Mathematische Zeitschrift最新文献

The aim of this article devotes to establishing the (L^1)-decay of cubic order spatial derivatives of solutions to the Navier–Stokes equations, which is a long-time challenging problem. To solve this problem, new tools have to be found to overcome these main difficulties: (L^1-L^1) estimate fails for the Stokes flow; the projection operator (P:,L^1(mathbb {R}^n_+)rightarrow L^1_sigma (mathbb {R}^n_+)) becomes unbounded; the steady Stokes’s estimates does not work any more in (L^1(mathbb {R}^n_+)). We first give the asymptotic behavior with weights of negative exponent for the Stokes flow and Navier–Stokes equations in (L^1(mathbb {R}^n_+)), and these are also independent of interest by themselves. Secondly, we decompose the convection term into two parts, and translate the unboundedness of projection operator into studying an (L^1)-estimate for an elliptic problem with homogeneous Neumann boundary conditions, which is established by using the weighted estimates of the Gaussian kernel’s convolution. Finally, a crucial new formula is given for the fundamental solution of the Laplace operator, which is employed for overcoming the strong singularity in studying the cubic order spatial derivatives in (L^1(mathbb {R}^n_+)).

In this paper, we study the hyperbolicity in the sense of Gromov of domains in (mathbb {R}^d) ((dge 3)) with respect to the minimal metric introduced by Forstnerič and Kalaj (Anal PDE 17(3):981–1003, 2024). In particular, we prove that every bounded strongly minimally convex domain is Gromov hyperbolic and its Gromov compactification is equivalent to its Euclidean closure. Moreover, we prove that the boundary of a Gromov hyperbolic convex domain does not contain non-trivial conformal harmonic disks. Finally, we study the relation between the minimal metric and the Hilbert metric in convex domains.

For a noetherian ring (Lambda ), the stabilization functor yields an embedding of the singularity category of (Lambda ) into the homotopy category of acyclic complexes of injective (Lambda )-modules. When (Lambda ) contains a semisimple artinian subring E, we give an explicit description of the stabilization functor using the Hom complexes in the E-relative singular Yoneda dg category of (Lambda ). As an application to an artin algebra, we obtain an explicit compact generator for the mentioned homotopy category, whose dg endomorphism algebra turns out to be quasi-isomorphic to the associated dg Leavitt algebra.

The aim of this work is to present new spectral tools for studying the orbital stability of standing waves solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on looping edge graphs, namely, a graph consisting of a circle with several half-lines attached at a single vertex. The main novelty of this paper is at least twofold: by considering (delta )-type boundary conditions at the vertex, the extension theory of Krein &von Neumann, and a splitting eigenvalue method, we identify the Morse index and the nullity index of a specific linearized operator around of a priori positive single-lobe state profile for every positive power, this information will be main for a local stability study; and so via a bifurcation analysis on the phase plane we build at least two families of positive single-lobe states and we study the stability properties of these in the subcritical, critical, and supercritical cases. Our results recover some spectral studies in the literature associated to the NLS on looping edge graphs which were obtained via variational techniques.

Consider a holomorphic map (F: D rightarrow G) between two domains in ({{mathbb {C}}}^N). Let ({mathscr {F}}) denote a family of geodesics for the Kobayashi distance, such that F acts as an isometry on each element of ({mathscr {F}}). This paper is dedicated to characterizing the scenarios in which the aforementioned condition implies that F is a biholomorphism. Specifically, we establish this when D is a complete hyperbolic domain, and ({mathscr {F}}) comprises all geodesic segments originating from a specific point. Another case is when D and G are (C^{2+alpha })-smooth bounded pseudoconvex domains, and ({mathscr {F}}) consists of all geodesic rays converging at a designated boundary point of D. Furthermore, we provide examples to demonstrate that these assumptions are essentially optimal.

