We consider a damped plate equation on an open bounded subset of R, or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskĭı-Šapiro condition. The damping term acts on a region without imposing a geometrical condition. We derive a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first though microlocal estimates, then local estimates, and finally a global estimate.
{"title":"Stabilization of the damped plate equation under general boundary conditions","authors":"J. Rousseau, E. Zongo","doi":"10.5802/jep.213","DOIUrl":"https://doi.org/10.5802/jep.213","url":null,"abstract":"We consider a damped plate equation on an open bounded subset of R, or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskĭı-Šapiro condition. The damping term acts on a region without imposing a geometrical condition. We derive a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first though microlocal estimates, then local estimates, and finally a global estimate.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127237307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a dynamical residue which generalizes the Guillemin-Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott-Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.
{"title":"Dynamical residues of Lorentzian spectral zeta functions","authors":"N. V. Dang, M. Wrochna","doi":"10.5802/jep.205","DOIUrl":"https://doi.org/10.5802/jep.205","url":null,"abstract":"We define a dynamical residue which generalizes the Guillemin-Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott-Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123650466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surface supports no nontrivial holomorphic vector bundles. This solves an open problem on the existence of matrix holomorphic integrating factors on simple surfaces and is applied to give a range characterisation for the non-Abelian X-ray transform. The main theorem is proved using the inverse function theorem of Nash and Moser and the required tame estimates are obtained from recent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators.
{"title":"The Transport Oka-Grauert principle for simple surfaces","authors":"Jan Bohr, G. Paternain","doi":"10.5802/jep.231","DOIUrl":"https://doi.org/10.5802/jep.231","url":null,"abstract":"This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surface supports no nontrivial holomorphic vector bundles. This solves an open problem on the existence of matrix holomorphic integrating factors on simple surfaces and is applied to give a range characterisation for the non-Abelian X-ray transform. The main theorem is proved using the inverse function theorem of Nash and Moser and the required tame estimates are obtained from recent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122172627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the exponential stability in the $H^{2}$ norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the Input-to-State stability of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.
{"title":"PI controllers for the general Saint-Venant equations","authors":"Amaury Hayat","doi":"10.5802/jep.210","DOIUrl":"https://doi.org/10.5802/jep.210","url":null,"abstract":"We study the exponential stability in the $H^{2}$ norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the Input-to-State stability of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132386493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alberto Abbondandolo, Christian Lange, M. Mazzucchelli
A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.
{"title":"Higher systolic inequalities for 3-dimensional contact manifolds","authors":"Alberto Abbondandolo, Christian Lange, M. Mazzucchelli","doi":"10.5802/jep.195","DOIUrl":"https://doi.org/10.5802/jep.195","url":null,"abstract":"A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122179740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a degeneration of compact projective complex manifolds $X$ over the punctured disc, with meromorphic singularities, and a relatively ample line bundle $L$ on $X$, we study spaces of plurisubharmonic metrics on $L$, with particular focus on (relative) finite-energy conditions. We endow the space $hat cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise Darvas $d_1$-distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_K$, $L_K$ over the discretely-valued field $K=mathbb{C}((t))$ of complex Laurent series, we show that the space $cE^1(L_Kan)$ of non-Archimedean finite-energy metrics over $L_Kan$ embeds isometrically and geodesically into $hat cE^1(L)$, and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case. We investigate consequences regarding convexity of non-Archimedean functionals.
给定穿孔盘上具有亚纯奇点的紧射复流形$X$的退化，以及$X$上相对充裕的线束$L$，我们研究了$L$上的多次谐波度量空间，特别关注了(相对)有限能量条件。我们赋予空间$ $ hat $ $ cE^1(L)$的相对最大的，相对有限能量的度量$ $d_1$型的距离由纤维Darvas $d_1$-距离集合在零处的Lelong数给出。我们证明了这个度量结构是完备的和测地线的。将$X$和$L$作为复洛朗级数离散值域$K=mathbb{C}((t))$上的$X_K$， $L_K$方案，我们证明了$L_Kan$上的非阿基米德有限能量度量空间$cE^1(L_Kan)$等距和测地嵌入到$hat cE^1(L)$中，并刻画了它的象。这概括了Berman-Boucksom-Jonsson先前的工作，处理了平凡值的情况。我们研究关于非阿基米德泛函的凸性的结果。
{"title":"The space of finite-energy metrics over a degeneration of complex manifolds","authors":"R'emi Reboulet","doi":"10.5802/jep.229","DOIUrl":"https://doi.org/10.5802/jep.229","url":null,"abstract":"Given a degeneration of compact projective complex manifolds $X$ over the punctured disc, with meromorphic singularities, and a relatively ample line bundle $L$ on $X$, we study spaces of plurisubharmonic metrics on $L$, with particular focus on (relative) finite-energy conditions. We endow the space $hat cE^1(L)$ of relatively maximal, relative finite-energy metrics with a $d_1$-type distance given by the Lelong number at zero of the collection of fibrewise Darvas $d_1$-distances. We show that this metric structure is complete and geodesic. Seeing $X$ and $L$ as schemes $X_K$, $L_K$ over the discretely-valued field $K=mathbb{C}((t))$ of complex Laurent series, we show that the space $cE^1(L_Kan)$ of non-Archimedean finite-energy metrics over $L_Kan$ embeds isometrically and geodesically into $hat cE^1(L)$, and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case. We investigate consequences regarding convexity of non-Archimedean functionals.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133681506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. — This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues. Résumé (Une théorie des résidus pour les fractions rationnelles tordues) Cet article constitue un premier pas en direction du développement de méthodes analytiques pour les polynômes tordus. Précisément, notre principal objectif est de développer une théorie des résidus pour les fractions rationnelles tordues (qui sont, par définition, les quotients de deux polynômes tordus). Nous démontrons en particulier des analogues tordus de la formule des résidus et de la formule classique de changement de variables.
