Poisson's equation
Fluid dynamics

For the incompressible Navier–Stokes equations, given by ∂ v ∂ t + (v ⋅ ∇) v = − 1 ρ ∇ p + ν Δ v + g, ∇ ⋅ v = 0. {\displaystyle {\begin{aligned}{\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla)\mathbf {v} &=-{\frac {1}{\rho }}\nabla p+\nu \Delta \mathbf {v} +\mathbf {g},\\\nabla \cdot \mathbf {v} &=0.\end{aligned}}}

The equation for the pressure field p {\displaystyle p} is an example of a nonlinear Poisson equation: Δ p = − ρ ∇ ⋅ (v ⋅ ∇ v) = − ρ Tr ((∇ v) (∇ v)). {\displaystyle {\begin{aligned}\Delta p&=-\rho \nabla \cdot (\mathbf {v} \cdot \nabla \mathbf {v})\\&=-\rho \operatorname {Tr} {\big (}(\nabla \mathbf {v})(\nabla \mathbf {v}){\big)}.\end{aligned}}} Notice that the above trace is not sign-definite.
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