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

• DOI: 10.1007/s00023-016-0519-0
• Publication Date: 2016-10-20T07:16:55Z
• Authors: Zachary Bradshaw; Tai-Peng Tsai
• Keywords: Weak solution

For any discretely self-similar, incompressible initial data $v_0$ which satisfies $\|v_0 \|_{L^3_w(\mathbb R^3)}\leq c_0$ where $c_0$ is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemari��-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 $\mathbb R^3\setminus \{0\}$. The same method is used to construct self-similar solutions for any $-1$-homogeneous initial data in $L^3_w$.