REMARK ON WELL-POSEDNESS FOR THE FOURTH ORDER NONLINEAR SCHRÖDINGER TYPE EQUATION

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 12, Pages 3559 3568 S 0002-9939(04)07620-8 Article electronically published on July 12, 2004 REMARK ON WELL-POSEDNESS FOR THE FOURTH ORDER NONLINEAR SCHRÖDINGER TYPE EQUATION JUN-ICHI SEGATA (Communicated by David S. Tartakoff) Abstract. We consider the initial value problem for the fourth order nonlinear Schrödinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper. 1. Introduction We consider the initial value problem for the fourth order nonlinear Scrödinger type equation (4NLS) of the form: { i t u + 2 (1.1) xu + ν xu 4 = F (u, u, x u, x u, xu, 2 xu), 2 (t, x) R R, u(0,x)=u 0 (x), x R, where u(x, t) :R R C is an unknown function. The nonlinear term F is given by F (u, u, x u, x u, x 2 u, 2 x u)= 1 2 u 2 u + λ 1 u 4 u + λ 2 ( x u) 2 u + λ 3 x u 2 u (1.2) + λ 4 u 2 xu 2 + λ 5 u 2 xu, 2 where ν, µ are real constants satisfying λ 1 =3µ/4,λ 2 =2µ ν/2,λ 3 =4µ+ν, λ 4 = µ, λ 5 =2µ ν. The equation in (1.1) describes the three-dimensional motion of an isolated vortex filament embedded in an inviscid incompressible fluid filling an infinite region. This equation is proposed by Fukumoto and Moffatt [8] as some detailed model taking account of the effect from the higher order corrections of the Da Rios model (cubic nonlinear Schrödinger equation): i t u + 2 x u = 1 2 u 2 u. For the physical background we refer to [7] and [8]. To motivate our problem in this paper, we state briefly our previous result associated with the well-posedness of the initial value problem (1.1). The notion of well-posedness used here includes the existence, uniqueness of a solution and local well- Received by the editors April 30, 2003. 2000 Mathematics Subject Classification. Primary 35Q55. Key words and phrases. Fourth order nonlinear Schrödinger type equation, posedness. c 2004 American Mathematical Society 3559

3560 JUN-ICHI SEGATA continuous dependence upon the initial data. In [14], we proved the time local well-posedness as the initial value problem (1.1) in the usual Sobolev spaces H s (R) with s 1/2 by imposing the condition λ 5 =2µ ν =0onthecoefficients. Itwas not clear whether this restriction has any physical interpretations. In the present paper, we eliminate this restriction and guarantee the time local well-posedness for (1.1) in Sobolev spaces not lacking the term u 2 x u, i.e., λ 5 =2µ ν 0. Consequently our result improved to include the non-completely integrable case, which appears in the real model. To state our main result precisely, we introduce some notation and function spaces. For a function u(x, t), we denote by û = F x u the Fourier transforms in the x variable. We denote by û(τ,ξ) =F t F x u(τ,ξ) the space-time Fourier transform. The operator D x and D x are given by D x = Fx 1 ξ F x and D x = Fx 1 ξ F x, respectively, where x =(1+ x 2 ) 1/2. We abbreviate L p t (R; Lq x (R)) as Lp t (Lq x )and H p t (R; Hx q(r)) as Hp t (Hx q ), respectively. Let ψ(t) be a smooth cut-off function to the interval [ 1, 1], i.e., ψ C0 (R), ψ(t) 1for t 1, and ψ(t) 0for t 2. For δ>0, we set ψ δ (t) =ψ(t/δ). W ν (t) is the unitary group generated by the linear equation of (1.1). For a real number s, lets+ denotes a fixed constant larger than s. The equation (1.1) is rewritten as the following integral equation: (1.3) u(t) =W ν (t)u 0 i t for t [ T,T]. Then our main result is the following: 0 W ν (t t )F (u, u, x u, x u, 2 x u, 2 x u)(t )dt, Theorem 1.1. Let ν<0. Ifs>7/12, b (1/2, 