Existence for the α-patch model and the QG sharp front in Sobolev spaces

arXiv:math/0701447v1 [math.AP] 16 Jan 2007 Existence for the α-patch model and the QG sharp front in Sobolev spaces Francisco Gancedo Abstract We consider a family of contour dynamics equations depending on a parameter α with 0 < α ≤1. The vortex patch problem of the 2-D Euler equation is obtained taking α→0, and the case α= 1 corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces. 1 Introduction The 2-D QG equation provides particular solutions of the evolution of the temperature from a general quasi-geostrophic system for atmospheric and oceanic flows. This equation is derived considering small Rossby and Ekman numbers and constant potential vorticity (see [12] for more details). It reads θt(x, t) + u(x, t)· ∇θ(x, t) = 0, x ∈R2, θ(x, 0) = θ0(x).(1) Here θis the temperature of the fluid, the incompressible velocity uis expressed by means of the stream function as follows u=∇⊥ψ= (−∂x2ψ, ∂x1ψ), and the relation between the stream function and the temperature is given by θ=−(−∆)1/2ψ. This system have been considered in frontogenesis, where the dynamics of hot and cold fluids is studied together with the formation and the evolution of fronts (see [4], [5], [8], [11]). From a mathematical point of view, this equation have been presented as a two dimensional model of the 3-D Euler equation due to their strong analogies (see [4]), being the formation of singularities for a regular initial data an open problem (see [4], [6], [7]). Nevertheless the QG equation has global in time weak solutions due to an extra cancellation (see [13]). A few sparse results are known about weak solutions of the 2-D and 3-D Euler equation in its primitive-variable form. 1 An outstanding kind of weak solutions for the QG equation are those in which the temperature takes two different values in complementary domains, modelling the evolution of a sharp front as follows θ(x1, x2, t) = θ1,Ω(t) θ2,R2rΩ(t).(2) In this work we study a problem similar to the 2-D vortex patch problem, where the vorticity of the 2-D Euler equation is given by a characteristic function of a domain, and it is considered the regularity of the free boundary of such domain. For this equation the vorticity satisfies wt(x, t) + u(x, t)· ∇w(x, t) = 0, x ∈R2, w(x, 0) = w0(x),(3) in a weak sense, and the velocity is given by the Biot-Savart law or analogously u=∇⊥ψ, and w= ∆ψ. Chemin [3] proved global-in-time regularity for the free boundary using paradifferential calculus. A simpler proof can be found in [1] due to Bertozzi and Constantin. We point out that in the QG equation, the velocity is determined from the temperature by singular integral operators (see [15]) as follows u= (−R2θ, R1θ),(4) where R1and R2are the Riesz transforms, making the system more singular than (3). Rodrigo [14] proposed the problem of the evolution of a sharp front for the QG equation. He derived the velocity on the free boundary in the normal direction, and proved localexistence and uniqueness for a periodic C∞front, i.e. θ(x1, x2, t) = θ1,{f(x1, t)> x2} θ2,{f(x1, t)≤x2}, with f(x1, t) periodic, using the Nash-Moser iteration. In this paper we study a family of contour dynamics equation given by weak solutions of the following system θt+u· ∇θ= 0, x ∈R2, u=∇⊥ψ, θ =−(−∆)1−α/2ψ, 0< α ≤1, (5) where the active scalar θ(x, t) satisfies (2). We notice that the case α= 0 is the 2-D vortex patch problem, and α= 1 correspond to the sharp front for the QG equation. This system was introduced by C´ordoba, Fontelos, Mancho and Rodrigo in [9], where they present a proof of local-existence for a periodic C∞front, and show evidence of singularities in finite time. The singular scenario is due to two patches collapse point-wise. 2 Here we give a proof of local-existence of the system (5) where the solution satisfies (2), with the boundary ∂Ω(t) given by the curve ∂Ω(t) = {x(γ, t) = (x1(γ, t), x2(γ, t)) : γ∈[−π, π]}, and x(γ, t) belongs to a Sobolev space. In the cases 0 < α < 1 we show uniqueness. It is well-known (see [10] and [14]) that in this kind of contour dynamics equations, the velocity in the tangential direction only moves the particles on the boundary. Therefore we do not alter the shape of the contour if we change the tangential component of the velocity; i.e., we are making a change on the parametrization. In the most singular case, α= 1 or the QG equation, we need to change the velocity in the tangential direction in order to get existence in the Sobolev spaces. We take a tangential velocity in such a way that |∂γx(γ, t)| satisfies |∂γx(γ, t)|2=A(t), and does not depend on γ. We would like to cite the work of Hou, Lowengrub and Shelley [10] in which this idea was used to study a contour dynamics problem. We notice that in order to get a non-singular normal velocity of the curve for 0 < α ≤1 (see [9] and [14]), we need a one to one curve, and parameterized in such a way that |∂γx(γ, t)|2>0. Rigorously, we need that |x(γ, t)−x(γ−η, t)| |η|>0,∀γ, η ∈[−π, π],(6) therefore we give an initial data satisfying this property, and we prove that this condition is satisfied locally in time. We point out the importance to take into account the evolution of this quantity due to the numerical simulations in [9]. Finally, I wish to thank Antonio C´ordoba and my thesis advisor Diego C´ordoba for their strong influence in this work, their advices and suggestions. The author was partially supported by the grants PAC-05-005-2 of the JCLM (Spain) and MTM2005-05980 of the MEC (Spain). 2 The Contour Equation In this section we deduce the family of contour equations in term of the free boundary x(γ, t). We consider the equations given by the system (1), with a velocity satisfying u(x, t) = ∇⊥ψ(x, t),(7) for the stream function it follows θ=−(−∆)1−α/2ψ, (8) and the active scalar fulfills 3 θ(x1, x2, t) = θ1,Ω(t) θ2,R2rΩ(t).(9) The boundary of Ω(t) is given by the curve ∂Ω(t) = {x(γ, t) = (x1(γ, t), x2(γ, t)) : γ∈[−π, π] = T}, with x(γ, t) one to one. Due to the identity (9), we find that ∇⊥θ= (θ1−θ2)∂γx(γ, t)δ(x−x(γ, t)), where δis the Dirac distribution. Using (7) and (8), we got that u=−(−∆)α/2−1∇⊥θ. Due to the integral operators −(−∆)α/2−1are Riesz potentials (see [15]), using the last to identities we obtain that u(x, t) = −Θα 2πZT ∂γx(γ−η, t) |x−x(γ−η, t)|αdη, (10) for x6=x(γ, t), and Θα= (θ1−θ2)Γ(α/2)/21−αΓ(2 −α/2). We notice that for α= 1, if x→x(γ, t) the integral in (10) is divergent. As we have showed before, we are interested in the normal velocity of the systems. Then we have that using the identity (10), and taking the limit as follows u(x, t)·∂⊥ γx(γ, t), x →x(γ, t),(11) we obtain u(x(γ, t), t)·∂⊥ γx(γ, t) = −Θα 2πZT ∂γx(γ−η, t)·∂⊥ γx(γ, t) |x(γ, t)−x(γ−η, t)|αdη. (12) This identity is well defined for 0 < α ≤1 and a one to one curve x(γ, t). Due to the fact that tangential velocity does not change the shape of the boundary, we fix the contour α-patch equations as follows xt(γ, t) = Θα 2πZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|αdη, 0< α ≤1, x(γ, 0) = x0(γ). (13) Seeing the equation (10), we show that the velocity in QG presents a logarithmic divergence in the tangential direction on the boundary. Nevertheless it belongs to Lp(R2) for 1 < p < ∞, and to the bounded mean oscillation space (see [15] for the definition of the BMO space). In QG the velocity is given by (4), and writing the temperature in the following way θ(x, t) = (θ1−θ2)XΩ(t)(x) + θ2, we find that u(x, t) = (θ1−θ2)(−R2(XΩ(t)), R1(XΩ(t))). Using that XΩ(t)∈Lp(R2) for 1 ≤p≤ ∞, we conclude de argument. In particular the energy of the system is conserved due to kukL2(t) = |θ1−θ2| |Ω(t)|1/2, and the area of Ω(t) is constant in time. 4 3 Weak solutions for the α-system In this section we show that if θ(x, t) is defined by (9) and the curve x(γ, t) is convected by the normal velocity (12), then θ(x, t) is a weak solution of the system (5) and conversely. We give the definition of weak solution below. Definition 3.1 The active scalar θis a weak solution of the α-system if for any function ϕ∈C∞ c(R2×(0, T)), we have ZT 0ZR2 θ(x, t)(∂tϕ(x, t) + u(x, t)· ∇ϕ(x, t))dxdt = 0,(14) where the incompressible velocity uis given by (7), and the stream function satisfies (8). Then Proposition 3.2 If θ(x, t)is defined by (9), and the curve x(γ, t)satisfies (6) and (12), then θ(x, t)is a weak solution of the α-system. Furthermore, if θ(x, t)is a weak solution of the α-system given by (9), and x(γ, t)satisfies (6), then x(γ, t)verifies (12). Proof: Let θ(x, t) be a weak solution of the α-system defined by (9). Integrating by parts we have I=ZT 0ZR2 θ(x, t)∂tϕ(x, t)dxdt =θ1ZT 0ZΩ(t) ∂tϕ(x, t)dxdt +θ2ZT 0ZΩ(t)rR2 ∂tϕ(x, t)dxdt =−(θ1−θ2)ZT 0ZT ϕ(x(γ, t), t)xt(γ, t)·∂⊥ γx(γ, t)dγdt. On the other hand, we obtain J=ZT 0ZR2 θ u · ∇ϕ dxdt =θ1ZT 0ZΩ u· ∇ϕ dxdt +θ2ZT 0ZR2rΩ u· ∇ϕ dxdt. Taking Ωε 1(t) = {x∈Ω : dist(x, Ω(t)) ≥ε}, and Ωε 2(t) = {x∈R2rΩ : dist(x, R2rΩ(t)) ≥ε}, we have that Jε→Jif ε→0, where Jεis given by Jε=θ1ZT 0ZΩε 1(t) u· ∇ϕ dxdt +θ2ZT 0ZΩε 2(t) u· ∇ϕ dxdt. Integrating by part in Jε, using that the velocity is divergence free, and taking the limit as in (11), we obtain J= (θ1−θ2)ZT 0ZT ϕ(x(γ, t), t)u(x(γ, t), t)·∂⊥ γx(γ, t)dγdt =−(θ1−θ2)Θα 2πZT 0ZT ϕ(x(γ, t), t)ZT ∂γx(γ−η, t)·∂⊥ γx(γ, t) |x(γ, t)−x(γ−η, t)|αdηdγdt. 5 We have that I+J= 0 using (14), and it follows ZT 0ZT f(γ, t)xt(γ, t)·∂⊥ γx(γ, t) + Θα 2πZT ∂γx(γ−η, t)·∂⊥ γx(γ, t) |x(γ, t)−x(γ−η, t)|αdηdγdt = 0, for f(γ, t) periodic in γ. We find that (12) is satisfied. Following the same arguments it is easy to check that if x(γ, t) satisfies (12), then θis a weak solution given by (9). 