On a Motion of a Perfect Fluid in a Domain with Sources and Sinks

Transcription

1 J. math. fluid mech. 4 (22) /2/21-17 $ / c 22 Birkhäuser Verlag, Basel Journal of Mathematical Fluid Mechanics On a Motion of a Perfect Fluid in a Domain with Sources and Sinks N. V. Chemetov and V. N. Starovoitov Communicated by V. A. Kazhikhov Abstract. We prove the global weak solvability of the problem on a flow of an ideal incompressible fluid in a domain containing various types of sources and sinks. In the first part of the paper the case of point sources and sinks is considered. This situation is characterized by singularities of the velocity vector field, whose second powers are not integrable. We suggest a new generalized formulation of the problem, where the singularities are described by corresponding measures. As a development of this technique, we solve in the second part of the paper the problem on a flow of an ideal fluid through a given domain. Mathematics Subject Classification (2). 76B3, 35D5. Keywords. Perfect fluid, source, flow. 1. Introduction In this paper we consider a problem on a singular flow of an ideal incompressible fluid in a simply connected bounded domain R 2 with a boundary Γ of the class C 2+α, <α<1. We suppose that = Γ contains a finite number of sources and sinks placed at points M i (t) (,...,) at the instant of time t. The sets of indexes {1,...,l} and {l +1,...,} correspond to the sources and to the sinks respectively. This problem will be referred as Problem A. Let us denote by (t) the domain \ {M i(t)} and by Γ (t) the set Γ \ {M i(t)}. The fluid flow is governed by the Euler equations: v t +(v )v= p, (1.1) div v = (1.2) for (x,t) T, where x =(x 1,x 2 ), T = {(x,t) R2+1 : x (t), t (,T)}, <T <,v=(v 1,v 2 ) is the velocity field of the flow, p is the pressure. This work was supported by NATO Grant PST.CLG Mathematical models of complex media

2 2 N. V. Chemetov and V. N. Starovoitov JMFM We also prescribe the initial data: and the no flux condition at Γ : v(x, ) = v (x), x (), (1.3) v n =, (1.4) where n is the external normal to Γ. The point sources and sinks are singularities of the velocity field. The function v can be represented as v(x,t)=u(x,t)+ a i 2π x M i (t) x M i (t) 2, (1.5) where u is a regular part of the velocity, div u =,a i is the strength of the i-th source if M i Γ. If M i Γ then the strength of the i th source is equal to a i /2. We assume that a i,,...,, are functions of t and a i (t), a i (t), i =1,...,l, i = l+1,..., for all t [,T]. As a consequence of (1.3) and (1.4), we have the following condition for {a i }: ai + a i =, (1.6) 2 where we take the sum over all i {1,...,} such that M i and over i {1,...,} with M i Γ. There are many works devoted to singular flows of an ideal fluid. We refer the reader to the papers [2], [3], [5], [8], dealing with the case, where v L 2 and curl v is a bounded measure. As one can see from (1.5), in our problem the velocity field does not belong to the space L 2. The same order of singularity arises in the problem on a motion of a point vortex in an ideal fluid. This situation was studied in the papers [9], [17], [14], [15], [16]. A problem similar to that studied in the current paper, was investigated in [19], [7], [12]. In these papers the velocity is a regular function, and the fluid is allowed to flow in or out the domain through parts of its boundary. In particular, V. I. Yudovich in his paper [19] mentions (without proof) that it is possible to obtain the solution of the problem with point sources by shrinking the components of the boundary where the fluid flows into the domain. Here we use a different technique. Moreover, in Section 5, we obtain solvability of the problem of the flow of an ideal fluid through a given domain, by passing to the limit in a sequence of problems with point sources and sinks.

