## Global Smoothness of the Plane Wave Solutions for Landau-Lifshitz Equation

Zhong Penghong, Chen Xingfa,

 基金资助: 国家自然科学基金青年基金.  11601092广东省青年创新人才项目.  2014KQNCX228广东省科技厅博士启动基金.  2014A030310330广东省特色创新人才项目.  2018KTSCX161广州市科技计划项目.  201607010352

 Fund supported: NSF for Young Scientists of China.  11601092the Project for Young Creative Talents of Ordinary University of Guangdong Province.  2014KQNCX228the PhD Start-up Fund of NSF of Guangdong Province.  2014A030310330the Special Innovation Projects of Universities in Guangdong Province.  2018KTSCX161the Funds of Guangzhou Science and Technology.  201607010352

Abstract

In this paper, we study the n-dimensional plane wave solution of the Landau-Lifshtz equation on $\mathbb{S}^2$. Based on the Hasimoto transformation, the equivalent plane wave type Schrödinger equation is obtained. By the Strichartz estimation and energy method under Fourier transform, the global existence of this solution is proved under a small initial value. The global solution obtained here is smooth and the norm falls in any order Hilbert space. The results in this paper improve the regularity of the solution in the paper[3].

Keywords： Landau-Lifshitz equation ; Existence ; Smooth solution ; Regularity

Zhong Penghong, Chen Xingfa. Global Smoothness of the Plane Wave Solutions for Landau-Lifshitz Equation. Acta Mathematica Scientia[J], 2021, 41(3): 729-739 doi:

## 1 引言

$$$\frac{\partial}{\partial t} S = S {\times} \Delta S ,$$$

$$${\rm i}Z_t -\Delta Z = \frac{2\mathop Z\limits^\_ }{1+\left| Z \right|^2}(\nabla Z)^2.$$$

## 2 LL方程的等价薛定谔方程

$$$S_{t} - S {\times} S_{\overline{r}\, \overline{r}} = 0.$$$

$$$Q = \frac{\kappa}{2} \exp \left[{\rm i} \int^{{\overline{r}}}_{0} \tau (t, \tilde{{{r}}}) {\rm d} \tilde{{{r}}} \right],$$$

$$$\kappa = \left( S_{\overline{r}} \cdot S_{\overline{r}} \right)^\frac{1}{2} \quad \rm{和} \quad \tau = \frac{S \cdot (S_{\overline{r}} {\times} S_{{\overline{r}}\, {\overline{r}}})}{\kappa ^2}.$$$

$$$e_{1t} - e_1 {\times} e_{1\overline{r}\, \overline{r}} = 0,$$$

$$$P_{M}u(t, \overline{r}) = (2{\rm i}t)^{M}e^{{\rm i}\frac{|\overline{r}|^{2}}{4t}}{\partial^M_{\overline{r}}}(e^{-{\rm i}\frac{|\overline{r}|^{2}}{4t}}u),$$$

$$$[P_{M}, \ {\rm i}\partial_{t}+{\partial^2_{\overline{r}}}] = 0,$$$

(3.2)式表明, 如果$u$是线性薛定谔方程

$$$\left\{\begin{array}{l} {\rm i}u_{t}+u_{{{ \overline{r}\, \overline{r}}}} = 0, \\ u(0) = \varphi \end{array}\right.$$$

${\cal U}(t)$表示如下的半群

$$$({\cal U}(t) \varphi )(\overline{r}) = \int_{-\infty}^{+\infty} {\Bbb K} (\overline{r}-x', t)\varphi(x'){\rm d}x',$$$

$$$u^{M}(t, \overline{r}) = (2{\rm i}t)^{M}e^{{\rm i}\frac{|\overline{r}|^{2}} {4\mathrm{t}}}\partial_{\overline{r}}^{M}(e^{-{\rm i}\frac{|\overline{r}|^{2}}{4t}}u(t, \ \overline{r}))$$$

