Showing papers in "Rendiconti del Seminario Matematico della Università di Padova in 2014"
36 citations
TL;DR: In this paper, a simple proof of the local existence of (finite energy) solutions in L3 for initial data u0∈L2∩L3, based on energy estimates and regularisation of the initial data with the heat semigroup was given.
Abstract: We consider the three-dimensional Navier–Stokes equations on the whole space R3 and on the three-dimensional torus T3. We give a simple proof of the local existence of (finite energy) solutions in L3 for initial data u0∈L2∩L3, based on energy estimates and regularisation of the initial data with the heat semigroup.
We also provide a lower bound on the existence time of a strong solution in terms of the solution v(t) of the heat equation with such initial data: there is an absolute constant e>0 such that solutions remain regular on [0,T] if ∥u0∥3L3∫T0∫R3|∇v(s)|2|v(s)|dxdt≤e. This implies the u∈C0([0,T];L3) regularity criterion due to von Wahl. We also derive simple a priori estimates in Lp for p>3 that recover the well known lower bound ∥u(T−t)∥Lp≥ct−(p−3)/2p on any solution that blows up in Lp at time T.
The key ingredients are a calculus inequality ∥u∥pL3p≤c∫|u|p−2|∇u|2 (valid on R3 and for functions on bounded domains with zero average) and the bound on the pressure ∥p∥Lr≤cr∥u∥2L2r, valid both on the whole space and for periodic boundary conditions.
Keywords: Navier-Stokes equations, critical spaces, calculus inequalities
31 citations
TL;DR: In this paper, a Tannakian description for l-adic perverse sheaves on semiabelian varieties is given, which combines a construction of Gabber and Loeser for algebraic tori with a generic vanishing theorem for the cohomology of constructible sheaves.
Abstract: We give a Tannakian description for categories of l-adic perverse sheaves on semiabelian varieties which combines a construction of Gabber and Loeser for algebraic tori with a generic vanishing theorem for the cohomology of constructible sheaves on abelian varieties. As an application we explain how the arising Tannaka groups on abelian varieties can be studied in terms of semiabelian degenerations via the functor of nearby cycles.
22 citations
TL;DR: In this article, it was shown that for all most every t > 0, locally, the solution is in SBV (Special functions of bounded variations) in space variable, and that for almost everywhere in t ≥ 0, the entropy solution cannot be removed.
Abstract: Consider a scalar conservation law in one space dimension with initial data
in LI : If the flux f is in C 2 and locally uniformly convex, then for all t > 0, the
entropy solution is locally in BV (functions of bounded variation) in space variable.
In this case it was shown in [5], that for all most every t > 0, locally, the solution is in
SBV (Special functions of bounded variations). Furthermore it was shown with an
example that for almost everywhere in t > 0 cannot be removed. This paper deals
with the regularity of the entropy solutions of the strict convex C 1 flux f which need
not be in C 2 and locally uniformly convex. In this case, the entropy solution need not
be locally in BV in space variable, but the composition with the derivative of the flux
function is locally in BV. Here we prove that, this composition is locally is in SBV on
all most every t > 0. Furthermore we show that this is optimal.
19 citations
19 citations
TL;DR: In this article, the authors give a simple criterium (in terms of those classes) for relative Poincare-Birkhoff-Witt type results to hold, inspired by the recent work of Chen-Stienon-Xu.
Abstract: Inspired by the recent work of Chen-Stienon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincare-Birkhoff-Witt type results to hold. The tools we use (e.g. the first infinitesimal neighbourhood Lie algebroid) are straightforward generalizations of the ones previously developped by Caldararu, Tu and the author for Lie algebra inclusions.
14 citations
13 citations
TL;DR: In this paper, an undirected simple graph D(G) whose vertices are the proper subgroups of G which are not contained in the Frattini subgroup of G and two vertices H and K are joined by an edge if and only if GhH ;Ki.
Abstract: For a finite group G different from a cyclic group of prime power order, we introduce an undirected simple graph D(G) whose vertices are the proper subgroups of G which are not contained in the Frattini subgroup of G and two vertices H and K are joined by an edge if and only if GhH ;Ki. In this paper we study D(G) and show that it is connected and determine the clique and chromatic number of D(G) and obtain bounds for its diameter and girth. We classify finite groups with complete graphs and also classify finite groups with domination number 1. Also we show that if the independence number of the graph D(G) is at most 7, then G is solvable. MATHEMATICS SUBJECT CLASSIFICATION (2010). 20D99; 05C25, 05C83.
