Vanishing viscosity and the accumulation of vorticity on the boundary
aa r X i v : . [ m a t h - ph ] M a y VANISHING VISCOSITY AND THE ACCUMULATION OFVORTICITY ON THE BOUNDARY
JAMES P. KELLIHER
Abstract.
We say that the vanishing viscosity limit holds in the clas-sical sense if the velocity for a solution to the Navier-Stokes equationsconverges in the energy norm uniformly in time to the velocity for asolution to the Euler equations. We prove, for a bounded domain indimension 2 or higher, that the vanishing viscosity limit holds in theclassical sense if and only if a vortex sheet forms on the boundary.
As is well known, for radially symmetric initial vorticity in a disk, the velocityfor a solution to the Navier-Stokes equations converges in the energy normuniformly in time to the velocity for a solution to the Euler equations. Itwas shown recently in [8] that for such initial data in a disk, it also happensthat a vortex sheet forms on the boundary as the viscosity vanishes. Bya vortex sheet, we mean a velocity field whose vorticity, as a finite Borelsigned measure (an element of the dual space of C (Ω)), is supported alonga curve—the boundary, in this case.It turns out that this phenomenon is in a sense more universal: eitherboth types of limits hold or neither holds for an arbitrary bounded domainin R d , d ≥
2, with C -boundary and with no particular assumption on thesymmetry of the initial data. More precisely, the vanishing viscosity limit inthe classical sense (condition ( B ) of Section 2) holds if and only if a vortexsheet of a particular type forms on the boundary (conditions ( E ) and ( E )of Section 2). Now, however, the vortex sheet has vorticity belonging to thedual space of H (Ω) rather than C (Ω). We show this in Theorem 2.1 forno-slip boundary conditions and in Theorem 4.1 for characteristic boundaryconditions on the velocity. 1. Introduction
Let Ω be a bounded domain in R d , d ≥
2, with C -boundary Γ, and let n bethe outward unit normal vector to Γ. A classical solution ( u, p ) to the Eulerequations satisfies( EE ) (cid:26) ∂ t u + u · ∇ u + ∇ p = f and div u = 0 on [0 , T ] × Ω ,u · n = 0 on [0 , T ] × Γ and u = u on { } × Ω . Mathematics Subject Classification.
Primary 76D05, 76B99, 76D99.
Key words and phrases.
Laplacian, Stokes operator, eigenvalue problems.
These equations describe the motion of an incompressible fluid of constantdensity and zero viscosity. The initial velocity u must at least lie in H = n u ∈ ( L (Ω)) d : div u = 0 in Ω , u · n = 0 on Γ o endowed with the L -norm, which, along with V = n u ∈ ( H (Ω)) d : div u = 0 in Ω , u = 0 on Γ o endowed with the H -norm, are the classical spaces of fluid mechanics.We assume that u is in C k + ǫ (Ω) ∩ H , ǫ >
0, where k = 1 for twodimensions and k = 2 for 3 and higher dimensions, and that f is in C ([0 , t ] × Ω) for all t >
0. Then as shown in [7] (Theorem 1 and the remarks on p.508-509), there is some