3/4), then for u 0 H s (R), there exist T = T ( u 0 H s) > 0 and a unique solution u(t) of the initial value problem (1.1) satisfying u C([ T,T]; H s (R)), ψ T W ν ( t)u Ht b (R; Hx(R)), s ψ T W ν ( t)f Ht b 1 (R; Hx(R)). s Moreover, given T (0,T), two maps u 0 u from H s (R) to C([ T,T ]; H s (R)) and u 0 ψ T W ν ( t)u from H s (R) to Ht b (R; Hx(R)) s are Lipschitz continuous, respectively. Remark. By employing an analogous method in Molinet-Saut-Tzvetkov [13], we showed in [14] that the initial value problem for (1.1) cannot be solved by the Picard iterative succession via the corresponding integral equation in the Sobolev space H s (R) withs<1/2. Therefore there is a gap between the index s =7/12+ of the Sobolev spaces in Theorem 1.1 and the s =1/2 suggested by the counterexample. However, assuming that λ 5 =2µ ν = 0, we could solve the local well-posedness of the initial value problem (1.1) in H s (R) withs 1/2 ([14]). When µ + ν/2 = 0, it should be remarked that (1.1) is the completely integrable equation (see [7]) and has infinitely many conserved quantities (see [9]); for example, Φ 1 (u) = 1 u 2 dx, Φ 2 (u) = i ( x u)udx, 2 R 2 R Φ 3 (u) = 1 x 2 R( 2 u)udx 1 u 4 dx,. 8 R

REMARK ON WELL-POSEDNESS FOR THE 4NLS 3561 Therefore, if µ + ν/2 = 0, combining the above properties and the Gagliardo- Nirenberg inequality: u 4 L 4 C u 3 L 2 xu L 2 C2 2 u 6 L + 1 2 2 xu 2 L 2, we can show an a priori bound of the solution in H 1 (R) for all t>0. Hence we have the global well-posedness of the solution in our theorem. To prove Theorem 1.1, we use the method of Fourier restriction norm introduced by Bourgain [4] and Kenig-Ponce-Vega [11], [12]. We define the Fourier restriction space X ν with b, s, ν R associated with the equation (1.1) as follows: We denote φ ν (ξ) =ξ 2 νξ 4.Thenτ + φ ν (ξ) represents the symbol of the linearized equation of (1.1). Let X ν {u S (R 2 ); u X ν < }, u X ν τ + φ ν (ξ) b ξ s û(ξ,τ) L 2 ξ (L 2 τ (1.4) ) = W ν ( t)u(t) H b t (Hx s). Then we introduce a new estimate of the maximal function related to the unitary group of the fourth order Schrödinger equation (see Proposition 2.2 in Section 2 below). This estimate enables us to handle the worst term u 2 xu 2 in the nonlinear terms, and we can show the crucial trilinear estimate relevant to this term. In the next section, we list some linear estimates including the estimate for the maximal function. In the last section, we show the crucial nonlinear estimate and prove Theorem 1.1. 2. Linear estimates In this section, we give the linear estimates needed for the proof of crucial nonlinear estimates (see Propositions 3.1 and 3.2 below). It is convenient to use the following notation for the proof of Proposition 3.2 below: For b R let us define F b by (2.1) for f S(R 2 ). F b (τ,ξ) = f(τ,ξ) τ ± φ ν (ξ) b, Lemma 2.1. Let b>1/2 and b > 1/4. For any f L 2 τ (L2 ξ ), we have (2.2) Dx 1/4 F b L 4 x (L t ) f L 2 τ (L 2 ξ ) (Kenig-Ruiz estimate), (2.3) Dx 3/2 b L x (L 2 t ) f L 2 τ (L 2 ξ ) (Kato type smoothing effect), (2.4) Dx 3/4 b L 4 x (L2 t ) f L 2 τ (L 2 ξ ) (Kato type smoothing effect), where F b is defined by (2.1). Proof of Lemma 2.1. The estimates (2.2) and (2.3) are due to Kenig-Ponce-Vega [10]. For the proof of those estimates, see Theorem 2.5 and Theorem 4.1 in [10], respectively. The inequality (2.4) follows from the interpolation between (2.3) and the Plancherel identity F 0 L 2 x (L 2 t ) = f L 2 ξ (L 2 τ ). The next proposition plays an important role in the proof of our main theorem (see the proof of Proposition 3.2 below).