4 Local well-posedness for 0< α < 1 In this section we prove existence and uniqueness for the contour equation in the cases 0< α < 1. We denote the Sobolev spaces by Hk(T), with norms kxk2 Hk=kxk2 L2+k∂k γxk2 L2, and the spaces Ck(T) with kxkCk= max j≤kk∂j γxkL∞. We need that the curve satisfies |x(γ, t)−x(γ−η, t)| |η|>0,∀γ, η ∈[−π, π],(15) then we define F(x)(γ, η, t) = |η| |x(γ, t)−x(γ−η, t)|∀γ, η ∈[−π, π],(16) with F(x)(γ, 0, t) = 1 |∂γx(γ, t)|. The main theorem in this section is the following Theorem 4.1 Let x0(γ)∈Hk(T)for k≥3with F(x0)(γ, η)<∞. Then there exists a time T > 0so that there is a unique solution to (13) for 0< α < 1in C1([0, T]; Hk(T)) with x(γ, 0) = x0(γ). Proof: We can choose Θα= 2πwithout loss of generality, obtaining the following equation xt(γ, t) = ZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|αdη, 0< α < 1, x(γ, 0) = x0(γ). (17) We present the proof for k= 3, being analogous for k > 3, using energy estimates (see [2] for more details). We ignore the time dependence to simplify the notation in some terms. Considering the quantity 6 ZT x(γ)·xt(γ)dγ =ZTZT x(γ)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|αdηdγ =−ZTZT x(η)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|αdηdγ =1 2ZTZT (x(γ)−x(η)) ·(∂γx(γ)−∂γx(η)) |x(γ)−x(η)|αdηdγ =1 2(2 −α)ZTZT ∂γ|x(γ)−x(γ−η)|2−αdγdη = 0, (18) we obtain d dtkxkL2(t) = 0.(19) We decompose as follows ZT ∂3 γx(γ)·∂3 γxt(γ)dγ =I1+I2+I3+I4, where I1=ZTZT ∂3 γx(γ)·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|αdηdγ, I2= 3 ZTZT ∂3 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η))∂γ(|x(γ)−x(γ−η)|−α)dηdγ, I3= 3 ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))∂2 γ(|x(γ)−x(γ−η)|−α)dηdγ, I4=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))∂3 γ(|x(γ)−x(γ−η)|−α)dηdγ. Operating as in (18), the term I1becomes I1=1 2ZTZT (∂3 γx(γ)−∂3 γx(γ−η)) ·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|αdηdγ =1 4ZTZT ∂γ|∂3 γx(γ)−∂3 γx(γ−η)|2 |x(γ)−x(γ−η)|αdηdγ =α 4ZTZT |∂3 γx(γ)−∂3 γx(γ−η)|2(x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)) |x(γ)−x(γ−η)|α+2 dηdγ. 7 One finds that I1≤α 4ZTZT |∂3 γx(γ)−∂3 γx(γ−η)|2|∂γx(γ)−∂γx(γ−η)| |x(γ)−x(γ−η)|α+1 dηdγ, and due to the inequality |∂γx(γ)−∂γx(γ−η)||η|−1≤ kxkC2,it follows I1≤α 4kxkC2ZTZT |η|−α|F(x)(γ, η)|1+α|∂3 γx(γ)−∂3 γx(γ−η)|2dηdγ ≤1 2kF(x)k1+α L∞kxkC2ZT |η|−αZT (|∂3 γx(γ)|2+|∂3 γx(γ−η)|2)dγdη ≤ kF(x)k1+α L∞kxkC2k∂3 γxk2 L2ZT |η|−αdη ≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2. (20) As before, we can obtain I2=−6I1, and it yields I2≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2.(21) In order to estimate the term I3, we consider I3=J1+J2+J3, where J1=−3αZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) A(γ, η) |x(γ)−x(γ−η)|α+2 dηdγ, J2=−3α ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|α+2 dηdγ, J3= 3α(2 + α) ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) (B(γ, η))2 |x(γ)−x(γ−η)|α+4 dηdγ, with A(γ, η) = (x(γ)−x(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)), and B(γ, η) = (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)). The identity ∂2 γx(γ)−∂2 γx(γ−η) = ηZ1 0 ∂3 γx(γ+ (s−1)η)ds, (22) yields J1≤3Z1 0ZTZT |η|(|∂2 γx(γ)|+|∂2 γx(γ−η)|)|∂3 γx(γ)||∂3 γx(γ+ (s−1)η)| |x(γ)−x(γ−η)|α+1 dγdηds ≤3kF(x)k1+α L∞kxkC2Z1 0ZT |η|−αZT (|∂3 γx(γ)|2+|∂3 γx(γ+ (s−1)η)|2)dγdηds ≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2. 