Mixing solutions for the Muskat problem with variable speed

AbstractWe provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

A damage initiation criterion for a class of viscoelastic solids

10.1098/rspa.2018.0064 ◽

2018 ◽

Vol 474 (2214) ◽

pp. 20180064

Author(s):

P. Alagappan ◽

K. R. Rajagopal ◽

K. Kannan

Keyword(s):

Ad Hoc ◽

The Body ◽

Relevant Parameter ◽

Elastic Solids ◽

Viscoelastic Solids ◽

Damage Initiation ◽

Link Type ◽

Deformed State ◽

Voigt Model ◽

We extend the methodology introduced for the initiation of damage within the context of a class of elastic solids to a class of viscoelastic solids (Alagappan et al. 2016 Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 472 , 20160231. ( doi:10.1098/rspa.2016.0231 )). In a departure from studies on damage that consider the body to be homogeneous, with initiation of damage being decided by parameters that are based on a quantity such as the strain, that requires information concerning a special reference configuration, or the use of ad hoc parameters that have no physically meaningful origins, in this study we use a physically relevant parameter that is completely determined in the current deformed state of the body to predict the initiation of damage. Damage is initiated due to the inhomogeneity of the body wherein certain regions in the body are unable to withstand the stresses, strains, etc. The specific inhomogeneity that is considered is the variation of the density in the body. We consider damage within the context of the deformation of two representative viscoelastic solids, a generalization of a model proposed by Gent (1996 Rubber Chemistry and Technology 69 , 59–61. ( doi:10.5254/1.3538357 )) for polymeric solids and a generalization of the Kelvin–Voigt model. We find that the criterion leads to results that are in keeping with the experiments of Gent & Lindley (1959 Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 249 , 195–205. ( doi:10.1098/rspa.1959.0016 )).

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

Meixner polynomials of the second kind and quantum algebras representing su(1, 1)

10.1098/rspa.2009.0497 ◽

2009 ◽

Vol 466 (2117) ◽

pp. 1409-1428 ◽

Cited By ~ 4

Author(s):

Gábor Hetyei

Keyword(s):

Orthogonal Polynomials ◽

Laguerre Polynomials ◽

Combinatorial Theory ◽

Quantum Algebras ◽

Combinatorial Approach ◽

Meixner Polynomials ◽

Link Type ◽

Polynomial Coefficients ◽

The Matrix ◽

We show how Viennot’s combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges (Hodges & Sukumar 2007 Proc. R. Soc. A 463 , 2401–2414 ( doi:10.1098/rspa.2007.0001 ); Sukumar & Hodges 2007 Proc. R. Soc. A 463 , 2415–2427 ( doi:10.1098/rspa.2007.0003 )) on the matrix entries in powers of certain operators in a representation of su(1, 1). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequence of results by Koelink, Groenevelt and Van Der Jeugt (Van Der Jeugt 1997 J. Math. Phys. 38 , 2728–2740 ( doi:10.1063/1.531984 ); Koelink & Van Der Jeugt 1998 SIAM J. Math. Anal. 29 , 794–822 ( doi:10.1137/S003614109630673X ); Groenevelt & Koelink 2002 J. Phys. A 35 , 65–85 ( doi:10.1088/0305-4470/35/1/306 )), for the related operators, substitutions into essentially the same Laguerre polynomials and Meixner polynomials of the second kind may be used to express their eigenvectors. Our combinatorial approach explains and generalizes this ‘coincidence’.

Magic informationally complete POVMs with permutations

Royal Society Open Science ◽

10.1098/rsos.170387 ◽

2017 ◽

Vol 4 (9) ◽

pp. 170387 ◽

Cited By ~ 9

Author(s):

Michel Planat ◽

Zafer Gedik

Keyword(s):

Quantum Computation ◽

Finite Geometries ◽

Link Type ◽

Universal Quantum ◽

Math Phys ◽

Stabilizer States

Eigenstates of permutation gates are either stabilizer states (for gates in the Pauli group) or magic states, thus allowing universal quantum computation (Planat, Rukhsan-Ul-Haq 2017 Adv. Math. Phys. 2017, 5287862 ( doi:10.1155/2017/5287862 )). We show in this paper that a subset of such magic states, when acting on the generalized Pauli group, define (asymmetric) informationally complete POVMs. Such informationally complete POVMs, investigated in dimensions 2–12, exhibit simple finite geometries in their projector products and, for dimensions 4 and 8 and 9, relate to two-qubit, three-qubit and two-qutrit contextuality.

