Abstract: Publication date: Available online 18 May 2017 Source:Comptes Rendus Mathematique Author(s): Claude Bardos, Kim Dang Phung We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker–Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof relies on the well-known diffusion approximation under convenience scaling and on the basic properties of this diffusion. Eventually, we propose a direct proof for the observation at one time of parabolic equations. It is based on the analysis of the heat kernel.

Abstract: Publication date: Available online 10 May 2017 Source:Comptes Rendus Mathematique Author(s): Raphaël Ponge In this note, we produce explicit quasi-isomorphisms computing the cyclic homology of crossed-product algebras associated with group actions on manifolds. We obtain explicit relationships with equivariant cohomology. On the way, we extend the results of the first part to the setting of group actions on locally convex algebras.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Khadijeh Baghaei, Ali Khelghati In this paper, we study the chemotaxis system: { u t = ∇ ⋅ ( ξ ∇ u − χ u ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 1 , with smooth boundary. Here, ξ and χ are some positive constants. We prove that the classical solutions to the above system are uniformly in-time-bounded provided that: ‖ v 0 ‖ L ∞ ( Ω ) < { 1 χ ξ 2 ( n + 1 ) [ π + 2 arctan ( ( 1 − ξ ) 2 2 ( n + 1 ) ξ ) ] , if 0 < ξ < 1 , π χ 2 ( n + 1 ) , if ξ = 1 , 1 χ ξ 2 ( n + 1 ) [ π − 2 arctan ( ( ξ − 1 ) 2 2 ( n + 1 ) ξ ) ] , if ξ > 1 . In the case of ξ = 1 , the recent results show that the classical solutions are global and bounded provided that 0 < ‖ v 0 ‖ L ∞ ( Ω ) ≤ 1 6 ( n PubDate: 2017-05-12T17:34:58Z

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Michael Ruzhansky, Durvudkhan Suragan, Nurgissa Yessirkegenov In this paper, we give an extension of the classical Caffarelli–Kohn–Nirenberg inequalities with respect to the range of parameters. We also establish best constants for large families of parameters. Moreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. We also establish sharp Hardy type inequalities in L p , 1 < p < ∞ , with superweights.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Raphaël Ponge In this note we produce explicit quasi-isomorphisms computing the cyclic homology of crossed-product algebras.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Cagri Sert The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramér on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers S n = { g 1 . … . g n g i ∈ S } of a subset S of a semisimple linear Lie group G (e.g., SL ( d , R ) ). This limit set has applications, among others, in the study of large deviations.

Abstract: Publication date: Available online 9 May 2017 Source:Comptes Rendus Mathematique Author(s): Ibrahim Ekren, Igor Kukavica, Mohammed Ziane We address the long-time behavior of solutions to damped dispersive stochastic partial differential equations, namely the KdV equation and the nonlinear Schrödinger equation on the whole space. We prove that the transition semigroup is Feller and establish the existence of an invariant measure using the asymptotic compactness property of the transition semigroup and the Aldous criterion.

Abstract: Publication date: Available online 5 May 2017 Source:Comptes Rendus Mathematique Author(s): Nguyen Xuan Hong In this note, we prove a semi-continuity theorem for certain weighted log canonical thresholds of toric plurisubharmonic functions.

Abstract: Publication date: Available online 4 May 2017 Source:Comptes Rendus Mathematique Author(s): Diomba Sambou, Amal Taarabt We investigate the discrete spectrum behaviour for the 2d Pauli operator with nonconstant magnetic field, perturbed by a sign-indefinite self-adjoint electric potential that decays polynomially at infinity. A localisation of the eigenvalues and new asymptotics are established.

Abstract: Publication date: Available online 2 May 2017 Source:Comptes Rendus Mathematique Author(s): Grégoire Allaire, Charles Dapogny, Alexis Faure, Georgios Michailidis The purpose of this article is to introduce a new functional of the domain, to be used in shape optimization problems as a means to enforce the constructibility of shapes by additive manufacturing processes. This functional aggregates the self-weights of all the intermediate structures of the shape appearing in the course of its layer-by-layer assembly. Its mathematical analysis is performed and an algorithm is proposed to accelerate the significant computational effort entailed by the implementation of these ideas. Eventually, a numerical validation and a concrete example are discussed.

Abstract: Publication date: Available online 28 April 2017 Source:Comptes Rendus Mathematique Author(s): Zhi Jiang, Qizheng Yin Let f : X → A be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that f ⁎ induces an isomorphism between the rational cohomology rings H • ( A , Q ) and H • ( X , Q ) if and only if f ⁎ induces an isomorphism between the Chow rings with rational coefficients CH • ( A ) Q and CH • ( X ) Q .