8 Using (22), we have for J2 J2=−3α Z1 0ZTZT |F(x)(γ, η)|2+α|∂γx(γ)−∂γx(γ−η)|2 η ∂3 γx(γ)·∂3 γx(γ+(s−1)η) |η|αdγdηds ≤3kF(x)k2+α L∞kxk2 C2Z1 0ZT |η|−αZT (|∂3 γx(γ)|2+|∂3 γx(γ+ (s−1)η)|2)dγdηds ≤CαkF(x)k2+α L∞kxk2 C2k∂3 γxk2 L2. The term J3is estimated by J3≤9 Z1 0ZTZT |η||∂γx(γ)−∂γx(γ−η)|2|∂3 γx(γ)||∂3 γx(γ+(s−1)η)| |x(γ)−x(γ−η)|α+2 dγdηds ≤CαkF(x)k2+α L∞kxk2 C2k∂3 γxk2 L2. We get finally I3≤Cα(kF(x)k1+α L∞kxkC2+kF(x)k2+α L∞kxk2 C2)k∂3 γxk2 L2.(23) We decompose the term I4=J4+J5+J6+J7+J8as follows J4=−αZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) C(γ, η) |x(γ)−x(γ−η)|α+2 dηdγ, J5=−3α ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) D(γ, η) |x(γ)−x(γ−η)|α+2 dηdγ, J6= 5α(α+ 2) ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) A(γ, η)B(γ, η) |x(γ)−x(γ−η)|α+4 dηdγ, J7= 5α(α+ 2) ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))B(γ, η)|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|α+4 dηdγ, J8=−2α(α+ 2)(α+ 4) ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) (B(γ, η))3 |x(γ)−x(γ−η)|α+6 dηdγ, with C(γ, η) = (x(γ)−x(γ−η)) ·(∂3 γx(γ)−∂3 γx(γ−η)), D(γ, η) = (∂γx(γ)−∂γx(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)). The most singular term is J4, in such a way that J4≤ kF(x)k1+α L∞kxkC2ZT |η|−αZT |∂3 γx(γ)||∂3 γx(γ)−∂3 γx(γ−η)|dγdη ≤CαkF(x)k1+α L∞kxkC2k∂3 γxk2 L2. For J5, we have J5≤3kF(x)k2+α L∞kxk2 C2ZT |η|−αZT |∂3 γx(γ)||∂2 γx(γ)−∂2 γx(γ−η)|dγdη ≤CαkF(x)k2+α L∞kxk2 C2k∂2 γxkL2k∂3 γxkL2. 9 Theorem 5.1 Let x0(γ)∈Hk(T)for k≥3with F(x0)(γ, η)<∞. Then there exists a time T > 0so that there is a solution to (31) in C1([0, T]; Hk(T)) with x(γ, 0) = x0(γ)and λ(γ, t) given by (36). Proof: Being analogous for k > 3, we give the proof for k= 3. We have showed before that (33) is satisfied if x(γ, t) is a solution of (31). Then we can rewrite λ(γ, t) as follows λ(γ, t) = γ+π 2πA(t)ZT ∂γx(γ, t)·∂γZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|dηdγ −1 A(t)Zγ −π ∂γx(η, t)·∂ηZT ∂γx(η, t)−∂γx(η−ξ, t) |x(η, t)−x(η−ξ, t)|dξdη. (37) We obtain ZT x(γ)·xt(γ)dγ =ZTZT x(γ)·∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ +ZT λ(γ)x(γ)·∂γx(γ)dγ =I1+I2, One finds that I1= 0, since I1=ZTZT x(γ)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|dηdγ =−ZTZT x(η)·∂γx(γ)−∂γx(η) |x(γ)−x(η)|dηdγ =1 2ZTZT (x(γ)−x(η)) ·(∂γx(γ)−∂γx(η)) |x(γ)−x(η)|dηdγ =1 2ZTZT ∂γ|x(γ)−x(γ−η)|dγdη = 0. For the term I2, one obtains that I2≤ kλkL∞kxkL2k∂γxkL2, and kλkL∞≤2 A(t)ZT |∂γx(γ)|∂γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ ≤2 A(t)ZT |∂γx(γ)|ZT |∂2 γx(γ)−∂2 γx(γ−η)| |x(γ)−x(γ−η)|dηdγ +2 A(t)ZT |∂γx(γ)|ZT |∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|2dηdγ =J1+J2. Due to 1/A(t)≤ kF(x)k2 L∞(t), we have J1≤2kF(x)k3 L∞Z1 0ZTZT |∂3 γx(γ+ (s−1)η)||∂γx(γ)|dγdηds ≤2kF(x)k3 L∞kxk2 H3, and J2≤2kF(x)k4 L∞kxkC1Z1 0ZTZT |∂2 γx(γ+ (s−1)η)|2dγdηds ≤2kF(x)k4 L∞kxk3 H3. 16 Therefore we obtain that d dtkxk2 L2(t)≤CkF(x)k4 L∞(t)kxk5 H3(t).(38) We decompose as follows ZT ∂3 γx(γ)·∂3 γxt(γ)dγ =ZT ∂3 γx(γ)·∂3 γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ +ZT ∂3 γx(γ)·∂3 γ(λ(γ)∂γx(γ))dγ =I3+I4. We take I3=J3+J4+J5+J6where J3=ZTZT ∂3 γx(γ)·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|dηdγ, J4= 3 ZTZT ∂3 γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η))∂γ(|x(γ)−x(γ−η)|−1)dηdγ, J5= 3 ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))∂2 γ(|x(γ)−x(γ−η)|−1)dηdγ, J6=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))∂3 γ(|x(γ)−x(γ−η)|−1)dηdγ. The term J3can be written