turkmath.org

It is well-known that upon taking an initial data in Lp (3 ≤ p < ∞) we have a unique (mild) solution to the Cauchy Problem for the 3D incompressible Navier-Stokes (NS) equations, up to finite time. This is linked to the fact that the above functional space corresponds to a class of regularity for the 3D incompressible NS system. However, for a solution in the weak−L3 space (L 3,∞) it is still unknown whether or not we have reg- ularity. Therefore in order to exhibit some non-uniqueness behavior for the Cauchy problem for the NS system, the weak−L3 space appears to be a right place to pick an initial data. A typical example of functions belonging to such space is (−1)−homogeneous functions. In their groundbreaking work ”Local-in-space estimates near initial time for weak solutions of the Navier- Stokes equations and forward self-similar solutions” Jia and Sverak showed the existence of a global scale-invariant solution to the Cauchy problem with (−1)−homogeneous initial data, which is smooth for positive times. Unique- ness, however, is unknown; since that same initial data might generate mul- tiple solutions. Jia-Sverak (in ”Are the incompressible 3D Navier-Stokes equations locally ill-posed in the natural energy space?”) proved that, upon assuming some technical conditions, this problem admits multiple solutions; and later Gillod-Sverak (in ”Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderlines spaces”) verified this numerically. In this talk, I will present two constructions of forward self-similar solu- tions to the 3D incompressible Navier-Stokes system, as the singular limit of forward self-similar solutions to certain parabolic systems. This we hope will bring us new insights into the non-uniqueness problem for the Navier-Stokes system.