Abstract: Publication date: Available online 27 April 2017 Source:Comptes Rendus Mathematique Author(s): Mohammad Daher According to a previous result of the author, if ( A 0 , A 1 ) is an interpolation couple, if A 0 ⁎ is weakly LUR, then the complex interpolation spaces ( A 0 ⁎ , A 1 ⁎ ) θ have the same property. Here we construct an interpolation couple ( B 0 , B 1 ) where B 0 is LUR, but where the complex interpolation spaces ( B 0 , B 1 ) θ are not strictly convex.

Abstract: Publication date: Available online 26 April 2017 Source:Comptes Rendus Mathematique Author(s): Nabiullah Khan, Talha Usman, Junesang Choi Many extensions and variants of the so-called Apostol-type polynomials have recently been investigated. Motivated mainly by those works and their usefulness, we aim to introduce a new class of Apostol-type Laguerre–Genocchi polynomials associated with the modified Milne–Thomson's polynomials introduced by Derre and Simsek and investigate its properties, including, for example, various implicit formulas and symmetric identities in a systematic manner. The new family of polynomials introduced here, being very general, contains, as its special cases, many known polynomials. So the properties and identities presented here reduce to yield those results of the corresponding known polynomials.

Abstract: Publication date: Available online 21 April 2017 Source:Comptes Rendus Mathematique Author(s): Quốc Anh Ngô In this note, we mainly study the relation between the sign of ( − Δ ) p u and ( − Δ ) p − i u in R n with p ⩾ 2 and n ⩾ 2 for 1 ⩽ i ⩽ p − 1 . Given the differential inequality ( − Δ ) p u < 0 , first we provide several sufficient conditions so that ( − Δ ) p − 1 u < 0 holds. Then we provide conditions such that ( − Δ ) i u < 0 for all i = 1 , 2 , … , p − 1 , which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ( − Δ ) p u = e 2 p u and ( − Δ ) p u = u q with q > 0 in R n .

Abstract: Publication date: Available online 20 April 2017 Source:Comptes Rendus Mathematique Author(s): Radu Ignat, Robert L. Jerrard We study a variational Ginzburg–Landau-type model depending on a small parameter ε > 0 for (tangent) vector fields on a 2-dimensional Riemannian surface. As ε → 0 , the vector fields tend to be of unit length and will have singular points of a (non-zero) index, called vortices. Our main result determines the interaction energy between these vortices as a Γ-limit (at the second order) as ε → 0 .

Abstract: Publication date: Available online 20 April 2017 Source:Comptes Rendus Mathematique Author(s): Alexander Blokh, Lex Oversteegen, Ross Ptacek, Vladlen Timorin W. Thurston constructed a combinatorial model of the Mandelbrot set M 2 such that there is a continuous and monotone projection of M 2 to this model. We propose the following related model for the space MD 3 of critically marked cubic polynomials with connected Julia set and all cycles repelling. If ( P , c 1 , c 2 ) ∈ MD 3 , then every point z in the Julia set of the polynomial P defines a unique maximal finite set A z of angles on the circle corresponding to the rays, whose impressions form a continuum containing z. Let G ( z ) denote the convex hull of A z . The convex sets G ( z ) partition the closed unit disk. For ( P , c 1 , c 2 ) ∈ MD 3 let c 1 ⁎ be the co-critical point of c 1 . We tag the marked dendritic polynomial ( P , c 1 , c 2 ) with the set G ( c 1 ⁎ ) × G ( P ( c 2 ) ) ⊂ D ‾ × D ‾ . Tags are pairwise disjoint; denote by MD 3 comb their collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD 3 to MD 3 comb so that MD 3 comb serves as a model for MD 3 .

Abstract: Publication date: Available online 18 April 2017 Source:Comptes Rendus Mathematique Author(s): Tokio Matsuyama, Michael Ruzhansky The aim of this note is to present the almost global well-posedness result for the Cauchy problem for the Kirchhoff equation with large data in Gevrey spaces. We also briefly discuss the corresponding results in bounded and in exterior domains.

Abstract: Publication date: Available online 18 April 2017 Source:Comptes Rendus Mathematique Author(s): Cristian Lăzureanu In this note, we construct integrable deformations of the three-dimensional real valued Maxwell–Bloch equations by modifying their constants of motions. We obtain two Hamilton–Poisson realizations of the new system. Moreover, we prove that the obtained system has infinitely many Hamilton–Poisson realizations. Particularly, we present a Hamilton–Poisson approach of the system obtained considering two concrete deformation functions.