3 Vol. 4 (22) On a Motion of a Perfect Fluid 3 2. Statement of the problem and main result As we mentioned in the Introduction, the function v in our problem does not belong to the space L 2. Thus, there arise some difficulties with definition of a generalized solution because of the presence of the nonlinear term (v )v in the momentum equation (1.1). By this reason, we make use of the vorticity formulation of the problem. Applying formally the differential operator curl to equation (1.1), we obtain the following system ω t + div(ωv) =, (2.1) curl v = ω, (2.2) div v = (2.3) for (x,t) T, where curl v = v = v 1 / x 2 v 2 / x 1. For the function v the boundary condition (1.4) is preserved, and the function ω satisfies the initial following condition ω(x, ) = ω (x) = curl v (x), x (). (2.4) The statement of the problem (2.1) (2.4) is incomplete. Similarly to the problem of a flow of an ideal fluid through a given domain, we should set an additional condition at the sources. We shall prescribe the influx of the vorticity ω. This condition will be described later. Equations (2.1) (2.3) are valid in a punctured domain. However, it is more convenient to consider the problem in the whole domain. To this end, us introduce the following measure µ a = a i δ(x M i ), where δ(x M i ) is the Dirac function concentrated at the point M i and a i,i= 1,...,, are functions of t. That is, for any continuous function ϕ. It is not difficult to see that µ a,ϕ = div v = µ a. a i ϕ(m i ) In order to describe the influx of the vorticity we introduce one more measure: l µ ab = a i b i δ(x M i ), where the coefficients b i are functions of t.

4 4 N. V. Chemetov and V. N. Starovoitov JMFM Definition 2.1. A pair of functions {ω, v} is said to be a generalized solution of Problem A, if ω L ( T ), v satisfies (1.5) with u L ( T ), where T = [,T], and the following integral identities T ω (φ t + v φ)dxdt + ω φ dx = T µ ab,φ dt, (2.5) v ζdx= µ a,ζ for a.e. t [,T], (2.6) v ξdx= ωξdx for a.e. t [,T] (2.7) hold for arbitrary functions ζ C 1 (), ξ C 1 () and φ C 1 ( T ) such that φ(x,t) = for all x and φ(x,t)=asx i=l+1 {M i(t)}, t [,T]. Remark. In this definition we require that ω is a measurable function (not a measure). This condition plays an important physical role. It implies that the vorticity flows out the domain through the sinks and it is not being accumulated there. Remark. Equation (2.6) implies that v n = almost everywhere on the boundary, even if sources or sinks lie on Γ. Nevertheless, this equation contains information about the influx of the fluid into the domain. This can be verified by standard calculations similar to those used in the theory of singular integrals. The main result of this work is the following theorem. Theorem 2.2. Let M i (t), i =1,...,, be Lipschitz continuous functions and there exists δ > such that for every t [,T] the following conditions hold: (i) for all i =1,..., either dist(m i (t), Γ) δ or dist(m i (t), Γ) = ; (ii) M i (t) M j (t) δ, for all i l and j l +1. If ω L (), a i,b j L (,T) for all i =1,...,, j =1,...,l and (1.6) is fulfilled for a.e. t [,T], then there exists at least one generalized solution of Problem A. Moreover, the functions ω and u satisfy the estimates: ω L ( T ) ω L () + max b i L (,T ), (2.8),...,l u L (,T ;C α () ( ω C L ( T ) + a i with a constant C depending on δ and. L (,T ) ) (2.9) Remark. Condition (i) in the formulation of Theorem 2.2 means that there exist two types of sources and of sinks. The sources (the sinks) of the first type always

5 Vol. 4 (22) On a Motion of a Perfect Fluid 5 lie on the boundary of the flow domain. The sources (the sinks) of the second type move inside the flow domain and do not come to the boundary. We prove Theorem 2.2 in Sections 3 and Construction of approximate solutions Let us introduce the following notation B iρ (t) ={x R 2 : x M i (t) <ρ}, B ρ (t) = i=l+1 B iρ (t), i =1,...,, ρ (t) =\B ρ (t), ρ T ={(x,t) T : x ρ (t)}, {, x 1, η(x)= ( Cexp 1 1 x ), x < 1, 2 where C is a constant such that η(x) dx =1. R 2 Let ω, ε a ε i and bε i be smooth approximations of the functions ω, a i and b i such that ω(x) ε ω (x), a ε i(t) a i (t), b ε i(t) b i (t) as ε for a.e. x and a.e. t [,T]. We require also that ω ε L () ω L (), a ε i L (,T ) a i L (,T ), b ε i L (,T ) b i L (,T ). Taking χ ε i (x,t)=ε 2 η ( (x M i (t)) ε 1), i =1,...,l, l µ ε ab(x,t)= a ε i (t) b ε i (t) χ ε i (x,t), µ ε a(x,t)= a ε i (t) χ ε i (x,t) we construct functions ω ε and v ε as a solution of the following problem, which will be referred below as Problem A ε. t ω ε + div (v ε ω ε )=µ ε ab, (x,t) ε T, (3.1) ω ε (x, ) = ω(x), ε x ε (), (3.2) ( ω ε (x,t)=ε 1 x M i (t) ω ε M i (t)+ε x M ) i(t) x M i (t),t (3.3)

