REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID


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1 REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), ] the authors consider the two dimensional NavierStokes equations in the exterior of an obstacle shrinking to a point and determine the limit velocity. Here we consider the same problem in the three dimensional case, proving that the limit velocity is a solution of the NavierStokes equations in the full space. 1. Introduction The investigation of small obstacle limits in an incompressible fluid was initiated in [3]. In that paper, the authors consider the Euler equations in the exterior of a twodimensional obstacle that shrinks homothetically (that is, by dilation) to a point. It is assumed that the initial vorticity is the restriction to the exterior of the obstacle of a smooth vorticity field compactly supported in R 2 \ {0}, and that the circulation of the velocity on the boundary of the obstacle is independent of the size of the obstacle. It is then proved in [3] that the limit velocity is a solution of a PDE that looks like an Euler equation that embeds the Dirac mass of the point the obstacle shrinks to. The vorticity also acquires a Dirac mass at this point. The case of several obstacles was treated in [5] and the twodimensional viscous case in [4], where it is proved that the limit equation is also NavierStokes but there is still formation of an additional Dirac mass in the limit vorticity. This is due to the circulation of the velocity on the boundary of the obstacle not vanishing. Here we consider the same problem in the threedimensional case: pass to the limit in the NavierStokes equations in the exterior of an obstacle that shrinks to a point. In contrast to the twodimensional case, we do not have to prescribe the circulation of the velocity on the boundary since the domain is simply connected. We prove that the limit equation is the NavierStokes equation in the full space and that the vorticity of the limit velocity at time t = 0 is simply the initial vorticity that we give for the obstacledependent problem. Therefore, there is no formation of additional vorticity as in the case of R 2. We are also able to consider more general obstacles than in [4]: Instead of assuming that the obstacle homothetically shrinks to a point we assume only that the diameter of the obstacle goes to zero Mathematics Subject Classification. Primary 76D05. Key words and phrases. NavierStokes equations. 1
2 More precisely, let Π ε = R 3 \ Ω ε be a simply connected exterior domain with C boundary such that Ω ε B(0, Mε), where the constant M is independent of ε. We assume that the initial vorticity ω 0 is smooth, divergencefree, and compactly supported in R 3. Since the domain Π ε is simply connected, there exists a unique squareintegrable, divergencefree velocity field u ε 0 in Π ε tangent to the boundary of Π ε and whose curl is ω 0 Πε (see Proposition 6 below). We also denote by u 0 the velocity defined on R 3 which is associated to the vorticity ω 0 : (1) u 0 (x) = R 3 x y 4π x y 3 ω 0(y) dy, where denotes the standard cross product of vectors in R 3. Let u ε = u ε (t, x) be a weak Leray solution of the NavierStokes equations in Π ε with initial velocity u ε 0 and homogeneous Dirichlet boundary conditions: t u ε ν u ε + u ε u ε = p ε in Π ε (0, ), div u ε = 0 in Π ε [0, ), (2) u ε = 0 on Π ε (0, ), u ε (0, ) = u ε 0 in Π ε. Such a weak solution is wellknown to exist, see for example [2]. The aim of this paper is to prove the following theorem. Theorem 1. Let u 0, ω 0, and u ε 0 be as defined above. Let u ε be a weak Leray solution of the NavierStokes equations on Π ε with initial velocity u ε 0 and denote by ũ ε the extension of u ε to R 3 with values 0 on Ω ε. There exists a subsequence of ũ ε that converges strongly in L 2 loc ([0, ) R3 ) to a weak Leray solution of the NavierStokes equations in R 3 with initial velocity u 0. The proof of this result consists of two parts. We prove first that u ε 0 converges to u 0 strongly in L 2 see Proposition 6 below. We then conclude by showing in Theorem 7 that strong convergence in L 2 for the initial velocity implies convergence of solutions in the vanishing obstacle limit. The regularity of the initial vorticity can be lowered considerably and still obtain convergence as in Theorem 1. We briefly discuss this in Section Notation and preliminary results For a function f defined on Π ε, we denote by f the function defined on R 3 which vanishes on Ω ε and equals f on Π ε. If f is regular enough and vanishes on Ω ε, then one has that f = f in R 3. If v is a regular enough vector field defined on Π ε and tangent to Ω ε, then one has that div ṽ = div v in R 3. In particular, we have that div ũ ε 0 = 0 in R 3. We denote by H ε the space of squareintegrable vector fields on Π ε which are divergencefree and tangent to the boundary. We will also use the classical Sobolev space H m and the space C m b of bounded functions having bounded derivates up to the order m. 2
3 Definition 2. We say that u ε is a weak Leray solution of (2) if ( u ε Cw 0 [0, ); Hε ) ) L ( ) ( [0, ); H ε L 2 loc [0, ); H 1 0 (Π ε ) ) verifies the equation in the sense of distributions, i.e. (3) u ε t ϕ + ν u ε ϕ + u ε u ε ϕ = u ε 0 ϕ(0) 0 Π ε 0 Π ε 0 Π ε Π ε for every divergencefree vector field ϕ C0 ([0, ) Π ε ), and moreover u ε verifies the following energy inequality: t (4) u ε (t) 2 L 2 (Π + 2ν ε) u ε 2 L 2 (Π ε) uε 0 2 L 2 (Π ε) t 0. 0 Above Cw 0 denotes the space of weakly continuous functions. We will use a similar definition for weak Leray solutions on R 3. For a vector field ϕ in R 3 we define a stream function ψ = T ϕ by x y ψ(x) = (T ϕ)(x) = (Sϕ)(x) (Sϕ)(0), (Sϕ)(x) = ϕ(y) dy. R 4π x y 3 3 We will use the following properties of the operator T. Lemma 3. Suppose that ϕ is a vector field belonging to L 2 (R 3 ) L 4 (R 3 ). Then T ϕ Cb 0(R3 ) and T ϕ L (R 3 ) C ϕ L 2 L4. Moreover, if div ϕ = 0 then curl T ϕ = ϕ. Finally, for all m 1, the operator T is bounded from H m (R 3 ) into C m 1 b (R 3 ). Proof. We decompose x y (Sϕ)(x) = ϕ(y) dy = + I 4π x y 3 1 (x) + I 