3562 JUN-ICHI SEGATA Proposition 2.2 (Estimate for the maximal function). Let ρ>1, T>0. For any u 0 L 2 x(r) and F L 2 x(l 1 T ), we have (2.5) D x ρ W (t)u 0 L 2 x (L T ) C u 0 L 2 x, t (2.6) D x 2ρ W ν (t t )F (t )dt L 2 x (L T ) C F L 2 x (L 1 T ), 0 where C>0 is a constant depending on T and ρ. Corollary 2.3. Let ρ>1/2, T (0, 1), b > 1/2, and f L 2 τ (L2 ξ ) with suppft 1 Fx 1 f ( T,T). Then, for any F b defined by (2.1), we have (2.7) D x ρ F b L 2 x (L t ) C f L 2 τ (L 2 ξ ). A similar result to Proposition 2.2 for the Schrödinger equation is obtained by Constantin-Saut [6], Sjölin [15] and Vega [16]. The estimate (2.5) is proved by applying the duality argument to the estimate (2.6). For the purpose of the proof for the inequality (2.6), it suffices to show that the integral kernel of D x 2ρ W ν (t t ) belongs to L 1 x(l T ). More precisely, we require the following lemma. Lemma 2.4. Let ρ>1. We define the integral kernel of D x 2ρ W ν (t t ) as K(t t,x y). Then for some ε>0, we have (2.8) K(t t,x y) x y 1 ε, where C>0is a constant depending only on T, ρ and independent of t, t [0,T]. A simple application of Young s inequality and Lemma 2.4 yield (2.6). Proof of Lemma 2.4. For simplicity, we only show the case ν = 1. Let φ(ξ) φ 1 (ξ) =ξ 2 + ξ 4. We note that the integral kernel of D x 2ρ W (t s) isgivenby K(σ, z) = 1 e izξ iσφ(ξ) ξ 2ρ dξ, 2π R where σ = t t and z = x y. By differentiating the phase, we have d ( dξ (zξ σφ(ξ)) = z σ(φ (ξ)) = 4σ ξ 3 + ξ 2 z ) = 4σα 3( η 3 + η ) 4σ 2α 2 1, whereweput z 4σ = α3, ξ = αη, α R, and φ = dφ dξ. Let p j (j =0, 1, 2) be the roots of the algebraic equation η 3 + 1 2α η 1 = 0, and 2 let p 0 be the unique real root. We note that the p j s are depending on α, and it is easy to see that p 0 1, p 1 e πi 3,p2 e 2πi 3 as α. Indeed, we have a more precise estimate by Rouché s Theorem as follows: Let α > 2. Then, we have (2.9) p j e πji 1 3 <, j =0, 1, 2. α 2 Remark. Since η 3 + 1 2α η 1=(η p 2 0 )(η p 1 )(η p 2 ), (2.9) yields (2.10) C 1 η p 0 η 2 η 3 + 1 2α 2 η 1 C2 η p 0 η 2 for η R, α > 2 where C 1,C 2 are independent of η and α. We often use the above inequality when we consider the estimate of K(σ, z).

