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For a time dependent vector field $v:\mathbb{R}^+\times \mathbb{R}^d\to\mathbb{R}^d$, and a say a (time dependent) probability density $u$, why do people call

$$ \partial_tu=\text{div }(uv), $$ the continuity equation, and

$$ \partial_t u=v\cdot \nabla u, $$ the transport equation?

Surely the terminology distinguishes between two fundamentally different dynamics can anyone broadly explain the main difference?

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  • $\begingroup$ The equation $\partial_t u=\text{div }(u\otimes v)$ usually appears in the formular of Euler equations; we can see it at the introduction of several papers. However, when we are going deeply into research, we will find that the form $\partial_tu=v\cdot\nabla u$ is more convenient to use. $\endgroup$
    – Feng
    2 days ago
  • $\begingroup$ On the other hand, $\partial_t u=\text{div }(u\otimes v)$ derives from some physical assumptions, so it has a name 'the continuity equation' due to physists; however, mathematicans found that the form $\partial_tu=v\cdot\nabla u$ is more convenient to use, which has a 'transport' feature. Also, when we call it 'a transport equation', it means that we will use some knowledge about transpot equations, which will help the readers to understand. $\endgroup$
    – Feng
    2 days ago

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