2 (x). R 3 x y 1 x y >1 x One has that χ x 3 x 1 L 4/3 (R 3 ) and x χ x 3 x >1 L 2 (R 3 ) so we obtain from Young s inequality that I 1 L (R 3 ) C x x χ x 1 3 L 4/3 (R ) ϕ 3 L 4 (R 3 ) C ϕ L 4 (R 3 ) and I 2 L (R 3 ) C x x χ x >1 3 L 2 (R 3 ) ϕ L 2 (R 3 ) C ϕ L 2 (R 3 ) together with the continuity of I 1 and I 2. Since T ϕ L (R 3 ) 2 Sϕ L (R 3 ) we obtain the desired continuity and uniform bound for T ϕ. Suppose now that ϕ H m (R 3 ). For any multiindex α of order 0 < α m 1 we have α T ϕ = S α ϕ and α ϕ H 1 (R 3 ) L 2 L 4. From the first part of the proof, we deduce that S α ϕ Cb 0, that is T ϕ Cm 1 (R 3 ) together with the desired bound. Finally, assume that ϕ is divergencefree. If ϕ is compactly supported, then clearly Sϕ = curl F where F = 1 1 ϕ. One has that div F = div ϕ = 0. Then 4π x 4π x 1 curl T ϕ = curl Sϕ = curl curl F = F + div F = ( 4π x ) ϕ = δ ϕ = ϕ. 3
4 If ϕ is not compactly supported, then there exists a sequence of divergence free compactly supported vector field ϕ n ϕ in L 2 L 4. Passing to the limit n in curl T ϕ n = ϕ n implies that curl T ϕ = ϕ. This completes the proof of Lemma 3. We will use in Section 3 the following approximation of smooth compactly supported divergencefree vector fields. Let ϕ be a vector field and η C (R 3 ) be such that η 0 on B(0, M) and η 1 on R 3 \ B(0, 2M). We define η ε (x) = η(x/ε) and ϕ ε = curl(η ε ψ) with ψ = T ϕ as before. We collect in the following lemma several properties relating ϕ ε to ϕ. Lemma 4. Let ϕ C 0 (R 3 ) be a divergencefree vector field and define ϕ ε as above. Then (i) ϕ ε is smooth, compactly supported, divergencefree, and vanishes in a neighborhood of the obstacle Ω ε ; (ii) ϕ ε ϕ strongly in H 1 (R 3 ); (iii) there exists a constant C independent of ε such that ϕ ε L (R 3 ) + ϕ ε H 1 (R 3 ) C ϕ H 3 (R 3 ); (iv) one can decompose ϕ ε = ξ ε + Ξ ε with ξ ε ϕ weak in L (R 3 ) and Ξ ε 0 strongly in L 2 (R 3 ) and there exists a compact set L independent of ε such that supp ξ ε, supp Ξ ε L for all ε 1. Remark 5. It will be clear from the proof below that we can allow a time dependence in ϕ. The results of this lemma hold true uniformly with respect to the time variable. Proof. We will repeatedly use in this proof that ψ(0) = 0 so (5) ψ L (B(0,2εM)) 2εM ψ L (R 3 ) Cε ϕ H 2. Part (i) follows immediately from the localization properties of η ε and from Lemma 3. To prove (ii) we observe from the explicit expression for η ε that η ε 1 and η ε converge to 0 in L 2 and 2 η ε L 2 (R 3 ) = ε η L 2 (R 3 ). Since ϕ ε = η ε ϕ + η ε ψ we have that ϕ ε ϕ L 2 (R 3 ) (η ε 1)ϕ L 2 (R 3 ) + η ε ψ L 2 (R 3 ) and similarly, η ε 1 L 2 (R 3 ) ϕ L (R 3 ) + η ε L 2 (R 3 ) ψ L (R 3 ) ε 0 0, (ϕ ε ϕ) L 2 (R 3 ) η ε 1 L 2 (R 3 ) ϕ L (R 3 ) +C η ε L 2 (R 3 )( ϕ L (R 3 ) + ψ L (R 3 )) + C 2 η ε L 2 (R 3 ) ψ L (B(0,2εM)) ε 0 0. The H 1 bound in part (iii) follows from the estimates above while the uniform bound is an immediate consequence of (5) and of the decomposition ϕ ε = η ε ϕ + η ε ψ. To prove (iv) we set ξ ε = η ε ϕ and Ξ ε = ϕ ε η ε ϕ so that supp ξ ε supp ϕ. The term Ξ ε is similar to the expressions estimated above, so it can be proved in the same way that it converges to 0 in L 2 as ε 0. The sequence ξ ε is bounded in L and converges to 4
