Journal•ISSN: 0862-7940
Applications of Mathematics
Springer Science+Business Media
About: Applications of Mathematics is an academic journal. The journal publishes majorly in the area(s): Finite element method & Boundary value problem. It has an ISSN identifier of 0862-7940. Over the lifetime, 1438 publications have been published receiving 10461 citations.
Topics: Finite element method, Boundary value problem, Nonlinear system, Mixed finite element method, Uniqueness
Papers published on a yearly basis
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TL;DR: A survey of regularity results for both minima of variational integrals and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations can be found in this article.
Abstract: I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...
410 citations
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TL;DR: In this paper, the authors discuss implicit constitutive theories for the Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process.
Abstract: In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained. In general, for such materials the material moduli that characterize the extra stress could depend on the constraint reaction. (E.g., in an incompressible fluid, the viscosity could depend on the constraint reaction associated with the constraint of incompressibility. In the linear case, this would be the pressure.) Here we discuss such implicit constitutive theories. We also discuss a class of bodies described by an implicit constitutive relation for the specific Helmholtz potential that depends on both the stress and strain, and which does not dissipate in any admissible process. The stress in such a material is not derivable from a potential, i.e., the body is not hyperelastic (Green elastic).
376 citations
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TL;DR: The mathematical interest of analyzing another degenerate parabolic system proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels is mentioned.
Abstract: Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia coli or amoeba like Dictyostelium discoideum exhibiting pointwise concentrations. For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted. The two systems can be unified by a kinetic approach that was proposed for Escherichia coli, based on more precise experiments showing a movement by ‘jump and tumble’. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells. We also mention the mathematical interest of analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels.
156 citations
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136 citations
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TL;DR: In this article, the authors study the nonstationary Navier-Stokes equations in the entire 3D space and give some criteria on certain components of gradient of the velocity which ensure its global-in-time smoothness.
Abstract: We study the nonstationary Navier-Stokes equations in the entire three-dimensional space and give some criteria on certain components of gradient of the velocity which ensure its global-in-time smoothness.
124 citations