． —This paper . constitutes rstfi试探性to do with sq analysis ols)。真正客观,our hand is to a theory of残留物for不得不“sq rational功能(which are two by the商数fi极权、景sq (ols)。prove We的谈话sq圣母残渣,模拟了formula and a sq圣母”,类似formula for残留物of change of变数。摘要(扭曲有理分数的残差理论)本文是发展扭曲多项式分析方法的第一步。恰恰是我们的主要目标是开发一个稳妥的残留馏分弯(理论受到挑战,谁fi淫行定义,两个多项式系数)。特别地，我们证明了残差公式和经典变变量公式的扭曲类比。
{"title":"A theory of residues for skew rational functions","authors":"X. Caruso","doi":"10.5802/jep.169","DOIUrl":"https://doi.org/10.5802/jep.169","url":null,"abstract":". — This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues. Résumé (Une théorie des résidus pour les fractions rationnelles tordues) Cet article constitue un premier pas en direction du développement de méthodes analytiques pour les polynômes tordus. Précisément, notre principal objectif est de développer une théorie des résidus pour les fractions rationnelles tordues (qui sont, par définition, les quotients de deux polynômes tordus). Nous démontrons en particulier des analogues tordus de la formule des résidus et de la formule classique de changement de variables.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116449362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes $N$ and $F$ in a finite graph $mathcal G$. The ants perform a sequence of random walks on this graph, starting from the nest and stopped when first hitting the source of food. At each step of its random walk, the $n$-th ant chooses to cross a neighbouring edge with probability proportional to the number of preceding ants that crossed that edge at least once. We say that {it the ants find the shortest path} if, almost surely as the number of ants grow to infinity, almost all the ants go from the nest to the source of food through one of the shortest paths, without loosing time on other edges of the graph. Our contribution is three-fold: (1) We prove that, if $mathcal G$ is a tree rooted at $N$ whose leaves have been merged into node $F$, and with one edge between $N$ and $F$, then the ants indeed find the shortest path. (2) In contrast, we provide three examples of graphs on which the ants do not find the shortest path, suggesting that in this model and in most graphs, ants do not find the shortest path. (3) In all these cases, we show that the sequence of normalised edge-weights converge to a {it deterministic} limit, despite a linear-reinforcement mechanism, and we conjecture that this is a general fact which is valid on all finite graphs. To prove these results, we use stochastic approximation methods, and in particular the ODE method. One difficulty comes from the fact that this method relies on understanding the behaviour at large times of the solution of a non-linear, multi-dimensional ODE.
{"title":"The trace-reinforced ants process does not find shortest paths","authors":"Daniel Kious, Cécile Mailler, Bruno Schapira","doi":"10.5802/jep.188","DOIUrl":"https://doi.org/10.5802/jep.188","url":null,"abstract":"In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes $N$ and $F$ in a finite graph $mathcal G$. The ants perform a sequence of random walks on this graph, starting from the nest and stopped when first hitting the source of food. At each step of its random walk, the $n$-th ant chooses to cross a neighbouring edge with probability proportional to the number of preceding ants that crossed that edge at least once. We say that {it the ants find the shortest path} if, almost surely as the number of ants grow to infinity, almost all the ants go from the nest to the source of food through one of the shortest paths, without loosing time on other edges of the graph. \u0000Our contribution is three-fold: (1) We prove that, if $mathcal G$ is a tree rooted at $N$ whose leaves have been merged into node $F$, and with one edge between $N$ and $F$, then the ants indeed find the shortest path. (2) In contrast, we provide three examples of graphs on which the ants do not find the shortest path, suggesting that in this model and in most graphs, ants do not find the shortest path. (3) In all these cases, we show that the sequence of normalised edge-weights converge to a {it deterministic} limit, despite a linear-reinforcement mechanism, and we conjecture that this is a general fact which is valid on all finite graphs. To prove these results, we use stochastic approximation methods, and in particular the ODE method. One difficulty comes from the fact that this method relies on understanding the behaviour at large times of the solution of a non-linear, multi-dimensional ODE.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115842030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Valentina Franceschi, A. Pratelli, Giorgio Stefani
In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using two densities. In particular, the perimeter density may also depend on the direction of the normal vector. The classical"Steiner property"for the Euclidean case (which corresponds to both densities being equal to $1$) says that minimal clusters are made by finitely many ${rm C}^{1,gamma}$ arcs, meeting in finitely many"triple points". We can show that this property holds under very weak assumptions on the densities. In the parallel paper"On the Steiner property for planar minimizing clusters. The isotropic case"we consider the isotropic case, i.e., when the perimeter density does not depend on the direction, which makes most of the construction much simpler. In particular, in the present case the three arcs at triple points do not necessarily meet with three angles of $120^circ$, which is instead what happens in the isotropic case.
