Adiabatic Approximation for the Motion of Ginzburg-Landau Vortex Filaments

Abstract

In this paper, we consider the concentration property of solutions to the dispersive Ginzburg-Landau (or Gross-Pitaevskii) equation in three dimensions. On a spatial domain, it has long been conjectured that such a solution concentrates near some curve evolving according to the binormal curvature flow, and conversely, that a curve moving this way can be realized in a suitable sense by some solution to the dispersive Ginzburg-Landau equation. Some partial results are known with rather strong symmetry assumptions. Our main theorems here provide affirmative answer to both conjectures under certain small curvature assumption. The results are valid for small but fixed material parameter in the equation, in contrast to the general practice to take this parameter to its zero limit. The advantage is that we can retain precise description of the vortex filament structure. The results hold on a long but finite time interval, depending on the curvature assumption.

Appendix A Properties of the Fréchet Derivatives

In this section we consider the Fréchet derivatives of the immersion f defined in (22).

1.1 Basic variational calculus

First, we recall some basic elements of variational derivatives that we used repeatedly. For details, see for instance [28, Appendix C]

Fréchet derivative. Let $X,\,Y$ be two Banach spaces, U be an open set in X. For a map $g:U\subset X\rightarrow Y$ and a vector $u\in U$, the Fréchet derivative dg(u) is a linear map from $X\rightarrow Y$ such that $g(u+v)-g(u)-dg(u)v=o(\left\Vert v\right\Vert _X)$ for every $v\in X$. If dg(u) exists at u, then it is unique. If dg(u) exists for every $u\in U$, and the map $u\mapsto dg(u)$ is continuous from U to the space of linear operators L(X, Y), then we we say g is $C^1$ on U. In this case, dg(u) is uniquely given by

$$\begin{aligned} v\mapsto \tfrac{dg(u+tv)}{dt}\vert _{t=0}\quad (v\in X). \end{aligned}$$

Iteratively, we can define higher order derivatives this way.

Gradient and Hessian. If X is a Hilbert space over scalar field Y, then by Riesz representation, we can identify dg(u) as an element in X, denoted $g'(u)$. The vector $g'(u)$ is called the X-gradient of g. Similarly, we denote $g''(u)$ the second-order Fréchet derivative $d^2g(u)$. If g is $C^2$, then $g''$ can be identified as a symmetric linear operator uniquely determined by the relation

$$\begin{aligned} \langle g''(u)v,\,w\rangle = \tfrac{\partial {^{2}}g(u+tv+sw)}{\partial {t^{}}\partial {s^{}}} \vert _{s=t=0}\quad (v,w\in X). \end{aligned}$$

Expansion. Let X be a Hilbert space over scalar field Y. Suppose g is $C^2$ on $U\subset X$. Define a scalar function $\phi (t):=g(u+tv)$ for vectors v, w such that $v+tw\in U$ for every $0\le t\le 1$. Then the elementary Taylor expansion at $\phi (1)$ gives

$$\begin{aligned} g(v+w)=g(v)+\langle g'(v),\,w\rangle +\frac{1}{2}\langle g''(v)w,\,w\rangle +o(\left\Vert w\right\Vert ^2). \end{aligned}$$

Here we have used the definition of $g'$ and $g''$ from the last paragraph.

Composition. Let $\Omega \subset \mathbb {R}^d$ be a bounded domain with smooth boundary. Fix $r>d/2,\,f\in C^{r+1}(\mathbb {R}^n)$. For $u:\Omega \rightarrow \mathbb {R}^n$, define a map $g:u\mapsto f\circ u$. Then $g:H^r(\Omega )\rightarrow H^r(\Omega )$, and is $C^1$. The Fréchet derivative is given by $v\mapsto \nabla f\cdot v$.

1.2 Various uniform estimates

In this section we assume $\epsilon \ll 1$ in (1). For two complex numbers, we use the real inner product $\langle u,\,v\rangle =\mathfrak {R}(\bar{u}v)$.