as J3=1 2ZTZT (∂3 γx(γ)−∂3 γx(γ−η)) ·∂4 γx(γ)−∂4 γx(γ−η) |x(γ)−x(γ−η)|dηdγ =1 4ZTZT ∂γ|∂3 γx(γ)−∂3 γx(γ−η)|2 |x(γ)−x(γ−η)|dηdγ =1 4ZTZT |∂3 γx(γ)−∂3 γx(γ−η)|2(x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)) |x(γ)−x(γ−η)|3dηdγ. If we define B(γ, η) = (x(γ)−x(γ−η)) ·(∂γx(γ)−∂γx(γ−η)), due to (32), we obtain that J3=1 4ZTZT |F(x)(γ, η)|3|∂3 γx(γ)−∂3 γx(γ−η)|2B(γ, η)η−2−∂γx(γ)·∂2 γx(γ) |η|dηdγ. 17 Using that  B(γ, η)η−2−∂γx(γ)·∂2 γx(γ) η≤2kxk2 C2,1 2|η|−1/2, we find J3≤ kF(x)k3 L∞kxk2 C2,1 2ZT |η|−1/2ZT (|∂3 γx(γ)|2+|∂3 γx(γ−η)|2)dγdη ≤CkF(x)k3 L∞kxk2 C2,1 2k∂3 γxk2 L2 ≤CkF(x)k3 L∞kxk4 H3. (39) We obtain that J4=−6J3, and it yields J4≤CkF(x)k3 L∞kxk4 H3.(40) In order to estimate the term J5, we consider J5=K1+K2+K3, where K1=−3ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) C(γ, η) |x(γ)−x(γ−η)|3dηdγ, K2=−3ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|3dηdγ, K3= 9 ZTZT ∂3 γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η)) (B(γ, η))2 |x(γ)−x(γ−η)|5dηdγ, with C(γ, η) = (x(γ)−x(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)). The inequality |∂2 γx(γ)−∂2 γx(γ−η)||η|−1/2≤ kxkC2,1 2,(41) yields K1≤3kF(x)k2 L∞kxkC2,1 2Z1 0ZT |η|−1/2ZT |∂3 γx(γ)||∂3 γx(γ+ (s−1)η)|dγdηds ≤CkF(x)k2 L∞kxk3 H3. As before, we have for K2that K2≤CkF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤CkF(x)k3 L∞kxk4 H3. The term K3is estimated by K3≤CkF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤CkF(x)k3 L∞kxk4 H3. We get finally J5≤CkF(x)k3 L∞kxk4 H3.(42) 18 We decompose the term J6=K4+K5+K6+K7+K8as follows K4=−ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) D(γ, η) |x(γ)−x(γ−η)|3dηdγ, K5=−3ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) E(γ, η) |x(γ)−x(γ−η)|3dηdγ, K6= 15 ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) B(γ, η)C(γ, η) |x(γ)−x(γ−η)|5dηdγ, K7= 15 ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))B(γ, η)|∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|5dηdγ, K8=−30 ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) (B(γ, η))3 |x(γ)−x(γ−η)|7dηdγ, with D(γ, η) = (x(γ)−x(γ−η)) ·(∂3 γx(γ)−∂3 γx(γ−η)), E(γ, η) = (∂γx(γ)−∂γx(γ−η)) ·(∂2 γx(γ)−∂2 γx(γ−η)). We obtain K5≤3kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤3kF(x)k3 L∞kxk4 H3, K6≤15kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤15kF(x)k3 L∞kxk4 H3, K7≤15kF(x)k4 L∞kxk3 C2k∂3 γxkL2k∂2 γxkL2≤15kF(x)k4 L∞kxk5 H3, and K8≤30kF(x)k4 L∞kxk3 C2k∂3 γxkL2k∂2 γxkL2≤30kF(x)k4 L∞kxk5 H3. For the most singular term, we have K4=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))η ∂γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η)) −D(γ, η) |x(γ)−x(γ−η)|3dηdγ −ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))η ∂γx(γ)·(∂3 γx(γ)−∂3 γx(γ−η)) |x(γ)−x(γ−η)|3dηdγ =L1+L2. One finds that L1≤ kF(x)k3 L∞kxk2 C2ZTZT |∂3 γx(γ)||∂3 γx(γ)−∂3 γx(γ−η)|dγdη ≤CkF(x)k3 L∞kxk4 H3. The term L2is decomposed, and it yields 19 L2=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η))η(∂γx(γ)−∂γx(γ−η)) ·∂3 γx(γ−η) |x(γ)−x(γ−η)|3dηdγ −ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) η∂γx(γ)·∂3 γx(γ)−∂γx(γ−η)·∂3 γx(γ−η) |x(γ)−x(γ−η)|3dηdγ =M1+M2. We estimate the term M1as follows M1≤ kF(x)k3 L∞kxk2 C2ZTZT |∂3 γx(γ)||∂3 γx(γ−η)|dγdη ≤ kF(x)k3 L∞kxk4 H3. Taking the derivative in (32), we find that ∂γx(γ)·∂3 γx(γ) = −|∂2 γx(γ)|2, and we rewrite M2=ZTZT ∂3 γx(γ)·(∂γx(γ)−∂γx(γ−η)) η|∂2 γx(γ)|2− |∂2 γx(γ−η)|2 |x(γ)−x(γ−η)|3dηdγ. The inequality ||∂2 γx(γ)|2− |∂2 γx(γ−η)|2| ≤ 2kxkC2|η|Z1 0 |∂3 γx(γ+ (s−1)η)|ds, (43) yields M2≤2kF(x)k3 L∞kxk2 C2Z1 0ZTZT |∂3 γx(γ)||∂3 γx(γ+ (s−1)η)|dγdηds ≤CkF(x)k3 L∞kxk4 H3. We recall that K4=L1+L2=L1+M1+M2≤CkF(x)k3 L∞kxk4 H3,and finally it follows J6≤CkF(x)k4 L∞kxk5 H3.