Existence and Uniqueness of Local Regular Solution to the Schrödinger Flow from a Bounded Domain in ${\mathbb {R}}^3$ into ${\mathbb {S}}^2$

Abstract

In this paper, we show the existence and uniqueness of local regular solutions to the initial-Neumann boundary value problem of the Schrödinger flow from a smooth bounded domain $\Omega \subset \mathbb {R}^3$ into $\mathbb {S}^2$(namely Landau–Lifshitz equation without dissipation). The proof is built on a parabolic perturbation method, an intrinsic geometric energy argument, the symmetric (algebraic) properties of ${\mathbb {S}}^2$ and some observations on the behaviors of some geometric quantities on the boundary of the domain manifold.It is based on methods from Ding and Wang (one of the authors of this paper) for the Schrödinger flows of maps from a closed Riemannian manifold into a Kähler manifold as well as on methods by Carbou and Jizzini for solutions of the Landau–Lifshitz equation.

Appendices

Appendix A Locally Regular Estimates of $u_\varepsilon$

In this section, we establish the regular estimates of the solution $v:\Omega \times [0,T]\rightarrow \mathbb {R}^3$ to the following uniform parabolic equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial v}{\partial t}-\varepsilon \Delta v-u\times \Delta v=f(x, t), \quad (x,t)\in \Omega \times [0,T],\\ v(x,0)=v_0:\Omega \rightarrow \mathbb {R}^3,\quad \frac{\partial v}{\partial \nu }|_{\partial \Omega \times [0, T]}=0. \end{array}\right. } \end{aligned}$$

(A1)

where

$$\begin{aligned} u\in L^\infty ([0,T], H^3(\Omega ))\cap C^0(\Omega \times [0,T]) \end{aligned}$$

(A2)

and

$$\begin{aligned} f(x,t)\in L^{2}([0, T], H^1(\Omega )). \end{aligned}$$

(A3)

Our main result on locally regular estimates of solution v to the above equation (A1) is as follows.

Theorem A.1

Let $v\in W^{2,1}_{2}(\Omega \times [0,T])$ is a strong solution to (A1) satisfying conditions (A2) and (A3). Then $v\in L^\infty _{loc} ((0,T],H^2(\Omega ))\cap L^2_{loc}((0,T],H^3(\Omega ))$. Moreover, for any $\delta >0$, there exists a positive constant $C(\delta )$ depending on $\Vert u\Vert _{L^\infty ([0,T], H^3(\Omega ))}$ such that there holds

$$\begin{aligned}&\Vert v\Vert _{L^2([\delta ,T],H^2(\Omega ))}+\Vert \frac{\partial v}{\partial t}\Vert _{L^{2}([\delta ,T],L^2(\Omega ))}\\&\quad \le C(\delta )\left( \Vert v\Vert _{L^2([0,T]\times \Omega )}+\Vert f\Vert _{L^2(\Omega \times [0,T])}\right) \end{aligned}$$

and

$$\begin{aligned}&\Vert v\Vert _{L^2([\delta ,T],H^3(\Omega ))}+\Vert \frac{\partial v}{\partial t}\Vert _{L^{2}([\delta ,T],H^1(\Omega ))}\\&\quad \le C(\delta )\left( \Vert v\Vert _{L^2([0,T], H^1(\Omega ))}+\Vert f\Vert _{L^2([0,T], H^1(\Omega ))}\right) . \end{aligned}$$

Proof

Let $\eta (t)$ be a smooth cut-off function such that $\text{ supp }\,\eta \subset (0,T]$ and $\eta \equiv 1$ on $[\delta ,T]$ for any $\delta >0$. Then $\eta v$ is a strong solution of the following equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial \omega }{\partial t}-\varepsilon \Delta \omega -u\times \Delta \omega =\tilde{f},\quad &{}\text {(x,t)}\in \Omega \times [0,T],\\ \frac{\partial \omega }{\partial \nu }|_{\partial \Omega \times [0,T]}=0,\quad \omega (x,0)=0,\quad &{}\text {x}\in \Omega .\\ \end{array}\right. } \end{aligned}$$

(A4)