9 citations
8 citations
TL;DR: In this paper, a variational proof of Matthew Grayson's convexification theorem for simple closed curves moving by curvature in the plane is given, where the authors show that the convexity theorem holds for all closed curves.
Abstract: In this note we show a variational proof of Matthew Grayson’s convexification theorem for simple closed curves moving by curvature in the plane. CONTENTS
7 citations
TL;DR: In this article, it was shown that if the slope of f is strictly less than min{k1, k2}− 2, then f is a classical Hilbert modular form of level Γ00(N)∩Γ0(p).
Abstract: Let F be a real quadratic field, p be a rational prime inert in F , and N ≥ 4 be an integer coprime to p. Consider an overconvergent p-adic Hilbert eigenform f for F of weight (k1, k2) ∈ Z and level Γ00(N). We prove that if the slope of f is strictly less than min{k1, k2}− 2, then f is a classical Hilbert modular form of level Γ00(N)∩Γ0(p).
TL;DR: In this article, the authors deduit la fidelite de l'algebre d'Iwasawa des matrices unipotentes superieures de GL2(O_F ) sur une representation lisse irreductible admissible de GL 2(F).
Abstract: Soit F une extension finie de Q_p, d'anneau des entiers O_F et E une extension finie de F_p. L'action naturelle de O_F× sur O_F se prolonge alors en une action continue sur l'algebre d'Iwasawa E[[O_F ]]. Dans ce travail, on deemontre que les ideaux non nuls de E[[O_F ]] stables par O_F× sont ouverts. En particulier, on en deduit la fidelite de l'action de l'algebre d'Iwasawa des matrices unipotentes superieures de GL2(O_F ) sur une representation lisse irreductible admissible de GL2(F).
TL;DR: In this paper, the authors present graphs with this property, embeddable in various lattices, and of remarkably small order, in which every vertex is missed by some longest cycle.
Abstract: No hypohamiltonian graphs are embeddable in the planar square lattice. This lattice contains, however, graphs in which every vertex is missed by some longest cycle. In this paper we present graphs with this property, embeddable in various lattices, and of remarkably small order.
TL;DR: A subgroup A of a group G is said to be a CAPsubgroup of G if for any chief factor H/K of G, there holds H ∩ A = K ∩ HA or HA = KA as discussed by the authors.
Abstract: A subgroup A of a group G is said to be a CAPsubgroup of G if for any chief factor H/K of G, there holds H ∩ A = K ∩ A or HA = KA. We investigate the influence of CAP-subgroups on the structure of finite groups. Some recent results are generalized. MSC(2000): 20D10, 20D15
TL;DR: In this paper, the authors analyzed the Einstein-Maxwell equations for an irrotational stiff fluid under the spherical symmetry assumption on the space-time, in Bondi co- ordinates, and reduced the considered model to a nonlinear evolution system of partial integrodifferential equations.
Abstract: We analyze the Einstein-Maxwell equations for an irrotational stiff fluid. Under the spherical symmetry assumption on the space-time, in Bondi co- ordinates, the considered model is reduced to a nonlinear evolution system of partial integrodifferential equations. Assuming regularity at the center of symmetry and that the matter content of the initial light cone is the so-called null dust, the characteristic initial value problem associated to the obtained system is solved globally by a contraction mapping argument. In future work we will address the issue of global well-posedness for the considered model in other physically interesting cases where the matter content of the initial light cone is not the null dust.
TL;DR: In this article, the computation for the Ramachandran index for Galois coverings and foliations is reduced to a solely boundary computation, which is a reminescence of the classical theory.
Abstract: The computation for the Ramachandran index for Galois coverings and foliations is reduced to a solely boundary computation. This is a reminescence of the classical theory. MATHEMATICS SUBJECT CLASSIFICATION (2012). 19K56, 58J32.
TL;DR: In this paper, the modulus and real part of the linear combination of f'(z) and f(z)/z are studied and conditions when f is with bounded turning are obtained.
Abstract: Let f be an analytic function in the open unit disk normalized such that f(0) = f′(0)- 1 = 0. In this paper the modulus and the real part of the linear combination of f'(z) and f(z)/z is studied and conditions when
f is with bounded turning are obtained.
TL;DR: In this article, it was shown that if a group is a group such that nse=nse, then nse = nse and nse is unique determined by nse.
Abstract: Let be a group and be the set of element orders of . Let and be the number of elements of order in . Let nse. In Khatami et al and Liu's works, the groups , and are unique determined by nse. In this paper, we prove that if is a group such that nse=nse, then .
