On the equations rot v=g and div u=f with zero boundary conditions

Borchers, Wolfgang; Sohr, Hermann

doi:10.14492/hokmj/1381517172

Cited by 220 publications

(162 citation statements)

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“…The first main result of the present paper is the existence of finite energy weak martingale solutions to (2). The precise statement is given in Theorem 1.…”

Section: Introductionmentioning

confidence: 89%

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Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

Mensah

2017

Annali di Matematica

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We give an existence and asymptotic result for the so-called finite energy weak martingale solution of the compressible isentropic Navier-Stokes system driven by some random force in the whole spatial region. In particular, given a general nonlinear multiplicative noise, we establish the convergence to the incompressible system as the Mach number, representing the ratio between the average flow velocity and the speed of sound, approaches zero.

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“…The first main result of the present paper is the existence of finite energy weak martingale solutions to (2). The precise statement is given in Theorem 1.…”

Section: Introductionmentioning

confidence: 89%

“…In estimating J 2 , we use instead the Bogovskiȋ operator in negative spaces which can be found in [18, Proposition 2.1], [2,10].…”

Section: Remarkmentioning

confidence: 99%

Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

Mensah

2017

Annali di Matematica

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“…The sequence u n is therefore bounded in W 1,4 . By the compact embedding W 1,4 → L 2 , we finally deduce that there is a subsequence, again denoted by u n , such that u n u in W weakly.…”

Section: Existence Of H 1 Solutionsmentioning

confidence: 94%

On steady third grade fluids equations

Busuioc¹,

Iftimie²,

Paicu³

2008

Nonlinearity

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Abstract. Let Ω be a simply connected, bounded, smooth domain of R 2 or R 3 . We consider the equation of steady motion of a third grade fluid in Ω with homogeneous Dirichlet boundary conditions. We prove that the monotonicity technique used by Paicu [17] in the full space for unsteady motion allows to obtain the existence of a W 1,4 0 solution provided that the forcing belongs to W −1, 4 3 . The size of the forcing is arbitrary.

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“…Its first published proof for domains with smooth boundaries appeared in Duvaut & Lions [46], p. 111; another proof was also given by Tartar [84]. Various extensions to "genuine" domains, i.e., with Lipschitzcontinuous boundaries, are given in Bolley & Camus [14], Geymonat & Suquet [Part 1 [52], and Borchers & Sohr [15]; Amrouche & Girault [6], Proposition 2.10 even proved that the more general implication {v ∈ D (Ω) and…”

Section: A Fundamental Lemma Of Jl Lionsmentioning

confidence: 99%

An Introduction to Shell Theory

Ciarlet

Mardare

2008

Series in Contemporary Applied Mathematics

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Abstract. These notes are intended to provide a thorough introduction to the mathematical theory of elastic shells.The main objective of shell theory is to predict the stress and the displacement arising in an elastic shell in response to given forces. Such a prediction is made either by solving a system of partial differential equations or by minimizing a functional, which may be defined either over a three-dimensional set or over a two-dimensional set, depending on whether the shell is viewed in its reference configuration as a three-dimensional or as a two-dimensional body (the latter being an abstract idealization of the physical shell when its thickness is "small").The first part of this article is devoted to the three-dimensional theory of elastic bodies, from which the three-dimensional theory of shells is obtained simply by replacing the reference configuration of a general body with that of a shell. The particular shape of the reference configuration of the shell does not play any rôle in this theory.The second part is devoted to the two-dimensional theory of elastic shells. In contrast to the three-dimensional theory, this theory is specific to shells, since it essentially depends on the geometry of the reference configuration of a shell.For a more comprehensive exposition of the theory of elastic shells, we refer the reader to Ciarlet [18] and the references therein for the first part of the article, and to Ciarlet [20] and the references therein for the second part. Une introduction a la théorie des coquesResumé. Ces notes sont destinéesà fournir une introduction détailléeà la théorie mathématique des coquesélastiques.L'objectif principal de la théorie des coques est de prédire les contraintes et les dépla-cements survenant dans une coqueélastique en réponseà des forces appliquées. Une telle prédiction est faite soit en résolvant un système d'équations aux dérivées partielles, soit en minimisant une fonctionnelle, qui peuventêtre définis soit dans un ensemble tridimensionnel, soit dans un ensemble bidimensionnel, selon que la coque est vue dans sa configuration de référence comme un corps tridimensionnel ou comme un corps bidimensionnel (ce dernierétant alors une idéalisation de la coque physique lorsque sonépaisseur est "petite").La première partie de cet article est consacréeà la théorie tridimensionnelle des corpś elastiques,à partir de laquelle la théorie tridimensionnelle des coques est obtenue en remplaçant simplement la configuration de référence d'un corpsélastique général par celle d'une coque. La forme particulière de la configuration de référence d'une coque ne joue aucun rôle dans cette théorie.La deuxième partie de cet article est consacréeà la théorie bidimensionnelle des coqueś elastiques. Contrairementà la théorie tridimensionnelle, cette théorie est spécifique aux coques, puisqu'elle dépend de façon essentielle de la géométrie de la configuration de référence de la coque.Pour un exposé plus complet de la théorie de coquesélastiques, nous renvoyons le lecteurà Ciarlet [18] et ses réf...

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On the equations rot v=g and div u=f with zero boundary conditions

Cited by 220 publications

References 0 publications

Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

On steady third grade fluids equations

An Introduction to Shell Theory

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