5 ϕ in L 2. Since ξ ε ϕ in L 2 we deduce first that (ξ ε ϕ) h 0 for all h C0 (R 3 ) 9. Since ξ ε is bounded in L and C0 is dense in L 1 we infer that (ξ ε ϕ) h 0 for all h L 1 (R 3 ) 9, i.e. ξ ε ϕ weak in L. Finally, the condition on the supports is trivially verified by construction. This completes the proof of the lemma. We prove next a convergence result for the initial velocities. Proposition 6. Let ω 0 be a divergencefree vector field in C 0 (R 3 ) with u 0 given by Equation (1). For all ε > 0 there exists exactly one vector field u ε 0 H ε such that curl u ε 0 = ω 0 Πε and there exists a constant C independent of ε such that ũ ε 0 u 0 L 2 (R 3 ) Cε 3 2. Proof. By the LerayHelmholtzWeyl decomposition there exists u ε 0 in H ε and p in Ḣ1 (Π ε ) = {p L 2 loc (Π ε) ; p L 2 (Π ε )} such that u 0 Πε = u ε 0 + p on Π ε with u ε 0 and p unique in these spaces (see, for instance, Theorem 1.1 p. 107 of [1]). Since the curl of a gradient is zero, curl u ε 0 = curl u 0 Πε = ω 0 Πε in Π ε. Now let w be any vector field in H ε with curl w = ω 0 Πε in Π ε. Then w u ε 0 is in H ε with curl(w u ε 0) = 0, so since Π ε is simply connected, w u ε 0 = q for some q in Ḣ1 (Π ε ). Also, q = div w div u ε 0 = 0 on Π ε and q n = w n = 0 on Π ε, where n is the outward unit normal to the boundary. By the uniqueness of the solution to the Neumann problem, q is a constant, so q = 0 and u ε 0 = w, giving the uniqueness of u ε 0. Noting that u ε 0 is the L 2 orthogonal projection of u 0 Πε on H ε, we have (6) u ε 0 u 0 Πε L 2 (Π ε) w ε u 0 Πε L 2 (Π ε) for all w ε H ε. Making the particular choice, w ε = curl(η ε ψ) = η ε u 0 + η ε ψ, ψ = T u 0, we see that w ε vanishes on Π ε and since w ε is a curl it is also divergencefree. Equation (6) with Lemma 3 then yield ũ ε 0 u 0 L 2 (R 3 ) u ε 0 u 0 Πε L 2 (Π ε) + u 0 L 2 (Ω ε) (7) η ε ψ L 2 (Π ε) + (1 η ε )u 0 L 2 (Π ε) + u 0 L 2 (Ω ε) η ε L 2 ψ L (B(0,2Mε)) + 2 u 0 L 2 (B(0,2Mε)) We used above that ψ(0) = 0. Cε 3/2 ψ L (R 3 ) + Cε 3/2 u 0 L (R 3 ) Cε 3/2 ω 0 H 3 (R 3 ). 3. Proof of the convergence of solutions The aim of this section is to prove a general convergence result: strong convergence in L 2 for the initial data implies convergence of weak Leray solutions in the vanishing obstacle limit. Such a convergence result is classical on a fixed domain; the difficulty here is to deal with the singularity induced by the obstacle that shrinks to a point. Throughout this 5
6 section we drop the previous assumptions on the initial vorticity and the special forms of the initial velocities u ε 0 and u 0. We will prove the following independent result. Theorem 7. Suppose that u ε 0 H ε and u 0 L 2 (R 3 ) is a divergencefree vector field such that ũ ε 0 u 0 strongly in L 2 (R 3 ). Let u ε be a weak Leray solution of the NavierStokes equations on Π ε with initial velocity u ε 0. There exists a subsequence of ũ ε that converges strongly