Fix some $\alpha >0$. Using the definition of the immersion f in Sect. 2.2, for $\sigma =(\lambda ,\gamma )\in \Sigma _{\epsilon ^\alpha }$ and $(\mu ,\xi )\in Y^k$, we compute the Freéchet derivative of f as

$$\begin{aligned} df(\sigma )(\mu ,\xi )=e^{i\lambda }(i\mu \psi _\gamma -\nabla _x\psi _\gamma \cdot \xi ). \end{aligned}$$

(A1)

This is uniformly bounded in $\sigma$ as an operator from $Y^0\rightarrow X^0$, since

$$\begin{aligned} \begin{aligned} \left\Vert df(\sigma )(\mu ,\xi )\right\Vert _{X^0}&\le \mu \left\Vert \psi _\gamma \right\Vert _{X^0}+\left\Vert \nabla _x\psi _\gamma \cdot \xi \right\Vert _{X^0}\\&\le \mu \left( \left\Vert \psi ^{(1)}\right\Vert _{L^2(\omega )}+O(\epsilon ^{2+\alpha /2})\right) \\&\qquad +\left( \left\Vert \nabla \psi ^{(1)}\right\Vert _{L^2(\omega )}+O(\epsilon ^{1+\alpha /2})\right) \left\Vert \xi \right\Vert _{C^0}\\&\le C(\Omega )\left|\log \epsilon \right|^{1/2}\left\Vert \sigma \right\Vert _{Y^0}. \end{aligned}\nonumber \\ \end{aligned}$$

Here we have used (14) and (25). Using (A1) and (39), one can get a uniform estimate for $d_\sigma df(\sigma ):Y^0\rightarrow L(Y^0, X^0).$

Write an element in $Y^k$ as $\sigma =([\sigma ]_\lambda ,[\sigma ]_\xi )$. The adjoint operator $df(\sigma )^*$ is determined by the relation

$$\begin{aligned} \begin{aligned}&\langle df(\sigma )(\mu ,\xi ),\,\phi \rangle \\&\quad =\int _x\int _z\langle e^{i\lambda }(i\mu \psi _\gamma -\nabla _x\psi _\gamma )\cdot \xi ,\,\phi \rangle \\&\quad =\mu [df(\sigma )^*\phi ]_\lambda +\int _z\xi \cdot [df(\sigma )^*\phi ]_\gamma \quad (\phi \in X^0). \end{aligned} \end{aligned}$$

Here and in the remaining of this section, it is understood that various integrals are taken over $(x,z)\in \omega \times I=\Omega$.

By Fubini’s Theorem and the identity $\langle v\cdot w,\, u\rangle =\langle u,\,v\rangle \cdot w$ for real vector w, the above relation implies

$$\begin{aligned} df(\sigma )^*\phi =\left( \int _x\int _z\langle \phi ,\,ie^{i\lambda }\psi _\gamma \rangle , -\int _x\langle {e^{i\lambda }\nabla _x\psi _\gamma (x,\cdot )},\,\phi (x,\cdot )\rangle \right) . \end{aligned}$$

(A2)

This adjoint operator is also uniformly bounded in $\sigma$ with $\left\Vert df(\sigma )^*\right\Vert _{X^0\rightarrow Y^0} \le C\left|\log \epsilon \right|^{1/2}.$ Moreover, by Sobolev embedding, $df(\sigma )^*$ maps $X^2$ into $Y^0$. Using (A2) and (39), one can get a uniform estimate for $d_\sigma df(\sigma )^*:Y^0\rightarrow L(X^0, Y^0).$

The operator $\mathcal {J}_\sigma$ is defined in (26). It induces a symplectic form w.r.t. the inner product (19) on the tangent bundle $T\Sigma$, since

$$\begin{aligned} \langle \mathcal {J}_\sigma \chi ,\,\chi \rangle =\langle g_\sigma ^*J^{-1} g_\sigma \chi ,\,\chi \rangle =\langle J^{-1}g_\sigma \chi ,\,g_\sigma \chi \rangle =0\quad (\chi \in Y^0). \end{aligned}$$

