# Transport vs Continuity equation

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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• 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.
– Feng
2 days ago
• 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.
– Feng
2 days ago