The Space of Inextensional Displacements for a Partially Clamped Linearly Elastic Shell with an Elliptic Middle Surface

We consider a linearly elastic shell with an “elliptic” middle surface, clamped along a portion of its lateral face and subjected to body forces. Under weak regularity assumptions on the middle surface, we prove that the space of linearized inextensional displacements is reduced to zero, by using unique continuation results. Consequently, when the thickness of the shell goes to zero, the limit of the average with respect to the thickness of the three-dimensional displacement vector solves the “generalized membrane” shell model, according to the terminology introduced by P.G. Ciarlet and the first author. This revised version was published online in August 2006 with corrections to the Cover Date.     

Explosion en temps fini de la solution d'un problème d'évolution de surface de film mince

In this Note, we are interested in the evolution of a surface of a crystal structure, constituted by an elastic substrate and a thin film. If the crystal is constrained, some morphological instabilities may appear. To study these instabilities, we made use of the model developped in Phys. Rev. B 47 (1993) 9760–9777. There, the map f of the free surface of the film satisfies a parabolic partial differential equation, depending on the elastic displacement of the substrate. For simplicity, the substrate is assumed to be linearly elastic and the structure to be infinite in one direction. Then, under some formal asymptotic assumptions, a formal expansion of the displacement can be determined after some appropriate scalings, allowing to derive a simplified parabolic nonlinear equation as in Lods et al. (Asymptotic Anal. 33 (2003) 67–91). We give here some results about the finite-time blow-up and the existence and uniqueness of the solution in an appropriate space. To validate the theoretical results, we also performed some numerical simulations using a pseudo-spectral method and also compute the initial-profile dependent critical value of the parameter θ involved in the nonlinear equation. To cite this article: M. Boutat et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.     

A New Cell for Electrical Conductivity Measurement on Saturated Samples at Upper Crust Conditions

Electrical resistivity soundings are used by geophysicists to determine the structure and composition of the Earth’s crust and mantle and to explore natural resources (ore, oil, gas, water). Their interpretations in terms of composition and in-situ physical conditions depend mainly on laboratory measurements of electrical conductivity of rocks at simulated crustal conditions of temperature, pressure, saturation and pore pressures. These measurements present a numbers of limitations, in particular, in the case where conductive pore fluids are present, as in the case of deep reservoir conditions, where temperature exceeds 250 °C. Here, we present a new cell capable of measuring electrical conductivity of large saturated samples at confining pressure up to 200 MPa, pore pressure up to 50 MPa, and temperature up to 500 °C. The measurement cell has been developed in a commercial, internally heated, gas pressure apparatus (Paterson press). It is based on the concept of “guard ring” electrode, which is adapted to samples that are jacketed by a very conductive, metallic material. Numerical modeling of the current flow in the electrical cell allowed defining the optimal cell geometry. Calibration tests have been performed on Fontainebleau sandstones saturated with electrolytes of different conductivities, up to 350 °C. The resulting electrical formation factor and temperature dependence of electrical conductivity are in very good agreement with previous studies. This new cell will improve the exploration and exploitation of deep fluid reservoirs, as in unconventional, high enthalpy geothermal fields. In particular, the investigations address possible effects of fluid-rock interactions on electrical resistivity of a reservoir host rock.     

Asymptotic analysis of linearly elastic shells: ‘Generalized membrane shells’