REMARK ON WELL-POSEDNESS FOR THE 4NLS 3563 We separate into two cases: z > 64T and z 64T. The case z 64T. It directly follows from the definition of K(σ, z) that (2.11) K(z,σ) 1 ξ 2ρ dξ C. 2π The case z > 64T. We note that α = z 1/3 4α > 64T 1/3 8T =2. Bytheidentity, d dξ (ξ αp 0)e izξ iσφ(ξ) = {1 4σi(ξ αp 0 ) and integrating by parts, we have (2.12) K(σ,z) = 1 e izξ iσφ(ξ) (ξ αp 0 ) d { 2π R dξ = 1 { e izξ iσφ(ξ) 2π R 1 2π R M 1 (σ, z)+m 2 (σ, z). R 2 (ξ αp j )}e izξ iσφ(ξ), j=0 1 1 4σi(ξ αp 0 ) 2 j=0 (ξ αp j) ξ 2ρ 8σi 2 j=0 (ξ αp j) {1 4σi(ξ αp 0 ) 2 j=0 (ξ αp j)} 2 + 4σi(ξ αp 0) 2 (2ξ αp 1 αp 2 ) {1 4σi(ξ αp 0 ) 2 j=0 (ξ αp j)} 2 e izξ iσφ(ξ) ξ αp 0 1 4σi(ξ αp 0 ) 2 } ξ 2ρ dξ d j=0 (ξ αp j) dξ ξ 2ρ dξ } dξ For M 1 (σ, z), we apply the inequality (2.10) and separate the result into two terms: σα 4 (η p 0 ) 2 (η 2 +1) M 1 (σ, z) R {1+Cσα 4 (η p 0 ) 2 (η 2 +1) } 2 αη 2ρ αdη σα 4 (η p 0 ) 2 η p 0 <1/4 {1+Cσα 4 (η p 0 ) 2 } 2 αη 2ρ αdη (2.13) σα 4 (η 2 +1) 2 + C η p 0 >1/4 {1+Cσα 4 (η 2 +1) 2 } 2 αη 2ρ αdη M 1,1 (σ, z)+m 1,2 (σ, z). Recalling ρ>1and4σα 3 = z, the first term in the right-hand side of (2.13) is estimated as follows: M 1,1 (σ, z) α 2ρ σα 4 η 2 α (1 + cσα 4 η 2 ) 2 dη (2.14) η <1/4 =Cσ 1/2 α 2ρ α 1 σ 1/2+(2ρ+1)/3 z ( 2ρ 1)/3 T 1/2+(2ρ+1)/3 z 1 ε, for ρ>1.

3564 JUN-ICHI SEGATA Similarly for the second term, (2.15) M 1,2 (σ, z) σα 4 (η 2 +1) 2 R {1+Cσα 4 (η 2 +1) 2 } 2 αη 2ρ αdη σα 4 η <1 (1 + Cσα 4 ) 2 αη 2ρ αdη σα 4 η 4 α + C η >1 (1 + Cσα 4 η 4 ) 2 dη (αη) 2ρ σ 1 α 4 + Cσ 1 α 3 2ρ T 2ρ/3 z 4/3. Combining (2.13)-(2.15), we obtain (2.16) M 1 (σ, z) z 1 ε. Next, we estimate M 2 (σ, z). We apply an integration by parts to have (2.17) M 2 (σ, z) = 1 e izξ iσφ(ξ) (ξ αp 0 ) 2π R d { dξ { R ξ αp 0 {1 4σi(ξ αp 0 ) 2 j=0 (ξ αp j)} 2 d dξ ξ 2ρ α(η p 0 ) {1+Cσα 4 (η p 0 ) 2 (η 2 +1)} 2 σα 5 (η p 0 ) 3 (η 2 +1) + {1+Cσα 4 (η p 0 ) 2 (η 2 +1)} 3 + C R } αη 2ρ 1 αdη α 2 (η p 0 ) 2 {1+Cσα 4 (η p 0 ) 2 (η 2 +1)} 2 αη 2ρ 2 αdη M 2,1 (σ, z)+m 2,2 (σ, z). } dξ The evaluations of M 2,1 (σ, z)andm 2,2 (σ, z) are similar to the estimates of M 1,1 (σ, z) and M 1,2 (σ, z), and we proceed by decomposing the integral interval into η p 0 < 1/4 and η p 0 > 1/4. Then we have (2.18) (2.19) M 2,1 (σ, z) z 1 2ρ/3, M 2,2 (σ, z) z 2. Combining (2.17)-(2.19), we obtain (2.20) M 2 (σ, z) z 1 ε. Combining (2.12), (2.16), (2.20) and (2.11) we have Lemma 2.4.

