Homemade Conslaws

Conservation laws

Let \(\boldsymbol U\) be a quantity defined on a domain \(\Omega \subset \R^n\). For any subdomain \(\omega \subset \Omega\), the temporal rate of change of \(\bm U\) is equal to the amount of \(\bm U\) created or destroyed and the flux going through the boundary \(\partial \omega\). It can be described mathematically by

\[ \dv{t} \int_{\omega} \bm U \dd \x = - \int_{\partial \omega} \bm F \cdot \bm \nu \dd \s + \int_{\omega} \bm S \dd \x, \]

where \(\bm F\) is the flux and \(\bm S\) is the source term. By the Gauss divergence theorem, we can rewrite this as

\[ \dv{t} \int_{\omega} \bm U \dd \x + \int_{\omega} \div {\bm F} \dd \x = \int_{\omega} \bm S \dd \x. \]

Since this equation holds for all subdomains \(\omega \subseteq \Omega\), we can write

\[ \bm U_t + \div {\bm F} = \bm S \quad \text{ in } \Omega \times \R_+ \]

We call this equation a conservation law.