We consider a family of linearly elastic shells indexed by their half-thickness , all having the same middle surface % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadofacqGH9aqpcqaHvpGAcaGGOaGafqyYdCNbaebacaGGPaaa% aa!4317!\[S = \varphi (\bar \omega )\], with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabew9aQjaacQdacuaHjpWDgaqeaiabgkOimlaadkfadaahaaWc% beqaaiaaikdaaaGccqGHsgIRcaWGsbWaaWbaaSqabeaacaaIZaaaaa% aa!4812!\[\varphi :\bar \omega \subset R^2 \to R^3 \], and clamped along a portion of their lateral face whose trace on S is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabew9aQjaacIcacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccaGG% Paaaaa!41EB!\[\varphi (\gamma _0 )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacIcacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa!401F!\[(\gamma _0 )\] is a fixed portion of with length % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacIcacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyOp% a4JaaGimaaaa!41E1!\[(\gamma _0 ) > 0\]. Let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacIcacqaHZoWzdaWgaaWcbaGaeqySdeMaeqOSdigabeaakiaa% cIcacqaH3oaAcaGGPaGaaiykaaaa!45AA!\[(\gamma _{\alpha \beta } (\eta ))\] be the linearized strain tensor of S. We make an essential geometric and kinematic assumption, according to which the semi-norm % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \] defined by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqaH3oaAcaGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaGccqGH9aqpdaGadeqaamaaqababaGaaiiFaiaacYhaaSqaaiabeg% 7aHfrbbjxAHXgaiuaacaWFSaGaeqOSdigabeqdcqGHris5aOGaeq4S% dCMaeqySdeMaeqOSdiMaaiikaiabeE7aOjaacMcacaGG8bGaaiiFam% aaDaaaleaacaWGmbWaaWbaaWqabeaacaaIYaaaaSGaaiikaiabeM8a% 3jaacMcaaeaacaaIYaaaaaGccaGL7bGaayzFaaWaaWbaaSqabeaaca% aIXaGaai4laiaaikdaaaaaaa!61F1!\[|\eta |_\omega ^M = \left\{ {\sum\nolimits_{\alpha ,\beta } {||} \gamma \alpha \beta (\eta )||_{L^2 (\omega )}^2 } \right\}^{1/2} \] is a norm over the space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadAfacaGGOaGaeqyYdCNaaiykaiabg2da9maacmqabaGaeq4T% dGMaeyicI4SaamisamaaCaaaleqabaGaaGymaaaakiaacIcacqaHjp% WDcaGGPaGaai4oaiabeE7aOjabg2da9iaab+gacaqGUbGaeq4SdC2a% aSbaaSqaaiaabcdaaeqaaaGccaGL7bGaayzFaaaaaa!5361!\[V(\omega ) = \left\{ {\eta \in H^1 (\omega );\eta = {\text{on}}\gamma _{\text{0}} } \right\}\], excluding however the already analyzed membrane shells, where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeo7aNnaaBaaaleaacaqGWaaabeaakiabg2da9iabgkGi2kab% eM8a3baa!42F8!\[\gamma _{\text{0}} = \partial \omega \] and S is elliptic. This new assumption is satisfied for instance if % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeo7aNnaaBaaaleaacaqGWaaabeaakiabgcMi5kabgkGi2kab% eM8a3baa!43B9!\[\gamma _{\text{0}} \ne \partial \omega \] and S is elliptic, or if S is a portion of a hyperboloid of revolution.We then show that, as 0, the averages % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiabew7aLbaa% aaGcdaWdXaqaaiaadwhadaqhaaWcbaGaamyAaaqaaiabew7aLbaaki% aabsgacaWG4bWaa0baaSqaaiaaiodaaeaacqaH1oqzaaaabaGaeyOe% I0IaeqyTdugabaGaeqyTduganiabgUIiYdaaaa!4E28!\[\frac{1}{{2^\varepsilon }}\int_{ - \varepsilon }^\varepsilon {u_i^\varepsilon {\text{d}}x_3^\varepsilon } \] across the thickness of the shell of the covariant components u _i of the displacement of the points of the shell strongly converge in the completion V _M ^#() of V() with respect to the norm % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \], toward the solution of a generalized membrane shell problem. This convergence result also justifies the recent formal asymptotic approach of D. Caillerie and E. Sanchez-Palencia.The limit problem found in this fashion is sensitive, according to the terminology recently introduced by J.L. Lions and E. Sanchez-Palencia, in the sense that it possesses two unusual features: it is posed in a space that is not necessarily contained in a space of distributions, and its solution is highly sensitive to arbitrarily small smooth perturbations of the data.Under the same assumption, we also show that the average % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiabew7aLbaa% aaGcdaWdXaqaaiaadwhadaqhaaWcbaGaamyAaaqaaiabew7aLbaaki% aabsgacaWG4bWaa0baaSqaaiaaiodaaeaacqaH1oqzaaaabaGaeyOe% I0IaeqyTdugabaGaeqyTduganiabgUIiYdaaaa!4E28!\[\frac{1}{{2^\varepsilon }}\int_{ - \varepsilon }^\varepsilon {u_i^\varepsilon {\text{d}}x_3^\varepsilon } \] where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadwhadaahaaWcbeqaaiabew7aLbaakiabg2da9iaacIcacaWG% 1bWaa0baaSqaaiaadMgaaeaacqaH1oqzaaGccaGGPaaaaa!452C!\[u^\varepsilon = (u_i^\varepsilon )\], and the solution % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4naaCaaaleqabaGaeqyTdugaaOGaeyicI4SaamOvamaa% BaaaleaacaWGlbaabeaakiaacIcacqaHjpWDcaGGPaaaaa!465B!\[\xi ^\varepsilon \in V_K (\omega )\] of Koiter's equations have the same principal part as 0 in the same space V _M () as above. For such generalized membrane shells, the two-dimensional shell model of W.T. Koiter is thus likewise justified.We also treat the case where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \] is no longer a norm over V(), but is a norm over the space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadAfadaWgaaWcbaGaam4saaqabaGccaGGOaGaeqyYdCNaaiyk% aiabg2da9maacmqabaGaeq4TdGMaeyypa0JaaiikaiabeE7aOnaaBa% aaleaacaWGPbaabeaakiaacMcacqGHiiIZcaWGibWaaWbaaSqabeaa% caaIXaaaaOGaaiikaiabeM8a3jaacMcacqGHxdaTcaWGibWaaWbaaS% qabeaacaaIYaaaaOGaaiikaiabeM8a3jaacMcacaGG7aGaeq4TdG2a% aSbaaSqaaiaadMgaaeqaaOGaeyypa0JaeyOaIy7aaSbaaSqaaiaadA% haaeqaaOGaeq4TdG2aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGim% aiGac+gacaGGUbGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaaGccaGL7b% GaayzFaaaaaa!68B8!\[V_K (\omega ) = \left\{ {\eta = (\eta _i ) \in H^1 (\omega ) \times H^2 (\omega );\eta _i = \partial _v \eta _3 = 0\operatorname{on} \gamma _0 } \right\}\], thus also excluding the already analyzed flexural shells. Then a convergence theorem can still be established, but only in the completion of the quotient space V()/V ₀() with repect to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadAfadaWgaaWcbaGaaGimaaqabaGccaGGOaGaeqyYdCNaaiyk% aiabg2da9maacmqabaGaeq4TdGMaeyicI4SaamOvaiaacIcacqaHjp% WDcaGGPaGaai4oaiabeo7aNjabeg7aHjabek7aIjaacIcacqaH3oaA% caGGPaGaeyypa0JaaeimaiaabMgacaqGUbGaeqyYdChacaGL7bGaay% zFaaaaaa!5997!\[V_0 (\omega ) = \left\{ {\eta \in V(\omega );\gamma \alpha \beta (\eta ) = {\text{0in}}\omega } \right\}\].These convergence results, together with those that we already obtained for membrane and flexural shells, jointly with B. Miara in the second case, thus constitute an asymptotic analysis of linearly elastic shells in all possible cases.     