T > u in C ([0 , T ]; C k + ǫ (Ω)) , (1.1)to ( EE ). In two dimensions, T can be arbitrarily large, though it is onlyknown that some nonzero T exists in three and higher dimensions.The Navier-Stokes equations describe the motion of an incompressiblefluid of constant density and positive viscosity ν . A classical solution to theNavier-Stokes equations with no-slip boundary conditions can be defined inanalogy to ( EE ) by( N S ) (cid:26) ∂ t u + u · ∇ u + ∇ p = ν ∆ u + f and div u = 0 on [0 , T ] × Ω ,u = 0 on [0 , T ] × Γ and u = u ν on { } × Ω , where u ν is in H and f is in L ([0 , T ]; L (Ω)). We will work, however, withweak solutions to the Navier-Stokes equations. (See, for instance, ChapterIII of [10].) Such weak solutions lie in L ∞ ([0 , T ]; H ) ∩ L ([0 , T ]; V ).In Section 2 we prove various equivalent forms of the vanishing viscositylimit for no-slip boundary conditions, including the formation of a vortexsheet on the boundary, and remark briefly on their derivation in Section 3.In Section 4 we extend the results of Section 2 to characteristic boundaryconditions. We discuss, in Section 5, our results in relation to those in [8]on vortex sheet formation for a disk. Finally, in Section 6, we include sometechnical lemmas employed in Section 2.2. Equivalent forms of the vanishing viscosity limit
Let u be a classical solution to ( EE ) in Ω and u = u ν be a weak solutionto ( N S ) in Ω as in Section 1, and assume that u ν → u in H and f → f in L ([0 , T ]; L (Ω)) as ν → γ n be the boundary trace operator for the normal component of avector field (see Lemma 6.1). Let M (Ω) be the space of finite Borel signedmeasures on Ω— M (Ω) is the dual space of C (Ω). Let µ in M (Ω) be themeasure supported on Γ for which µ | Γ corresponds to Lebesgue measure onΓ (arc length for d = 2, area for d = 3, etc.). Then µ is also a member of H (Ω) ′ . ANISHING VISCOSITY AND VORTICITY ON THE BOUNDARY 3
We define the vorticity ω ( u ) to be the d × d antisymmetric matrix ω ( u ) = 12 (cid:2) ∇ u − ( ∇ u ) T (cid:3) . (2.1)When working specifically in two dimensions, we can alternately define thevorticity as the scalar curl of u : ω ( u ) = ∂ u − ∂ u . (2.2)Letting ω = ω ( u ) and ω = ω ( u ), we define the following conditions:( A ) u → u weakly in H uniformly on [0 , T ] , ( A ′ ) u → u weakly in ( L (Ω)) d uniformly on [0 , T ] , ( B ) u → u in L ∞ ([0 , T ]; H ) , ( C ) ∇ u → ∇ u − h γ n · , uµ i in (( H (Ω)) d × d ) ′ uniformly on [0 , T ] , ( D ) ∇ u → ∇ u in ( H − (Ω)) d × d uniformly on [0 , T ] , ( E ) ω → ω − (cid:10) γ n ( · − · T ) , uµ (cid:11) in (( H (Ω)) d × d ) ′ uniformly on [0 , T ] . In conditions ( C ), ( D ), and ( E ), the convergence is in the weak ∗ topologyof the given spaces. In ( C ) and ( E ), (( H (Ω)) d × d ) ′ is the dual space of( H (Ω)) d × d ; in ( D ), ( H − (Ω)) d × d is the dual space of ( H (Ω)) d × d . Thus,condition ( C ) means that( ∇ u ( t ) , M ) → ( ∇ u ( t ) , M ) − Z Γ ( M · n ) · u ( t ) in L ∞ ([0 , T ])for any M in ( H (Ω)) d × d , condition ( D ) means that( ∇ u ( t ) , M ) → ( ∇ u ( t ) , M ) in L ∞ ([0 , T ])for any M in ( H (Ω)) d × d , and condition ( E ) means that( ω ( t ) , M ) → ( ω ( t ) , M ) − Z Γ (( M − M T ) · n ) · u ( t ) in L ∞ ([0 , T ])for any M in ( H (Ω)) d × d .In two dimensions, defining the vorticity as in Equation (2.2), we alsodefine the following two conditions:( E ) ω → ω − ( u · τ ) µ in ( H (Ω)) ′ uniformly on [0 , T ] , ( F ) ω → ω in H − (Ω) uniformly on [0 , T ] . Here, τ is the unit tangent vector on Γ that is obtained by rotating theoutward unit normal vector n counterclockwise by 90 degrees.Condition ( E ) means that( ω ( t ) , f ) → ( ω ( t ) , f ) − Z Γ ( u ( t ) · τ ) f in L ∞ ([0 , T ])for any f in H (Ω), while condition ( F ) means that( ω ( t ) , f ) → ( ω ( t ) , f ) in L ∞ ([0 , T ]) JAMES P. KELLIHER for any f in H (Ω). Theorem 2.1.
