Petr Kaplický

A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain, posed with homogeneous Dirichlet boundary conditions. The distinguishing feature of this class is the indefinite superlinear (but subcritical) growth of the nonlinearity at infinity. We present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria. In addition, we can prove the existence of connecting orbits in many cases. One advantage of our approach is that the parabolic semiflow is naturally order preserving, in contrast to pseudo-gradient flows considered when using variational methods. Therefore we can obtain much information on nodal properties of equilibria that was not known before.

An L 2-maximal regularity result for the evolutionary Stokes–Fourier system

We establish an L 2-regularity result for a weak solution of the evolutionary Stokes–Fourier system. Although this system does not contain the convective terms, the fact that the viscosity depends on the temperature makes the considered system of partial differential equations nonlinear. The result holds for a class of the viscosities that includes the Arrhenius formula as a special case. For simplicity, we restrict ourselves to a spatially periodic setting in this study.

ON GLOBAL EXISTENCE OF SMOOTH TWO-DIMENSIONAL STEADY FLOWS FOR A CLASS OF NON-NEWTONIAN FLUIDS UNDER VARIOUS BOUNDARY CONDITIONS

We study steady two-dimensional flows of shear dependent fluids in a bounded domain subjected to three kinds of boundary conditions: (i) general nonhomogeneous Dirichlet, (ii) nonhomogeneous Dirichlet with zero normal component at the boundary (fixed wall) and (iii) free-stick (slippery boundary). The existence of a C1,α-solution is proved: while condition (i) requires smallness of a given function at boundary, conditions (ii) provide smooth solutions for all choice of data. Some results regarding a special construction of an extension operator are interesting on their own.

Boundary regularity of flows under perfect slip boundary conditions

We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ϕ-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.

Evolutionary, symmetric

We consider the evolutionary symmetric

p

p

-Laplacian. Interior regularity of time derivatives and its consequences

-Laplacian with safety

Gradient Lq theory for a class of non-diagonal nonlinear elliptic systems

1

ISOMETRIC REPRESENTATION OF LIPSCHITZ-FREE SPACES OVER CONVEX DOMAINS IN FINITE-DIMENSIONAL SPACES

. By symmetric we mean that the full gradient of

BMO estimates for the p-Laplacian

p