$$$u^M(t) = {\cal U}(t)u^M(0) = {\cal U}(t)x^{M}\varphi,$$$

$$$\Vert P_{M}u(t, \overline{r})\Vert_{L^2_{\overline{r}}} = \Vert u^M \Vert_{L^2_{\overline{r}}} = \Vert \overline{r}^{M}\varphi\Vert_{L^2_{\overline{r}}}.$$$

$$$(2|t|)^{M}\Vert \partial_{\overline{r}}^{M}(e^{-{\rm i}\frac{|\overline{r}|^{2}}{4t}}u(t)) \Vert_{L^2_{\overline{r}}} = \Vert \overline{r}^{M}\varphi\Vert_{L^2_{\overline{r}}}.$$$

$$${\rm i}Q_{{t}}+ Q_{{{\it \overline{r}\, \overline{r}}}} = {\Bbb F},$$$

(3.4) 式可得到线性传播子的色散不等式[2]

$$$\left\| {\cal U}(t) \varphi \right\|_{L_{\overline{r}}^{\infty} } \leq |t|^{-1/2} \| \varphi \|_{ L_{\overline{r}}^{1} }.$$$

(3.4) 式也可得到如下的Hardy-Littlewood-Sobolev不等式[2]

$$$\left\| {\cal U}(t) \varphi \right\|_{L_{\overline{r}}^{4} } \leq C \| \varphi \|_{ L_{\overline{r}}^{2} }.$$$

$$$Q = {\cal U}(t) Q_0 - \int^{t}_{0} {\cal U}(t-t'){\Bbb F}(t') \, {\rm d}t',$$$

$$$\| {\cal U}(t) Q_0 \|_{ L^4 ( [0, t], L^{\infty}_{\overline{r}} ) }\leq C \| Q_0 \|_{ L_{\overline{r}}^{2} }.$$$

$\begin{eqnarray} \bigg \| \int^{t}_{0} {\cal U}(t-t'){\Bbb F}(t') \, {\rm d}t'\bigg \|_{ L^{\infty}_{\overline{r}} } & \leq & C \int^{t}_{0} |t-t'|^{-1/2} \| Q |Q|^2 \|_{ L^{1}_{\overline{r}} } \, {\rm d}t'{}\\ &\leq &C \int^{t}_{0} |t-t'|^{-1/2} \| Q \|_{L^{\infty}_{\overline{r}}}\|Q \|^2_{L^{2}_{\overline{r}}} \, {\rm d}t' . \end{eqnarray}$

$$$\bigg\| \int^{t}_{0} U(t-t'){\Bbb F}(t') \, {\rm d}t'\bigg \|_{ L^4 ( [t_0, t], L^{\infty}_{\overline{r}} ) } \leq C E(Q_0) t ^{1/2} \| Q \| _{ L^4 ( [0, t], L^{\infty}_{\overline{r}} ) }.$$$

$$$\| Q \|_{ L^4 ( [0, T], L^{\infty}_{\overline{r}} ) } \leq C \| Q_0 \|_{ L_{\overline{r}}^{2} }.$$$

$$$\| \partial_{\overline{r}} S \|_{ L^4 ( [0, T], L^{\infty}_{\overline{r}} ) } \leq C_T \| S_0 \|_{ H_{\overline{r}}^{1} },$$$

(3.17)式有助于推导出高阶导数估计, 从而很好地解决古典解的存在性问题. 为了得到高阶估计, 我们首先对方程(3.9)进行微分(记$V = \partial_{\overline{r}} Q$), 从而得到

$$${\rm i}V_{{t}}- \left( {\rm i}\beta-\alpha \right) \left( V_{{{\it \overline{r}\, \overline{r}}}} \right) = {\widetilde {\mathbb{F}}},$$$

$$$V = {\cal U}(t) V_0 + \int^{t}_{0} {\cal U}(t) {\widetilde {\mathbb{F}}}(t') \, {\rm d}t',$$$

${\cal W} = { L^{\infty} ( [0, T], L^{2}_{\overline{r}} ) } \bigcap { L^4 ( [0, T], L^{\infty}_{\overline{r}} ) }$, 结合$V_0\in L^{2}_{\overline{r}}$, 我们可从(3.11)式得到估计

$$$\| {\cal U}(t) V_0 \|_{ {\cal W} }\leq C \| V_0 \|_{ L_{\overline{r}}^{2} }$$$

