# A Unified Theory for Some Non-Newtonian Fluids Under Singular Forcing

@article{Bulek2016AUT,
title={A Unified Theory for Some Non-Newtonian Fluids Under Singular Forcing},
author={Miroslav Bul{\'i}{\vc}ek and Jan Burczak and Sebastian Schwarzacher},
journal={SIAM J. Math. Anal.},
year={2016},
volume={48},
pages={4241-4267}
}
• Published 28 April 2016
• Mathematics
• SIAM J. Math. Anal.
We consider a model of steady, incompressible non-Newtonian flow with neglected convective term under external forcing. Our structural assumptions allow for certain nondegenerate power-law or Carreau-type fluids. Within our setting, we provide the full-range theory, namely, existence, optimal regularity, and uniqueness of solutions, not only with respect to forcing belonging to Lebesgue spaces, but also with respect to their refinements, namely, the weighted Lebesgue spaces, with weights in a…
Well posedness of nonlinear parabolic systems beyond duality
• Mathematics
Annales de l'Institut Henri Poincaré C, Analyse non linéaire
• 2019
Existence of steady solutions of Newtonian and Non-Newtonian fluids with right hand sides beyond duality
• Mathematics
• 2019
We provide new a-priori estimates for classical Navier-Stokes equations as well as non-Newtonian fluids of power law type with right-hand sides that are not in the natural existence class. The
Weighted estimates for generalized steady Stokes systems in nonsmooth domains
• Mathematics
• 2016
We consider a generalized steady Stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted Lq space for 2<q<∞. We prove the global weighted
A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow
We propose a semismooth Newton method for non-Newtonian models of incompressible flow where the constitutive relation between the shear stress and the symmetric velocity gradient is given implicitly;
On the analysis and approximation of some models of fluids over weighted spaces on convex polyhedra
• Mathematics
Numerische Mathematik
• 2022
This work studies the Stokes problem over convex, polyhedral domains on weighted Sobolev spaces, and provides well--posedness and approximation results to some classes of non--Newtonian fluids.
VARIABLE LORENTZ ESTIMATE FOR GENERALIZED STOKES SYSTEMS IN NON-SMOOTH DOMAINS
• Mathematics
• 2019
We prove a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for the variable power of the gradient of weak solution pair (u, P ) to the generalized steady Stokes system over a
Stability of the Stokes projection on weighted spaces and applications
• Mathematics
Math. Comput.
• 2020
It is shown that, on convex polytopes and two or three dimensions, the finite element Stokes projection is stable on weighted spaces where the weight belongs to a certain Muckenhoupt class and the integrability index can be different from two.

## References

SHOWING 1-10 OF 32 REFERENCES
SOLENOIDAL LIPSCHITZ TRUNCATION FOR PARABOLIC PDEs
• Mathematics
• 2013
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
On steady flows of incompressible fluids with implicit power-law-like rheology
• Mathematics
• 2009
Abstract We consider steady flows of incompressible fluids with power-law-like rheology given by an implicit constitutive equation relating the Cauchy stress and the symmetric part of the velocity
Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems
• Mathematics
• 2016
We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of
Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids
• Mathematics
• 2001
Introduction 1 Weak solutions to boundary value problems in the deformation theory of perfect elastoplasticity 1.0. Preliminaries 1.1. The classical boundary value problem for the equilibrium state
The Navier-Stokes Equations Ii-Theory and Numerical Methods: Proceedings of a Conference Held in Oberwolfach, Germany, August 18-24, 1991
• Mathematics
• 1993
Analyticity of a free boundary in plane quasi-steady flow of a liquid form subject to variable surface tension.- On a free boundary problem for the stationary navier-stokes equations with a dynamic
Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces
We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the
Regularity for a class of non-linear elliptic systems
where ~ is a smooth positive function satisfying the ellipticity condition o(Q) + 2~'(Q)q > 0, V denotes the gradient, and [Vs[2 = ~ = 1 ]Vskl 2. This type of system arises as the EulerLagrange
Existence of very weak solutions to elliptic systems of p-Laplacian type
• Mathematics
• 2016
We study vector valued solutions to non-linear elliptic partial differential equations with p-growth. Existence of a solution is shown in case the right hand side is the divergence of a function
On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications
• Mathematics
• 2008
We study properties of Lipschitz truncations of Sobolev functions with constant and vari- able exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of
Lq theory for a generalized Stokes System
• Mathematics
• 2013
Regularity properties of solutions to the stationary generalized Stokes system are studied. The extra stress tensor is assumed to have a growth given by some N-function, which includes the situation