in L 2 loc ([0, ) R3 ) to a weak Leray solution of the NavierStokes equations in R 3 with initial velocity u 0. We proceed now with the proof of this theorem. We will use the notation introduced in Section 2. Since u ε is a weak Leray solution and u ε 0 is bounded in L 2 (Π ε ), the energy inequality (4) implies that u ε is bounded in L (R + ; L 2 (Π ε )) L 2 loc ([0, ); H1 (Π ε )). We require now some temporal estimates for u ε Temporal estimates. Let φ C0 (R 3 ) be a divergencefree vector field. We construct φ ε as in Section 2. It follows from Definition 2, using a standard approximation argument for the test function φ ε and taking advantage of the weak continuity in time of u in L 2 (Π ε ), that u ε (t), φ ε u ε (s), φ ε t t ν = u ε φ ε + u ε u ε φ ε s Π ε s Π ε (8) t t ν u ε L 2 (Π ε) φ ε L 2 (R 3 ) + u ε L 2 (Π ε) u ε L 2 (Π ε) φ ε L (R 3 ) s s ) C(t s) 1 2 φ H 3 (R 3 ) u ε L 2 (R + Π ε)( 1 + u ε L (R + ;L 2 (Π ε)) C(t s) 1 2 φ H 3 (R 3 ) u ε 0 L 2 (Π ε)(1 + u ε 0 L 2 (Π ε)) where we used (4), Lemma 4 (iii) and the constant C is independent of ε, s and t (though it depends on ν). For t R + let us define F ε (t) D (R 3 ) by means of We deduce from (4) and (5) that C 0 (R 3 ) 3 h F ε (t), h = ũ ε (t), η ε T h. F ε (t), h u ε (t) L 2 (Π ε) η ε L 2 (R 3 ) T h L (B(0,2εM)) Cε 3 2 u ε 0 L 2 (Π ε) h H 2 (R 3 ). Remembering that u ε C 0 w( [0, ); Hε ) ) we infer that F ε belongs to C 0 w([0, ); H 2 (R 3 )) and is bounded by Cε 3 2 in L (R + ; H 2 (R 3 )). From (8) one has that so η ε ũ ε (t) + F ε (t) η ε ũ ε (s) F ε (s), φ C(t s) 1 2 φ H 3 (R 3 ) P[η ε ũ ε (t) + F ε (t) η ε ũ ε (s) F ε (s)] H 3 (R 3 ) C(t s) 1 2, where P denotes the usual Leray projector in R 3, i.e. the L 2 orthogonal projection on the subspace of divergencefree vector fields. We conclude that the set P(η ε ũ ε + F ε ) is equicontinuous (in time) in C 0 ([0, ); H 3 (R 3 )). 6
7 Next, we observe that div ũ ε = 0 so P(ũ ε ) = ũ ε. Therefore P(η ε ũ ε + F ε ) = ũ ε + P[(η ε 1)ũ ε + F ε ] ũ ε + v ε. We argue now that (9) v ε L (R + ;H 3 (R 3 )) Cε 3 2. Indeed, we know that P is bounded in any H s so PF ε L (R + ;H 3 (R 3 )) C F ε L (R + ;H 3 (R 3 )) Cε 3 2. and P[(η ε 1)ũ ε ] H 3 (R 3 ) (η ε 1)ũ ε H 3 (R 3 ) C (η ε 1)ũ ε L 1 (R 3 ) uniformly with respect to t. C η ε 1 L 2 (R 3 ) ũ ε L 2 (R 3 ) Cε 3 2 u ε 0 L 2 (R 3 ) 3.2. Passing to the limit. Given the bounds (4) and by the Ascoli theorem, we can extract from the sequence ũ ε a subsequence ũ ε k such that (10) (11) (12) ũ ε k u ũ ε k u P(η εk ũ ε k + F εk ) = ũ ε k + v εk w for some limit vector fields u and w in L (R + ; L 2 (R 3 )) weak in L 2 loc([0, ); H 1 (R 3 )) weakly u L (R + ; L 2 (R 3 )) L 2 loc([0, ); H 1 (R 3 )), in C 0 ([0, ); H 4 loc (R3 )) strongly w C 0 ([0, ); H 4 loc (R3 )). Since div ũ ε k = 0 we necessarily have that div u = 0. Next, from (9) and (12) we infer that ũ ε k w in L loc([0, ); H 4 loc (R3 )) strongly. Next, using that ũ ε k is bounded in L 2 loc ([0, ); H 1 (R 3 )) and the interpolation inequality L 2 (W ) C 1 5 H 4 (W ) 4 5 H 1 (W ) that holds true for every bounded open