2018/11/01 by Dallas Albritton, Albritton, Dallas, Tobias Barker +1
Engineering · Mathematics · #35Q30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1811.00507
openalex publication_date 2018/11/01 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
We generalize two results in the Navier-Stokes regularity theory whose proofs\nrely on `zooming in' on a presumed singularity to the local setting near a\ncurved portion \Γ \⊂ \∂\Ω of the boundary. Suppose that\nu is a boundary suitable weak solution with singularity z^* = (x^*,T^*),\nwhere x^* \∈ \Ω \∪ \Γ. Then, under weak background assumptions,\nthe L3 norm of u tends to infinity in every ball centered at x^*:\n\
limt
to T^*-
lVert u(
cdot,\nt)
rVert_L3
left(
Omega
cap B(x^*,r)
right) =
infty
quad
forall r gt; 0.\n Additionally, u generates a non-trivial `mild bounded ancient\nsolution' in \ℝ3 or \ℝ3+ through a rescaling procedure\nthat `zooms in' on the singularity. Our proofs rely on a truncation procedure\nfor boundary suitable weak solutions. The former result is based on energy\nestimates for L3 initial data and a Liouville theorem. For the latter\nresult, we apply perturbation theory for L_\∞ initial data based on\nlinear estimates due to K. Abe and Y. Giga.\n