# 弱解

## 一个具体的例子

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

(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

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

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

## 更一般的情况

$P(x, \partial)u(x)=\sum a_{\alpha_1, \alpha_2, \dots, \alpha_n}(x) \partial^{\alpha_1}\partial^{\alpha_2}\cdots \partial^{\alpha_n} u(x)$

$\int_W u(x) Q(x, \partial) \varphi (x) \, \mathrm{d} x=0$

$Q(x, \partial)\varphi (x)=\sum (-1)^{| \alpha |} \partial^{\alpha_1} \partial^{\alpha_2} \cdots \partial^{\alpha_n} \left[a_{\alpha_1, \alpha_2, \dots, \alpha_n}(x) \varphi(x) \right].$

$(-1)^{| \alpha |} = (-1)^{\alpha_1+\alpha_2+\cdots+\alpha_n}$

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

(所谓的强解), 那么可积函数 u 被称作弱解如果

$\int_W u(x) Q(x, \partial) \varphi (x)\, \mathrm{d} x = 0$

## 参考资料

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