Spectral analysis of transport equations with bounce‐back boundary conditions

We investigate the spectral properties of the time‐dependent linear transport equation with bounce‐back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from Sbihi (J. Evol. Equations 2007; 7 :689–711), we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all L^p‐spaces with 1<p<∞. Application to the linear Boltzmann equation for granular gases is provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particle Bath

We show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host medium with a fixed distribution. This is achieved by controlling the L ^p -norms, the moments and the regularity of the solutions to the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states.     

Exponential convergence to equilibrium for subcritical solutions of the Becker–Döring equations

We prove that any subcritical solution to the Becker–Döring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin and Niethammer (2003) [17]. Our approach is based on a careful spectral analysis of the linearized Becker–Döring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted ?¹${?}^1$ spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.     

Kinetic Description of a Rayleigh Gas with Annihilation

In this paper, we consider the dynamics of a tagged point particle in a gas of moving hard-spheres that are non-interacting among each other. This model is known as the ideal Rayleigh gas. We add to this model the possibility of annihilation (ideal Rayleigh gas with annihilation), requiring that each obstacle is either annihilating or elastic, which determines whether the tagged particle is elastically reflected or removed from the system. We provide a rigorous derivation of a linear Boltzmann equation with annihilation from this particle model in the Boltzmann–Grad limit. Moreover, we give explicit estimates for the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. The estimates show that the system can be approximated by the Boltzmann equation on an algebraically long time scale in the scaling parameter.

     

Substochastic semigroups for transport equations with conservative boundary conditions

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C₀-semigroup in L¹. With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring to be stochastic. We apply these results to examples from kinetic theory.