{"title":"On the Steiner property for planar minimizing clusters. The anisotropic case","authors":"Valentina Franceschi, A. Pratelli, Giorgio Stefani","doi":"10.5802/jep.238","DOIUrl":"https://doi.org/10.5802/jep.238","url":null,"abstract":"In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using two densities. In particular, the perimeter density may also depend on the direction of the normal vector. The classical\"Steiner property\"for the Euclidean case (which corresponds to both densities being equal to $1$) says that minimal clusters are made by finitely many ${rm C}^{1,gamma}$ arcs, meeting in finitely many\"triple points\". We can show that this property holds under very weak assumptions on the densities. In the parallel paper\"On the Steiner property for planar minimizing clusters. The isotropic case\"we consider the isotropic case, i.e., when the perimeter density does not depend on the direction, which makes most of the construction much simpler. In particular, in the present case the three arcs at triple points do not necessarily meet with three angles of $120^circ$, which is instead what happens in the isotropic case.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124714628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
—We investigate the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely, we consider the limit where the thickness of these slender rigid bodies tends to zero with a common rate ε, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves. While for each positive ε, the bodies’ dynamics are given by the Newton equations and correspond to some coupled second-order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first-order ODEs whose coefficients only depend on the limit curves and on the background flow. We also determine the limit effect due to the limit curves on the fluid, in the spirit of the immersed boundary method. Résumé (Mouvement de filaments rigides minces dans un fluide de Stokes) Nous étudions la dynamique de filaments rigides minces se déplaçant dans un fluide qui est décrit par un état de base régulier perturbé par la présence des solides selon les équations du système de Stokes stationnaire tridimensionnel. Plus précisément, nous considérons la limite dans laquelle l’épaisseur des corps solides tend vers zéro avec un taux commun ε, tandis que leur densité de masse volumétrique est maintenue constante, de sorte que les solides limites occupent des courbes que l’on suppose d’intersections deux à deux vides, et ont une masse nulle. Pour ε ą 0, la dynamique des solides est donnée par les équations de Newton et correspondent à des équations différentielles ordinaires du second ordre, couplées. Nous prouvons que les équations limites sont des équations différentielles ordinaires du premier ordre, découplées, dont les coefficients ne dépendent que des courbes limites et du flot de base. Nous déterminons également l’effet limite des courbes limites sur le fluide, dans l’esprit de la méthode des frontières immergées. Mathematical subject classification (2020). — 74F10.
我们研究了在平滑背景流存在的情况下，由三维稳定斯托克斯系统驱动的流动中几个细长刚体的动力学。更准确地说，我们考虑了这些细长刚体的厚度以共同速率ε趋近于零的极限，而它们的体积质量密度保持固定，从而使它们收缩成分离的无质量曲线。对于每一个正ε，物体的动力学由牛顿方程给出，并对应于物体位置的一些耦合二阶微分方程，我们证明了极限方程是解耦的一阶微分方程，其系数只依赖于极限曲线和背景流。我们还根据浸入边界法的精神，确定了由于流体的极限曲线而产生的极限效应。系统系统系统(系统系统系统): 系统系统系统系统(系统系统系统系统):系统系统系统(系统系统系统):系统系统(系统系统):三维空间。加上prassicesimement, nous认为samons la limited dans laquelle l ' samisur des corps solides趋向于samsamuise的平均数，samsamuise的平均数，samuise的平均数，samuise的平均数，samuise的平均数，samuise的平均数，samuise的平均数，samuise的平均数，samuise的平均数，samuise的平均数，以及samuise的平均数。2008年，关于固体的动力学，关于牛顿的和其他的，关于二阶的，偶联的，关于牛顿的和其他的。一般情况下，有不同的限制条件、不同的限制条件、不同的限制条件、不同的限制条件、不同的限制条件、不同的限制条件、不同的限制条件、不同的限制条件、不同的限制条件。现有的薪金薪金决定了薪金薪金的效力，限制了薪金薪金的效力，限制了薪金薪金的流动，也就决定了薪金薪金决定了薪金薪金的性质。数学学科分类(2020)。- 74 f10。
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