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Currently, Navier-Stokes Equation, its solution's existence and smoothness is not well established, making the problem as one of famous Millennium Prize Problems. On the other hand, I noticed that Boltzmann Transport Equation also can be used to describe fluid motion, but its existence and smoothness was not in the Millennium Prize Problems.

  • Is Boltzmann Transport Equation's solution existence and smoothness problem already solved?
  • If so, can it be applied to Navier-Stokes existence and smoothness problem?
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You can obtain the Navier-Stokes equation from the Boltzmann equation assuming that the distribution of velocities follows a Maxwellian. See, for example, this notes for the complete derivation. Basically, in the framework of the Boltzmann equation, you think of velocities as an arbitrary random variable and you try to obtain its distribution. In the Navier-stokes formalism, the fluctuations of the velocities are neglected and you think of them as a deterministic field. So both problems are related, but really taking care about different aspects.

I am not an expert in partial differential equations, but I know that the subject is subtle. For example, it is not true that a linear partial differential equation that involves smooth functions has well-defined solutions (see Wikipedia: existence of solutions for PDEs). So even if the Boltzmann equation is linear, I don't think that proving the existence and uniqueness of solutions in a rigorous fashion is trivial without imposing strong constraints on the functions that appear in the equation. It is especially delicate the collision term, which can be non-analytical depending on the case.

Having said so, I really don't know what the current state of the art on this topic is... And any sensate idea could be a good starting point to face an unsolved problem ;).

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