Using (A1)–(A2), we can compute $\mathcal {J}_\sigma$ explicitly as

$$\begin{aligned} \begin{aligned}&{[}\mathcal {J}_\sigma (\mu ,\xi )]_\lambda =-\int _x\int _z\langle \nabla _x\psi _\gamma \cdot \xi ,\,\psi _\gamma \rangle ,\\&{[}\mathcal {J}_\sigma (\mu ,\xi )]_\gamma =\mu \int _x \langle \nabla _x\psi _\gamma (x,\cdot ),\,\psi _\gamma (x,\cdot )\rangle +\int _x \langle \nabla _x\psi _\gamma (x,\cdot )\cdot J\xi (\cdot ),\,\nabla _x\psi _\gamma (x,\cdot )\rangle . \end{aligned}\nonumber \\ \end{aligned}$$

(A3)

Here we have used the identity that for any complex-valued $C^1$ function $\phi$ and vector field $\chi$ in $\mathbb {R}^2$, by the Cauchy-Riemann equation,

$$\begin{aligned} -i\nabla \phi \cdot \chi =\nabla {\phi }\cdot J\chi . \end{aligned}$$

One can also write $\mathcal {J}_\sigma =\mathcal {J}_\sigma (\mu ,\xi (z))$ as the multiplication operator by the matrix $B_{ij}$, where

$$\begin{aligned} \begin{aligned}&B_{ii}=0\quad (i=1,2,3),\\&B_{12}=-\int _x\int _z\langle \tfrac{\partial \psi _\gamma }{\partial x_1}(x,z),\,\psi _\gamma (x,z)\rangle ,\\&B_{13}=-\int _x\int _z\langle \tfrac{\partial \psi _\gamma }{\partial x_2}(x,z),\,\psi _\gamma (x,z)\rangle ,\\&B_{21}=\int _x\langle \tfrac{\partial \psi _\gamma }{\partial x_1}(x,z),\,\psi _\gamma (x,z)\rangle ,\\&B_{23}=\int _x\langle \tfrac{\partial \psi _\gamma }{\partial x_1}(x,z),\,\tfrac{\partial \psi _\gamma }{\partial x_1}(x,z)\rangle ,\\&B_{31}=\int _x\langle \tfrac{\partial \psi _\gamma }{\partial x_2}(x,z),\,\psi _\gamma (x,z)\rangle ,\\&B_{32}=-\int _x\langle \tfrac{\partial \psi _\gamma }{\partial x_2}(x,z),\,\tfrac{\partial \psi _\gamma }{\partial x_2}(x,z)\rangle . \end{aligned} \end{aligned}$$

(A4)

Here we have used the Cauchy-Riemann equation to eliminate certain cross-terms of the form $\langle \tfrac{\partial \psi _\gamma }{\partial x_1},\,\tfrac{\partial \psi _\gamma }{\partial x_2}\rangle$.

Using (A4), the fact $\left\Vert \nabla \psi ^{(1)}\right\Vert _{L^2(\omega )}\sim C(\omega )\left|\log \epsilon \right|^{1/2}$ (see (14) and [24, Chap. V.1]), and the asymptotics (10) for $\psi ^{(1)}$, one can check that $\mathcal {J}_\sigma$ is invertible and satisfies the uniform estimates

$$\begin{aligned} \left\Vert \mathcal {J}_\sigma \right\Vert _{Y^k\rightarrow Y^k}&\le C(\omega )\left|\log \epsilon \right|^2, \end{aligned}$$

(A5)

$$\begin{aligned} \left\Vert \mathcal {J}_\sigma ^{-1}\right\Vert _{Y^k\rightarrow Y^k}&\le C(\omega )\left|\log \epsilon \right|^{-2} \end{aligned}$$

(A6)

for every $k\in \mathbb {N}$.