Angew Math Research Articles

On the inner automorphisms of a singular foliation

A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give a proof of the (surprisingly non-trivial) fundamental fact that the time-one flow of an element of a singular foliation (i.e. its exponential) is an automorphism of the singular foliation. This fact was previously proven in Androulidakis and Skandalis (J Reine Angew Math 626:1–37, 2009) using an infinite dimensional argument (involving differential operators), and the purpose of this note is to complement that proof with a finite dimensional proof in which the problem is reduced to solving an elementary ODE.

Long-time behavior of solutions for the compressible quantum magnetohydrodynamic model in $${\mathbb {R}}^3$$ R 3

In this paper, long-time behavior of solutions for the compressible viscous quantum magnetohydrodynamic model in three-dimensional whole space is studied. We establish the optimal time decay rates for higher-order spatial derivatives of density, velocity and magnetic field, which improve the work of Pu and Xu (Z Angew Math Phys 68:18, 2017).

Stability of thermoelastic Bresse systems

In this paper, we study thermoelastic Bresse systems where the heat conductions, acting on the axial force, are modeled for both Fourier’s and Cattaneo’s laws. For the Fourier’s case, we prove exponential stability of solutions if and only if the condition of equal wave speeds is satisfied. For the Cattaneo’s case, we characterize the exponential stability by a new condition on the coefficients of the system. We also prove, in the general case, polynomial stability of solutions. These results complement previous results obtained by Liu and Rao (Z Angew Math Phys 60:54–69, 2009), Fatori and Rivera (IMA J Appl Math 75:881–904, 2010) and Dell’Oro (J Differ Equ 258:3902–3927, 2015).

Domains in Metric Measure Spaces with Boundary of Positive Mean Curvature, and the Dirichlet Problem for Functions of Least Gradient

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincare inequality. We propose a notion of domain with boundary of positive mean curvature and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here least gradient is defined as minimizing total variation (in the sense of BV functions), and boundary conditions are satisfied in the sense that the boundary trace of the solution exists and agrees with the given boundary data. This extends the result of Sternberg et al. (J Reine Angew Math 430:35–60, 1992) to the non-smooth setting. Via counterexamples, we also show that uniqueness of solutions and existence of continuous solutions can fail, even in the weighted Euclidean setting with Lipschitz weights.

Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

This paper presents a homogenisation-based constitutive model to describe the effective transient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick’s second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981–998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.

Structure of prescribed gradient domains for non-integrable vector fields

Let $$F \in C^1(\Omega ,{\mathbb {R}}^{n})$$ and $$f\in C^2(\Omega )$$ , where $$\Omega $$ is an open subset of $${\mathbb {R}}^{n}$$ with n even. We describe the structure of the set of points in $$\Omega $$ at which the equality $$D f = F$$ and a certain non-integrability condition on F hold. This result generalizes the second statement of Balogh (J Reine Angew Math 564:63–83, 2003, Theorem 3.1).

Flagged Grothendieck polynomials

We show that the flagged Grothendieck polynomials defined as generating functions of flagged set-valued tableaux of Knutson et al. (J Reine Angew Math 630:1–31, 2009) can be expressed by a Jacobi–Trudi-type determinant formula generalizing the work of Hudson–Matsumura (Eur J Comb 70:190–201 2018). We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide.

Blow-Up Criteria and Regularity Criterion for the Three-Dimensional Magnetic Bénard System in the Multiplier Space

This study is devoted to investigating the blow-up criteria of strong solutions and regularity criterion of weak solutions for the magnetic Benard system in $$\mathbb {R}^3$$ in a sense of scaling invariant by employing a different decomposition for nonlinear terms. Firstly, the strong solution $$(u,b,\theta )$$ of magnetic Benard system is proved to be smooth on (0, T] provided the velocity field u satisfies $$\begin{aligned} u\in {L}^{\frac{2}{1-r}}(0,T;\dot{\mathbb {X}}_r(\mathbb {R}^3))\quad ~~with\quad 0\le {r}<1, \end{aligned}$$ or the gradient field of velocity $$\nabla {u}$$ satisfies $$\begin{aligned} \nabla {u}\in {L}^{\frac{2}{2-\gamma }}(0,T;\dot{\mathbb {X}}_\gamma (\mathbb {R}^3))\quad ~~with\quad 0\le {\gamma }\le {1}. \end{aligned}$$ Moreover, we prove that if the following conditions holds: $$\begin{aligned} u\in {L}^\infty (0,T;\dot{\mathbb {X}}_1(\mathbb {R}^3))\quad and \quad \Vert u\Vert _{L^\infty (0,T;\dot{\mathbb {X}}_1(\mathbb {R}^3))}<\varepsilon , \end{aligned}$$ where $$\varepsilon >0$$ is a suitable small constant, then the strong solution $$(u,b,\theta )$$ of magnetic Benard system can also be extended beyond $$t=T$$ . Finally, we show that if some partial derivatives of the velocity components, magnetic components and temperature components (i.e. $$\tilde{\nabla }\tilde{u}$$ , $$\tilde{\nabla }\tilde{b}$$ , $$\tilde{\nabla }\theta $$ ) belong to the multiplier space, the solution $$(u,b,\theta )$$ actually is smooth on (0, T). Our results extend and generalize the recent works (Qiu et al. in Commun Nonlinear Sci Numer Simul 16:1820–1824, 2011; Tian in J Funct Anal, 2017. https://doi.org/10.1155/2017/3795172 ; Zhou and Gala in Z Angew Math Phys 61:193–199, 2010; Zhang et al. in Bound Value Probl 270:1–7, 2013) respectively on the blow-up criteria for the three-dimensional Boussinesq system and MHD system in the multiplier space.