6 6 N. V. Chemetov and V. N. Starovoitov JMFM for x B iε (t), i=l+1,...,, curl v ε = ω ε, (3.4) div v ε = µ ε a, (3.5) for (x,t) T, v ε n=, (x,t) Γ T =Γ [,T]. (3.6) Let us elaborate on this point. We define the function ω ε in the domain ε T as a solution of the problem (3.1), (3.2) and then, by using (3.3), we extend it to the whole domain T as a continuous function such that max ω ε (x,t) max ω ε (x,t). (x,y) T (x,y) ε T (3.7) Instead of (3.3) we could use other extensions satisfying (3.7). Equations (3.1), (3.4) and (3.5) are understood in the sense of Definition 2.1. We prove solvability of Problem A ε by the Schauder fixed point theorem. Let z = z(x,t) be a function from C( T ) and z C(T ) D, where D = ω L () + max i L (,T ). The unique solution of the problem,...,l curl v = z, div v = µ ε a, (x,t) T (3.8) v n =, (x,t) Γ T, (3.9) can be represented as follows: v = v 1 + v 2 + v 3, (3.1) where v 1 = ψ, ψ = z, x and ψ =, x Γ, (3.11) a ε i v 2 (x,t)= 2π χ ε x y i (y,t) dy, x y 2 (3.12) B iε(t) ϕ v 3 = ϕ, ϕ =, x and n = v 2 n, x Γ. (3.13) By the classical theory of elliptic equations, v C( T ) and for all t (,T)we have: v 1 (,t) C β (), <β<1, v 2 (,t) C (), v 3 (,t) C 1+α (), where α (, 1) is such that Γ C 2+α. Moreover (see [18], [19], [6]), there exists a constant C 1 independent of ε such that v 1 (,t) C() C 1 z(,t) C(), (3.14) v 1 (x,t) v 1 (y,t) C 1 z(,t) C() x y ( 1+ log x y ) (3.15) for all x, y, t [,T].

7 Vol. 4 (22) On a Motion of a Perfect Fluid 7 We wish to obtain an additional information about the functions v 2 and v 3. At first we notice that there exists a positive, infinitely differentiable function θ : R + R + such that θ () =, θ(ε) =1, θ(ρ) Cas ρ [,ε] for some constant C and where v 2 (x,t)= a ε i (t) 2π V ε(x M i (t)), (3.16) x V ε (x) = ε 2 θ( x ), x ε, x x 2, x ε. (3.17) Representation (3.16) can be obtained by direct calculations. We denote by V the function x x 2. Lemma 3.1. Let Σ be a curve of the class C 2+α, α (, 1], and the point x = belongs to Σ. Then there exists a constant K independent of ε such that V ε n C α (Σ) + V n C α (Σ) K, (3.18) where n is the unit normal to Σ. The function V n is defined at the point x = by continuity and lim V (x) n(x) = k() x 2, (3.19) x Σ where k is the curvature of Σ. Let the curve Σ be prescribed parametrically by a function x = x(s), where s is the natural parameter on Σ. We take x() =. Due to the Frene formula, (x(s) n(s)) = x(s) n (s) =k(s)x(s) x (s)= k(s) ( x(s) 2 ). (3.2) 2 This relation with s = yields (3.19). Since the function V is smooth everywhere on the set Σ \{x=}, in order to prove estimate (3.18), we need only to consider some neighborhood of the point x =. Let s > be such that ( x(s) 2) for s [,s ]. Then, by integrating (3.2), we obtain for s [,s ]: 2 x(s) n(s) k(s) x(s) 2 = = k(ξ) s s k(p) ( x(p) 2) dp k(s) x(s) 2 ( x(p) 2 ) dp k(s) x(s) 2 = x(s) 2 k(ξ) k(s) Cs α x(s) 2, (3.21)

