Weak Navier–Stokes solution

On an open space-time cylinder $Q\subset\mathbb R^n\times\mathbb R$, a weak Navier–Stokes solution with viscosity $\nu>0$ and locally integrable force $f$ is a pair $u\in L^2_{\rm loc}(Q)$, $p\in L^1_{\rm loc}(Q)$ satisfying the integral identities

for all compactly supported smooth vector tests $\phi$ and scalar tests $\psi$. Here the outer product has entries $u_i u_j$, and $(u\otimes u):\nabla\phi=\sum_{i,j}u_i u_j\partial_j\phi_i$. The indicated local integrability makes these products meaningful. These are the distributional momentum and incompressibility equations.

Here $L^2_{\rm loc}$ uses Lebesgue square integrability on compact subsets. The tests are from the smooth test-function space; $\nabla$, $\operatorname{div}$, and $\Delta$ are the spatial gradient, divergence, and Laplacian.