# Mark SteinhauerUniversität Koblenz-Landau · Department of Mathematics

Mark Steinhauer

## About

19

Publications

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654

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Citations since 2016

Introduction

**Skills and Expertise**

## Publications

Publications (19)

We consider weak solutions of nonlinear elliptic systems in a \(W^{1,p}\)-setting which arise as Euler–Lagrange equations of certain variational integrals with pollution term, and we also consider minimizers of a variational problem. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes w...

We obtain everywhere 𝒞α-regularity for vector solutions to a class of nonlinear elliptic systems whose principal part is the Euler operator to a variational integral ∫F(u,∇u)dx with quadratic growth in ∇u and which satisfies a generalized splitting condition that cover the case F(u,∇u):=∑_i Qi, where Q_i:=∑_{αβ}A_{αβ}^i(u,∇u)∇u_α⋅∇u_β, or the case...

We consider a two-dimensional generalized Kelvin-Voigt model describing a motion of a compressible viscoelastic body. We establish the existence of a unique classical solution to such a model in the spatially periodic setting. The proof is based on Meyers' higher integrability estimates that guarantee the Holder continuity of the gradient of veloci...

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two dimensional case and show the existence
of a weak solution in the case of mixed boundary conditions. The proof is based on an analysis of the stream function, exploiting
the convective term.
Mathematics Subject Classification (1991)Primary: 35D05–35Q30–Secon...

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two-dimensional case and show the existence
of a weak solution for homogeneous Dirichlet boundary conditions.

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three dimensional case and show the existence of a weak solution for homogeneous Dirichlet (no-slip) boundary conditions under the assumption that the adiabatic exponent satisfies γ>43. In particular we cover with our existence result the cases of a monoatomic ga...

We study properties of Lipschitz truncations of Sobolev functions
with constant and variable exponent.
As non-trivial applications we use the
Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in
[Frehse et al., SIAM J. Math. Anal 34 (2003) 1064–1083]. We also establish ne...

no. 347 Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Abstract. We consider the Navier–Stokes equations for compressible isentropic flow in the steady three dimensional case and show the exis-tence...

TÃ ecnico; Av. Rovisco Pais; 1049-001 Lisboa; Portugal Communicated by P. Colli SUMMARY This note bridges the gap between the existence and regularity classes for the third-grade Rivlin– Ericksen uid equations. We obtain a new global a priori estimate, which conveys the precise regularity conditions that lead to the existence of a global in time re...

We study time-dependent flows of incompressible degenerate power-law fluids
characterized by the power-law index $p-2$ with $p>2$. In this case, the generalized
viscosity vanishes as (the modulus of) the shear rate tends to zero. We prove
global-in-time existence of a weak solution if $p>\max\{\frac{3d-4}{d},2\}$. This improves
the range $p>\frac{3...

We consider the Navier–Stokes equations for compressible isentropic flow in the steady three-dimensional case. The pressure ρ
γ
and the kinetic energy
are estimated uniformly in L
q
with
ρ being the density. This is an improvement of known estimates in the case

The p-Laplace equation is considered for p > 2 on a n-dimensional convex polyhedral domain under a Dirichlet boundary value condition. Global regularity of weak solutions in weighted Sobolev spaces and in fractional order Nikolskij and Sobolev spaces are proven.

We deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for p > 2/d+2. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test f...

We survey and improve some results concerning uniqueness and regularity of solutions to the instationary Navier-Stokes equations in three (and higher) dimensions. In particular we show that the class of weak solutions which additionally belong to the space L
2(0,T; BMO) guarantees uniqueness as well as regularity. The method of proof which we prese...

We give an example of a bounded discontinuous divergence-free solution of a linear elliptic system with measurable bounded coefficients in R 3 and a corresponding example for a Stokes-like system.

In this thesis we consider systems of partial differential equations of continuum mechanics and analyze regularity properties of their weak solutions. The first chapter contains a detailed introduction and reviews the contents of chapter two, three and four. We start in chapter 2 with the local regularity problem related to the equations modelling...

## Projects

Project (1)