8 8 N. V. Chemetov and V. N. Starovoitov JMFM where C is a constant and ξ [,s] is such that s k(p) ( x(p) 2) s dp = k(ξ) It is not difficult to verify the following identity: ( ) x n x 2 = x x x 2 ( x(p) 2 ) dp. ( k 2 x n x 2 Integrating this identity from s 1 to s 2 with s 1,s 2 (,s ) and taking into account (3.21), we find: x(s 2) n(s 2 ) x(s 2 ) 2 x(s 1) n(s 1 ) C x(s 1 ) 2 s 2 s 1 ). s α 1 ds C s 2 s 1 α. (3.22) This estimate implies (3.18) since V ε is a mollification of the function V. Due to (3.13) and Lemma 3.1, there exists a constant C 3 depending on δ but not on ε such that We notices also that v 3 (,t) C α () C 3 a i (t). (3.23) x M i (t) v 2 (x,t) x M i (t) C as x M i (t) = ε (3.24) ε for some positive constant C and i = l +1,...,. For all s, t [,T] and y we can define an operator U s,t : R 2 such that U s,t (y) is the unique solution of the problem: dx(t) = v(x(t),t), x(s)=y. (3.25) dt Due to (3.9), if y Γ then U s,t (y) Γ for all s, t [,T]. Thus, U s,t :. Moreover, from (3.14), (3.24) and (3.23), it follows that there exists ε <δ /2 such that x M i (t) v(x,t) x M i (t) > dm i (t) as x M i (t) = ε (3.26) dt for any ε<ε,i=l+1,...,. Therefore, if y B iε (s) then U s,t (y) B iε (t) as ε<ε for all t>s. As it was shown by T. Kato [6], the function U s,t (y) satisfies the estimate: U s,t (y) U s,t (y ) C ε (D) ( y y σ + t t σ + s s σ) (3.27) for any (y,s),(y,s ) ε T and t s, t s, where σ = σ(d).

9 Vol. 4 (22) On a Motion of a Perfect Fluid 9 Let us consider the problem ω t + div (ω v) =µ ε ab, (x,t) ε T, (3.28) ω(x, ) = ω(x), ε x ε (), where v is the solution of the problem (3.8) (3.9). From results obtained in [4], we have that this problem has a unique bounded solution. It is not difficult to verify that the solution has the form: ω(x,t)=ω( ε Ut, (x) ) ( t exp t + µ ε a( Ut,s (x),s ) ) ds µ ε ab( Ut,s (x),s ) ( exp t s µ ε a( Ut,θ (x),θ ) ) dθ ds (3.29) for (x,t) ε T. Since µε a(x,t) and µ ε ab (x,t) βµε a(x,t)as(x,t) ε T, where β = max,...,l bε i L (,T ), we have the following estimate ω(x,t) ω ε C() t ( t ) + β µ ε a(u t,s (x),s) exp µ ε a(u t,θ (x),θ)dθ ds = ω ε C() + β = ω ε C() + β t d ( ds exp ( ( 1 exp s t s t ) µ ε a(u t,θ (x),θ)dθ ds ) ) µ ε a(u t,θ (x),θ)dθ ω L () + max b i L (,T ) = D (3.3),...,l for (x,t) ε T. Notice, that this estimate is uniform with respect to ε. Thus, ω C(T ) ω C( ε T ) D. (3.31) Making use of (3.27), (3.29) and the smoothness of the functions ω ε, a ε i, bε i,we obtain: ω(x,t) ω(x,t ) C ε (D) ( x x σ + t t σ) (3.32) for any (x,t), (x,t ) ε T, where the constant C ε(d) depends on ε. Due to (3.3) this estimate holds for any (x,t), (x,t ) T. Therefore, we have constructed a mapping F such that F : Z Z and ω = F (z), where Z = {z C( T ): z C(T ) D}.

