In this paper, we analyze the three-dimensional exterior Stokes problem with the Navier slip boundary conditions, describing the flow of a viscous and incompressible fluid past an obstacle where it ...is assumed that the fluid may slip at the boundary. Because the flow domain is unbounded, we set the problem in weighted spaces in order to control the behavior at infinity of the solutions. This functional framework also allows to prescribe various behaviors at infinity of the solutions (growth or decay). Existence and uniqueness of solutions are shown in a Hilbert setting which gives the tools for a possible numerical analysis of the problem. Weighted Korn's inequalities are the key point in order to study the variational problem.
Nonlinear Korn inequalities Ciarlet, Philippe G.; Mardare, Cristinel
Journal de mathématiques pures et appliquées,
12/2015, Letnik:
104, Številka:
6
Journal Article
Recenzirano
Odprti dostop
Let Ω be a bounded and connected open subset of Rn with a Lipschitz-continuous boundary Γ, the set Ω being locally on the same side of Γ, and let Θ:Ω‾→Rn and Φ:Ω‾→Rn be two smooth enough ...“deformations” of the set Ω‾. Then the classical Korn inequality asserts that, when Θ=id, there exists a constant c such that‖v‖H1(Ω)≤c(‖v‖L2(Ω)+‖∇v+∇vT‖L2(Ω)) for all v∈H1(Ω), where v:=(Φ−id):Ω‾→Rn denotes the corresponding “displacement” vector field, and where the symmetric tensor field ∇v+∇vT:Ω‾→Sn is nothing but the linear part with respect to v of the difference between the metric tensor fields ∇ΦT∇Φ and I that respectively correspond to the deformations Φ and Θ=id.
Assume now that the identity mapping id is replaced by a more general orientation-preserving immersion Θ∈C1(Ω‾;Rn). We then show in particular that, given any 1<p<∞ and any q∈R such that max{1,p2}≤q≤p, there exists a constant C=C(p,q,Θ) such that‖Φ−Θ‖W1,p(Ω)≤C(‖Φ−Θ‖Lp(Ω)+‖∇ΦT∇Φ−∇ΘT∇Θ‖Lq(Ω)q/p) for all Φ∈W1,2q(Ω) that satisfy det∇Φ>0 almost everywhere in Ω. Such an inequality thus constitutes an instance of a “nonlinear Korn inequality”, in the sense that the symmetric tensor field ∇ΦT∇Φ−∇ΘT∇Θ:Ω‾→Sn appearing in its right-hand side is now the exact difference between the metric tensor fields corresponding to the deformations Φ and Θ.
We also show that, like in the linear case, an analogous nonlinear Korn inequality holds, but without the norm ‖Φ−Θ‖Lp(Ω) in its right-hand side, if the difference Φ−Θ vanishes on a subset Γ0 of Γ with dΓ-meas Γ0>0.
The key to providing such nonlinear Korn inequalities is a generalization of the landmark “geometric rigidity lemma in H1(Ω)” established in 2002 by G. Friesecke, R.D. James, and S. Müller, as later extended to W1,p(Ω) by S. Conti.
Soit Ω un ouvert borné connexe de Rn de frontière Γ lipschitzienne, l'ensemble Ω étant localement d'un seul coté de Γ, et soit Θ:Ω‾→Rn et Φ:Ω‾→Rn deux “déformations” suffisamment régulières de l'ensemble Ω‾. Alors l'inégalité de Korn classique exprime que, lorsque Θ=id, il existe une constante c telle que‖v‖H1(Ω)≤c(‖v‖L2(Ω)+‖∇v+∇vT‖L2(Ω)) pour tout v∈H1(Ω), où v:=(Φ−id):Ω‾→Rn represente le champ de “déplacements” correspondant, et où le champ de tenseurs symétriques ∇v+∇vT:Ω‾→Sn n'est autre que la partie linéaire en v de la différence entre les champs de tenseurs métriques ∇ΦT∇Φ et I, qui correspondent respectivement aux déformations Φ et Θ=id.
