Strong solution of modified anistropic 3D-Navier–Stokes equations

This study delves into a comprehensive examination of modified three-dimensional, incompressible, anisotropic Navier–Stokes equations. The modification involves incorporating a power term in the nonlinear convection component, a particularly relevant adjustment in porous media scenarios, especially when the fluid adheres to the Darcy–Forchheimer law instead of the conventional Darcy law. Our main contributions include establishing global existence over time and demonstrating the uniqueness of solutions. Importantly, these achievements are realized without the need to assume smallness conditions on the initial data, but with the condition \(\beta >3\). However, when \(\beta =3\), the problem is limited to the case \(0<\alpha <4\) as the above inequality is unsolvable for these \(\alpha \) values using our method. To address our statement, we will add a “slight disturbance” the function \(\log (e+|u|^{2})\) to \(|u|^{2}u\). The primary objective of our research is to affirm that the solution, denoted as u in this equation, exhibits continuity in \(L^{2}(\mathbb {R}^{3})\).