedure in greater detail, for example, in the case of the terms containing the product U k U k ψ k : q sup k=l+ ψ k ψ k,2;ω q CΩ ) h U k 2 dx Ω k CΩ ) h [ CΩ ) k=l+ h [ CΩ ) + q k=l+ q k=l+ h Ω h U k U k ψ k dx k ) /2 q h k=l+ U k 3/2 2;Ω k U k 3/2 6;Ω k ) /3 Ω k U k 3 dx U k 3/2 2;Ω k U k 3/2 2;Ω k + U k 3/2 2;Ω k ) ] /3 n k= U k 2 2;Ω k ) 3/4 ] /3 CΩ ). ) /3 Here the constant CΩ ) also depends on the right hand sides of 7) and 35). In order to estimate S 2, we use the identities U k x) U k Xtk ; t k,x) ) = = tk tk t k d dξ U k Xξ ; tk,x) ) dξ t k a ξ Xξ ; t k,x) ) U k Xξ ; tk,x) ) dξ. Then the sum S 2 can be estimated by means of ) and 35) as follows: S 2 CΩ ) sup ψ k CΩ )[ q CΩ ). k=l+ ψ k 6;Ω ψ k,2;ω [ tk t k tk t k q [ tk k=l+ t k Ω Ω Uk Xξ ; tk,x) ) ] /2 2 dxdξ a ξ Xξ ; t k,x) ) 3 dx) 2/3 dξ ] /2 Uk x) ] /2 [ 2 dxdξ a ξ x) 5 dxdξ Ω k Ω ξ Substituting the estimates of S and S 2 to 47), we finally obtain ] /5 ϑ γ ẑ nϑ),2;ω CΩ ) ϑ γ. 48) The constant CΩ ) is independent of n. Recall that inequality 48) holds for ϑ >. Since the exponent γ satisfies < γ < 2, the right hand side of 48) is integrable on, ),+ ) with power 2. This, together with 46), implies

21 A Weak Solvability of the Navier Stokes Equation around a Ball Striking the Wall 2 that the sequence { ϑ γ ẑn ϑ)} is bounded in L2 R; W,2,σ Ω )). Consequently, the sequence {wn } is bounded in Hγ I ; Wσ,2Ω ), W,2,σ Ω )). This space is reflexive, hence there exists a subsequence we denote it again by {wn }) which converges weakly in H γ I ; Wσ,2Ω ), W,2,σ Ω )). Due to 44), the limit must be w. Applying now Lemma 3, we have: w n w = P σ u strongly in L2 I ; L 2 Ω )3). This strong convergence, together with the weak convergence 44), enables us to pass to the limit in the first three terms on the left hand side of 43). The procedure is standard see e.g. J. L. Lions [8] or R. Temam [26]), therefore we omit the details. Using also the equation Ω ϕ ) ϕ φ dx =, following from the inclusion φ Lσ 2 Ω ) and from the identity ϕ ) ϕ = 2 ϕ 2), we can verify the validity of 43), and consequently also the validity of 4). This confirms that u is a weak solution of the weak problem 7). The proof of Theorem is thus completed. 7 Concluding remarks Energy inequality for the weak solution. The limit processes 36) 38) and Lemma 2 imply that the limit function u, which is a solution of 7), satisfies the same estimates 33) and 34) as the approximations. Inequality 33) thus provides an estimate of the kinetic energy associated with the flow u at a.a. times t,t ) and the first inequality in 34) estimates the dissipation of this energy in the time interval, T ). The question whether u also satisfies the energy inequality 8), formally derived in Section 2 see Lemma ), is open. To obtain 8), it would be necessary to make the limit transition in inequality 32) which is a discrete equivalent of 8)). Here we need a piece of information on the strong convergence of a subsequence {u n } in L 2,T ; Lσ 2 Ω t )) in order to control the second term on the right hand side of 32). This is, however, a problem because we have only obtained the strong convergence of appropriate local interior Helmholtz proections of u n in Section 6. It was sufficient for the limit transition 4), but it does not enable us to treat the integral on the right hand side of 32) in a similar way. The condition of smallness iv). Condition iv) see Lemma ) requires a sufficient smallness of the speed of the ball B t at times close to the critical instant t c of the collision of the ball with the wall. We need this condition because estimate 3), based on the continuous imbedding W,2 Ω t )) L 6 Ω t ), cannot be used in order to estimate the approximations at times close to t c. The constant in the imbedding inequality increases too rapidly to infinity as t t c.) Thus, we use estimate ) instead of 3) at times close to t c and since we need the right hand side to be absorbed by the viscous term, it must be sufficiently small.

22 22 Jiří Neustupa and Patrick Penel Flow around a body of a general shape striking the wall. We have mainly used the information on the shape of B t i.e. that it is a ball) in the region close to the point of the collision of B t with the wall. Particularly, the shape of B t influences the form of function g t in Section 2. With another function g t, we would obtain other inequalities than 7) 3) for function a t.) Thus, Theorem could be generalized in such a way that instead of the ball B t we would speak on a compact body of another however sufficiently smooth) shape, which coincides with a ball in the neighborhood of the point of the collision. Acknowledgements The research was supported by the Grant Agency of the Czech Academy of Sciences Grant No. IAA962), by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AVZ953, and by the University of Sud Toulon Var. References. Adams R.: Sobolev Spaces. Academic Press, New York San Francisco London 975) 2. Conca C., San Martín J., Tucsnak M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. in Partial Diff. Equations 25 5&6), ) 3. Desardins B., Esteban M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Rat. Mech. Anal. 46, ) 4. Desardins B., Esteban M.J.: On weak solutions for fluid rigid structure interaction: compressible and incompressible models. Comm. in Partial Diff. Equations 25 7&8), ) 5. Dautray R., Lions M.J.: Mathematical Analysis and Numerical Methods for Science and Technology II. Springer, Berlin Heidelberg 2) 6. Feireisl E.: On the motion of rigid bodies in a viscous incompressible fluid. J. of Evolution Equations 3, ) 7. Fuita H., Sauer N.: On existence of weak solutions of the Navier Stokes equations in regions with moving boundaries. J. Fac. Sci. Univ. Tokyo, Sec. A, 7, ) 8. Galdi G.P.: An Introduction to the Mathematical Theory of the Navier Stokes Equations, Vol. I, Linear Steady Problems. Springer Tracts in Natural Philosophy ) 9. Galdi G.P.: An Introduction to the Mathematical Theory of the Navier Stokes Equations, Vol. II, Nonlinear Steady Problems. Springer Tracts in Natural Philosophy ). Galdi G.P.: An Introduction to the Navier Stokes initial boundary value problem. In Fundamental Directions in Mathematical Fluid Mechanics, ed. G.P.Galdi, J.Heywood, R.Rannacher, series Advances in Mathematical Fluid Mechanics, Vol., Birkhauser Verlag, Basel, 98 2). Galdi G.P.: On the motion of a rigid body in a viscous fluid: a mathematical analysis with applications. In Handbook of Mathematical Fluid Dynamics I, Ed. S. Friedlander and D. Serre, Elsevier 22) 2. Gunzburger M.D., Lee H.C., Seregin G.: Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2, ) 3. Hoffmann K.H., Starovoitov V.N.: On a motion of a solid body in a viscous fluid. Two dimensional case. Adv. Math. Sci. Appl. 9, ) 4. Hoffmann K.H., Starovoitov V.N.: Zur Bewegung einer Kugel in einen zähen Flüssigkeit. Documenta Mathematica 5, 5 2 2) 5. Hopf E.: Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen. Math. Nachr. 4, )

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