REMARK ON WELL-POSEDNESS FOR THE 4NLS 3565 3. Crucial nonlinear estimates In this section, we first state the nonlinear estimates obtained in the paper [14]. Proposition 3.1. Let ν<0, s > 7/12, a < 1/4 and b>1/2. Then for any u j X ν with supp u j ( T,T), T (0, 1), we have (3.1) (3.2) (3.3) (3.4) (3.5) u 1 u 2 u 3 X ν s,a u 1 u 2 u 3 u 4 u 5 X ν s,a x u 1 u 2 x u 3 X ν s,a u 1 x u 2 x u 3 X ν s,a u 1 xu 2 2 u 3 X ν s,a 3, 5, 3, 3, 3. For the proof of Proposition 3.1, see [14]. The next proposition is the crucial estimate in this paper. Proposition 3.2. Let ν<0, s > 7/12, a < 1/4 and b>1/2. Then for any u j X ν with supp u j ( T,T), T (0, 1), we have (3.6) u 1 u 2 xu 2 3 Xs,a 3. Remark. Concerning the estimates (3.1)-(3.5), we can show still smaller s. However we do not need those estimates for s 7/12 because of the worst term u 2 2 xu. Proof of Proposition 3.2. From the definition of X ν in (1.4) and duality, the inequality (3.6) is reduced to the following estimate: For any 0 f 4 L 2 τ (L2 ξ ), (3.7) I Γ τ ξ 4 Γξ s ξ 3 2 4 f j(τ j,ξ j ) ξ 1 s ξ 2 s ξ 3 s 3 τ j +( 1) j φ ν (ξ j ) b τ 4 + φ ν (ξ 4 ) a 4 f j L 2 τ (L 2 ξ ). Here we set f j (τ,ξ) = ξ s τ +( 1) j φ ν (ξ) b û(( 1) j τ,( 1) j ξ) for j =1, 2, 3, and Γ τ, Γ ξ denote the hyperplanes on R 4 : Γ τ = {(τ 1,τ 2,τ 3,τ 4 ) R 4 ; τ 1 + τ 2 + τ 3 + τ 4 =0}, Γ ξ = {(ξ 1,ξ 2,ξ 3,ξ 4 ) R 4 ; ξ 1 + ξ 2 + ξ 3 + ξ 4 =0}, respectively. We split the domain of integration I into ξ 4 1and ξ 4 1.

3566 JUN-ICHI SEGATA The case ξ 4 1. We only prove (3.7) for the case 7/12 <s<3/4. The case s 3/4 is shown in the same manner. It will be convenient to define ξ max ξ med ξ min to be the maximum, median and minimum of ξ 1, ξ 2, ξ 3, respectively. Then (3.8) ξ 3 2 ξ 4 s ξ 1 s ξ 2 s ξ 3 s C ξ 3 2 ξ 4 3/4 ξ 1 s ξ 2 s ξ 3 s C ξ max 2 ξ 4 3/4 ξ med 1/4 ξ min 3s 3/4 ξ max. 1/2 Without loss of generality, we may assume ξ 1 = ξ min, ξ 2 = ξ med and ξ 3 = ξ max.let (3.9) ˆF j,b (τ,ξ)= f j (τ j,ξ j ) τ j +( 1) j, for j =1,, 4. φ ν (ξ j ) b Plugging those inequalities (3.8) into I in (3.7) and applying Lemma 2.1 (2.2), (2.3), (2.4), and Corollary 2.3 (2.7), the integral I restricted to this case is bounded by the Hölder inequality so that (3.10) f 1 (τ 1,ξ 1 ) f 2 (τ 2,ξ 2 ) ξ 3 3 2 f 3 (τ 3,ξ 3 ) ξ 4 3 4 f 4 (τ 4,ξ 4 ) Γ τ Γ ξ τ 1 φ ν (ξ 1 ) b ξ 1 3s 3 4 τ 2 + φ ν (ξ 2 ) b ξ 2 1 4 τ 3 φ ν (ξ 3 ) b τ 4 + φ ν (ξ 4 ) a D x 3s+ 3 4 F1,b (t, x) D 1 4 x F 2,b (t, x) D 3 2 x F 3,b (t, x) D 3 4 x F 4, a (t, x) dtdx R 2 D x 3s+ 3 4 F1,b L 2 x (L t ) D 1 4 x F 2,b L 4 x (L t ) D 3 2 x F 3,b L x (L 2 t ) D 3 4 x F 4, a L 4 x (L 2 t ) 4 f j L 2 τ (L 2 ξ ). Here, we used the fact that 3s +3/4 < 1. The Case ξ 4 1. This case is simpler than the case ξ 4 1. By the same manner as the preceding case, we may assume ξ 1 ξ 2 ξ 3. Then, we easily see that ξ 3 2 ξ 4 s (3.11) ξ 1 s ξ 2 s ξ 3 s C ξ 3 2 ξ 1 s ξ 2 s ξ 3 s C ξ max 2 ξ med 1/4 ξ min 1/4 ξ max. 1/2 Combining Lemma 2.1 (2.2), (2.3) and (3.11), the integral I in this case again is estimated by (3.12) Γ τ f 1 (τ 1,ξ 1 ) f 2 (τ 2,ξ 2 ) ξ 3 3/2 f 3 (τ 3,ξ 3 ) Γ ξ τ 1 φ ν (ξ 1 ) b ξ 1 1/4 τ 2 + φ ν (ξ 2 ) b ξ 2 1/4 τ 3 φ ν (ξ 3 ) b f 4(τ 4,ξ 4 ) Dx 1/4 F 1,b (t, x) D 1/4 R 2 x F 2,b (t, x) Dx 3/2 F 3,b (t, x) F 4,0 (t, x) dtdx Dx 1/4 F 1,b L 4 x (L t ) Dx 1/4 F 2,b L 4 x (L t ) Dx 3/2 F 3,b L x (L 2 t ) F 4,0 L 2 x (L 2 t ) 4 f j L 2 τ (L 2 ξ ). By collecting (3.10) and (3.12), we obtain the desired estimate (3.7).

