# 弱解

## 一个具体的例子

${\displaystyle {\frac {\partial u}{\partial t}}+{\frac {\partial u}{\partial x}}=0\quad \quad (1)}$

(其中的记号请参阅偏导数)其中 u = u(t, x) 是两个变量的函数. 假设 u欧式空间R2连续可微 , 在方程的两侧同时乘以一个具紧支集光滑函数 φ 并积分. 得到

${\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\frac {\partial u}{\partial t}}(t,x)\varphi (t,x)\,\mathrm {d} t\mathrm {d} x+\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\frac {\partial u}{\partial x}}(t,x)\varphi (t,x)\,\mathrm {d} t\mathrm {d} x=0.}$

${\displaystyle -\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }u(t,x){\frac {\partial \varphi }{\partial t}}(t,x)\,\mathrm {d} t\mathrm {d} x-\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }u(t,x){\frac {\partial \varphi }{\partial x}}(t,x)\,\mathrm {d} t\mathrm {d} x=0.\quad \quad (2)}$

## 更一般的情况

${\displaystyle P(x,\partial )u(x)=\sum a_{\alpha _{1},\alpha _{2},\dots ,\alpha _{n}}(x)\partial ^{\alpha _{1}}\partial ^{\alpha _{2}}\cdots \partial ^{\alpha _{n}}u(x)}$

${\displaystyle \int _{W}u(x)Q(x,\partial )\varphi (x)\,\mathrm {d} x=0}$

${\displaystyle Q(x,\partial )\varphi (x)=\sum (-1)^{|\alpha |}\partial ^{\alpha _{1}}\partial ^{\alpha _{2}}\cdots \partial ^{\alpha _{n}}\left[a_{\alpha _{1},\alpha _{2},\dots ,\alpha _{n}}(x)\varphi (x)\right].}$

${\displaystyle (-1)^{|\alpha |}=(-1)^{\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}}$

${\displaystyle P(x,\partial )u(x)=0{\mbox{ for all }}x\in W}$

(所谓的强解), 那么可积函数 u 被称作弱解如果

${\displaystyle \int _{W}u(x)Q(x,\partial )\varphi (x)\,\mathrm {d} x=0}$

## 参考资料

• L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2