Supposons maintenant que l'application identique id est remplacée par une immersion plus générale Θ∈C1(Ω‾;Rn) qui préserve l'orientation. Alors on montre en particulier que, pour tout 1<p<∞ et pour tout q∈R tels que max{1,p2}≤q≤p, il existe une constante C=C(p,q,Θ) telle que‖Φ−Θ‖W1,p(Ω)≤C(‖Φ−Θ‖Lp(Ω)+‖∇ΦT∇Φ−∇ΘT∇Θ‖Lq(Ω)q/p) pour tout Φ∈W1,2q(Ω) satisfaisant det∇Φ>0 presque partout dans Ω. Une telle inégalité constitue donc un exemple d'“inégalité de Korn non linéaire”, au sens que le champ de tenseurs symétriques ∇ΦT∇Φ−∇ΘT∇Θ:Ω‾→Sn apparaissant dans son membre de droite est maintenant la différence exacte entre les champs de tenseurs métriques correspondant aux déformations Φ et Θ.
On montre également que, comme dans le cas linéaire, on peut établir une inégalité de Korn non linéaire analogue, mais sans la norme ‖Φ−Θ‖Lp(Ω) dans son membre de droite, si la différence Φ−Θ s'annule sur une partie Γ0 de Γ telle que dΓ-mes Γ0>0.
La clef pour établir de telles inégalités de Korn non linéaires est une généralisation du remarquable “lemme de rigidité géométrique dans H1(Ω)” établi en 2002 par G. Friesecke, R.D. James, et S. Müller, tel qu'il a été ensuite étendu à W1,p(Ω) par S. Conti.
We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic ...to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.
For a bounded domain Ω⊂R3 with Lipschitz boundary Γ and some relatively open Lipschitz subset Γt≠∅ of Γ, we prove the existence of some c>0, such ...that(0.1)c‖T‖L2(Ω,R3×3)≤‖symT‖L2(Ω,R3×3)+‖CurlT‖L2(Ω,R3×3) holds for all tensor fields in H(Curl;Ω), i.e., for all square-integrable tensor fields T:Ω→R3×3 with square-integrable generalized rotation CurlT:Ω→R3×3, having vanishing restricted tangential trace on Γt. If Γt=∅, (0.1) still holds at least for simply connected Ω and for all tensor fields T∈H(Curl;Ω) which are L2(Ω)-perpendicular to so(3), i.e., to all skew-symmetric constant tensors. Here, both operations, Curl and tangential trace, are to be understood row-wise.
For compatible tensor fields T=∇v, (0.1) reduces to a non-standard variant of the well known Korn's first inequality in R3, namelyc‖∇v‖L2(Ω,R3×3)≤‖sym∇v‖L2(Ω,R3×3) for all vector fields v∈H1(Ω,R3), for which ∇vn, n=1,…,3, are normal at Γt. On the other hand, identifying vector fields v∈H1(Ω,R3) (having the proper boundary conditions) with skew-symmetric tensor fields T, (0.1) turns to Poincaré's inequality since2c‖v‖L2(Ω,R3)=c‖T‖L2(Ω,R3×3)≤‖CurlT‖L2(Ω,R3×3)≤2‖∇v‖L2(Ω,R3). Therefore, (0.1) may be viewed as a natural common generalization of Korn's first and Poincaré's inequality. From another point of view, (0.1) states that one can omit compatibility of the tensor field T at the expense of measuring the deviation from compatibility through CurlT. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions and the Maxwell estimate.
We develop a lowest-order nonconforming virtual element method for planar linear elasticity, which can be viewed as an extension of the idea in Falk (1991) to the virtual element method (VEM), with ...the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence. The crucial step is to establish the discrete Korn's inequality, yielding the coercivity of the discrete bilinear form. We also provide a unified locking-free scheme both for the conforming and nonconforming VEMs in the lowest-order case. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.
We consider a viscous fluid in a finite-depth domain of two dimensions, with a free moving boundary and a fixed solid boundary. The fluid dynamics are governed by gravity-driven incompressible ...micropolar equations, and the surface tension is neglected on the free surface. The main result of this paper is to prove the global well-posedness of the surface wave problem when the fluid domain is horizontally periodic. And the solutions decay to equilibrium at an almost exponential rate. This is achieved by establishing delicate estimates when dealing with the strong coupling of fluid velocity and micro-rotation velocity. Our proof relies on two-tier nonlinear energy method developed by Guo and Tice 17–19.