REMARK ON WELL-POSEDNESS FOR THE 4NLS 3567 Proof of Theorem 1.1. We put r = u 0 H s.nowfort (0, 1), we define B(r) ={u S : u X ν 2Cr}, Φ(u) =ψ(t)w ν (t)u 0 iψ(t) t 0 W ν (t t )ψ T (t )F (t )dt. By similar arguments as in [4], [11] and [12], we have for b, b with 1/2 <b<b < 3/4 and for u B(r), (3.13) Φ(u) X ν C 0 r + C 1 ψ T F X ν C 0 r + C 1 T b b F X ν 1. Combining Proposition 3.1 with 3.2, the right-hand side of (3.13) is bounded by C 0 r + C 1 T b b ( u 3 X ν + u 5 X ν ) C 0r + C 1 T b b (1 + r 2 )r 3. Therefore, if we choose T b b C 0 {(1 + r 2 )r 2 C 1 } 1,thenΦ(u) B(r). Similarly, we can show that Φ is a contraction on B(r) bychoosingt>0sufficiently small. Therefore Banach s Fixed Point Theorem guarantees the existence of a solution in B(r) X ν. Concerning the uniqueness of the solution in the whole of Xν,we refer to section 4 in [3]. Similar to [3], we introduce the norm: u XT =inf { w X w ν : w Xν such that u(t) =w(t), t [ T,T]inH s (R)}. If u u XT =0,wehaveu(t) =u (t) inh s (R) fort [ T,T]. By similar arguments as in [3], we reduce the uniqueness. The persistency of a solution follows directly from the Sobolev embedding Ht b(r; Hs x (R)) C(R; Hs x (R)). Acknowledgments I wish to express my sincere gratitude to Professor Takayoshi Ogawa for several discussions and valuable advice. I would also like to thank Professor Yasuhide Fukumoto for letting one know of the paper [9]. I would also like to thank Professor Hideo Takaoka for help and encouragement. References [1] Ben-Artzi M., Koch H. and Saut J. C., Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. ParisSér. I Math. 330 (2000), 87 92. MR 2001a:35149 [2] Bekiranov D., Ogawa T. and Ponce G., Weak solvability and well posedness of a coupled Schrödinger Korteweg-de Vries equation in the capillary-gravity wave interactions, Proc. Amer. Math. Soc. 125 no.10 (1997), 2907-2919. MR 97m:35238 [3] Bekiranov D., Ogawa T. and Ponce G., Interaction equations for short and long dispersive waves, J. Funct. Anal. 158 (1998), 357-388. MR 99i:35143 [4] Bourgain J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I Schrödinger equations, II The KdV equation, Geom. Funct. Anal. 3 (1993), 107-156, 209-262. MR 95d:35160a [5] Colliander J., Keel M., Staffilani G., Takaoka H. and Tao T., A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal. 34 (2002), 64-86. MR 2004c:35381 [6] Constantin P. and Saut J. C., Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. MR 89d:35150 [7] Fukumoto Y., Three dimensional motion of a vortex filament and its relation to the localized induction hierarchy, Eur. Phys. J. B. 29 (2002), 167 171. [8] Fukumoto Y. and Moffatt H. K., Motion and expansion of a viscous vortex ring. Part I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech. 417 (2000), 1-45. MR 2002g:76049

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