On the stability of $$L^p$$ L p -norms of Riemannian curvature at rank one symmetric spaces

We study stability and local minimizing property of $$L^p$$ -norms of Riemannian curvature tensor denoted by $$\mathcal {R}_p$$ by variational methods. We compute the Hessian of $$\mathcal {R}_p$$ at compact rank 1 symmetric spaces and prove that they are stable for $$\mathcal {R}_p$$ for certain values of $$p\ge 2$$ . A similar result also holds for compact quotients of rank 1 symmetric spaces of non-compact type. Consequently, we obtain stability of $$L^{\frac{n}{2}}$$ -norm of Weyl curvature at these metrics using results from Gursky and Viaclovsky (J Reine Angew Math 400:37–91, 2015).

Local energy inequalities for mean curvature flow into evolving ambient spaces

We establish a local monotonicity formula for mean curvature flow into a curved space whose metric is also permitted to evolve simultaneously with the flow, extending the work of Ecker (Ann Math (2) 154(2):503–525, 2001), Huisken (J Differ Geom 31(1):285–299, 1990), Lott (Commun Math Phys 313(2):517–533, 2012), Magni, Mantegazza and Tsatis (J Evol Equ 13(3):561–576, 2013) and Ecker et al. (J Reine Angew Math 616:89–130, 2008). This formula gives rise to a monotonicity inequality in the case where the target manifold’s geometry is suitably controlled, as well as in the case of a gradient shrinking Ricci soliton. Along the way, we establish suitable local energy inequalities to deduce the finiteness of the local monotone quantity.

The $$C^{2,\alpha }$$ C 2 , α -estimate for conical Kähler–Ricci flow

In this note, we establish a parabolic version of Tian’s $$C^{2,\alpha }$$ -estimate for conical complex Monge–Ampere equations (Tian in Chin Ann Math Ser B 38(2):687–694, 2017), which includes conical Kahler–Einstein metrics. Our estimate will complete the proof of the existence of unnormalized conical Kahler–Ricci flow in Shen (J Reine Angew Math, [28]).

A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling

In this paper, we give a targeted review of the state of the art in the study of planar elastic beams in large deformations, also in the presence of geometric nonlinearities. The main scope of this work is to present the different methods of analysis available for describing the possible equilibrium forms and the motions of elastic beams. For the sake of completeness, we start by giving an overview of the nonlinear theories introduced for approaching this argument and then we account for the variational principles and deformation energies introduced for modelling beams undergoing large deformations and displacements. We then consider different kinds of loads treated in the literature and the corresponding induced beam deformations. We conclude by accounting for the available analysis for stability and some considerations about problems where live loads are applied, as well as by describing some relevant numerical methods of use in the applications we have in mind. The selection criterion for the reviewed papers is dictated by the need to study large deformations and the dynamics of pantographic sheets. (Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A 2016; 472(2185): 20150790), dell’Isola et al. (Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 2015; 66(6): 3473–3498), Turco et al. (Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67(4): 1–28)].

Large time behavior of a generalized Oseen evolution operator, with applications to the Navier–Stokes flow past a rotating obstacle

Consider the motion of a viscous incompressible fluid in a 3D exterior domain D when a rigid body $$\mathbb R^3{\setminus } D$$ moves with prescribed time-dependent translational and angular velocities. For the linearized non-autonomous system, $$L^q$$ - $$L^r$$ smoothing action near $$t=s$$ as well as generation of the evolution operator $$\{T(t,s)\}_{t\ge s\ge 0}$$ was shown by Hansel and Rhandi (J Reine Angew Math 694:1–26, 2014) under reasonable conditions. In this paper we develop the $$L^q$$ - $$L^r$$ decay estimates of the evolution operator T(t, s) as $$(t-s)\rightarrow \infty $$ and then apply them to the Navier–Stokes initial value problem.

The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

We construct the $$\Lambda $$ -adic de Rham analogue of Hida’s ordinary $$\Lambda $$ -adic etale cohomology and of Ohta’s $$\Lambda $$ -adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of $$\mathbf {Q}_p$$ , we give a purely geometric proof of the expected finiteness, control, and $$\Lambda $$ -adic duality theorems. Following Ohta, we then prove that our $$\Lambda $$ -adic module of differentials is canonically isomorphic to the space of ordinary $$\Lambda $$ -adic cuspforms. In the sequel (Cais, Compos Math, to appear) to this paper, we construct the crystalline counterpart to Hida’s ordinary $$\Lambda $$ -adic etale cohomology, and employ integral p-adic Hodge theory to prove $$\Lambda $$ -adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and (Cais, Compos Math, to appear), we will be able to provide a “cohomological” construction of the family of $$(\varphi ,\Gamma )$$ -modules attached to Hida’s ordinary $$\Lambda $$ -adic etale cohomology by Dee (J Algebra 235(2), 636–664, 2001), as well as a new and purely geometric proof of Hida’s finiteness and control theorems. We are also able to prove refinements of the main theorems in Mazur and Wiles (Compos Math 59(2):231–264, 1986) and Ohta (J Reine Angew Math 463:49–98, 1995).

