Conditions For Hyperbolicity Research Articles

A combinatorial higher-rank hyperbolicity condition

We investigate a coarse version of a 2(n+1) -point inequality characterizing metric spaces of combinatorial dimension at most n due to Dress. This condition, experimentally called (n,\delta) -hyperbolicity, reduces to Gromov’s quadruple definition of \delta -hyperbolicity in case n = 1 . The l_{\infty} -product of n \delta -hyperbolic spaces is (n,\delta) -hyperbolic. Every (n,\delta) -hyperbolic metric space, without any further assumptions, possesses a slim (n+1) -simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. In connection with recent work in geometric group theory, we show that every Helly group and every hierarchically hyperbolic group of (asymptotic) rank n acts geometrically on some (n,\delta) -hyperbolic space.

Mild solutions for some nonautonomous evolution equations with state-dependent delay governed by equicontinuous evolution families

In this work, we study the existence solutions and the dependence continuous with the initial data for some nondensely nonautonomous partial functional differential equations with state-dependent delay in Banach spaces. We assume that the linear part is not necessarily densely defined, satisfies the well-known hyperbolic conditions and generate a noncompact evolution family. Our existence results are based on Sadovskii fixed point Theorem. An application is provided to a reaction-diffusion equation with state-dependent delay.

On the hyperbolicity of the Krein space numerical range

In this paper a necessary and sufficient condition for hyperbolicity of the indefinite numerical range is established. As a consequence, an indefinite version of Brown–Spitkovsky theorem stating the ellipticity of the numerical range of certain tridiagonal matrices is revisited. This result leads to necessary and sufficient conditions for hyperbolicity of indefinite numerical ranges of new classes of tridiagonal matrices.

Spherically symmetric black hole spacetimes on hyperboloidal slices

Gravitational radiation and some global properties of spacetimes can only be unambiguously measured at future null infinity (ℐ+). This motivates the interest in reaching it within simulations of coalescing compact objects, whose waveforms are extracted for gravitational wave modeling purposes. One promising method to include future null infinity in the numerical domain is the evolution on hyperboloidal slices: smooth spacelike slices that reach future null infinity. The main challenge in this approach is the treatment of the compactified asymptotic region at ℐ+. Evolution on a hyperboloidal slice of a spacetime including a black hole entails an extra layer of difficulty in part due to the finite coordinate distance between the black hole and future null infinity. Spherical symmetry is considered here as the simplest setup still encompassing the full complication of the treatment along the radial coordinate. First, the construction of constant-mean-curvature hyperboloidal trumpet slices for Schwarzschild and Reissner-Nordström black hole spacetimes is reviewed from the point of view of the puncture approach. Then, the framework is set for solving hyperboloidal-adapted hyperbolic gauge conditions for stationary trumpet initial data, providing solutions for two specific sets of parameters. Finally, results of testing these initial data in evolution are presented.

Heliocentric transfer trajectory design for multiple spacecraft using lunar gravity assists

This paper presents the methodologies to design the transfer trajectories from Earth nearby to heliocentric orbits. For multiple spacecraft with the same initial condition but aiming to different phases in the heliocentric orbit, several lunar gravity assists are used to divert them into different interplanetary orbits. Firstly, based on the patched-conic approximation, a complete design process for deep-space transfer orbits was proposed for multiple spacecraft performing lunar gravity assist maneuvers. In this design process, it was assumed that each spacecraft had the same initial conditions. The periodic matching method was introduced to design the interplanetary transfer orbit, satisfying the initial and terminal constraints. An iterative method was proposed to search lunar gravity assist parameters that satisfied the hyperbolic excess velocity condition for the escape trajectories. Furthermore, a semi-analytical correction method was proposed to further approximate the precise transfer trajectory. This correction method was found to have a reduced time cost and a better error convergence compared with conventional correction methods.