Here

$$\begin{aligned} \tilde{f}=\eta f+\frac{\partial \eta }{\partial t}v, \end{aligned}$$

it is easy to see that the assumptions in Theorem A.1 imply

$$\begin{aligned} \tilde{f}\in L^2([0,T], H^{1}(\Omega )). \end{aligned}$$

By the Galerkin approximation method, we claim that there exists a solution $w\in L^\infty ([0,T],H^2(\Omega ))\cap L^2([0,T],H^3(\Omega ))$ to the above equation (A4). Namely, we consider the following Galerkin approximation equation to (A4)

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial \omega ^n}{\partial t}-\varepsilon \Delta \omega ^n -P_n(u\times \Delta \omega ^n)=P_n(\tilde{f}),\quad &{}\text {(x,t)}\in \Omega \times [0,T],\\ \omega ^n(x,0)=0,\quad &{}\text {x}\in \Omega ,\\ \end{array}\right. } \end{aligned}$$

(A5)

where the Galerkin projection $P_n$ is defined in Sect.  2. By the assumptions satisfied by u and $\tilde{f}$, one can show there exists a unique solution $w^n(x,t)=\sum _{i=1}^{n}g_i(t)f_i(x)$ to (A5) on $\Omega \times [0,T]$ (cf. [9]).

Then by taking $w^n$, $\Delta w^n$ and $\Delta ^2 w^n$ as test functions to (A5), a simple calculation shows

$$\begin{aligned}&\frac{\partial }{\partial t}\int _{\Omega }|\omega ^n|^2dx+\varepsilon \int _{\Omega }|\nabla \omega ^n|^2dx\\&\quad \le C(\varepsilon ) \Vert u\Vert _{L^\infty ([0,T], H^3)}\int _{\Omega }|\omega ^n|^2dx+C(\varepsilon )\int _{\Omega }|\tilde{f}|^2dx,\\&\quad \frac{\partial }{\partial t}\int _{\Omega }|\nabla \omega ^n|^2dx+\varepsilon \int _{\Omega }|\Delta \omega ^n|^2dx\le C(\varepsilon )\int _{\Omega }|\tilde{f}|^2dx,\\&\quad \frac{\partial }{\partial t}\int _{\Omega }|\Delta \omega ^n|^2dx+\varepsilon \int _{\Omega }|\nabla \Delta \omega ^n|^2dx\\&\quad \le C(\varepsilon ) \Vert u\Vert _{L^\infty ([0,T], H^3)}\int _{\Omega }|\Delta \omega ^n|^2dx+C(\varepsilon )\int _{\Omega }|\nabla \tilde{f}|^2dx. \end{aligned}$$

The Gronwall inequality gives the following inequalities

$$\begin{aligned}&\sup _{0\le t\le T}\Vert w^n\Vert ^2_{H^1}+\varepsilon \int _{0}^{T}\int _{\Omega }|\Delta \omega ^n|^2dxdt\\&\quad \le C(\varepsilon , \Vert u\Vert _{L^\infty ([0,T],H^3)}, T)\int _{0}^{T}\int _{\Omega }|\tilde{f}|^2dxdt,\\ \end{aligned}$$

and

$$\begin{aligned}&\sup _{0\le t\le T}\int _{\Omega }|\Delta \omega ^n|^2dx+\varepsilon \int _{0}^{T}\int _{\Omega }|\nabla \Delta \omega ^n|^2dxdt\\&\quad \le C(\varepsilon , \Vert u\Vert _{L^\infty ([0,T],H^3)}, T)\int _{0}^{T}\int _{\Omega }|\nabla \tilde{f}|^2dxdt. \end{aligned}$$

By using Equation (A5) again and then applying Lemma 2.1, one obtains

$$\begin{aligned}&\sup _{0\le t\le T}\Vert w^n\Vert ^2_{H^1}+\varepsilon \left( \int _{0}^{T}\int _{\Omega }|\frac{\partial w^n}{\partial t}|^2dxdt+\int _{0}^{T}\Vert w^n\Vert ^2_{H^2}dt\right) \\&\quad \le C(\varepsilon , \Vert u\Vert _{L^\infty ([0,T],H^3)}, T)\int _{0}^{T}\int _{\Omega }|\tilde{f}|^2dxdt,\\&\sup _{0\le t\le T}\Vert w^n\Vert ^2_{H^2}+\varepsilon \left( \int _{0}^{T}\int _{\Omega }|\nabla \frac{\partial w^n}{\partial t}|^2dxdt+\int _{0}^{T}\Vert w^n\Vert ^2_{H^3}dt\right) \\&\quad \le C(\varepsilon , \Vert u\Vert _{L^\infty ([0,T],H^3)}, T)\int _{0}^{T}\int _{\Omega }|\tilde{f}|^2+|\nabla \tilde{f}|^2dxdt. \end{aligned}$$