(44) Due to (39), (40), (42) and (44), we obtain I3≤CkF(x)k4 L∞kxk5 H3.(45) We take I4=J7+J8+J9+J10, where J7=ZT λ(γ)∂3 γx(γ)·∂4 γx(γ)dγ, J8= 3 ZT ∂γλ(γ)|∂3 γx(γ)|2dγ, J9= 3 ZT ∂2 γλ(γ)∂3 γx(γ)·∂2 γx(γ)dγ, J10 =ZT ∂3 γλ(γ)∂3 γx(γ)·∂γx(γ)dγ. We integrate by parts in the term J7, and we get 20 J7=−1 2ZT ∂γλ(γ)|∂3 γx(γ)|2dγ ≤1 2k∂γλkL∞k∂3 γxk2 L2. Using (37), we find that ∂γλ(γ, t) = 1 2πA(t)ZT ∂γx(γ, t)·∂γZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|dηdγ −1 A(t)∂γx(γ, t)·∂γZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|dη =K9+K10. (46) The term K9is estimated as J1and J2, obtaining K9≤ kF(x)k4 L∞kxk3 H3. We have for K10 that K10 ≤kxkC2 A(t)ZT|∂2 γx(γ, t)−∂2 γx(γ−η, t)| |x(γ, t)−x(γ−η, t)|+|∂γx(γ, t)−∂γx(γ−η, t)|2 |x(γ, t)−x(γ−η, t)|2dη ≤2kF(x)k4 L∞kxk3 C2,1 2ZT |η|−1/2dη ≤CkF(x)k4 L∞kxk3 H3, and therefore J7≤CkF(x)k4 L∞kxk5 H3.(47) Due to the identity J8=−6J7, one finds that J8≤CkF(x)k4 L∞kxk5 H3.(48) Using that ∂2 γλ(γ, t) = −1 A(t)∂2 γx(γ, t)·∂γZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|dη −1 A(t)∂γx(γ, t)·∂2 γZT ∂γx(γ, t)−∂γx(γ−η, t) |x(γ, t)−x(γ−η, t)|dη, one gets J9=−1 A(t)ZT ∂3 γx(γ)·∂2 γx(γ)∂2 γx(γ)·∂γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ −1 A(t)ZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·∂2 γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ =L3+L4. 21 Therefore L3≤kxk2 C2 A(t)ZTZT |∂3 γx(γ)||∂2 γx(γ, t)−∂2 γx(γ−η, t)| |x(γ, t)−x(γ−η, t)|+|∂γx(γ, t)−∂γx(γ−η, t)|2 |x(γ, t)−x(γ−η, t)|2dηdγ ≤ kF(x)k4 L∞kxk3 C2Z1 0ZTZT |∂3 γx(γ)|(|∂3 γx(γ+ (t−1)η)|+|∂2 γx(γ+ (t−1)η)|)dγdηds ≤CkF(x)k4 L∞kxk5 H3. Moreover L4=−1 A(t)ZTZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·∂3 γx(γ)−∂3 γx(γ−η) |x(γ)−x(γ−η)|dηdγ +2 A(t)ZTZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·(∂2 γx(γ)−∂2 γx(γ−η))B(γ, η) |x(γ)−x(γ−η)|3dηdγ −1 A(t)ZTZT ∂3 γx(γ)·∂2 γx(γ)∂γx(γ)·(∂γx(γ)−∂γx(γ−η))∂2 γ(|x(γ)−x(γ−η)|−1)dηdγ =M3+M4+M5. The terms M4and M5are estimated as before, and we obtain M4+M5≤CkF(x)k5 L∞kxk6 H3. The most singular term is M3, but we find that M3=1 A(t)ZTZT ∂3 γx(γ)·∂2 γx(γ)∂3 γx(γ−η)·∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ −1 A(t)ZTZT ∂3 γx(γ)·∂2 γx(γ)∂3 γx(γ)·∂γx(γ)−∂3 γx(γ−η)·∂γx(γ−η) |x(γ)−x(γ−η)|dηdγ =N1+N2. We obtain N1≤ kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤ kF(x)k3 L∞kxk4 H3, and using (32) N2=1 A(t)ZTZT ∂3 γx(γ)·∂2 γx(γ)|∂2 γx(γ)|2− |∂2 γx(γ−η)|2 |x(γ)−x(γ−η)|dηdγ. Due to (43), we conclude that N2≤2kF(x)k3 L∞kxk2 C2k∂3 γxk2 L2≤2kF(x)k3 L∞kxk4 H3. We have J9=L3+L4=L3+M3+M4+M5=L3+N1+N2+M4+M5, and therefore J9≤ kF(x)k5 L∞kxk6 H3.(49) 22 The identity (32) yields J10 =−ZT ∂3 γλ(γ)|∂2 γx(γ)|2dγ = 2 ZT ∂2 γλ(γ)∂3 γx(γ)·∂2 γx(γ)dγ =2 3J9, and therefore J10 ≤ kF(x)k5 L∞kxk6 H3.(50) Due to the inequalities (47), (48), (49), and (50), we get I4≤CkF(x)k5 L∞kxk6 H3. Using (45) and the last estimate, we have d dtk∂3 γxk2 L2(t)≤CkF(x)k5 L∞(t)kxk6 H3(t). This inequality and (38) bound the evolution of the Sobolev norms of the curve as follows d dtkxkH3(t)≤CkF(x)k5 L∞(t)kxk5 H3(t).