The u‐p approximation versus the exact dynamic equations for anisotropic fluid‐saturated solids. I. Hyperbolicity

AbstractThe numerical solution of dynamic problems for porous fluid‐saturated solids is often performed with the use of simplified equations known as the u‐p approximation. The simplification of the equations consists in neglecting some acceleration terms, which is justified for a certain class of problems related, in particular, to geomechanics and earthquake engineering. There exist two u‐p approximations depending on how many acceleration terms are neglected. All comparative studies of the exact and u‐p formulations are focused on the question of how well the u‐p solutions approximate those obtained with the exact equations. In this paper, the equations are compared from a different point of view, addressing the question of well‐posedness of boundary value problems. The exact equations must be hyperbolic and satisfy the corresponding hyperbolicity conditions for the boundary value problems to be well posed. The u‐p equations are not of the form to which the conventional definition of hyperbolicity applies. A slight extension of the approach makes it possible to derive hyperbolicity conditions as necessary conditions for well‐posedness for the u‐p approximations. The hyperbolicity conditions derived in this paper for the u‐p approximations are formulated in terms of the acoustic tensor of the skeleton. They differ essentially from the hyperbolicity conditions for the exact equations.

Conditions for hyperbolicity and approximate Riemann invariants in one dimensional nonlinear nonlocal elasticity

We consider a generic constitutive law of nonlocal nonlinear elasticity in one space dimension corresponding to an integrodifferential equation and convert it to a system of first order. We find conditions for hyperbolicity as well as the approximate Riemann invariants for this system.

Improved Regularity of Harmonic Diffeomorphic Extensions on Quasihyperbolic Domains

Let \(\mathbb{X}\) be a Jordan domain satisfying certain hyperbolic growth conditions. Assume that φ is a homeomorphism from the boundary \(\partial \mathbb{X}\) of \(\mathbb{X}\) onto the unit circle. Denote by h the harmonic diffeomorphic extension of φ from \(\mathbb{X}\) onto the unit disk. We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of h. These generalize the Sobolev regularity of h in [A. Koski, J. Onninen, Sobolev homeomorphic extensions, J. Eur. Math. Soc. 23 (2021) 4065–4089, Theorem 3.1].

Topological states and continuum model for swarmalators without force reciprocity

Swarmalators are systems of agents which are both self-propelled particles and oscillators. Each particle is endowed with a phase which modulates its interaction force with the other particles. In return, relative positions modulate phase synchronization between interacting particles. In the present model, there is no force reciprocity: when a particle attracts another one, the latter repels the former. This results in a pursuit behavior. In this paper, we derive a hydrodynamic model of this swarmalator system and show that it has explicit doubly periodic traveling wave solutions in two space dimensions. These special solutions enjoy non-trivial topology quantified by the index of the phase vector along a period in either dimension. Stability of these solutions is studied by investigating the conditions for hyperbolicity of the model. Numerical solutions of both the particle and hydrodynamic models are shown. They confirm the consistency of the hydrodynamic model with the particle one for small times or large phase noise but also reveal the emergence of intriguing patterns in the case of small phase noise.

Flame dynamics of premixed CH4/H2/air flames in a microchannel with a wall temperature gradient

The effect of hydrogen (H2) addition on the flame dynamics of premixed methane/air mixtures in a microchannel was investigated through two-dimensional numerical computations using a detailed chemistry model. Detailed numerical simulations were performed in a 2 mm diameter tube with 120 mm length and a hyperbolic wall temperature gradient condition. All numerical computations were performed at stoichiometric mixture conditions with a fixed inlet velocity of 15 cm/s. Flame repetitive extinction and ignition (FREI) has been observed to appear at various mixture conditions. The frequency of FREI shows a non-monotonic variation for CH4/air mixtures with H2 addition. The time taken for completion of one FREI cycle increases with H2 addition. The maximum temperature and heat of the reaction were observed to decrease with hydrogen addition. The effect of diameter on the FREI cycle was studied by comparing the numerical results for 1, 1.5, and 2 mm diameter tubes. As the diameter is reduced from 2 mm to 1 mm, the FREI frequency increased, and the maximum temperature decreased owing to increased heat loss through channel walls. The location of the ignition and extinction shifted downstream for 1 mm tube, as compared to a 2 mm diameter tube.