Therefore by letting $n\rightarrow \infty$, $w^n$ converges to a solution w of (A4) as we claimed, which satisfies

$$\begin{aligned}&\Vert \omega \Vert _{L^\infty ([0, T], H^1(\Omega ))}+\Vert \omega \Vert _{L^2([0,T],H^2(\Omega ))}+\Vert \frac{\partial \omega }{\partial t}\Vert _{L^{2}([0,T],L^2(\Omega ))}\\&\quad \le C(\varepsilon )\Vert \tilde{f}\Vert _{L^2([0,T]\times \Omega )},\\&\Vert \omega \Vert _{L^\infty ([0, T], H^2(\Omega ))}+\Vert \omega \Vert _{L^2([0,T],H^3(\Omega ))}+\Vert \frac{\partial \omega }{\partial t}\Vert _{L^{2}([0,T],H^1(\Omega ))}\\&\quad \le C(\varepsilon )\Vert \tilde{f}\Vert _{L^2([0,T], H^1(\Omega ))}. \end{aligned}$$

Then the uniqueness of strong solutions to (A4) implies $\eta v=\omega$. Therefore the desired result is proved. $\square$

Now, we apply the above Theorem A.1 to show the local estimates of solution $\frac{\partial u_\varepsilon }{\partial t}$ to (3.7) which can be summarized as the following theorem.

Theorem A.2

The solution $u_\varepsilon$ to (3.1) obtained in Theorem 3.1 satisfies

$$\begin{aligned} \frac{\partial u_\varepsilon }{\partial t}\in L^2_{loc}((0,T_\varepsilon ), H^3(\Omega )) \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 u_\varepsilon }{\partial t^2}\in L^2_{loc}((0,T_\varepsilon ), H^1(\Omega )). \end{aligned}$$

Proof

Since $\frac{\partial u_\varepsilon }{\partial t}$ satisfies the following equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial v}{\partial t}-\varepsilon \Delta v-u_\varepsilon \times \Delta v=f(u_\varepsilon , \nabla v, v),\\ v(x,0)=\frac{\partial u_\varepsilon }{\partial t}|_{t=0},\quad \frac{\partial v}{\partial t}|_{\partial \Omega \times (0, T_\varepsilon )}=0, \end{array}\right. } \end{aligned}$$

(A6)

where

$$\begin{aligned} f(u_\varepsilon , \nabla v, v)=v\times \Delta u_\varepsilon +2\varepsilon \left\langle \nabla u_\varepsilon ,\nabla v\right\rangle u_\varepsilon +\varepsilon |\nabla u_\varepsilon |^2v, \end{aligned}$$

then, by the above Theorem A.1 we need only to check that $u_\varepsilon$ and $f(u_\varepsilon , \nabla v,v)$ satisfies the conditions (A2) and (A3) respectively.

Since $u_\varepsilon \in L^\infty ([0,T], H^3(\Omega ))$ and $\frac{\partial u_\varepsilon }{\partial t}\in L^2([0,T], L^2(\Omega ))$ for all $T<T_\varepsilon$, then (2) of Lemma 2.3 implies

$$\begin{aligned} u_{\varepsilon }\in C^0([0,T], H^2(\Omega )). \end{aligned}$$

Indeed, for any (x, t) and $(x_0,t_0)$ we have

$$\begin{aligned} \begin{aligned}&|u_\varepsilon (x,t)-u_\varepsilon (x_0,t_0)|\\&\quad \le |u_\varepsilon (x,t)-u_\varepsilon (x_0,t)|+|u_\varepsilon (x_0,t)-u_\varepsilon (x_0,t_0)|\\&\quad \le \sup _{\Omega }|\nabla u_\varepsilon |(\cdot ,t)|x-x_0|+|u_\varepsilon (x_0,t)-u_\varepsilon (x_0,t_0)|\\&\quad \le C\Vert u_\varepsilon \Vert _{L^\infty ([0,T], H^3(\Omega ))}|x-x_0|+C\Vert u_\varepsilon (\cdot , t)-u_\varepsilon (\cdot , t_0)\Vert _{H^2(\Omega )}. \end{aligned} \nonumber \\ \end{aligned}$$