TL;DR: In this paper, an odd degree rational valued character x of a solvable group is induced from a linear character of some subgroup of G. The authors extend this result to odd degree characters x of G that take values in certain cyclotomic extensions of Q.
Abstract: Let G be a solvable group. An odd degree rational valued character x of G is induced from a linear character of some subgroup of G. We extend this result to odd degree characters x of G that take values in certain cyclotomic extensions of Q. MATHEMATICS SUBJECT CLASSIFICATION (2010). 20C15.
TL;DR: In this paper, the affine group scheme whose category of finite dimensional representations is equivalent to a tensor category of vector spaces equipped with semi-stable (multiple) filtrations of slope zero is connected.
Abstract: We show that the affine group scheme whose category of finite dimensional representations is equivalent to a tensor category of finite dimensional vector spaces equipped with semi-stable (multiple) filtrations of slope zero is connected. MATHEMATICS SUBJECT CLASSIFICATION (2010). 20G05 (primary), 14L24, 18D10 (secondary).
TL;DR: In this article, the Lie transformation algebra of a monoid object A in a K-linear symmetric monoidal category (C; ; 1), where K is a field, is defined.
Abstract: We define the Lie transformation algebra of a (not necessarily associative) monoid object A in a K-linear symmetric monoidal category (C; ; 1), where K is a field. When A is associative and satisfies certain conditions, we describe explicity the Lie transformation algebra and inner derivations of A. Additionally, we also show that derivations preserve the nucleus of the monoid A MATHEMATICS SUBJECT CLASSIFICATION (2010). 17A36, 18D10.
TL;DR: In this article, a Sobolev inequality for elliptic systems of partial differential equations is established, which generalizes the ordinary Banach algebra property of such spaces, and for all f,c 2 W(V) that satisfy sptc Vs V and domains V R that are nonempty, open, and satisfy the cone condition.
Abstract: In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for p 2 [1;1), n;m 2 Z, and m 2 that satisfy m > n=p, kfckm;p;V K sup Vs jfj ! kckm;p;V kckmÿ1;q;V kckmÿ1;p;V kfkm;p;V \" # for all f;c 2 W(V) that satisfy sptc Vs V and domains V R that are nonempty, open, and satisfy the cone condition. Here q p if p > n, q 2 (n= ; pn=(nÿ p)] if n > p, q 2 (n= ;1) if p n, K K(n; p;m; q; C), where C is the cone from the cone condition, and : [[ n=p ]], the largest integer less than or equal to n=p. MATHEMATICS SUBJECT CLASSIFICATION (2010). 46E35, 35J57, 74B15. KEYWORDS. Elasticity, elliptic regularity, Sobolev estimate, systems of partial differential equations. 1. Introduction; Sobolev Spaces A standard classical methodology used to obtain a priori estimates for elliptic systems of partial differential equations is to first prove the required estimate when the system has constant coefficients and the region has smooth boundary and then use a partition of unity to extend the es(*) Indirizzo dell'A.: Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA. E-mail: hsimpson@math.utk.edu (**) Indirizzo dell'A.: Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA. E-mail: sspector@siu.edu timate to coefficients that depend on position and regions that are less regular. For example, Agmon, Douglis, and Nirenberg [3, 4] first establish the estimate (in the notation from Elasticity): for all u 2 C1(V; R) that satisfy u 0 in a neighborhood of D : @VnS, kukm1;p;V N Div C[ru] mÿ1;p;V C[ru]n mÿ1 p;p;S kukp;V ;
1:1 where C : M n ! M n is a constant linear mapping of the n n matrices M n and V is a ball with S [ or V is a half-ball and S is the flat portion of the boundary of V. Here m 2 Z, p 2 (1;1), n is the outward unit normal to the boundary @V,
TL;DR: In this paper, the authors study Cartan-Eilenberg Gorenstein categories by introducing CEprojective CE-generators and CE-injective CEcogenerators.
Abstract: We study Cartan-Eilenberg Gorenstein categories by introducing CEprojective CE-generators and CE-injective CE-cogenerators in the paper. We give a relationship between injective cogenerators (resp., projective generators) introduced by Sather-Wagstaff, Sharif and White and CE-injective CE-cogenerators (resp., CEprojective CE-generators). As applications, we prove some stability results of CartanEilenberg Gorenstein categories. Mathematics Subject Classification (2010). 18G10, 18G25, 18G35.