The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the ...constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity.
The fundamental theorem of surface theory asserts that a surface in the three-dimensional Euclidean space E3 can be reconstructed from the knowledge of its two fundamental forms under the assumptions ...that their components are smooth enough—classically in the space C2(ω) for the first one and in the space C1(ω) for the second one—and satisfy the Gauss and Codazzi–Mainardi equations over a simply-connected open subset ω of R2; the surface is then uniquely determined up to proper isometries of E3. Then S. Mardare showed in 2005 that this result still holds under the much weaker assumptions that the components of the first form are only in the space Wloc1,p(ω) and those of the second form only in the space Llocp(ω), the components of the immersion defining the reconstructed surface being then in the space Wloc2,p(ω), p>2.
The purpose of this paper is to complement this last result as follows. First, under the additional assumption that ω is bounded and has a Lipschitz-continuous boundary, we show that a similar existence and uniqueness theorem holds with the spaces Wm,p(ω) instead of Wlocm,p(ω). Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W2,p(ω)-norm, p>2, between two given surfaces is bounded, at least locally, by the distance in the W1,p(ω)-norm between their first fundamental forms and the distance in the Lp(ω)-norm between their second fundamental forms. Third, we show that the mapping that uniquely defines in this fashion a surface up to proper isometries of E3 in terms of its two fundamental forms is locally Lipschitz-continuous.
Le théorème fondamental de la théorie de surfaces affirme qu'une surface dans l'espace euclidien tri-dimensionnel E3 peut être reconstruite à partir de la connaissance de ses formes fondamentales sous l'hypothèse que leurs composantes sont suffisamment régulières—classiquement dans l'espace C2(ω) pour la première et dans l'espace C1(ω) pour la seconde—et satisfont les équations de Gauss et de Codazzi–Mainardi dans un ouvert simplement connexe ω de R2 ; la surface est alors déterminée uniquement à une isométrie propre de E3 près. Puis S. Mardare a démontré en 2005 que ce résultat reste vrai sous l'hypothèse beaucoup plus faible que les composantes de la première forme sont seulement dans l'espace Wloc1,p(ω) et celles de la seconde forme seulement dans l'espace Llocp(ω), les composantes de l'immersion définissant la surface étant alors dans l'espace Wloc2,p(ω), p>2.
L'objet de cet article est de compléter ce dernier résultat de la manière suivante : Pour commencer, sous l'hypothèse supplémentaire que l'ouvert ω est borné et a une frontière lipschitzienne, nous établissons un résultat analogue d'existence et d'unicité avec les espaces Wm,p(ω) au lieu des espaces Wlocm,p(ω). Ensuite, nous établissons une inégalité de Korn non linéaire sur une surface, montrant que la distance en norme de W2,p(ω), p>2, entre deux surfaces est majorée, au moins localement, par la distance en norme de W1,p(ω) entre leur premières formes fondamentales et la distance en norme de Lp(ω) entre leur deuxièmes formes fondamentales. Nous établissons enfin que l'application qui définit de cette façon une surface à une isométrie propre de E3 près en fonction de ses deux formes fondamentales est localement lipschitzienne.
For 1 < p < ∞, we prove an Lp‐version of the generalized Korn inequality for incompatible tensor fields P in
W01,p(Curl;Ω,ℝ3×3). More precisely, let
Ω⊂ℝ3 be a bounded Lipschitz domain. Then there ...exists a constant c = c(p, Ω) > 0 such that
‖P‖Lp(Ω,ℝ3×3)≤c‖symP‖Lp(Ω,ℝ3×3)+‖CurlP‖Lp(Ω,ℝ3×3)
holds for all tensor fields
P∈W01,p(Curl;Ω,ℝ3×3), that is, for all
P∈W1,p(Curl;Ω,ℝ3×3) with vanishing tangential trace
P×ν=0 on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω. For compatible
P=Du, this recovers an Lp‐version of the classical Korn's first inequality and for skew‐symmetric
P=A an Lp‐version of the Poincaré inequality.
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