Conditions ( A ), ( A ′ ), ( B ), ( C ), ( D ), and ( E ) are equiv-alent. In two dimensions, conditions ( E ) and ( F ) are equivalent to theother conditions.Proof. ( A ) ⇐⇒ ( A ′ ): Let v be in ( L (Ω)) d . By Lemma 6.3, v = w + ∇ p ,where w is in H and p is in H (Ω). Then assuming ( A ) holds,( u ( t ) , v ) = ( u ( t ) , w ) → ( u ( t ) , w ) = ( u ( t ) , v )uniformly over t in [0 , T ], so ( A ′ ) holds. The converse is immediate.( A ) ⇐⇒ ( B ): The forward implication is proved in Theorem 1 of [4]. Thebackward implication is immediate.( A ′ ) = ⇒ ( C ): Assume that ( A ′ ) holds and let M be in ( H (Ω)) d × d . Then( ∇ u ( t ) , M ) = − ( u ( t ) , div M ) → − ( u ( t ) , div M ) in L ∞ ([0 , T ]) . But, − ( u ( t ) , div M ) = ( ∇ u ( t ) , M ) − Z Γ ( M · n ) · u, giving ( C ).( C ) = ⇒ ( D ): This follows simply because H (Ω) ⊆ H (Ω).( D ) = ⇒ ( A ): Assume ( D ) holds, and let v be in H . Then v = div M forsome M in ( H (Ω)) d × d by Corollary 6.5, so( u ( t ) ,v ) = ( u ( t ) , div M ) = − ( ∇ u ( t ) , M ) + Z Γ ( M · n ) · u ( t )= − ( ∇ u ( t ) , M ) → − ( ∇ u ( t ) , M ) , uniformly over [0 , T ]. But, − ( ∇ u ( t ) , M ) = ( u ( t ) , div M ) − Z Γ ( M · n ) · u ( t ) = ( u ( t ) , v ) , from which ( A ) follows.Now assume that d = 2.( A ′ ) = ⇒ ( E ): Assume that ( A ′ ) holds and let f be in H (Ω). Then( ω ( t ) , f ) = − (div u ⊥ ( t ) , f ) = ( u ⊥ ( t ) , ∇ f ) = − ( u ( t ) , ∇ ⊥ f ) → − ( u ( t ) , ∇ ⊥ f ) in L ∞ ([0 , T ])where u ⊥ = − (cid:10) u , u (cid:11) and we used the identity ω ( u ) = − div u ⊥ . But, − ( u ( t ) , ∇ ⊥ f ) = ( u ⊥ ( t ) , ∇ f ) = − (div u ⊥ ( t ) , f ) + Z Γ ( u ( t ) ⊥ · n ) f = − (div u ⊥ ( t ) , f ) − Z Γ ( u ( t ) · τ ) f = ( ω ( t ) , f ) − Z Γ ( u ( t ) · τ ) f, giving ( E ). ANISHING VISCOSITY AND VORTICITY ON THE BOUNDARY 5 ( E ) = ⇒ ( F ): Follows for the same reason that ( C ) = ⇒ ( D ).( F ) = ⇒ ( A ): Assume ( F ) holds, and let v be in H . Then v = ∇ ⊥ f forsome f in H (Ω) ( f is called the stream function for v ), and( u ( t ) ,v ) = ( u ( t ) , ∇ ⊥ f ) = − ( u ⊥ ( t ) , ∇ f ) = (div u ⊥ ( t ) , f )= − ( ω ( t ) , f ) → − ( ω ( t ) , f ) in L ∞ ([0 , T ]) . But, − ( ω ( t ) ,f ) = (div u ⊥ ( t ) , f ) = − ( u ⊥ ( t ) , ∇ f ) = ( u ( t ) , ∇ ⊥ f )= ( u ( t ) , v ) , which shows that ( A ) holds.What we have shown so far is that ( A ), ( A ′ ), ( B ), ( C ), and ( D ) areequivalent, as are ( E ) and ( F ) in two dimensions. It remains to show that( E ) is equivalent to these conditions as well. We do this by establishing theimplications ( C ) = ⇒ ( E ) = ⇒ ( A ).( C ) = ⇒ ( E ): Follows directly from Equation (2.1).( E ) = ⇒ ( A ): Let v be in H and let x be the vector field in ( H (Ω) ∩ H (Ω)) d solving ∆ x = v on Ω ( x exists and is unique by standard elliptic theory).Then, utilizing Lemma 6.6 twice (and suppressing the explicit dependenceof u and u on t ),( u, v ) = ( u, ∆ x ) = − ( ∇ u, ∇ x ) + Z Γ ( ∇ x · n ) · u = − ( ∇ u, ∇ x )= − ω ( u ) , ω ( x )) − Z Γ ( ∇ ux ) · n = − ω ( u ) , ω ( x )) → − ω ( u ) , ω ( x )) + 2 12 Z Γ (( ω ( x ) − ω ( x ) T ) · n ) · u = − ω ( u ) , ω ( x )) + 2 Z Γ ( ω ( x ) · n ) · u = − ( ∇ u, ∇ x ) + Z Γ ( ∇ ux ) · n + 2 Z Γ ( ω ( x ) · n ) · u = − ( ∇ u, ∇ x ) + 2 Z Γ ( ω ( x ) · n ) · u = ( u, ∆ x ) − Z Γ ( ∇ x · n ) · u + 2 Z Γ ( ω ( x ) · n ) · u = ( u, v ) − Z Γ (( ∇ x ) T · n ) · u. (2.3)Thus, ( E ) = ⇒ ( A ) if and only if Z Γ (( ∇ x ) T · n ) · u = 0 . (2.4) JAMES P. KELLIHER