(51) We continue the argument considering the evolution of the quantity kF(x)kL∞(t). Taking p > 2, it yields d dtkF(x)kp Lp(t)≤pZTZT|η| |x(γ, t)−x(γ−η, t)|p+1 |xt(γ, t)−xt(γ−η, t)| |η|dγdη. We have xt(γ)−xt(γ−η) = ZT (∂γx(γ)−∂γx(γ−ξ) |x(γ)−x(γ−ξ)|−∂γx(γ)−∂γx(γ−ξ) |x(γ−η)−x(γ−η−ξ)|)dξ +ZT ∂γx(γ)−∂γx(γ−η) + ∂γx(γ−η−ξ)−∂γx(γ−ξ) |x(γ−η)−x(γ−η−ξ)|dξ + (λ(γ)−λ(γ−η))∂γx(γ) + λ(γ−η)(∂γx(γ)−∂γx(γ−η)) =I5+I6+I7+I8. The term I5yields I5≤ZT |∂γx(γ)−∂γx(γ−ξ)||x(γ)−x(γ−ξ)| − |x(γ−η)−x(γ−η−ξ)| |x(γ)−x(γ−ξ)||x(γ−η)−x(γ−η−ξ)|dξ ≤ kF(x)k2 L∞kxkC2ZT |ξ|−1|x(γ)−x(γ−η)−(x(γ−ξ)−x(γ−η−ξ))|dξ ≤ kF(x)k2 L∞kxkC2|η|Z1 0ZT∂γx(γ+ (s−1)η)−∂γx(γ+ (s−1)η−ξ) |ξ|dξds ≤2πkF(x)k2 L∞kxk2 C2|η|. 23 For I6we take I6≤ kF(x)kL∞|η|Z1 0ZT∂2 γx(γ+ (s−1)η)−∂2 γx(γ+ (s−1)η−ξ) |ξ|dξds ≤ kF(x)kL∞kxkC2,1 2|η|Z1 0ZT |ξ|−1/2dξds ≤CkF(x)kL∞kxkC2,1 2|η| We have for I7 I7≤2kxkC2 A(t)|η|max γ|∂γx(γ)||∂γZT ∂γx(γ)−∂γx(γ−η) |x(γ)−x(γ−η)|dη| ≤2kF(x)k2 L∞kxk2 C2|η|max γZT |∂2 γx(γ)−∂2 γx(γ−η)| |x(γ)−x(γ−η)|dη+ZT |∂γx(γ)−∂γx(γ−η)|2 |x(γ)−x(γ−η)|2dη ≤4kF(x)k4 L∞kxk4 H3|η|. Estimating kλkL∞as before, easily we get I8≤ kλkL∞kxkC2|η| ≤ 4kF(x)k4 L∞kxk4 H3|η|. The last four estimates show that d dtkF(x)kLp(t)≤Ckxk4 H3(t)kF(x)k5 L∞(t)kF(x)kLp(t), by integrating in time and taking p→ ∞, we obtain kF(x)kL∞(t+h)≤ kF(x)kL∞(t)exp CZt+h t kxk4 H3(s)kF(x)k5 L∞(s)ds. As in the previous section, it follows d dtkF(x)kL∞(t)≤Ckxk4 H3(t)kF(x)k6 L∞(t). Then, due to (51) and the above estimate, we find finally that d dt(kxkH3(t) + kF(x)kL∞(t)) ≤C(kxkH3(t) + kF(x)kL∞(t))10. Integrating, we have kxkH3(t) + kF(x)kL∞(t)≤kx0kH3+kF(x0)kL∞ 1−tCkx0kH3+kF(x0)kL∞91 9 , 24 where Cis a constant. We have used the equality (32) to obtain the a priori estimates. In order to get the solution of (31), we have to choose an appropriate regularized problem preserving (32). We propose the system xε,δ t(γ, t) = φε∗ZT ∂γ(φε∗xε,δ(γ, t)−φε∗xε,δ(γ−η, t)) |xε,δ(γ, t)−xε,δ(γ−η, t)|+δdη +λε,δ(γ, t)∂γxε,δ(γ, t), xε,δ(γ, 0) = x0(γ), (52) with λε,δ(γ, t) = γ+π 2πZT ∂γxε,δ(γ, t) |∂γxε,δ(γ, t)|2·∂γφε∗ZT ∂γ(φε∗xε,δ(γ, t)−φε∗xε,δ(γ−η, t)) |xε,δ(γ, t)−xε,δ(γ−η, t)|+δdηdγ −Zγ −π ∂γxε,δ(η, t) |∂γxε,δ(η, t)|2·∂ηφε∗ZT ∂γ(φε∗xε,δ(η, t)−φε∗xε,δ(η−ξ, t)) |xε,δ(η, t)−xε,δ(η−ξ, t)|+δdξdη. We can obtain energy estimates of the system (52) depending on εand δ, but without using (32), and therefore we obtain existence of (52). As long as the solution exists, we have that ∂γxε,δ(γ, t)·∂2 γxε,δ(γ, t) = 0. Using this property of the solution, we obtain energy estimates that depend only on δ, and taking ε→0 we get a solution of the following equation xδ t(γ, t) = ZT ∂γxδ(γ, t)−∂γxδ(γ−η, t)) |xδ(γ, t)−xδ(γ−η, t)|+δdη +λδ(γ, t)∂γxδ(γ, t), xδ(γ, 0) = x0(γ), (53) with λδ(γ, t) = γ+π 2πZT ∂γxδ(γ, t) |∂γxδ(γ, t)|2·∂γZT ∂γxδ(γ, t)−∂γxδ(γ−η, t) |xδ(γ, t)−xδ(γ−η, t)|+δdηdγ −Zγ −π ∂γxδ(η, t) |∂γxδ(η, t)|2·∂ηZT ∂γxδ(η, t)−∂γxδ(η−ξ, t)) |xδ(η, t)−xδ(η−ξ, t)|+δdξdη. Again we have that the solutions of this system satisfy ∂γxδ(γ, t)·∂2 γxδ(γ, t) = 0, and taking advantage of this, we find energy estimates independent of δ. If we tend δto 0, we conclude the existence result. References [1] A. L. Bertozzi and P. Constantin. Global regularity for vortex patches. Comm. Math. Phys. 152 (1): 19–28, 1993. 25