Higher derivatives estimate for the 3D Navier-Stokes equation
In this article, a non linear family of spaces, based on the energy dissipation, is introduced. This family bridges an energy space (containing weak solutions to Navier-Stokes equation) to a critical space (invariant through the canonical scaling of the Navier-Stokes equation). This family is used to get uniform estimates on higher derivatives to solutions to the 3D Navier-Stokes equations. Those estimates are uniform, up to the possible blowing-up time. The proof uses blow-up techniques. Estimates can be obtained by this means thanks to the galilean invariance of the transport part of the equation.
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