But, (div( ∇ x ) T ) j = ∂ j ∂ i x j = ∂ i div x or div( ∇ x ) T = ∇ div x . Similarly,div( ∇ u ) T = ∇ div u = 0. It follows that Z Γ (( ∇ x ) T · n ) · u = (( ∇ x ) T , ∇ u ) + ( ∇ div x, u )= ( ∇ x, ( ∇ u ) T ) − (div x, div u ) + Z Γ u · n div x = (( ∇ u ) T , ∇ x )= − (div( ∇ u ) T , x ) + Z Γ (( ∇ u ) T · n ) x = 0 . (cid:3) Remarks
The equivalent conditions of Theorem 2.1 complement those of [4], [11],[12], and [6].It is only in the proof of ( A ) = ⇒ ( B )—in which we quote a result ofKato’s in [4]—where the requirement that u be a classical solution to theEuler equations and that f → f in L ([0 , T ]; L ) is used; in fact, it is theonly place where the fact that u and u are solutions to the Navier-Stokes andEuler equations, respectively, appear in the proof at all. That is, assumingonly that u is a vector field parameterized by ν that lies in L ∞ ([0 , T ]; H ) ∩ L ([0 , T ]; V ) and that u is a vector field lying in L ∞ ([0 , T ]; H ∩ H (Ω)),all of the implications in the proof of Theorem 2.1 remain valid except for( A ) = ⇒ ( B ).The proof of ( A ) = ⇒ ( B ) in [4] consists, using our terminology, ofproving ( B ) = ⇒ ( A ) = ⇒ ( i ) = ⇒ ( ii ) = ⇒ ( B ), where ( i ) and ( ii ) arethe conditions, ( i ) ν Z T k∇ u k L (Ω) → , ( ii ) ν Z T k∇ u k L (Γ cν ) → , with Γ cν a boundary layer of width proportional to ν .If we weaken the regularity of Γ from C to only locally Lipschitz, thenthe proof of ( E ) = ⇒ ( A ) fails because we would only have x in ( H (Ω)) d .Kato’s proof that ( A ) = ⇒ ( B ) also requires a C boundary.In two dimensions, we need only have convergence of the vorticity awayfrom the boundary—condition ( F )—to insure that the vanishing viscositylimit holds. In particular, it follows that formation of a vortex sheet on theboundary of a type other than that given in ( E ) is inconsistent with u beinga solution to ( N S ). In higher dimensions it is an open problem whether theanalogous statement is true; that is, whether ( F ) = ⇒ ( A ), where ( F ) isthe condition,( F ) ω → ω in ( H − (Ω)) d × d uniformly on [0 , T ] . ANISHING VISCOSITY AND VORTICITY ON THE BOUNDARY 7
The remarks that follow attempt to give some insight into the nature of thisproblem.One approach to proving that ( F ) = ⇒ ( A ) is to prove that ( F ) = ⇒ ( D ),since we have ( D ) = ⇒ ( A ). So suppose that ( F ) holds, and let M be in( H (Ω)) d × d . For any vector field v ,( ∇ v, M ) = ( ∇ v − ( ∇ v ) T , M ) + (( ∇ v ) T , M ) = 2( ω ( v ) , M ) + ( ∇ v, M T ) . Thus, ( ∇ u, M − M T ) = 2( ω ( u ) , M ) → ω ( u ) , M ) = ( ∇ u, M − M T ) . If M is antisymmetric then M − M T = 2 M and we conclude that ( D ) holdsfor antisymmetric matrix fields in ( H (Ω)) d × d . But