$\begin{eqnarray} \bigg\| \int^{t}_{0} {\cal U}(t) {\widetilde {\mathbb{F}}}(t') \, {\rm d}t'\bigg \|_{ {\cal W} } & \leq & C \int^{t}_{0} \| {\widetilde {\mathbb{F}}}(t') \|_{L^{2}_{\overline{r}}}{}\\ &\leq & {C} \left[ e^{T}T + e^{T} \int^{T}_{0} \left( \| V(t') \|_{L^{2}_{\overline{r}}} + \| V(t') \|_{L^{\infty}_{\overline{r}}} \right)\, {\rm d}t' \right], \end{eqnarray}$

## 4 光滑解的全局存在性

$$${\rm i}u_{{t}}+ u_{{{\it \overline{r}\, \overline{r}}}} + 2\, \left| u \right|^{2}u = 0,$$$

$$${\rm i}u_{t}^{M}+u_{\overline{r}\, \overline{r}}^{M}+2(2{\rm i}t)^{M}e^{{\rm i}\frac{|\overline{r}|^{2}}{4t}}\partial_{\overline{r}}^{M} (|e^{-{\rm i}\frac{|\overline{r}|^{2}}{4t}}u|^{2}e^{-{\rm i}\frac{|\overline{r}|^{2}}{4t}}u) = 0.$$$

$$$v(t, \, \overline{r}) = e^{-{\rm i}\frac{|\overline{r}|^2}{4t}}u(t, \, \overline{r}).$$$

$$$u^{M}(t) = {\cal U}(t)(\overline{r}^{M}\varphi)+2{\rm i}\int_{0}^{t}{\cal U}(t-s)((2{\rm i}s)^{M}e^{{\rm i}\frac{|\overline{r}|^{2}}{4\mathrm{s}}} \sum\limits_{n+j+k = M}\partial_{\overline{r}}^{n}v(s)\partial_{\overline{r}}^{j}\overline{v(s)}\partial_{\overline{r}}^{k}v(s)){\rm d}s,$$$

$\begin{eqnarray} \Vert u^{M}(t)\Vert_{L^{2}_{\overline{r}}} &\leq & \Vert \overline{r}^{M}\varphi\Vert_{L^{2}_{\overline{r}}}+2\int_{0}^{t}(2s)^{M}\Vert\sum\limits_{n+j+k = M}\partial_{\overline{r}}^{n}v(s) \partial_{\overline{r}}^{j}\overline{v(s)}\partial_{\overline{r}}^{k}v(s)\Vert_{L^{2}_{\overline{r}}}{\rm d}s {}\\ &\leq & \Vert \overline{r}^{M}\varphi\Vert_{L^{2}_{\overline{r}}}+2\int_{0}^{t}(2s)^{M}\sum\limits_{n+j+k = M}\Vert\partial_{\overline{r}}^{n}v(s)\Vert_{ L^{ \frac{2M}{n} }_{\overline{r}} }\Vert\partial_{\overline{r}}^{j}v(s)\Vert_{ L^{ \frac{2M}{j} }_{\overline{r}} } \Vert\partial_{\overline{r}}^{k}v(s)\Vert_{ L^{ \frac{2M}{k} }_{\overline{r}} } {\rm d}s. {}\\ \end{eqnarray}$

