Dominic Breit

SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs

We consider functions u ∈ L∞(L2)∩Lp(W1, p) with 1 < p < ∞ on a time–space domain. Solutions to nonlinear evolutionary PDEs typically belong to these spaces. Many applications require a Lipschitz approximation uλ of u which coincides with u on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids of Diening, Ruzicka and Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010) 1–46. Since divuλ = 0, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of Breit, Diening and Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics, J. Differential Equations253 (2012) 1910–1942.

Finite Element Approximation of the

We study a priori estimates for the

p(\cdot)

p(\cdot)

-Laplacian

-Laplace Dirichlet problem,

New regularity theorems for non-autonomous anisotropic variational integrals

-\mathrm{div}(\vert{\nabla \mathbf{v}}\vert^{p(\cdot)-2} \nabla \mathbf{v}) = \mathbf{f}

Incompressible Limit for Compressible Fluids with Stochastic Forcing

. We show that the gradients of the finite element approximation with zero boundary data converge with rate

Existence Theory for Stochastic Power Law Fluids

O(h^\alpha)

Trace-free korn inequalities in orlicz spaces

if the exponent

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

p

Weak solutions for a non-Newtonian diffuse interface model with different densities

is