Abstract: Publication date: Available online 18 April 2017 Source:Comptes Rendus Mathematique Author(s): Michèle Vergne Let G be a torus with Lie algebra g and let M be a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let Λ ⊂ g ⁎ be the weight lattice of G. We consider a parameter k ≥ 1 and the multiplicity m ( λ , k ) of the quantized representation R R G ( M , L k ) . Define 〈 Θ ( k ) , f 〉 = ∑ λ ∈ Λ m ( λ , k ) f ( λ / k ) for f a test function on g ⁎ . We prove that the distribution Θ ( k ) has an asymptotic development 〈 Θ ( k ) , f 〉 ∼ k dim M / 2 ∑ n = 0 ∞ k − n 〈 D H n , f 〉 where the distributions D H n are the twisted Duistermaat–Heckman distributions associated with the graded equivariant Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.

Abstract: Publication date: Available online 14 April 2017 Source:Comptes Rendus Mathematique Author(s): Duván Cardona In this note we present some results on the action of global pseudo-differential operators on Besov spaces on compact Lie groups.

Abstract: Publication date: Available online 14 April 2017 Source:Comptes Rendus Mathematique Author(s): Fedor Sukochev, Aleksandr Veksler We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E. The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.

Abstract: Publication date: Available online 6 April 2017 Source:Comptes Rendus Mathematique Author(s): Fathi Haggui, Abdelwahed Chrih In this paper, we prove that if Ω is a bounded convex domain in C n , n ≥ 2 , and S is an affine complex hyperplane such that Ω ∩ S is not empty, then Ω ∖ S is not Gromov hyperbolic with respect to the Kobayashi distance. Next, we show that if X is a bounded convex domain in C n , then Ω = { ( z , w ) ∈ X × C ⁎ , w < e − φ ( z ) } is not Gromov hyperbolic, where φ is a strictly plurisubaharmonic function on X continuous up to X ‾ .

Abstract: Publication date: Available online 31 March 2017 Source:Comptes Rendus Mathematique Author(s): Belhassen Dehman, Sylvain Ervedoza The goal of this note is to prove observability estimates for the wave equation with a density which is only continuous in the domain, and satisfies some multiplier-type condition only in the sense of distributions. Our main argument is that one can construct suitable approximations of such density by a sequence of smooth densities whose corresponding wave equations are uniformly observable. The end of the argument then consists in a rather standard passage to the limit.

Abstract: Publication date: Available online 30 March 2017 Source:Comptes Rendus Mathematique Author(s): Hsuan-Yi Liao, Mathieu Stiénon, Ping Xu With any g -manifold M are associated two dglas tot ( Λ • g ∨ ⊗ k T poly • ( M ) ) and tot ( Λ • g ∨ ⊗ k D poly • ( M ) ) , whose cohomologies H CE • ( g , T poly • ( M ) → 0 T poly • + 1 ( M ) ) and H CE • ( g , D poly • ( M ) → d H D poly • + 1 ( M ) ) are Gerstenhaber algebras. We establish a formality theorem for g -manifolds: there exists an L ∞ quasi-isomorphism Φ : tot ( Λ • g ∨ ⊗ k T poly • ( M ) ) → tot ( Λ • g ∨ ⊗ k D poly • ( M ) ) whose first ‘Taylor coefficient’ (1) is equal to the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd cocycle of the g -manifold M, and (2) induces an isomorphism of Gerstenhaber algebras on the level of cohomology. Consequently, the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the g -manifold M is an isomorphi... PubDate: 2017-04-04T16:04:53Z

Abstract: Publication date: Available online 29 March 2017 Source:Comptes Rendus Mathematique Author(s): Indranil Biswas, Harish Seshadri Let X be a compact connected Riemann surface of genus at least two, and let Q X ( r , d ) be the quot scheme that parameterizes all the torsion coherent quotients of O X ⊕ r of degree d. This Q X ( r , d ) is also a moduli space of vortices on X. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of Q X ( r , d ) is not nef. Equivalently, Q X ( r , d ) does not admit any Kähler metric whose Ricci curvature is semipositive.

Abstract: Publication date: Available online 29 March 2017 Source:Comptes Rendus Mathematique Author(s): Dong Dong We prove that for a large class of functions P and Q, the discrete bilinear operator T P , Q ( f , g ) ( n ) = ∑ m ∈ Z ∖ { 0 } f ( n − P ( m ) ) g ( n − Q ( m ) ) 1 m is bounded from l 2 × l 2 into l 1 + ϵ , ∞ for any ϵ ∈ ( 0 , 1 ] .

