Seminar

Václav Mácha
 (Institute of Mathematics, CAS)

Global BMO estimates for nonNewtonian fluids with perfect slip boundary conditions

Abstract:

We study the generalized stationary Stokes system in a bounded domain in the plane equipped with perfect slip boundary conditions. We show natural stability results in oscillatory spaces, i.e. Hölder spaces and Campanato spaces including the border line spaces of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). Especially we show that under appropriate assumptions gradients of solutions are globally continues. Since the stress tensor is assumed to be governed by a general Orlicz function, our theory includes various cases of (possibly degenerate) shear thickening and shear thinning fluids; including the model case of power law fluids. It is a joint work with Sebastian Schwarzacher.

Ji?í Neustupa
 (Institute of Mathematics, CAS)

A contribution to the theory of regularity of a weak solution to the NavierStokes equations via one component of velocity and other related quantities

Abstract:

We deal with a suitable weak solution (v,p) to the NavierStokes equations, where v=(v_1,v_2,v_3). We give a brief survey of known criteria of regularity that use assumptions on just one component of v. We show that the regularity of (v,p) at a spacetime point (x_0,t_0) is essentially determined by the Serrintype integrability of the positive part of a certain linear combination of v_1^2, v_2^2, v_3^2 and p in a backward neighborhood of (x_0,t_0). An appropriate choice of coefficients in the linear combination leads to the Serrintype condition on one component of v or, alternatively, on the positive part of the Bernoulli pressure (1/2)v^2+p or the negative part of p, etc.
prof. RNDr. Eduard Feireisl, DrSc.
Šárka Nečasová, Milan Pokorný
chairmen