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I was reading Fluid Mechanics by I.G.Currie and after deriving the conservation of mass ($ \frac {\partial \rho}{\partial t} + \frac {\partial (\rho u_k)}{\partial x_k} = 0$), it said "since the equation is a partial differential equation, the implication is that the velocity is continuous." I haven't done any course on pde, so I am having difficulty in understanding how does a pde imply continuity.

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A formal rigorous mathematical framework PDE (particularly nonlinear PDE) is a subtle business. What the author is getting at is that in writing down the expression

$$ \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_k)}{\partial x_k} = 0 $$

The assumption is that $\rho u_k$ is continuous and in fact once differentiable. If $u_k$ were to be discontinuous, writing down $\partial (pu_k)/\partial x_k$ would make no sense. Or would it?

In fact, one does not actually need continuity or even differentiability of $u$, at least formally. One only needs that $u$ is arbitrarily close to a smooth function, say. One routinely solves PDEs with discontinuous functions all the time. (Consider a linear advection equation with a square wave initial condition.) In fact, we often solve PDEs with objects that are not strictly functions at all like the so-called delta "function".

Thus, the author here is not making a strictly accurate mathematical statement, per say, but it is a nice assumption to make about $u$ if one doesn't want to invoke a lot more complicated math.

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For short, if velocity is not continuous, its spatial derivatives may not exist or be infinite. Then the equation for the conservation of mass (containing derivatives of velocity w.r.t space) become meaningless.

In practice (numerical calculation), all partial differential equations will be discretized and finite difference schemes will be used to calculate the derivatives. Hence the continuous requirement can be relaxed and is only needed in theory.

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