Abstract: Publication date: Available online 24 March 2017 Source:Comptes Rendus Mathematique Author(s): Wassim Nasserddine Let G be a separable locally compact group with type-I left regular representation, G ˆ its dual and A ( G ) its Fourier algebra. We prove an analogue of Parseval's theorem and that the mapping T ⟶ u ( x ) : = ∫ G ˆ T r [ T ( π ) π ( x ) − 1 ] d μ ( π ) is an isometric isomorphism of Banach spaces from L 1 ( G ˆ ) onto A ( G ) .

Abstract: Publication date: Available online 22 March 2017 Source:Comptes Rendus Mathematique Author(s): Mejdi Azaïez, Faker Ben Belgacem, Tomás Chacón Rebollo, Macarena Gómez Mármol, Isabel Sánchez Muñoz In this work, we analyze the convergence of the POD expansion for the solution to the heat conduction parameterized with respect to the thermal conductivity coefficient. We obtain error bounds for the POD approximation in high-order norms in space that assure an exponential rate of convergence, uniformly with respect to the parameter whenever it remains within a compact set of positive numbers. We present some numerical tests that confirm this theoretical accuracy.

Abstract: Publication date: Available online 22 March 2017 Source:Comptes Rendus Mathematique Author(s): Qi'an Guan, Xiangyu Zhou In this note, we describe a relation between Lelong numbers and complex singularity exponents. As an application, we obtain a new proof of Siu's semicontinuity theorem for Lelong numbers.

Abstract: Publication date: Available online 21 March 2017 Source:Comptes Rendus Mathematique Author(s): Hyeseon Kim In this note, under an additional condition, we present an alternative proof of a stability theorem for the boundary asymptotics of the Bergman kernel due to T. Ohsawa. Our method relies on the localization of the minimum integral related to the weighted Bergman kernel.

Abstract: Publication date: Available online 21 March 2017 Source:Comptes Rendus Mathematique Author(s): Qingzhong Ji, Hourong Qin In this paper, we consider the function field analogue of the Lehmer's totient problem. Let p ( x ) ∈ F q [ x ] and φ ( q , p ( x ) ) be the Euler's totient function of p ( x ) over F q [ x ] , where F q is a finite field with q elements. We prove that φ ( q , p ( x ) ) ( q deg ( p ( x ) ) − 1 ) if and only if (i) p ( x ) is irreducible; or (ii) q = 3 , p ( x ) is the product of any 2 non-associate irreducibles of degree 1; or (iii) q = 2 , p ( x ) is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.

Abstract: Publication date: Available online 21 March 2017 Source:Comptes Rendus Mathematique Author(s): Benjamin Schmidt, Jon Wolfson A Riemannian manifold has CVC ( ϵ ) if its sectional curvatures satisfy sec ≤ ε or sec ≥ ε pointwise, and if every tangent vector lies in a tangent plane of curvature ε. We present a construction of an infinite-dimensional family of compact CVC ( 1 ) three-manifolds.

Abstract: Publication date: Available online 18 March 2017 Source:Comptes Rendus Mathematique Author(s): Assyr Abdulle, Martin E. Huber, Simon Lemaire A new optimization-based numerical method is proposed for the solution to diffusion problems with sign-changing conductivity coefficients. In contrast to existing approaches, our method does not rely on the discretization of a stabilized equation, and the convergence of the scheme can be proved without any symmetry assumption on the mesh near the interface where the conductivity sign changes.

Abstract: Publication date: Available online 15 March 2017 Source:Comptes Rendus Mathematique Author(s): Kevin Langlois Let G be a connected reductive linear algebraic group. We consider the normal G-varieties with horospherical orbits. In this short note, we provide a criterion to determine whether these varieties have at most canonical, log canonical or terminal singularities in the case where they admit an algebraic curve as rational quotient. This result seems to be new in the special setting of torus actions with general orbits of codimension 1. For the given G-variety X, our criterion is expressed in terms of a weight function ω X that is constructed from the set of G-invariant valuations of the function field k ( X ) . In the log terminal case, the generating function of ω X coincides with the stringy motivic volume of X. As an application, we discuss the case of normal k ⋆ -surfaces.

Abstract: Publication date: Available online 14 March 2017 Source:Comptes Rendus Mathematique Author(s): Guy David, Joseph Feneuil, Svitlana Mayboroda We introduce a new notion of a harmonic measure for a d-dimensional set in R n with d < n − 1 , that is, when the codimension is strictly bigger than 1. Our measure is associated with a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the d-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.