Words, Hausdorff dimension and randomly free groups

We bound the size of fibers of word maps in finite and residually finite groups, and derive various applications. Our main result shows that, for any word $$1 \ne w \in F_d$$ there exists $$\epsilon > 0$$ such that if $$\Gamma $$ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $$w:\Gamma ^d \rightarrow \Gamma $$ have Hausdorff dimension at most $$d -\epsilon $$ . We conclude that profinite groups $$G := {\hat{\Gamma }}$$ , $$\Gamma $$ as above, satisfy no probabilistic identity, and therefore they are randomly free, namely, for any $$d \ge 1$$ , the probability that randomly chosen elements $$g_1, \ldots , g_d \in G$$ freely generate a free subgroup (isomorphic to $$F_d$$ ) is 1. This solves an open problem from Dixon et al. (J Reine Angew Math (Crelle’s) 556:159–172, 2003). Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit–Thompson theorem.

On finite dimensional Nichols algebras of diagonal type

This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand–Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59–124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164:175–188, 2006; Heckenberger and Yamane in Math Z 259:255–276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in Angiono (J Eur Math Soc 17:2643–2671, 2015; J Reine Angew Math 683:189–251, 2013); the PBW-basis; the dimension or the Gelfand–Kirillov dimension; the associated Lie algebra as in Andruskiewitsch et al. (Bull Belg Math Soc Simon Stevin 24(1):15–34, 2017). Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory.

Noether’s problem for cyclic groups of prime order

Let k be a field and \(k(x_0,\ldots ,x_{p-1})\) be the rational function field of p variables over k where p is a prime number. Suppose that \(G=\langle \sigma \rangle \simeq C_p\) acts on \(k(x_0,\ldots ,x_{p-1})\) by k-automorphisms defined as \(\sigma :x_0\mapsto x_1\mapsto \cdots \mapsto x_{p-1}\mapsto x_0\). Denote by P the set of all prime numbers and define \(P_0=\{p\in P:\mathbb {Q}(\zeta _{p-1})\) is of class number one\(\}\) where \(\zeta _n\) a primitive n-th root of unity in \(\mathbb {C}\) for a positive integer n; \(P_0\) is a finite set by Masley and Montgomery (J Reine Angew Math 286/287:248–256, 1976). Theorem. Let k be an algebraic number field and \(P_k=\{p\in P: p\) is ramified in \(k\}\). Then \(k(x_0,\ldots ,x_{p-1})^G\) is not stably rational over k for all \(p\in P\backslash (P_0\cup P_k)\).

On the analytic systole of Riemannian surfaces of finite type

In (J Differ Geom 103(1):1–13, 2016) we introduced, for a Riemannian surface S, the quantity {Lambda(S):={rm inf}_Flambda_0(F)}, where {lambda_0(F)} denotes the first Dirichlet eigenvalue of F and the infimum is taken over all compact subsurfaces F of S with smooth boundary and abelian fundamental group. A result of Brooks (J Reine Angew Math 357:101–114, 1985) implies {Lambda(S)geqlambda_0(tilde{S})}, the bottom of the spectrum of the universal cover {tilde{S}}. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate {Lambda(S)} with the systole, improving the main result of (Enseign Math 60(2):1–23, 2014).

Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

In this paper, we study the limit cycles for m-piecewise discontinuous polynomial Lienard differential systems of degree n with m/2 straight lines passing through the origin whose slopes are $$\tan (\alpha + 2j\pi /m)$$ for $$j = 0, 1, \ldots , m/2 -1$$ , and prove that for any positive even number m, if $$\sin ( m\alpha /2)\ne 0$$ , then there always exists such a system possessing at least $$\left[ \frac{1}{2}(n-\frac{m-2}{2}) \right] $$ limit cycles. This result verifies a conjecture proposed by Llibre and Teixerira (Z Angew Math Phys 66:51–66, 2015).

The control of the boundary layer for the micropolar fluid equations with zero limits of angular and microrotational viscosities

In this paper, we consider an initial-boundary value problem to the two-dimensional incompressible micropolar fluid equations. Our main purpose is to study the boundary layer effects, and especially, we pay more attention to control the boundary layer as the angular and microrotational viscosities go to zero. It is shown that the boundary layer thickness can be controlled by the derivative of the boundary temperature with respect to time. As a matter of fact, the relationship between the boundary layer thickness (\(O(\gamma ^\beta )\)) and the time derivative of the boundary temperature can be found in (1.13). Meanwhile, we generalize the conclusion of Reference Chen et al. (Z Angew Math Phys 65:687–710, 2014).