A new proof of the Théorème de Structure related to a weak solution to the Navier-Stokes equations
Paolo Maremonti, Filippo Palma
TL;DR
The paper presents a new proof of the Théorème de Structure for Leray-type weak solutions of the Navier–Stokes initial-boundary value problem, by exploiting a priori estimates on carefully constructed approximating sequences (via mollified and Galerkin schemes). It derives uniform bounds, including a critical bound on the second derivatives D^2 v^m in L^α with α=2/3, and uses these to obtain strong convergence of gradients and to identify a finite-time θ after which the solution is regular on a large interval, plus a countable collection of maximal regularity intervals elsewhere. This approach accommodates Hopf-type weak solutions in bounded or unbounded domains without requiring a strong energy inequality, broadening the partial regularity framework. The results provide a flexible route to partial regularity and potentially extend to fluid-structure interaction problems. Overall, the work adds a robust approximating-sequences methodology to the partial regularity theory for Navier–Stokes IBVPs.
Abstract
It is well known that a Leray's weak solution to the Navier-Stokes Cauchy problem enjoys a partial regularity which is known in the literature as the Théorème de Structure of a Leray's weak solution. As well, this result has been extended by some authors to the case of the IBVP. In this note, we achieve the Théorème de Structure by means of a new proof. Our proof is based on a priori estimates for a suitable approximating sequence. In this way our result covers a more general setting in the sense that, e.g., we can also include the case of the weak solutions furnished by Hopf for an IBVP in bounded domains without requiring an energy inequality in a strong form, but just employing a priori estimates on the Galerkin approximation.