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 $\nabla p$, 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.