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A new regularity class for the Navier-Stokes equations in $$\mathbb{R}^ n$$. (English) Zbl 0837.35111
Summary: Consider the Navier-Stokes equations in $$\mathbb{R}^n\times (0, T)$$, for $$n\geq 3$$. Let $$1< \alpha\leq \min\{2, n/(n- 2)\}$$ and define $$\beta$$ by $$(2/\alpha)+ (n/\beta)= 2$$. Set $$\alpha'= \alpha/(\alpha- 1)$$. It is proved that $$Dv$$ belongs to $$C(0, T; L^{\alpha'})\cap L^{\alpha'}(0, T; L^{2\beta/(n- 2)})$$, whenever $$Dv\in L^\alpha(0, T; L^\beta)$$. In particular, $$v$$ is a regular solution. This result is the natural extension to $$\alpha\in (1, 2]$$ of the classical sufficient condition that establishes that $$L^\alpha(0, T; L^\gamma)$$ is a regularity class if $$(2/\alpha)+ (n/\gamma)= 1$$. Even the borderline case $$\alpha= 2$$ is significant. In fact, this result states that $$L^2(0, T; W^{1,n})$$ is a regularity class if $$n\leq 4$$. Since $$W^{1, n}\hookrightarrow L^\infty$$ is false, this result does not follow from the classical one that states that $$L^2(0, T; L^\infty)$$ is a regularity class.

MSC:
 35Q30 Navier-Stokes equations 35B65 Smoothness and regularity of solutions to PDEs 35K55 Nonlinear parabolic equations 76D05 Navier-Stokes equations for incompressible viscous fluids