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What's the difference between "balance laws" and "conservation laws" ? Can someone give me some examples?

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In the partial differential equations literature, the terms conservation law and balance law refer to particular types of first-order PDEs in space and time. Such equations are very common in physics since they express the conservation of a quantity over time (cf. the divergence theorem and the continuity equation). The use presented hereinafter is quite common.

Systems of conservation laws are equations of the form $$ \partial_t\boldsymbol{u} + \sum_{k=1}^d \partial_{x_k} \boldsymbol{f}_k(\boldsymbol{u}) = \boldsymbol{0}\, , $$ where $\boldsymbol{u} \in \Bbb R^p$ is the vector of conserved variables, $\boldsymbol{f}_k \in \Bbb R^p$ is the flux along the $k$th spatial direction, and $d$ denotes the spatial dimension. A single conservation law corresponds to the case $p=1$. Typical examples are the linear advection equation $u_t + u_x = 0$ and the inviscid Burgers equation $u_t + uu_x = 0$.

Systems of balance laws are systems of conservation laws with some additional terms, such as the relaxation term $\boldsymbol{r}(\boldsymbol{u})$ in $$ \partial_t\boldsymbol{u} + \sum_{k=1}^d \partial_{x_k} \boldsymbol{f}_k(\boldsymbol{u}) = \boldsymbol{r}(\boldsymbol{u})\, . $$ A typical example of a single such balance law is the Burgers equation with relaxation $u_t + uu_x = -u$.

Remark. Many authors use sometimes the term balance law for the particular case of conservation laws. In practice, the vocabulary may vary from one author to another, so it is hard to make a clear semantic distinction between conservation laws and balance laws.

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Balance laws refer to quantities that balance each other, like forces.

Conservation laws refer to quantities that are conserved in a closed system, like energy and momentum.

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    $\begingroup$ @Darry If this answers your question, you should accept it (and answers to all your other questions too) $\endgroup$ – Max Nov 2 '12 at 14:13
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Consider a continuum, say a fluid, in a domain $\Omega$. Let $V$ be a subset of $\Omega$. It is a portion of the continuum for which we are measuring some quantity, say $G$. Then the rate of change of $G$ in $V$ can be written as \begin{equation} \frac{d}{dt}\int GdV = -\int G_1\vec{v}\cdot\hat{n}dS + \int G_2dV + \int G_3 dV, \end{equation} where $G_1\vec{v}$ is the flux of $G$ through the boundary $\partial V$ of $V$, $G_2$ is the supply of $G$ from outside of $V$ and $G_3$ is the source or sink of $G$ within $V$. A balance law becomes a conservation law when $G_3 = 0$.

Refer to 'Stability Criteria for Fluid Flows' by Georgescu and Palese for examples of balance laws that are not conservation laws.

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