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Regularity criteria for solutions of the three-dimensional Navier-Stokes equations are derived in this paper. Let $$\Omega(t, q) := \left\{x:|u(x,t)| > C(t,q)\normVT{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\}, \tilde\Omega(t,q) := \left\{x:|u(x,t)| \le C(t,q)\normVT{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\},$$ where $$q\ge3$$ and $$C(t,q) := \left(\frac{\normVT{u}_{L^4(\mathbb{R}^3)}^2\normVT{|u|^{(q-2)/2}\,\nabla|u|}_{L^2(\mathbb{R}^3)}}{cq\normVT{u_0}_{L^2(\mathbb{R}^3)} \normVT{p+\mathcal{P}}_{L^2(\tilde\Omega)}\normVT{|u|^{(q-2)/2}\, \widehat{u}\cdot\nabla|u|}_{L^2(\tilde\Omega)}}\right)^{2/(q-2)}.$$ Here $$u_0=u(x,0)$$, $$\mathcal{P}(x,|u|,t)$$ is a pressure moderator of relatively broad form, $$\widehat{u}\cdot\nabla|u|$$ is the gradient of $$|u|$$ along streamlines, and $$c=(2/\pi)^{2/3}/\sqrt{3}$$ is the constant in the inequality $$\normVT{f}_{L^6(\mathbb{R}^3)}\le c\normVT{\nabla f}_{L^2(\mathbb{R}^3)}$$.

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9 - Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure

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9 - Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure

Summary