10 1 N. V. Chemetov and V. N. Starovoitov JMFM Let us verify that F is continuous. Let z n,z Z and z n z as n in C( T ). From (3.14), it follows that there exists a constant C depending only on such that v n v C(T ) C z n z C(T ), (3.33) where v n is a solution of (3.8) (3.9) with z replaced by z n. Let us denote by U n s,t(y) the solution of the problem dx dt = v n(x,t), Since the function v satisfies the estimate x(s)=y. v(x 1,t) v(x 2,t) Cγ( x 1 x 2 ), x 1,x 2 ε (t),t [,T], where γ(s) =s(1 + log s ) and the constant C depends on D and ε, we have the following inequality U n s,t(y) U s,t (y) (s t) v n v C(T ) +C s γ ( U n s,θ(y) U s,θ (y) ) dθ, (3.34) which holds for all s, t [,T], s t, and (y,s) ε T. Estimates (3.33) and (3.34) allow us to conclude that (see [1]) max U t,( ) n U t, ( ) C( ε (t)) as n. t [,T ] That is, ω n ω C( ε T ) and, due to (3.3), ω n ω C(T ) asn. Here ω n = F (z n ). Thus, F is continuous in Z. Due to (3.32), F is a compact mapping and, therefore, it has a fixed point in the set Z. So, we have proved that there exist functions ω ε and v ε that are a generalized solution of Problem A ε. Moreover, t ω ε L ( T ) D = ω L () + max,...,l b i L (,T ). (3.35) 4. Passage to the limit as ε From (3.35), it follows that from the sequence {ω ε } we can select a subsequence, denoted again by {ω ε }, such that ω ε ω as ε weakly in L ( T ). Let us investigate the sequence {v ε }. The function v ε can be represented as v ε = v 1ε + v 2ε + v 3ε, where the functions v iε, i =1,2,3,satisfy (3.11), (3.12) and (3.13) with v 1, v 2, v 3, ψ, z and ϕ replaced by v 1ε, v 2ε, v 3ε, ψ ε, ω ε and ϕ ε respectively. Due to (3.35), relation (3.11) implies that v 1ε L (,T ;W 1 p ()) CD, (4.1)

11 Vol. 4 (22) On a Motion of a Perfect Fluid 11 where 1 p< and the constant C depends only on. From estimate (4.1), we have that the sequence {v 1ε } has a subsequence, denoted again by {v 1ε }, such that v 1ε v 1 as ε weakly in L (,T;W 1 p()) for all p [1, ). Let us investigate the convergence of the sequences {v 2ε } and {v 3ε }.Itisnot difficult to see that v 2ε v 2 strongly in L q (,T;L r ()) for all q [1, ) and r [1, 2), where v 2 (x,t)= a i (t) 2π x M i (t) x M i (t) 2. (4.2) Besides that, v 2ε (x,t) v 2 (x,t) for almost all x Γ and t [,T]. Since, by Lemma 3.1, the sequence {v 2ε n} is uniformly bounded in L (Γ T ), we conclude that v 2ε n v 2 n in L 2 (Γ T ). By (3.13) and the classical theory of elliptic equations (see [11]), we have the estimate: v 3ε v 3ε L2() C v 2ε n v 2ε n L2(Γ), which implies that {v 3ε } is a Cauchy sequence in L 2 ( T ). Consequently, it converges to some function v 3 in this space. Now we can pass to the limit in the linear equations (3.4) and (3.5). The limit functions v = v 1 + v 2 + v 3 and ω satisfy the integral identities (2.6) and (2.7). Let us prove (2.5). Let ρ be an arbitrary number from the interval (,δ /2) and φ C 1 ( T )be such that φ(x,t) = for all x and φ(x,t)=asx B ρ (t), t [,T]. The generalized formulation of equation (3.1) looks as follows T ω ε (φ t + v ε φ)dxdt + ω ε φ dx = Let us consider the only nonlinear term in this identity: T ω ε v ε φdxdt = T T µ ε ab,φ dt. (4.3) ω ε (v 1ε + v 2ε + v 3ε ) φdxdt. (4.4) The passage to the limit in other terms causes no difficulties. The strong convergence of the sequences {v 2ε } and {v 3ε } implies lim T ε ω ε v kε φdxdt = T ω v k φdxdt (4.5) for k =2,3. The case k = 1 should be treated in a different way. Denote by B ρ (t 1,t 2 ) the set t [t 1,t 2] Bρ (t) and by ρ (t 1,t 2 ) the set \B ρ (t 1,t 2 ) for t 1,t 2 [,T].Equation (4.3) with an arbitrary test function φ vanishing outside