Abstract: Publication date: Available online 13 March 2017 Source:Comptes Rendus Mathematique Author(s): Thomas Budzinski A simple way to sample a uniform triangulation of the sphere with a fixed number n of vertices is the Monte-Carlo method: we start from an arbitrary triangulation and flip repeatedly a uniformly chosen edge. We give a lower bound of order n 5 / 4 on the mixing time of this Markov chain.

Abstract: Publication date: Available online 13 March 2017 Source:Comptes Rendus Mathematique Author(s): Hoai-Minh Nguyen, Marco Squassina We provide a new characterization of the logarithmic Sobolev inequality.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Erik Burman In this note, we prove error estimates in natural norms on the approximation of the boundary data in the elliptic Cauchy problem, for the finite element method first analysed in E. Burman, Error estimates for stabilised finite element methods applied to ill-posed problems, C. R. Acad. Sci. Paris, Ser. I 352 (7–8) (2014) 655–659.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Luca Baracco, Tran Vu Khanh, Stefano Pinton We show here a “weak” Hölder regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge–Ampère equation with data in the L p space and Ω satisfying an f-property. The f-property is a potential-theoretical condition that holds for all pseudoconvex domains of finite type and many examples of infinite-type ones.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Angelo R.F. de Holanda In this paper, we study the existence of positive blow-up solutions for a general class of the second-order differential equations and systems, which are positive radially symmetric solutions to many elliptic problems in R N . We explore fixed point arguments applied to suitable integral equations to get solutions.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Duván Cardona, Michael Ruzhansky In this note, we give embeddings and other properties of Besov spaces, as well as spectral and Fourier multiplier theorems, in the setting of graded Lie groups. We also present a Nikolskii-type inequality and the Littlewood–Paley theorem that play a role in this analysis and are also of interest on their own.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Vincent Cossart, Olivier Piltant, Bernd Schober Let X be an excellent scheme; we denote by H X the modified Hilbert–Samuel function. This function is upper semi-continuous along X and does not increase for the lexicographical ordering after permissible blowing ups. When X is embedded in a regular ambient scheme, for all x ∈ X , the “stable τ at x” (“τ stable de x”), denoted by τ st ( x ) , is the codimension of the ridge of the tangent cone of X at x in the tangent cone of W at x. It is well known that the function ι : X → N N × − N , x ↦ ( H X ( x ) , − τ st ( x ) ) , does not increase for the lexicographical ordering after permissible blowing ups. In this note, we show that ι is upper semi-continuous along X . This result is generalized to the non-embedded case.

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Indranil Biswas, Subramaniam Senthamarai Kannan, Donihakalu Shankar Nagaraj Let G ‾ be the wonderful compactification of a simple affine algebraic group G of adjoint type defined over C . Let T ‾ ⊂ G ‾ be the closure of a maximal torus T ⊂ G . We prove that the group of all automorphisms of the variety T ‾ is the semi-direct product N G ( T ) ⋊ D , where N G ( T ) is the normalizer of T in G and D is the group of all automorphisms of the Dynkin diagram, if G ≠ PSL ( 2 , C ) . Note that if G = PSL ( 2 , C ) , then T ‾ = C P 1 and so in this case Aut ( T ‾ ) = PSL ( 2 , C ) .

Abstract: Publication date: Available online 11 March 2017 Source:Comptes Rendus Mathematique Author(s): Yuri Berest, Alimjon Eshmatov, Wai-Kit Yeung In this paper, we give a new algebraic construction of knot contact homology in the sense of Ng [35]. For a link L in R 3 , we define a differential graded (DG) k-category A ˜ L with finitely many objects, whose quasi-equivalence class is a topological invariant of L. In the case when L is a knot, the endomorphism algebra of a distinguished object of A ˜ L coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng, and Sullivan in [13]. The input of our construction is a natural action of the braid group B n on the category of perverse sheaves on a two-dimensional disk with singularities at n marked points, studied by Gelfand, MacPherson, and Vilonen in [19]. As an application, we show that the category of finite-dimensional representations of the link k-category A ˜ L = H 0 ( A ˜ L ) defined as the 0-th homology of A ˜ L is equivalent to the category of perverse sheaves on R 3 that are singular along the link L. We also obtain several generalizations of the category A ˜ L by extending the Gelfand–MacPherson–Vilonen braid group action. Detailed proofs of results announced in this paper will appear in [4].

Abstract: Publication date: Available online 6 March 2017 Source:Comptes Rendus Mathematique Author(s): Nam Q. Le We show that for an L 2 drift b in two dimensions, if the Hardy norm of div b is small, then the weak solutions to Δ u + b ⋅ ∇ u = 0 have the same optimal Hölder regularity as in the case of divergence-free drift, that is, u ∈ C loc α for all α ∈ ( 0 , 1 ) .