set W, we conclude that ũ ε k w strongly in L 2 loc (R + R 3 ). By uniqueness of limits in the sense of distributions, we infer that u = w and therefore (13) ũ ε k u in L 2 loc(r + R 3 ) strongly. With these pieces of information, it is easy to pass to the limit in the equation of u ε k and obtain that u is a weak solution of the NavierStokes equations in R 3. Indeed, let ϕ C0 ([0, ) R 3 ) be a divergencefree test vector field and define ϕ εk as in Section 2. Equation (3) with ϕ εk instead of ϕ gives (14) ũ εk t ϕ εk + ν ũ εk ϕ εk 0 R 3 0 R 3 0 (ũ ε k ũ ε k ) ϕ εk = R 3 ũ ε k 0 ϕ εk (0). R 3 7
8 From Lemma 4 applied to t ϕ (see also Remark 5) we know that t ϕ εk t ϕ strongly in L 1 (R + ; L 2 (R 3 )), that ϕ εk (0) ϕ(0) strongly in L 2 (R 3 ) and that ϕ εk ϕ strongly in L 2 (R + R 3 ). Given (10), (11) and the convergence of the initial velocity (note that only weak convergence in L 2 is required at this point for ũ ε k 0 ), we deduce that the righthand side and the first two terms on the lefthand side of (14) converge to the expected limit. Using the decomposition ϕ εk = ξ εk + Ξ εk given in Lemma 4 (iv), we write (ũ ε k ũ ε k ) ϕ εk = (ũ ε k ũ ε k ) ξ εk + (ũ ε k ũ ε k ) Ξ εk. 0 R 3 0 R 3 0 R 3 Given (13) and that ξ εk ϕ weak in L (R + R 3 ) with supports contained in a compact set independent of ε k, one has that (ũ ε k ũ ε k ε ) ξ k 0 εk (u u) ϕ. 0 R 3 0 R 3 Next, we use the Sobolev embedding H 1 L 6 and a Hölder inequality to write T (ũ ε k ũ ε k ) Ξ εk C ũ ε k 2 L 6 (R 3 ) Ξ ε k 3 L 2 (R 0 R ) C ũ ε k 2 L 2 ((0,T );H 1 ) Ξ ε k L ((0,T );L 3 2 ) where T is such that supp ϕ [0, T ] R 3. We conclude that sending ε k 0 in (14) results in u t ϕ + ν u ϕ (u u) ϕ = u 0 ϕ(0), 0 R 3 0 R 3 0 R 3 R 3 ε k 0 0, which is the weak formulation of the NavierStokes equations in R 3. To finish the proof of Theorem 7, it remains to prove that the solution u is weakly continuous in time with values in L 2 and that it verifies the energy inequality. We show first that u(t) L 2 for all t 0. Recall that u C 0 ([0, ); H 4 loc ) so that u(t) is defined for all t 0. For fixed t, by the energy inequality (4) the sequence ũ ε k (t) is bounded in L 2 (R 3 ). Moreover, ũ ε k (t) u(t) in H 4 loc so the limit u(t) must belong to L 2 (R 3 ). Next, using again that ũ ε k (t) u(t) in H 4 loc we have that [ũ ε k (t) u(t)] h 0 for all h C0 (R 3 ) 3. On the other hand, C0 (R 3 ) is dense in L 2 (R 3 ) and ũ ε k (t) is bounded in L 2, so [ũ ε k (t) u(t)] h 0 for all h L 2 (R 3 ) 3. Therefore, for all t 0 one has that ũ ε k (t) u(t) in L 2 weakly. One can prove in a similar manner that ũ ε k u weakly in L 2( (0, t) R 3) for all t 0 and also that u is weakly continuous in time with values in L 2. We prove now that the energy inequality holds true for u. This is done by means of the following classical liminf argument. We apply the lim inf to (4) to obtain ε k 0 (15) lim inf ε k 0 ũε k (t) 2 L 2 (R 3 ) + 2ν lim inf ε k 0 t 0 ũ ε k 2 L 2 (R 3 ) u 0 2 L 2 (R 3 ) t 0. 8