12 12 N. V. Chemetov and V. N. Starovoitov JMFM of ρ (t 1,t 2 ) [t 1,t 2 ] yields the estimate: t 2 t 1 ω ε φ t dxdt C φ L2(t 1,t 2;Wp 1(ρ (t 1,t 2))), where p>2 and the constant C depends on D,,ρand δ. Consequently, ω ε ω in L 2 ( t1,t 2 ; [ W1 p ( ρ (t 1,t 2 )) ] ), (4.6) where W 1 p ( ρ (t 1,t 2 )) = {φ W 1 p () : φ(x) =asx B ρ (t 1,t 2 )} and X is a space adjoint to X. This fact follows from compact imbedding of L ( ρ (t 1,t 2 )) in [ W 1 p ( ρ (t 1,t 2 )) ] and from well known results obtained in [1], [13]. Let us define numbers t,t 1,...,t n [,T] such that t =,t n =T and B ρ (t κ,t κ+1 ) B 2ρ (t) for any t [t κ,t κ+1 ] for κ =,1,...,n 1. If φ C 2 ( T ) and φ(x,t)=asx B 2ρ (t),t [,T] then we have: lim T ε ω ε v 1ε φdxdt = = = n lim t κ ε κ=1 t κ 1 ρ (t κ 1,t κ) n t κ κ=1 t κ 1 ρ (t κ 1,t κ) T ω ε v 1ε φdxdt ω v 1 φdxdt ω v 1 φdxdt. (4.7) Here we used (4.6) and the fact that v 1ε φ v 1 φ weakly in L 2 (t 1,t 2 ; W 1 p( ρ (t 1,t 2 ))) for any t 1,t 2 [,T]. This fact follows from (4.1). Since ρ is arbitrary, relations (4.5), (4.7) and (4.3) give us (2.5). Estimates (2.8) follows from (3.35). Let us prove (2.9). The function u is the sum of v 1 and v 3, which are solutions of problems (3.11) with z = ω and (3.13) with (v 2 n)(x,t)= a i (t) 2π V (x M i(t)) n(x), respectively, where the function V n is defined in Lemma 3.1. Estimate (2.9) follows from (3.15), Lemma 3.1 and (3.23). Theorem 2.2 is proved.

13 Vol. 4 (22) On a Motion of a Perfect Fluid Flow of an ideal fluid through a given domain In this section we consider the problem on a flow of a perfect incompressible fluid through the domain. The problem will be referred as Problem B Statement of Problem B We apply the same approach as in the problem with point sources and sinks. To this end, let us redefine the measures µ a and µ ab introduced in Section 2. Namely, µ a,ϕ = a + ϕds + a ϕ ds, (5.1) Γ + Γ µ ab,ϕ = a + b ϕ ds, (5.2) Γ + where a +,a,b are functions of t and x, ϕ is a continuous function, Γ + and Γ are curves, depending on t. Suppose that the functions a + and ( a ) are nonnegative. That is, Γ + is a source and Γ is a sink. The curves Γ + and Γ can be prescribed parametrically as follows: Γ ± (t) : x=x + (τ,t), τ [, 1], t [,T]. (5.3) A decomposition of the velocity vector field similar to (1.5) is also valid for Problem B. Indeed, we can write v = u + u + + u, (5.4) where u is a continuous function and u ± (x,t)= 1 a ± x y (y,t) 2π x y 2 ds y. (5.5) Γ ± (t) Here we understand the integral in the Cauchy principal value sense. Definiton 5.1. A pair of functions {ω, v} is said to be a generalized solution of Problem B, if ω L ( T ), v satisfies (5.4) with u L ( T ) and integral identities (2.5) (2.7) (with µ a and µ ab defined by (5.1) and (5.2)) hold for arbitrary functions ζ C 1 (), ξ C 1 () and φ C 1 ( T ) such that φ(x,t) = for all x and φ(x,t)=asx Γ (t),t [,T]. The main result of this section is the following theorem. Theorem 5.2. Let the curves Γ ± satisfy the conditions: (i) the functions x ± are Lipschitz continuous with respect to τ and t; (ii) Γ (t) Γ for all t [,T];