On weak solutions of the boundary value problem within linear dilatational strain gradient elasticity for polyhedral Lipschitz domains

Mathematics and Mechanics of Solids ◽

10.1177/10812865211025576 ◽

2021 ◽

pp. 108128652110255

Author(s):

Victor A. Eremeyev ◽

Francesco dell’Isola

Keyword(s):

Strain Gradient ◽

Weak Solutions ◽

Existence And Uniqueness ◽

Numerical Solutions ◽

Gradient Elasticity ◽

Lipschitz Domains ◽

Elastic Model ◽

Strain Gradient Elasticity ◽

Existence And Uniqueness Theorem ◽

We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose potential energy density depends quadratically on linear strains and on the gradient of dilatation only. Such a model has many applications, e.g., to describe phenomena of interest in poroelasticity or in some situations where media with scalar microstructure are necessary. We present an extension of the previous results by Eremeyev et al. (2020 Z angew Math Phys 71(6): 1–16) to the case of domains with edges and when external line forces are applied. Let us note that the interest paid to Lipschitz polyhedra-type domains is at least twofold. First, it is known that geometrical singularity of the boundary may essentially influence singularity of solutions. On the other hand, the analysis of weak solutions in polyhedral domains is of great significance for design of optimal computations using a finite-element method and for the analysis of convergence of numerical solutions.

Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport

Archive for Rational Mechanics and Analysis ◽

10.1007/s00205-020-01579-3 ◽

2020 ◽

Cited By ~ 2

Author(s):

Matt Jacobs ◽

Inwon Kim ◽

Alpár R. Mészáros

Keyword(s):

Surface Tension ◽

Weak Solutions ◽

Mean Curvature Flow ◽

Optimal Transport ◽

Heat Content ◽

Threshold Dynamics ◽

Muskat Problem ◽

Equilibrium Configurations ◽

Wasserstein Space ◽

Energy Convergence

Abstract Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoḡlu–Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations.

SIG 15 Perspectives Vol. 20, No. 3, September 2015

Perspectives on Gerontology ◽

10.1044/gero20.3.c1 ◽

2015 ◽

Vol 20 (3) ◽

Keyword(s):

Learning Center ◽

Abstract Download the CE Questions PDF from the toolbar, above. Use the questions to guide your Perspectives reading. When you're ready, purchase the activity from the ASHA Store and follow the instructions to take the exam in ASHA's Learning Center. Available until August 13, 2018.

SIG 11 Perspectives Vol. 22, No. 2, July 2012

Perspectives on Administration and Supervision ◽

10.1044/aas22.2.1-ce ◽

2012 ◽

Vol 22 (2) ◽

Author(s):

Kathryn Taylor ◽

Emily White ◽

Rachael Kaplan ◽

Colleen M. O'Rourke

Keyword(s):

Sorry, this activity is no longer available for CEUs. Visit the SIG 11 page on the ASHA Store to see available CE activities. Use the CE questions PDF here as study questions to guide your Perspectives reading.

Perspectives on Communication Disorders and Sciences in Culturally and Linguistically Diverse Populations ◽

SIG 14 Perspectives Vol. 20, No. 3, December 2013

10.1044/cds20.3.1 ◽

2013 ◽

Vol 20 (3) ◽

Keyword(s):

Sorry, this activity is no longer available for CEUs. Visit the SIG 14 page on the ASHA Store to see available CE activities. Use the CE questions PDF here as study questions to guide your Perspectives reading.

SIG 17 Perspectives Vol. 2, No. 1, May 2012

Perspectives on Global Issues in Communication Sciences and Related Disorders ◽

10.1044/gics2.1.1-ce ◽

2012 ◽

Vol 2 (1) ◽

pp. 1-5

Author(s):

Celeste Domsch

Keyword(s):

Sorry, this activity is no longer available for CEUs. Visit the SIG 17 page on the ASHA Store to see available CE activities. Use the CE questions PDF here as study questions to guide your Perspectives reading.

SIG 12 Perspectives Vol. 21, No. 4, December 2012

Perspectives on Augmentative and Alternative Communication ◽

10.1044/aac21.4.1-ce ◽

2012 ◽

Vol 21 (4) ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Cathy Binger ◽

Jennifer Kent-Walsh

Keyword(s):

Sorry, this activity is no longer available for CEUs. Visit the SIG 12 page on the ASHA Store to see available CE activities. Use the CE questions PDF here as study questions to guide your Perspectives reading.