if M in ( H (Ω)) d × d issymmetric,2( ω ( u ) , M ) = ( ∇ u, M ) − (( ∇ u ) T , M ) = ( ∇ u, M − M T ) = 0 , so ( ω ( u ) , M ) = ( ω ( u ) , M ) = 0, and we can conclude nothing from thisapproach.But some use can still be made of this observation. Let v be any elementof H . Then from Corollary 6.5, for some M in ( H (Ω)) d × d ,( u, v ) = ( u, div M ) = − ( ∇ u, M ) . Now, if we could insure that M can be chosen to be antisymmetric, then if( F ) holds for M so does ( D ), as we just showed, and − ( ∇ u, M ) → − ( ∇ u, M ) = ( u, v ) , and ( A ) would follow.In two dimensions, we can choose M = (cid:18) − ff (cid:19) , (3.1)where f is the stream function for v as in the proof of ( F ) = ⇒ ( A ), whichgives a slight variation on the proof of that same implication.In three dimensions, however, it is not possible to find such an M . To seethis, suppose that M in ( H (Ω)) d × d is antisymmetric. Then can write M as M = a b − a c − b − c with a = b = c = 0 on Γ. In this form the condition div u = div div M = 0is automatically satisfied, and letting ˜ ω be the vector h c, − b, a i , we see that u = div M = curl ˜ ω, (3.2)where curl is the usual three-dimensional operator. But curl maps H ∩ C ∞ (Ω)bijectively onto itself when Γ is C ∞ (see, for instance, [2]), so in general weonly have ˜ ω · n = 0 on Γ. That is, the condition that M be antisymmetricis not compatible with the condition that it vanish on Γ. JAMES P. KELLIHER
Finally, let E (Ω) = n u ∈ ( L (Ω)) d : div u ∈ L (Ω) , u · n = 0 on Γ o , with k u k E (Ω) = k u k L (Ω) + k div u k L (Ω) . It is easy to see from the proofsof ( A ′ ) = ⇒ ( C ) and ( D ) = ⇒ ( A ) that condition ( D ) can be weakenedfrom convergence in the dual space of ( H (Ω)) d × d to convergence in the dualspace of ( E (Ω)) d . (This is advantageous as a sufficient condition, though notas a necessary one.)Returning to Equation (3.2), the condition ˜ ω · n = 0 on Γ does not translateto M · n = 0 on Γ. Hence, M does not lie in ( E (Ω)) d so we cannot use thisweakening of condition ( D ) to conclude that ( A ) holds.4. Characteristic boundary conditions on the velocity
We modify (
N S ) by allowing the velocity on the boundary to be equalto a nonzero time-varying vector field b , which is required, however, to betangential to the boundary. This gives( N S b ) (cid:26) ∂ t u + u · ∇ u + ∇ p = ν ∆ u + f and div u = 0 on [0 , T ] × Ω ,u = b on [0 , T ] × Γ and u = u ν on { } × Ω , where, as before, u ν is in H .We require sufficient regularity on b so that ( N S b ) is well-posed. Forsimplicity, we will assume that b · n = 0 on Γ with b ∈ L ∞ ([0 , T ]; H / (Γ)) , ∂ t b ∈ L ([0 , T ]; H − / (Γ)) (4.1)so that b lifts (extends) to a vector field, which we also call b , with b ∈ L ∞ ([0 , T ]; H ∩ H (Ω)) , ∂ t b ∈ L ([0 , T ]; L (Ω)) . The assumption on b in Equation (4.1) is not the weakest possible, but theassumption on ∂ t b can probably not be weakened.We can then use the equation that corresponds to u − b (which lies in L ∞ ([0 , T ]; H ) ∩ L ([0 , T ]; V ) for classical solutions) to define a weak