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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.

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