Optimality of Serrin type extension criteria to the Navier-Stokes equations
Reinhard Farwig, Ryo Kanamaru
Abstract
Abstract We prove that a strong solution u to the Navier-Stokes equations on (0, T ) can be extended if either u ∈ L θ (0, T ; <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msubsup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>U</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="normal">∞</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo>/</m:mo> </m:mrow> <m:mi>θ</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo>−</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msubsup> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$ ) for 2/ θ + α = 1, 0 < α < 1 or u ∈ L 2 (0, T ; <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msubsup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>V</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="normal">∞</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>0</m:mn> </m:mrow> </m:msubsup> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$ ), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msubsup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>U</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo>,</m:mo> <m:mi>σ</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>s</m:mi> </m:mrow> </m:msubsup> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msubsup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>V</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>s</m:mi> </m:mrow> </m:msubsup> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$ are Banach spaces that may be larger than the homogeneous Besov space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msubsup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mover> <m:mi>B</m:mi> <m:mo>˙</m:mo>