solutionto ( N S b ). (This is essentially what is done in Section 4 of [12].)With such solutions to ( N S b ) in place of those for ( N S )—but withoutchanging the formulation of ( EE )—we define the condition,( C b ) ∇ u → ∇ u − h γ n · , ( u − b ) µ i in (( H (Ω)) d × d ) ′ uniformly on [0 , T ] , ( E b ) ω → ω − (cid:10) γ n ( · − · T ) , ( u − b ) µ (cid:11) in (( H (Ω)) d × d ) ′ uniformly on [0 , T ] , and in two dimensions, the condition,( E b ) ω → ω − (( u − b ) · τ ) µ in ( H (Ω)) ′ uniformly on [0 , T ] . Theorem 2.1 then becomes:
ANISHING VISCOSITY AND VORTICITY ON THE BOUNDARY 9
Theorem 4.1.
Let u be a solution to ( N S b ) and u be a solution to ( EE ).Conditions ( A ), ( A ′ ), ( B ), ( C b ), ( D ), and ( E b ) are equivalent. In twodimensions, conditions ( E b ) and ( F ) are equivalent to the other conditions.Proof. The proof of ( A ′ ) = ⇒ ( C b ) is identical to the proof of ( A ′ ) = ⇒ ( C )except that a boundary term is included in the first step: this term leadsto the “ − b ” in condition ( C b ). A similar comment applies to the proof of( A ′ ) = ⇒ ( E b ) and ( C ) = ⇒ ( E b ). The proof of ( C ) = ⇒ ( D ) and( E ) = ⇒ ( F ) are unaffected by the presence of the vector field b , as areall other implications except for ( A ) = ⇒ ( B ) and ( E b ) = ⇒ ( A ).( A ) = ⇒ ( B ): In [12] it is shown, using our terminology, that ( B ) = ⇒ ( i ) = ⇒ ( ii ′ ) = ⇒ ( B ), where ( ii ′ ) is the condition( ii ′ ) ν Z T k∇ τ u τ k L (Γ δ ( ν ) ) → ν Z T k∇ τ u n k L (Γ δ ( ν ) ) → . Here, ∇ τ is the gradient only in the direction tangential to the boundary, u τ and u n are the components of the velocity tangential and normal to theboundary, respectively, and Γ δ ( ν ) is a boundary layer whose width δ ( ν ) is ofarbitrary order larger than ν .But, ( B ) = ⇒ ( A ) is immediate, and ( A ) = ⇒ ( i ) follows from com-bining the argument on pages 232 through 233 of [12] with the proof of theimplication ( A ) = ⇒ ( i ) on page 90 of [4] (in Kato’s terminology, this is (ii)implies (iii)). This gives ( A ) = ⇒ ( B ).( E b ) = ⇒ ( A ): Adapting the argument of ( E ) = ⇒ ( A ), we see that thefirst boundary integral in Equation (2.3) does not vanish, since now u = b on Γ. Also, u becomes u − b in the boundary integrals involving ω ( x ). Thisleads to( u, v ) = ( u, ∆ x ) + Z Γ ( ∇ x · n ) · b − Z Γ ( ∇ x · n ) · u + 2 Z Γ ( ω ( x ) · n ) · ( u − b )= ( u, v ) − Z Γ (( ∇ x ) T · n ) · ( u − b ) . The last boundary integral vanishes for the same reason that Equation (2.4)holds, u − b being in H , completing the proof. (cid:3) Radially symmetric initial vorticity in a disk
