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We consider a fluid-structure interaction system composed of a rigid ball immersed into a viscous incompressible fluid. The motion of the structure satisfies the Newton laws and the fluid equations are the standard Navier–Stokes system. At the boundary of the fluid domain, we use the Tresca boundary conditions, that permit the fluid to slip tangentially on the boundary under some conditions on the stress tensor. More precisely, there is a threshold determining if the fluid can slip or not and there is a friction force acting on the part where the fluid can slip. Our main result is the existence of contact in finite time between the ball and the exterior boundary of the fluid for this system in the bidimensional case, in presence of gravity and in the case of a symmetric configuration.
The moduli space of principally polarized abelian varieties of genus is defined over and admits a minimal compactification , also defined over . The Hodge bundle over has its Chern classes in the Chow ring of with -coefficients. We show that over , these Chern classes naturally lift to and do so in the best possible way: despite the highly singular nature of they are represented by algebraic cycles on which define elements in the bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky–Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.
In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry’s random wave conjecture. Using recent results of Bourgain, Buckley and Wigman, we will prove that some deterministic families of eigenfunctions on satisfy the conclusions of the random wave conjecture. We also show that on an arbitrary domain, a sequence of Laplace eigenfunctions always admits local weak limits. We explain why these local weak limits can be a powerful tool to study the asymptotic number of nodal domains.
Let . Define the average of over the square integers by
We show that satisfies a local scale-free -improving estimate, for :
provided is supported in some interval of length , and is the conjugate index. The inequality above fails for . The maximal function || satisfies a similar sparse bound. Novel weighted and vector valued inequalities for follow. A critical step in the proof requires the control of a logarithmic average over of a function counting the number of square roots of . One requires an estimate uniform in .
A new formula is obtained for the holomorphic bidifferential operators on tube-type domains which are associated to the decomposition of the tensor product of two scalar holomorphic representations, thus generalizing the classical Rankin–Cohenbrackets. The formula involves a family of polynomials of several variables which may be considered as a (weak) generalization of the classical Jacobipolynomials.
On a complex symplectic manifold, we prove a finiteness result for the global sections of solutions of holonomic -modules in two cases: (a) by assuming that there exists a Poisson compactification, (b) in the algebraic case. This extends our previous result in which the symplectic manifold was compact. The main tool is a finiteness theorem for -constructible sheaves on a real analytic manifold in a nonproper situation.
We describe the Griffiths group of the product of a curve and a surface as a quotient of the Albanese kernel of over the function field of . When is a hyperplane section of varying in a Lefschetz pencil, we prove the nonvanishing in Griff of a modification of the graph of the embedding for infinitely many members of the pencil, provided the ground field is of characteristic , the geometric genus of is , and is large or is “of motivated abelian type”.
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