9 Since ũ ε k (t) u(t) in L 2 weakly and ũ ε k u weakly in L 2 ( (0, t) R 3) we have that (16) u(t) L 2 (R 3 ) lim inf ε k 0 ũε k (t) L 2 (R 3 ). and (17) u(t) L 2 ((0,t) R 3 ) lim inf ε k 0 ũε k (t) L 2 ((0,t) R 3 ). The energy inequality for u now follows from Equations (15), (16) and (17). The proof of Theorem 7 is completed. Remark 8. It is clear from the proof that if we assume that the initial velocities u ε k 0 converge only weakly to u 0, then we can still prove convergence of u ε k to some solution u of the NavierStokes equation in the sense of Definition 2 but without the energy inequality. The strong convergence of u ε k 0 to u 0 is required only to prove the energy inequality. 4. Lower regularity of the initial velocity The main issue in this work is to prove convergence of the solutions, and not to consider the weakest possible regularity of the initial vorticity. As we are dealing with weak solutions, however, it is natural to ask that the initial vorticity have only enough regularity to obtain existence of the weak solutions while still obtaining convergence. We give next just an example of how one can improve these regularity assumptions if the support of ω 0 excludes the origin. Proposition 9. Let ω 0 be a divergencefree vector field in L 1 (R 3 ) Ḣ 1 (R 3 ) with compact support contained in R 3 \ {0} and let u 0 be the unique squareintegrable divergence free vector field in R 3 whose curl is ω 0. For sufficiently small ε there exists exactly one vector field u ε 0 H ε such that curl u ε 0 = ω 0 and there exists a constant C independent of ε such that ũ ε 0 u 0 L 2 (R 3 ) Cε 3 2. Proof. The operator that associates to every divergence free ω 0 the unique divergence free vector field u 0 of curl ω 0 is the Fourier multiplier 1. This operator clearly sends Ḣ 1 (R 3 ) into L 2 (R 3 ). Therefore, for every divergence free ω 0 Ḣ 1 (R 3 ) there exists a unique divergence free u 0 L 2 (R 3 ) such that curl u 0 = ω 0. Moreover, u 0 L 2 C ω 0 Ḣ 1. Next, let ε be sufficiently small such that supp ω 0 Π ε. We observe that the existence and uniqueness of u ε 0 follows exactly as in the proof of Proposition 6. Indeed, in that part of the proof the only assumption that was used is that u 0 L 2. We prove now the bound on ũ ε 0 u 0 L 2 (R 3 ). We consider first the case when ω 0 is smooth. More precisely, we show that for any δ > 0 there exists ε 0 = ε 0 (δ, M) and K = K(δ, M) such that for any divergence free ω 0 C0 (R 3 \ B(0, δ)) and ε ε 0, one has that ũ ε 0 u 0 L 2 (R 3 ) Kε 3 2 ω 0 L 1. In this case, relation (1) holds true and the proof is the same as that of Proposition 6 except that instead of ψ = T u 0 we use ψ = E ω 0 E ω 0 (0), where E = 1. This 4π x is a valid replacement because relation (1) immediately implies that curl ψ = u 0. In fact, 9