In the simplicial theory of hypercoverings we replace the indexing category (Delta ) by the symmetric simplicial category (Delta S) and study (a class of) (Delta _{textrm{inj}}S)-hypercoverings, which we call spaces admitting symmetric (semi)simplicial filtration—this special class happens to have a structure of a module over a graded commutative monoid of the form (textrm{Sym},M) for some space M. For (Delta S)-hypercoverings we construct a spectral sequence, somewhat like the Čech-to-derived category spectral sequence. The advantage of working with (Delta S) over (Delta ) is that various combinatorial complexities that come with working on (Delta ) are bypassed, giving simpler, unified proof of results like the computation of (in some cases, stable) singular cohomology (with (mathbb {Q}) coefficients) and étale cohomology (with (mathbb {Q}_{ell }) coefficients) of the moduli space of degree n maps (Crightarrow mathbb {P}^r) with C a smooth projective curve of genus g, of unordered configuration spaces, of the moduli space of smooth sections of a fixed (mathfrak {g}^r_d) that is m-very ample for some m etc. In the special case when a (Delta _{textrm{inj}}S)-object X admits a symmetric semisimplicial filtration by M, we relate these moduli spaces to a certain derived tensor.

We show that the direct counterpart of the Talenti symmetrization estimate for the Laplacian does not hold neither for the complex nor real Monge–Ampère equations. We also use this Talenti result to improve some known estimates for subharmonic functions in ({mathbb {C}},) where the constant depends on the area of the domain, instead of the diameter.

Let (mathfrak {g}) be a simple Lie algebra: its dual space (mathfrak {g}^*) is a Poisson variety. It is well known that for each nilpotent element f in (mathfrak {g}), it is possible to construct a new Poisson structure by Hamiltonian reduction which is isomorphic to some subvariety of (mathfrak {g}^*), the Slodowy slice (S_f). Given two nilpotent elements (f_1) and (f_2) with some compatibility assumptions, we prove Hamiltonian reduction by stages: the slice (S_{f_2}) is the Hamiltonian reduction of the slice (S_{f_1}). We also state an analogous result in the setting of finite W-algebras, which are quantizations of Slodowy slices. These results were conjectured by Morgan in his Ph.D. thesis. As corollary in type A, we prove that any hook-type W-algebra can be obtained as Hamiltonian reduction from any other hook-type one. As an application, we establish a generalization of the Skryabin equivalence. Finally, we make some conjectures in the context of affine W-algebras.

The concept of an abelian DG-category, introduced by the first-named author in Positselski (Exact DG-categories and fully faithful triangulated inclusion functors. arXiv:2110.08237 [math.CT]), unites the notions of abelian categories and (curved) DG-modules in a common framework. In this paper we consider coderived and contraderived categories in the sense of Becker. Generalizing some constructions and results from the preceding papers by Becker (Adv Math 254:187–232, 2014. arXiv:1205.4473 [math.CT]) and by the present authors (Positselski and Št’ovíček in J Pure Appl Algebra 226(#4):106883, 2022. arXiv:2101.10797 [math.CT]), we define the contraderived category of a locally presentable abelian DG-category (textbf{B}) with enough projective objects and the coderived category of a Grothendieck abelian DG-category (textbf{A}). We construct the related abelian model category structures and show that the resulting exotic derived categories are well-generated. Then we specialize to the case of a locally coherent Grothendieck abelian DG-category (textbf{A}), and prove that its coderived category is compactly generated by the absolute derived category of finitely presentable objects of (textbf{A}), thus generalizing a result from the second-named author’s preprint (Št’ovíček in On purity and applications to coderived and singularity categories. arXiv:1412.1615 [math.CT]). In particular, the homotopy category of graded-injective left DG-modules over a DG-ring with a left coherent underlying graded ring is compactly generated by the absolute derived category of DG-modules with finitely presentable underlying graded modules. We also describe compact generators of the coderived categories of quasi-coherent matrix factorizations over coherent schemes.

In this note we compute the tautological ring of Hilbert modular varieties at an unramified prime. This is the first computation of the tautological ring of a non-compactified Shimura variety beyond the case of the Siegel modular variety (mathcal {A}_{g}). While the method generalises that of van der Geer for (mathcal {A}_{g}), there is an added difficulty in that the highest degree socle has (d>1) generators rather than 1. To deal with this we prove that the d cycles obtained by taking closures of codimension one Ekedahl–Oort strata are linearly independent. In contrast, in the case of (mathcal {A}_{g}) it suffices to prove that the class of the p-rank zero locus is non-zero. The limitations of this method for computing the tautological ring of other non-compactified Shimura varieties are demonstrated with an instructive example.