14 14 N. V. Chemetov and V. N. Starovoitov JMFM (iii) there exists δ > such that for all t [,T] either Γ + (t) Γ or dist(γ + (t), Γ) δ. If ω L (), a ±,b C( T ) and κ + a + ds + a ds = (5.6) Γ + (t) Γ (t) for every t [,T], where κ + =2if Γ + and κ + =1if Γ + Γ, then there exists at least one generalized solution of Problem B. Moreover, the functions ω and u satisfy the estimates: ω L ( T ) ω L () + max b i(,t) C(Γ+ (t)), (5.7) t [,T ] u L (,T ;C α () ( ω C ( L ( T ) + max a + (t) L1(Γ + (t)) + a (t) L1(Γ (t))) ) t [,T ] (5.8) with a constant C depending on δ and. Remark. Just for the sake of simplicity, we have supposed that there are only one source curve Γ + and only one sink curve Γ. Of course, this result can be generalized to the case of any finite number of the source and sink curves. It is also possible to place point sources and sinks in Proof of Theorem 5.2 Let us approximate µ a and µ ab by sequences of measures µ k a and µ k ab, k N, which are defined as follows. We divide the curves Γ ± (t) intokarcs Γ ± i (t) of the length ± i (t), i =1,...,k, and take points M ± i (t) Γ± i (t). We can do it in such a way that the functions M ± i (t) will be Lipschitz continuous. Let us require also that max max (t) as k. t [,T ],...,k ± i For every continuous function ϕ we have: k µ a,ϕ = lim (a +( M + i (t),t) + i (t)ϕ( M + i (t)) +a ( M i (t),t) i (t)ϕ( M i (t))) where and k 2k = lim k a i (t) ϕ ( M i (t) ) lim k µk a,ϕ, a i (t) =a +( M + i (t),t) + i (t), M i(t)=m + i (t) for i =1,...,k a i (t) =a ( M i (t),t) i (t), M i(t)=m i (t) for i = k +1,...,2k.

15 Vol. 4 (22) On a Motion of a Perfect Fluid 15 Similarly, µ ab,ϕ = lim k k a i (t) b i (t) ϕ ( M i (t) ) lim k µk ab,ϕ, where b i (t) =b ( M + i (t),t),...,k. Problem B with µ k a and µ k ab instead of µ a and µ ab will be called Problem B k. By Theorem 2.2, it has a generalized solution {ω k, v k } for every finite k. Notice that estimates (2.8) and (2.9) do not depend on k. Furthermore, the sequence {v k } is bounded in L (,T;L q ()) for q [1, 2). Therefore, the sequences {ω k } and {v k } have subsequences such that ω k ω -weakly in L ( T ), v k v -weakly in L (,T;L q ()) as k. This fact allows us to obtain the linear equations (2.6) and (2.7). We prove identity (2.5) in a similar way as in Section 4. By this reason, we give only a sketch of the proof. The function v k can be represented as follows: v k = v k 1 + v k 2 + v k 3, where v k 1 = ψ, ψ = ω k, x and ψ =, x Γ, v k 2(x,t)= 2k a i (t) 2π x M i (t) x M i (t) 2, v k ϕ 3 = ϕ, ϕ =, x and n = vk 2 n, x Γ. Due to (2.8), the sequence {v k 1} is bounded in L (,T;Wp()), 1 1 p<. Let v 1 be its -weak limit in this space. The sequence {v k 2} converges to u + + u in L r (,T;L q ()) for all r [1, ) and q [1, 2). Moreover, by Lemma 3.1, v k 2(x,t) n(x) is a continuous function on Γ. Here n is the normal to Γ. Consequently, v k 2(x,t) n(x) ( u + (x,t)+u (x,t) ) n(x) for all x Γ and for almost all t [,T]. Since, by Lemma 3.1, the sequence {v k 2 n} is uniformly bounded in L (Γ T ), we conclude that v k 2 n (u + +u ) n in L 2 (Γ T ). This implies that the sequence {v k 3} converges in L 2 ( T ) to some function v 3. The validity of identity (2.5) and of the rest assertions of Theorem 5.2 can be proved now by simply repeating the arguments applied in Section 4. Acknowledgement. This work was completed while V. N. Starovoitov was author to Universidade Independente, Lisbon. His work was partially supported by Russian Foundation of Basic Researches (grant N ).

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17 Vol. 4 (22) On a Motion of a Perfect Fluid 17 N. V. Chemetov Universidade Independente de Lisboa Av. Marechal Gomes de Costa, Lisbon Portugal and CMAF Universidade de Lisboa Av. Gama Pinto, 2 Lisbon Portugal chemetov@ptmat.lmc.fc.ul.pt V. N. Starovoitov Lavrentyev Institute of Hydrodynamics Novosibirsk, 639 Russia star@hydro.nsc.ru (accepted: July 2, 21)