It seems you're working with a complex equation, which involves partial derivatives, vector calculus, functional spaces, and other concepts. Let me break it down to interpret the different components of the expression:

1. Numerator: $\left( \frac{\partial u}{\partial t} + (u \cdot \nabla)u + \nabla p - \nu \Delta u \right) \cdot H^{2p}(Q, Q) \cap H^{2p}(X) \cdot \text{Rank}(E(\mathbb{Q})) \cdot \text{old } E$
   • $\frac{\partial u}{\partial t}$ represents the time derivative of a vector field $u$.
   • $(u \cdot \nabla)u$ refers to the convective term, which is the advection of the field $u$ by itself.
   • $\nabla p$ represents the gradient of pressure, typically seen in fluid dynamics equations.
   • $- \nu \Delta u$ involves the diffusion term where $\nu$ is the viscosity constant, and $\Delta$ is the Laplacian operator, which indicates diffusion or smoothing of the field.
   • $H^{2p}(Q, Q)$ and $H^{2p}(X)$ are Sobolev spaces, commonly used in partial differential equations (PDEs). They define the smoothness and integrability of the functions.
   • The expression involves an intersection of these Sobolev spaces: ( H^{2p}(Q, Q) \cap H^{2p}(X)