We assume, in this section only, that Ω is the unit disk D and that theinitial vorticity ω is radially symmetric. In this case, the solution to ( EE )is stationary: ω ( t ) = ω for all time t .The vanishing viscosity limit in the classical sense of condition ( B ) holdsin this setting under fairly general circumstances. Under the assumptions ofSection 1 on the regularity of the initial velocity, and assuming that b = 0,this convergence is implicit in [4] at least for zero forcing (see [5]), but wasfirst explicitly proved by Matsui (see [9]). For nonzero b having the regu-larity assumed in Equation (4.1), the convergence is a simple consequence of the condition ( ii ′ ) in the proof of Theorem 4.1 as observed by Wang in[12]. For substantially lower regularity on u and on b than we assume, theconvergence is established in [8].More precisely, the authors of [8] assume that u = u and f = f = 0(which are not significant limitations, since one can handle u → u in H by using the triangle inequality, and nonzero forcing presents no realdifficulties), with u ∈ R ( D ) = n v ∈ ( L ( D )) : v ( x ) = s ( | x | ) x ⊥ for some s, ω ( v ) ∈ L ( D ) o = (cid:8) v ∈ H : ω ( v ) ∈ L ( D ) , ω ( v ) radially symmetric (cid:9) . They assume that b ( t, · ) = α ( t )—that is, b ( t ) is constant on the boundary—and that α ∈ BV([0 , T ]), the space of bounded variation functions. Theyprove (combining Propositions 9.6 and 9.7 of [8]) that ω → ω − ( B (2 π ) − − b · τ ) µ in M ( D ) uniformly on [0 , T ] , (5.1)where B = Z D ω . But, on Γ, u · τ is constant, so by Green’s theorem, B = Z Γ u · τ = 2 πu ( x ) · τ ( x )for any point x on Γ, and we see that Equation (5.1) is the same as condition( E b ), except that the convergence is stronger.That is, both conditions ( B ) and ( E b ) hold for a disk, except that theconvergence in ( E b ) is in M (Ω), which is stronger convergence than that of( E b ). What we have shown is that either both conditions ( B ) and ( E b ) holdor neither condition holds for a given initial velocity in a general boundeddomain in the plane—and in the analogous sense, in R d . It was the questionof whether this was, in fact, the case that motivated this paper.The regularity we assume in Equation (4.1) corresponds to α lying in H ([0 , T ]), which is considerably stronger than the assumption in [8] that α lie in BV([0 , T ]). And their assumption on the regularity of u is far lowerthan our assumption that u lies in C ǫ (Ω) ∩ H . Without the assump-tion of radial symmetry, however, it seems unlikely that one can weaken ourassumptions in Equation (4.1) on b to any significant degree, since theseassumptions go to the heart of establishing the existence of the correspond-ing weak solutions of ( N S b ). Weakening the regularity assumptions on u would seem equally impossible, since the boundedness of ∇ u on [0 , T ] × Ω isindispensable in Kato’s argument showing that ( A ) = ⇒ ( B ).6. Some technical lemmas
In this section we assume only that Ω is bounded and that Γ is locallyLipschitzian, which of course includes the case that Γ is C . ANISHING VISCOSITY AND VORTICITY ON THE BOUNDARY 11
The various integrations by parts that we make are justified by Lemma 6.1,which is Theorem 1.2 p. 7 of [10] for locally Lipschitz domains. (Temamstates the theorem for C boundaries but the proof for locally Lipschitzboundaries is the same, using a trace operator for Lipschitz boundaries inplace of that for C boundaries: see p. 117-119 of [3], in particular, Theorem2.1 p. 119.) Lemma 6.1.
