On Korn’s first inequality for mixed tangential and normal boundary conditions on bounded Lipschitz domains in $$\mathbb {R}^N$$ R N

Bauer, Sebastian; Pauly, Dirk

doi:10.1007/s11565-016-0247-x

Cited by 14 publications

(13 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…), and using the refined Helmholtz type decompositions (10) and (11) as well as the results of Lemma 2.2 we immediately see: Lemma 2.7 (fa-toolbox lemma 3). The following assertions are equivalent:…”

Section: Remark 23 (Sufficient Assumptions For the Fa-toolbox)mentioning

confidence: 85%

Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

Pauly¹,

Valdman²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Remark 23 (Sufficient Assumptions For the Fa-toolbox)mentioning

confidence: 85%

Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

Pauly¹,

Valdman²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…As r < 9 5 (< 3) and W 1,r 0 (Ω) is compactly embedded into L q (Ω) for any q ∈ [1, 3r 3−r ), we also have…”

Section: No-slip Casementioning

confidence: 98%

“…Note that the restriction r ≥ 9 5 is due to the requirement that for s ∈ N arbitrary . In this case, we consider, for > 0, the following approximating problem:…”

Section: No-slip Casementioning

confidence: 99%

On the Classification of Incompressible Fluids and a Mathematical Analysis of the Equations That Govern Their Motion

Blechta¹,

Málek²,

Rajagopal³

2020

SIAM J. Math. Anal.

View full text Add to dashboard Cite

In the first part of the paper we provide a new classification of incompressible fluids characterized by a continuous monotone relation between the velocity gradient and the Cauchy stress. The considered class includes Euler fluids, Navier-Stokes fluids, classical power-law fluids as well as stress power-law fluids, and their various generalizations including the fluids that we refer to as activated fluids, namely fluids that behave as an Euler fluid prior activation and behave as a viscous fluid once activation takes place. We also present a classification concerning boundary conditions that are viewed as the constitutive relations on the boundary. In the second part of the paper, we develop a robust mathematical theory for activated Euler fluids associated with different types of the boundary conditions ranging from no-slip to free-slip and include Navier's slip as well as stick-slip. Both steady and unsteady flows of such fluids in three-dimensional domains are analyzed.

show abstract

“…] d×d , and Z = [L 2 (Ω(t))] d , and the linear map A := ∇ s : X → Y and the natural embedding T := id : X → Z. Theorem 13 in [2] shows that A is an injection. The natural embedding T is compact by Rellich-Kondrachov Theorem.…”

Section: Well-posedness Of Velocity-pressure Systemmentioning

confidence: 99%

Numerical solution of a two dimensional tumour growth model with moving boundary

Droniou,

Flegg,

Remesan

2020

Preprint

View full text Add to dashboard Cite

We consider a biphasic continuum model for avascular tumour growth in two spatial dimensions, in which a cell phase and a fluid phase follow conservation of mass and momentum. A limiting nutrient that follows a diffusion process controls the birth and death rate of the tumour cells. The cell volume fraction, cell velocity-fluid pressure system, and nutrient concentration are the model variables. A coupled system of a hyperbolic conservation law, a viscoelastic system, and a parabolic diffusion equation governs the dynamics of the model variables. The tumour boundary moves with the normal velocity of the outermost layer of cells, and this time-dependence is a challenge in designing and implementing a stable and fast numerical scheme. We recast the model into a form where the hyperbolic equation is defined on a fixed extended domain and retrieve the tumour boundary as the interface at which the cell volume fraction decreases below a threshold value. This procedure eliminates the need to track the tumour boundary explicitly and the computationally expensive re-meshing of the time-dependent domains. A numerical scheme based on finite volume methods for the hyperbolic conservation law, Lagrange P 2 − P 1 Taylor-Hood finite element method for the viscoelastic system, and mass-lumped finite element method for the parabolic equations is implemented in two spatial dimensions, and several cases are studied. We demonstrate the versatility of the numerical scheme in catering for irregular and asymmetric initial tumour geometries. When the nutrient diffusion equation is defined only in the tumour region, the model depicts growth in free suspension. On the contrary, when the nutrient diffusion equation is defined in a larger fixed domain, the model depicts tumour growth in a polymeric gel. We present numerical simulations for both cases and the results are consistent with theoretical and heuristic expectations such as early linear growth rate and preservation of radial symmetry when the boundary conditions are symmetric. The work presented here could be extended to include the effect of drug treatment of growing tumours.

show abstract

On Korn’s first inequality for mixed tangential and normal boundary conditions on bounded Lipschitz domains in $$\mathbb {R}^N$$ R N

Abstract: We prove that for bounded Lipschitz domains in R N Korn's first inequality holds for vector fields satisfying homogeneous mixed tangential and normal boundary conditions. Date: June 14, 2018. 1991 Mathematics Subject Classification. 49J40 / 82C40 / 76P05.

Cited by 14 publications

References 10 publications

Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

On the Classification of Incompressible Fluids and a Mathematical Analysis of the Equations That Govern Their Motion

Numerical solution of a two dimensional tumour growth model with moving boundary

Contact Info

Product

Resources

About