Existence of local suitable weak solutions to the Navier-Stokes equations for initial data in $L^{2}_{\rm loc} (\mathbb{R}^3)$

We consider the Navier-Stokes equations in $\mathbb{ R}^3$ subject to the initial condition with initial velocity field in $L^{ 2}_{ \rm loc} (\mathbb{ R}^3)$ such that $\limsup_{ R \to +\infty } R^{ -1} \|u_{ 0} \|_{ L^{ 2}(B(R))} < +\infty$. Our aim is to show the local existence of a weak solution, global existence of weak solution if $C=0$ and the partial regularity in the sense of Caffarelli-Kohn-Nirenberg.

Categories: math.AP