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Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces (2411.06483v3)

Published 10 Nov 2024 in math.AP

Abstract: In this paper, we study the quantitative regularity and blowup criteria for classical solutions to the three-dimensional incompressible Navier-Stokes equations in a critical Besov space framework. Specifically, we consider solutions $u\in L\infty_t(\dot{B}_{p,\infty}{-1+\frac{3}{p}})$ such that $|D|{-1+\frac{3}{p}}|u|\in L\infty_t (Lp)$ with $3<p<\infty$. By deriving refined regularity estimates and substantially improving the strategy in \cite{Tao_20}, we overcome difficulties stemming from the low regularity of the Besov spaces and establish quantitative bounds for such solutions. These bounds are expressed in terms of a triple exponential of $| u (t)|{\dot{B}{p,\infty}{-1+\frac{3}{p}}}$ combined with a single exponential of $\bigl| |D|{-1+\frac{3}{p}}|u(t)| \bigr|{Lp}$. Consequently, we obtain a new blowup rate which can be interpreted as a coupling of triple logarithm of $| u(t) |{\dot{B}{p,\infty}{-1+\frac{3}{p}}}$ and a single logarithm of $\bigl| |D|{-1+\frac{3}{p}}|u(t)| \bigr|{Lp}$.

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