Uniform Boundedness of Homogeneous Incompressible Flows in ${\mathbb{R}}^3$

This paper investigates the extendability of local solutions for incompressible 3D Navier-Stokes and 3D Euler problems, with initial data ${\mathbf{u}}_0$ in the Sobolev space $H^s({\mathbb{R}}^3)$, where $s$ ensures the existence and uniqueness of classical solutions. A geometric decomposition of the configuration space, identified by the orthogonality between the solution $\mathbf{u}$ and the pressure forces $\mathrm{\nabla }p$, splits the problem into two simpler subproblems, which enable the uniform boundedness of the solution in each component of the partition, thereby ensuring the extendability of the solution.