Jörg Wolf

A New Criterion for Partial Regularity of Suitable Weak Solutions to the Navier-Stokes Equations

In the present paper we study local properties of suitable weak solutions to the Navier-Stokes equation in a cylinder Q = Ω × (0, T). Using the local representation of the pressure we are able to define a positive constant ɛ⋆ such that for every parabolic subcylinder QR ⊂ Q the condition

Existence of Weak Solutions to the Equations of Natural Convection with Dissipative Heating

R^{-2}\int_{Q_R}|u|^3dxdt\leq\varepsilon_{\ast}

Existence of weak solutions for unsteady motions of generalized Newtonian fluids

implies \({\bf U}\in L^{\infty}(Q_{R/2})\)). As one can easily check this condition is weaker then the well known Serrins condition as well as the condition introduced by Farwig, Kozono and Sohr in a recent paper. Since our condition can be verified for suitable weak solutions to the Navier-Stokes system it improves the known results substantially.

Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity

In this paper, we prove a Meyers’ type estimate for weak solutions to a Stokes system with bounded measurable coefficients in place of the usual constant viscosity. Besides the perturbation argument due to Meyers, we make use of the solvability of the classical Stokes problem in [W 1, q 0,σ (Ω)] n (n = 2 or n = 3, 2 < q < 3 + e, ∂Ω Lipschitz).

On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions

In this paper, we prove the existence of a weak solution to a system of PDE’s which model the non-stationary motion of a heat-conducting incompressible viscous fluid including the effects of dissipative and adiabatic heating. Our method of proof consists in approximating these heat sources by bounded nonlinearities.

On the local regularity of suitable weak solutions to the generalized Navier-Stokes equations

We study Beltrami flows in the setting of weak solution to the stationary Euler equations in

A direct proof of the Caffarelli-Kohn-Nirenberg theorem

\Bbb R^3

On the boundary regularity of suitable weak solutions to the Navier-Stokes equations

. For this weak Beltrami flow we prove the regularity and the Liouville property. In particular, we show that if tangential part of the velocity has certain decay property at infinity, then the solution becomes trivial. This decay condition of of the velocity is weaker than the previously known sufficient conditions for the Liouville property of the Betrami flows. For the proof we establish a mean value formula and other various formula for the tangential and the normal components of the weak solutions to the stationary Euler equations.