(A7)

This implies $u_\varepsilon \in C^0(\Omega \times [0,T])$.

Next, we want to show $f(u_\varepsilon ,\nabla v, v)\in L^2([0,T], H^1(\Omega ))$. A simple calculations shows

$$\begin{aligned} \int _0^T\int _{\Omega }|f|^2dxdt\le C(1+\varepsilon )(\Vert u_\varepsilon \Vert ^4_{L^\infty ([0,T], H^3)}+1)\Vert \frac{\partial u_\varepsilon }{\partial t}\Vert ^2_{L^2([0,T],H^1)}\le C(T), \end{aligned}$$

and

$$\begin{aligned} \nabla f=&\nabla \frac{\partial u_\varepsilon }{\partial t}\times \Delta u_\varepsilon +\frac{\partial u_\varepsilon }{\partial t}\times \nabla \Delta u_\varepsilon \\&+\nabla ^2\frac{\partial u_\varepsilon }{\partial t}\#\nabla u_\varepsilon \# u_\varepsilon +\nabla \frac{\partial u_\varepsilon }{\partial t}\# \nabla ^2 u_\varepsilon \# u_\varepsilon \\&+\nabla \frac{\partial u_\varepsilon }{\partial t}\# \nabla u_\varepsilon \# \nabla u_\varepsilon +\frac{ \partial u_\varepsilon }{\partial t}\# \nabla ^2 u_\varepsilon \#\nabla u_\varepsilon ,\\ \end{aligned}$$

where “$\#$” denotes the linear contraction. Thus, we have

$$\begin{aligned} \int _0^T\int _{\Omega }|\nabla f|^2dxdt\le C(\Vert u_\varepsilon \Vert ^4_{L^\infty ([0,T], H^3)}+1)\Vert \frac{\partial u_\varepsilon }{\partial t}\Vert _{L^2([0,T], H^2)}\le C(T). \end{aligned}$$

Here, we have used the estimate (3.2). Therefore, from Theorem A.1 we can obtain the desired results. $\square$

Appendix B The Schrödinger Flow in Moving Frame and Parallel Transportation

1.1 B.1 The Schrödinger flow in moving frame

Let $\Omega$ be a smooth bounded domain in $\mathbb {R}^3$. Suppose that $u: \Omega \times [0,T]\rightarrow \mathbb {S}^2$ is a solution to the Schrödinger flow

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu =J(u)\tau (u),\quad \quad &{}\text {(x,t)}\in \Omega \times \mathbb {R}^+,\\ \frac{\partial u}{\partial \nu }=0, &{}\text {(x,t)}\in \partial \Omega \times \mathbb {R}^+,\\ u(x,0)=u_0: \Omega \rightarrow \mathbb {S}^2, \end{array}\right. } \end{aligned}$$

(B1)

where $J(u)=u\times$. We are going to rewrite the above equation in a chosen gauge of the pull-back bundle $u^*T\mathbb {S}^2$ over $\Omega \times [0,T]$.

Let $\nabla ^{\mathbb {S}^2}$ be the connection on $\mathbb {S}^2$ and $\nabla =u^*\nabla ^{\mathbb {S}^2}$ be the pull-back connection on $u^*T\mathbb {S}^2$. Let $\{x_1,x_2,x_3, t\}$ be the canonical coordinates on $\Omega \times [0,T]$, denote $\nabla _{t}=\nabla _{\frac{\partial }{\partial t}}$ and $\nabla _{i}=\nabla _{\frac{\partial }{\partial x_1}}$ for $i=1,2,3$. Recall that the tension field $\tau (u)={{\,\textrm{tr}\,}}\nabla ^2 u=\nabla _i\nabla _i u$, we can write the equation (B1) in the form

$$\begin{aligned} \nabla _t u=J(u)\nabla _i\nabla _i u=\nabla _i(J(u)\nabla _i u). \end{aligned}$$