$\begin{eqnarray} & &\sum\limits_{n+j+k = M}\Vert\partial_{\overline{r}}^{n}v(s)\Vert_{ L^{ \frac{2M}{n} }_{\overline{r}} }\Vert\partial_{\overline{r}}^{j}v(s)\Vert_{ L^{ \frac{2M}{j} }_{\overline{r}} } \Vert\partial_{\overline{r}}^{k}v(s)\Vert_{ L^{ \frac{2M}{k} }_{\overline{r}} }{}\\ & \leq & C \sum\limits_{n+j+k = M} \Vert\partial_{\overline{r}}^{M}v(s)\Vert_{L^{2}_{\overline{r}}}^{\frac{n}{M}} \Vert v(s)\Vert_{L^{2}_{\overline{r}}}^{\frac{M-n}{M}} \Vert\partial_{\overline{r}}^{M}v(s)\Vert_{L^{2}_{\overline{r}}}^{\frac{j}{M}} \Vert v(s)\Vert_{L^{2}_{\overline{r}}}^{\frac{M-j}{M}} \Vert\partial_{\overline{r}}^{M}v(s)\Vert_{L^{2}_{\overline{r}}}^{\frac{k}{M}} \Vert v(s)\Vert_{L^{2}_{\overline{r}}}^{\frac{M-k}{M}} {}\\ & \leq & \frac{C}{s^{M}}\Vert u^{M}(s)\Vert_{L^{2}_{\overline{r}}}\Vert u(s)\Vert_{L^{\infty}_{\overline{r}}}^{2} {}\\ & \leq & \frac{C}{s^{M}}\Vert u^{M}(s)\Vert_{L^{2}_{\overline{r}}}. \end{eqnarray}$

$$$\Vert u(t)\Vert_{H^{M}_{\overline{r}}}\leq\Vert \varphi\Vert_{H^{M}_{\overline{r}}}+C\int_{0}^{t}\Vert u(s)\Vert_{L^{\infty}_{\overline{r}}}^{2}\Vert u(s)\Vert_{H^{M}_{\overline{r}}}{\rm d}s,$$$

$$$R_1 \leq C(P)\Vert {\overline{r}}^{P}\Psi^{M+1}\Vert_{L^{2}_{\overline{r}}}\Vert e^{-\epsilon {\overline{r}}^{2}}{\overline{r}}^{P+1}\Psi^{M}\Vert_{L^{2}_{\overline{r}}}\leq C(P, \ M)\Vert e^{-\epsilon {\overline{r}}^{2}}{\overline{r}}^{P+1}\Psi^{M}\Vert_{L^{2}_{\overline{r}}}.$$$

$\begin{eqnarray} R_2 & \leq & \Vert e^{-\epsilon {\overline{r}}^{2}}{\overline{r}}^{P+1}\Psi^{M}\Vert_{L^{2}_{\overline{r}}}\sum\limits_{n+j+k = M}\Vert e^{-\epsilon {\overline{r}}^{2}}{\overline{r}}^{P+1}\Psi^{n}\overline{\Psi^{j}}\Psi^{k}\Vert_{L^{2}_{\overline{r}}} {}\\& \leq & C(M)\Vert e^{-\epsilon {\overline{r}}^{2}}{\overline{r}}^{P+1}\Psi^{M}\Vert_{L^{2}_{\overline{r}}}\sum\limits_{k = 0}^{M}\Vert e^{-\epsilon {\overline{r}}^{2}}{\overline{r}}^{P+1}\Psi^{k}\Vert_{L^{2}_{\overline{r}}}\ {}\\& \leq & C(M) \sum\limits_{k = 0}^{M}\Vert e^{-\epsilon{\overline{r}}^{2}}{\overline{r}}^{P+1}\Psi^{k}\Vert_{L^{2}_{\overline{r}}}. \end{eqnarray}$

$\varphi\in {\cal S}'$满足$x^{M}\varphi\in L^{2}_{\overline{r}}$利用引理3.1将解延拓到$[\epsilon, \ \infty)$.

## 5 结论

{\qquad}本文研究了$n$维空间${\Bbb S}^2$中LL方程的光滑解的存在性. 我们使用一等价的复方程的解来研究原LL方程的解. 基于Strichartz估计和傅里叶变换下的Duhamel公式的能量估计, 得到了薛定谔方程解在小初值约束下的任意阶希尔伯特范数下的光滑解的存在性, 从而得到LL方程平面波型的小初值光滑解的全局存在性. 小初值的LL方程是否意味着解总是一个整体解依然没有完全清晰. 特别的, 高维解是否为全局光滑解是一个未知问题. 文章中的结论适用于任意空间维度, 对理解LL方程高维空间中的全局光滑性有很好的帮助.

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