Let E (Ω) = n v ∈ ( L (Ω)) d : div v ∈ L (Ω) o with k v k E (Ω) = k v k L (Ω) + k div v k L (Ω) . There exists an extension of thetrace operator γ n : ( C ∞ (Ω)) d → C ∞ (Γ) defined by u u · n on Γ to acontinuous linear operator from E (Ω) onto H − / (Γ) . The kernel of γ n isthe space E (Ω) —the completion of C ∞ (Ω) in the E (Ω) norm. For all u in E (Ω) and f in H (Ω) , ( u, ∇ f ) + (div u, f ) = Z Γ ( u · n ) f . (6.1) Lemma 6.2.
Assume that u is in ( D ′ (Ω)) d with ( u, v ) = 0 for all v in V .Then u = ∇ p for some p in D ′ (Ω) . If u is in ( L (Ω)) d then p is in H (Ω) ;if u is in H then p is in H (Ω) and ∆ p = 0 .Proof. For u in ( D ′ (Ω)) d see Proposition 1.1 p. 10 of [10]. For u in ( L (Ω)) d the result follows from a combination of Theorem 1.1 p. 107 and Remark4.1 p. 55 of [3] (also see Remark 1.4 p. 11 of [10]). (cid:3) Lemma 6.3.
For any u in ( L (Ω)) d there exists a unique v in H and p in H (Ω) such that u = v + ∇ p .Proof. This follows, for instance, from Theorem 1.1 p. 107 of [3], which holdsfor an arbitrary domain, along with Lemma 6.2. (cid:3)
Lemma 6.4.
For any f in L (Ω) and a in ( H / (Γ)) d satisfying the com-patibility condition, Z Ω f = Z Γ a · n there exists a (non-unique) solution v in ( H (Ω)) d to div v = f in Ω , v = a on Γ .Proof. This follows from Lemma 3.2 p. 126-127, Remark 3.3 p. 128-129,and Exercise 3.4 p. 131 of [3] (and see the comment on p. 67 of [1]). (cid:3)
Corollary 6.5.
For any v in H there exists a matrix-valued function M in ( H (Ω)) d × d such that v = div M .Proof. Let v be in H and observe that Z Ω v i = Z Ω v · ∇ x i = − Z Ω div v x i + Z Γ ( v · n ) x i = 0 . Thus, we can apply Lemma 6.4 to each component v i using a ≡ w i in H (Ω) satisfying div w i = v i on Ω, w i = 0 on Γ. Forming amatrix-valued function whose rows are w , w , . . . , w d gives M . (cid:3) Lemma 6.6.
Assume that u is in ( H (Ω)) d or ( C (Ω)) d with div u = 0 and v is in ( H (Ω)) d . Then ( ∇ u, ∇ v ) = 2( ω ( u ) , ω ( v )) + Z Γ ( ∇ uv ) · n . Proof.
Directly from Equation (2.1),2 ω ( u ) · ω ( v ) = 12 ( ∇ u − ( ∇ u ) T ) · ( ∇ v − ( ∇ v ) T )= ∇ u · ∇ v − ( ∇ u ) T · ∇ v. (6.2)Since div u = 0, we have ( ∇ u ) T · ∇ v = ∂ j v i ∂ i u j = ∂ j ( v i ∂ i u j ) = div( ∇ uv ),so if u and v are both in ( C ∞ (Ω)) d with div u = 0 then2 Z Ω ω ( u ) · ω ( v ) = Z Ω ∇ u · ∇ v − ( ∇ u ) T · ∇ v = Z Ω ∇ u · ∇ v − Z Ω div( ∇ uv ) = Z Ω ∇ u · ∇ v − Z Γ ( ∇ uv ) · n . The result then follows by the density of C ∞ (Ω) in H (Ω), H (Ω), and C (Ω). (cid:3) Acknowledgements
The author would like to thank Helena Nussenzveig Lopes, Milton LopesFilho, and Robert Pego for valuable comments on early versions of this paper.The author was supported in part by NSF grant DMS-0705586 during theperiod of this work.
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