Furthermore, let $\{e_\alpha \}_{\alpha =1}^2$ be a local frame of the pull-back bundle $u^*T\mathbb {S}^2$ over $\Omega \times [0,T]$, such that the complex structure J in this frame is reduced to $J_0=\sqrt{-1}$. If we denote $\phi :=\nabla u=u_i^\alpha e_\alpha \otimes dx^i$, where $i=1,2,3$, then

$$\begin{aligned} \nabla _t u=J_0\nabla _i \phi _i \quad \text {and}\quad \nabla _t \phi =\nabla (J_0\nabla _i \phi _i), \end{aligned}$$

The Neumann condition on boundary in (B1) is equivalent to

$$\begin{aligned} \sum _{i=1}^3\phi _i\cdot \nu _i|_{\partial \Omega \times [0,T]}=0, \end{aligned}$$

where $\nu =(\nu _1,\nu _2,\nu _3)$ is the outer normal vector of $\partial \Omega$.

1.2 B.2 Parallel transportation and some lemmas

Let $B\subset \mathbb {S}^2$ be an open geodesic ball with radius $<\frac{\pi }{2}$. Then for any $y_1,\, y_2\in B$, there exists a unique minimizing geodesic $\gamma (s):[0,1]\rightarrow \mathbb {S}^2$ connecting $y_1$ and $y_2$, and let $\mathcal {P}:T_{y_2}\mathbb {S}^2\rightarrow T_{y_1}\mathbb {S}^2$ be the linear map given by parallel transport along $\gamma$. Let $\{e_1(s),e_2(s)\}$ be the frame gotten by parallel transport along $\gamma$, set $e_\alpha (y_1)=e_\alpha (0)$ and $e_\alpha (y_2)=e_\alpha (1)$. Then, for any $X=X^\alpha (y_2)e_\alpha (1)\in T_{y_2}\mathbb {S}^2$, the above linear map $\mathcal {P}$ has the following formula

$$\begin{aligned} \mathcal {P}X=X^\alpha (y_2)e_\alpha (0). \end{aligned}$$

Let $d:\mathbb {S}^2\times \mathbb {S}^2\rightarrow \mathbb {R}$ be the distance function on $\mathbb {S}^2$, and $\tilde{\nabla }=\nabla ^{\mathbb {S}^2}\otimes \nabla ^{\mathbb {S}^2}$ be the product connection on $\mathbb {S}^2\times \mathbb {S}^2$. We have the following estimates for gradient and Hessian of the distance function, whose proof can be found in [12, 43, 44].

Lemma B.1

Suppose that $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ are two vectors in $T_{y_1}\mathbb {S}^2\times T_{y_2}\mathbb {S}^2$ where $d(y_1,y_2)<\frac{\pi }{2}$. Then, there hold true

• 1. $\frac{1}{2}\tilde{\nabla }d^2(X)=\left\langle \gamma ^\prime (0),\mathcal {P}X_2-X_1\right\rangle$,

• 2. $\frac{1}{2}|\tilde{\nabla }^2d^2(X,Y)|\le |\mathcal {P}X_2-X_1||\mathcal {P}Y_2-Y_1|+Cd^2(y_1,y_2)(|X_1|+|X_2|)(|Y_1|+|Y_2|)$.

On the other hand, let $u_l:\Omega \times [0,T]\rightarrow \mathbb {S}^2$, $l=1,2$, with

$$\begin{aligned} \sup _{\Omega \times [0,T]} d(u_1(x,t),u_2(x,t))<\frac{\pi }{2} \end{aligned}$$

