Forward Discretely Self-Similar Solutions of the Navier–Stokes Equations II

Abstract

For any discretely self-similar, incompressible initial data v 0 which satisfies v 0 L 3 w (R 3 ) ≤ c 0 where c 0 is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemarié-Rieusset to the 3D Navier-Stokes equations in the whole space. No further assumptions are imposed on the initial data; in particular, the data is not required to be continuous or locally bounded on R 3 \ {0}. The same method gives a third construction of self-similar solutions (after those in [9] and [14]) and works for any −1-homogeneous initial data in L 3 w .

Short Time Regularity of Navier–Stokes Flows with Locally L3 Initial Data and Applications

“…In [15], Jia andŠverák constructed forward self-similar solutions for large data where the data is assumed to be Hölder continuous away from the origin. This result has been generalized in a number of directions by a variety of authors [2,3,4,5,7,21,23,32]; see also the survey [17].…”

Forward Discretely Self-similar Solutions of the MHD Equations and the Viscoelastic Navier–Stokes Equations with Damping

“…In Sect. 2, we recall some results in [1] and construct a time-periodic solution to the Leray system for the MHD equations and the viscoelastic Navier-Stokes equations with damping. In Sect.…”

“…for given T -periodic divergence-free vector fields U 0 and A 0 . We first revisit the assumption for the background vector field U 0 and the corresponding results in [1].…”

“…where c(R 0 , U 0 ) depends on R 0 and quantities associated with U 0 which are finite by Assumption 2.1. Similar to the Navier-Stokes counterpart of time-periodic Leray system in [1], we define periodic weak solutions and suitable periodic weak solutions of (2.1) as follows.…”

“…More precisely, we construct DSS local Leray weak solutions for DSS initial data with possibly large L 3 w -norm, and SS local Leray solutions for (−1)-homogeneous initial data in L 3 w . Our method follows from [1] and is based on the a priori bounds (1.25) and (1.26), and the Galerkin method. To begin with, we briefly introduce the MHD equations and the viscoelastic Navier-Stokes equations.…”