10 this definition of ψ is equivalent to that of Proposition 6, but we don t need to prove this. Because the support of ω 0 excludes B(0, δ), ψ = (χ B(0,δ/2) c E) ω 0 on B(0, 2Mε) for all ε < δ/(4m). Here, χ B(0,δ/2) c denotes the characteristic function of R 3 \ B(0, δ/2). But then ψ L (B(0,2Mε)) (χb(0,δ/2) c E) ω L 0 χb(0,δ/2) c E (R 3 ) L ω 0 L 1 = 1 πδ ω 0 2 L 1. Similarly, under the same condition ε < δ/(4m) one can use (1) to bound u 0 L (B(0,2Mε)) = x y 4π x y ω 0(y) dy 1 3 L (B(0,2Mε)) πδ ω 0 2 L 1. x y >δ/2 With these bounds, Equation (7) can be rewritten as follows ũ ε 0 u 0 L 2 (R 3 ) η ε L 2 ψ L (B(0,2Mε)) + 2 u 0 L 2 (B(0,2Mε)) Cε 3/2 ψ L (B(0,2Mε)) + Cε 3/2 u 0 L (B(0,2Mε)) C ε3/2 δ 2 ω 0 L 1. This completes the proof when ω 0 is smooth. The general case classically follows by approximation. Let ω 0 L 1 Ḣ 1 and δ be such that supp ω 0 B(0, 2δ) =. Let ρ be a standard mollifying kernel and let us mollify ω 0 in a classical manner: ω 0,n = ρ 1/n ω 0. One has that ω 0,n ω 0 in L 1 Ḣ 1 and ω 0,n C 0 (R 3 \ B(0, δ)) for n sufficiently large. Denoting by u 0,n and u ε 0,n the velocities associated to the vorticity ω 0,n, we observe that u 0,n = ρ 1/n u 0 so u 0,n u 0 in L 2. Suppose that ε min(ε 0, δ). Applying the previous part of the proof to ω 0,n ω 0,m we get that u ε 0,n u ε 0,m L 2 (Π ε) ũ ε 0,n u 0,n ũ ε 0,m + u 0,m L 2 (R 3 ) + u 0,n u 0,m L 2 (R 3 ) Kε 3 2 ω0,n ω 0,m L 1 (R 3 ) + u 0,n u 0,m L 2 (R 3 ) 0 as m, n. Therefore, u ε 0,n is a Cauchy sequence in H ε so it converges in H ε. Moreover, curl lim u ε 0,n = n lim curl n uε 0,n = lim ω 0,n = ω 0 in the sense of distributions. By uniqueness of u 0 n ε, we conclude that u ε 0,n u ε 0 in H ε. We apply now the previous part of the proof to ω 0,n to deduce that ũ ε 0,n u 0,n L 2 (R 3 ) Kε 3 2 ω 0,n L 1. Letting n we finally deduce that ũ ε 0 u 0 L 2 (R 3 ) Kε 3 2 ω 0 L 1. Using Proposition 9 in place of Proposition 6 gives the convergence in Theorem 1 for initial vorticity in L 1 (R 3 ) Ḣ 1 (R 3 ) compactly supported in R 3 \ {0}. Acknowledgments: D.I. would like to thank Lorenzo Brandolese, Grzegorz Karch, Milton Lopes Filho and Helena Nussenzveig Lopes for several interesting discussions on the subject of this paper. Part of this work was done during the Special Semester in Fluid Mechanics at the Centre Interfacultaire Bernoulli, EPFL; D.I. wishes to express his gratitude for the hospitality received. J.P.K. was supported in part by NSF grant DMS during the period of this work. 10
11 References [1] G. P. Galdi. An introduction to the mathematical theory of the NavierStokes equations. Vol. I, volume 38 of Springer Tracts in Natural Philosophy. SpringerVerlag, New York, [2] E. Hopf. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, (1951), [3] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Comm. Part. Diff. Eqns. 28 (2003), no. 12, [4], Twodimensional incompressible viscous flow around a small obstacle, Math. Ann. 336 (2006), no. 2, [5] M. C. Lopes Filho, Vortex dynamics in a twodimensional domain with holes and the small obstacle limit, SIAM J. Math. Anal. 39 (2007), no. 2, D. Iftimie: Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan, Bâtiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F Villeurbanne Cedex, France. J. Kelliher: Brown University, Department of Math, Box 1917, Providence, RI 02912, USA. 11
arxiv: v1 [math.ap] 15 Nov 2018
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