and denote $\nabla _l=u_l^*\nabla ^{\mathbb {S}^2}$. Then, for any $(x,t)\in \Omega \times [0,T]$ there exists a unique minimizing geodesic $\gamma _{(x,t)}(s):[0,1]\rightarrow \mathbb {S}^2$ connecting $u_1(x,t)$ and $u_2(x,t)$. More precisely, we define a map $U:\Omega \times [0,T]\times [0,1]\rightarrow \mathbb {S}^2$ such that $U(x,t,s)=\gamma _{(x,t)}(s)$, then $u_l^*T\mathbb {S}^2=U^*T\mathbb {S}^2|_{s=l-1}$ and $\nabla _l=U^*\nabla ^{\mathbb {S}^2}|_{s=l-1}$. Therefore, we can define a global bundle isomorphism $\mathcal {P}: u^*_2T\mathbb {S}^2 \rightarrow u^*_1T\mathbb {S}^2$ by the parallel transportation along each geodesic. And hence, $\mathcal {P}$ can be extended naturally to a bundle isomorphism from $u^*_2T\mathbb {S}^2\otimes T^*\Omega$ to $u^*_1T\mathbb {S}^2\otimes T^*\Omega$.

Let $\{e_1,e_2\}$ be a fixed local frame of bundle $u_1^*T\mathbb {S}^2$ such that $J(u_1)=\sqrt{-1}$. For each point (x, t), we transport parallel this frame to get a moving frame $\{e_1(s),e_2(s)\}$ along the geodesic $\gamma _{(x,t)}(s)$, and set $e_{1,\alpha }=e_\alpha (0)$ and $e_{2,\alpha }=e_\alpha (1)$ for $\alpha =1,2$. Under this local frame $\{e_1(1), e_2(1)\}$ of $u^*_2 T\mathbb {S}^2$, we still have

$$\begin{aligned} J(u_2)=\sqrt{-1}, \end{aligned}$$

since $\nabla _{\frac{\partial \gamma }{\partial s}} J(\gamma )=\frac{\partial }{\partial s} J\circ \gamma =0$ and $J\circ \gamma (0, x,t)=\sqrt{-1}$.

On the other hand, if we denote $\nabla _l u_l=u^\alpha _{l,i}e_{l,\alpha }\otimes dx^i$ and set $\phi _l=u^\alpha _{l,i}e_{1,\alpha }\otimes dx^i$, then

$$\begin{aligned} \mathcal {P}\nabla _2 u_2=\mathcal {P}u^\alpha _{2,i} e_{2,\alpha }\otimes dx^i=u^\alpha _{2,i} e_{1,\alpha }\otimes dx^i=\phi _2, \end{aligned}$$

and hence

$$\begin{aligned} \Phi :=\mathcal {P}\nabla _2 u_2-\nabla _1 u_1=(u^\alpha _{2,i}-u^\alpha _{1,i})e_{1,\alpha }\otimes dx^i=\phi _2-\phi _1. \end{aligned}$$

Denote the difference of the two connections by

$$\begin{aligned} B=\nabla _2-\nabla _1=(\left\langle \nabla _2 e_\alpha (1), e_{\beta }(1)\right\rangle -\left\langle \nabla _1 e_\alpha (0), e_{\beta }(0)\right\rangle )e_\beta (0), \end{aligned}$$

which is a tensor. The following estimates for the difference of connections is essential to control the energy $\int _{\Omega }|\Phi |^2dx$ in the proof of the uniqueness, whose proof can be found in [44].

Lemma B.2

The exists constant C independing on $u_1$ and $u_2$, such that the following estimates hold true.

• 1. $|B_t|=|\nabla _{2,t}-\nabla _{1,t}|\le C(|\nabla _t u_1|+|\nabla _t u_2|)d(u_1,u_2)$,

• 2. $|B_i|=|\nabla _{2,i}-\nabla _{1,i}|\le C(|\nabla _i u_1|+|\nabla _i u_2|)d(u_1,u_2)$,

where $i=1,2,3$. Moreover, for any $i,k=1,2,3$, we have

$$\begin{aligned} |(\nabla _{2,k}\nabla _{2,i}-\nabla _{1,k}\nabla _{1,i})J_0\phi _{2,i}|\le C_1(|\Phi |+(|\nabla _1^2 u_1|+|\nabla _2^2 u_2|+1)d(u_1,u_2)), \end{aligned}$$

where $C_1$ depends only on $\Vert u_1\Vert _{H^3(\Omega )}$ and $\Vert u_2\Vert _{H^3(\Omega )}$.