Additional note on partial regularity of weak solutions of the Navier-Stokes equations in the class $L^\infty (0,T,L^3(\Omega )^3)$

Zdeněk Skalák

On the exterior problem in 2D for stationary flows of fluids with shear dependent viscosity

Michael Bildhauer, Martin Fuchs (2012)

Commentationes Mathematicae Universitatis Carolinae

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On the complement of the unit disk $B$ we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field $u$ is equal to zero provided ${u |}_{\partial B} = 0$ and ${lim}_{| x | \to \infty} {| x |}^{1 / 3} | u (x) | = 0$ uniformly. For slow flows the latter condition can be replaced by ${lim}_{| x | \to \infty} | u (x) | = 0$ uniformly. In particular, these results hold for the classical Navier-Stokes case.

Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case $q = \frac{3 d}{d + 2}$

Jörg Wolf (2007)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider weak solutions $𝐮 : Ω \to ℝ^{d}$ to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain $Ω \subset ℝ^{d}$ ( $d = 2$ or $d = 3$ ). For the critical case $q = \frac{3 d}{d + 2}$ we prove the higher integrability of $\nabla 𝐮$ which forms the basis for applying the method of differences in order to get fractional differentiability of $\nabla 𝐮$ . From this we show the existence of second order weak derivatives of $u$ .

Displaying similar documents to “Additional note on partial regularity of weak solutions of the Navier-Stokes equations in the class L ∞ ( 0 , T , L 3 ( Ω ) 3 ) ”

On the exterior problem in 2D for stationary flows of fluids with shear dependent viscosity

Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case q = 3 d d + 2

Displaying similar documents to “Additional note on partial regularity of weak solutions of the Navier-Stokes equations in the class $L^{\infty} (0, T, L^{3} {(Ω)}^{3})$ ”

Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case $q = \frac{3 d}{d + 2}$