Hyperbolicity of First Order Quasi-Linear Equations

The theorem about equivalence of the strong hyperbolicity concept and the Friedrichs hyperbolicity concept for partial quasi-linear differential equations of the first order is proved. On the basis of this theorem, the necessary and sufficient conditions of hyperbolicity are found in terms of the matrix of the corresponding linearized first order equations system.

On the Removability Property of φ‐Natural Domains in Real Banach Spaces

In this paper, we investigate the removability of φ‐natural domains in Banach spaces. Let E be a real Banach space with dimension at least 2, G a domain in E, and let be a countable subset of G which satisfies a quasi hyperbolic separation condition. Then, G is φ‐natural if and only if G is ψ‐natural, quantitatively.

Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection–diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank–Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the τ2 + hp+½ error estimates for the L2-norm under either the standard hyperbolic CFL condition, when piecewise affine (p = 1) approximation is used, or in the case of finite element approximation of order p ≥ 1, a stronger, so-called 4/3-CFL, i.e. τ ≤ Ch4/3. The theory is illustrated with some numerical examples.

Wave Equation with Hyperbolic Boundary Condition: a Frequency Domain Approach

In this paper, we investigate the stability of the linear wave equation where one part of the boundary, which is seen as a lower-dimensional Riemannian manifold, is governed by a coupled wave equation, while the other part is subject to a dissipative Robin velocity feedback. We prove that the closed-loop equations generate a semi-uniformly stable semigroup of linear contractions on a suitable energy space. Furthermore, under multiplier-related geometrical conditions, we establish a polynomial decay rate for strong solutions. This is achieved by estimating the growth of the resolvent operator on the imaginary axis.

Algorithmic Reduction of Biological Networks with Multiple Time Scales

We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting. The existence of invariant manifolds is subject to hyperbolicity conditions, for which we propose an algorithmic test based on Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.

Waves interacting with a partially immersed obstacle in the Boussinesq regime

This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for $2\times2$ hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a configuration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. To the authors' knowledge, this is the first time that these phenomena (likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems) are exhibited.

Nonlinear plane waves in saturated porous media with incompressible constituents

We consider the propagation of nonlinear plane waves in porous media within the framework of the Biot–Coussy biphasic mixture theory. The tortuosity effect is included in the model, and both constituents are assumed incompressible (Yeoh-type elastic skeleton, and saturating fluid). In this case, the linear dispersive waves governed by Biot’s theory are either of compression or shear-wave type, and nonlinear waves can be classified in a similar way. In the special case of a neo-Hookean skeleton, we derive the explicit expressions for the characteristic wave speeds, leading to the hyperbolicity condition. The sound speeds for a Yeoh skeleton are estimated using a perturbation approach. Then we arrive at the evolution equation for the amplitude of acceleration waves. In general, it is governed by a Bernoulli equation. With the present constitutive assumptions, we find that longitudinal jump amplitudes follow a nonlinear evolution, while transverse jump amplitudes evolve in an almost linearly degenerate fashion.

Asymptotic analysis of the eigenstructure of the two-layer model and a new family of criteria for evaluating the model hyperbolicity

Two-layer and multi-layer depth-averaged models have become popular for simulating exchange flows, seawater currents and geophysical flows. The partial differential equation systems associated with these models are similar to the single-layer shallow-water model. Yet, their eigenstructures are more complex owing to the pressure coupling between the layers. Such models occasionally lose their hyperbolic character, which may lead to numerical issues. A physical explanation is that Kelvin-Helmholtz type instabilities arise at the layers' interface, if the velocity difference between the layers becomes sufficiently large. A way to avoid the hyperbolicity loss is to locally introduce an extra momentum exchange between the layers, assessable from the system eigenstructure and aimed at roughly mimicking the dynamical effects of such instabilities. To better understand the hyperbolicity conditions, the eigenstructure of the two-layer model is methodically studied by an asymptotic analysis. The analysis for the limiting cases, where the layers' thicknesses are either comparable or very different from each other, reveals new stability criteria. These analytical criteria are, then, exploited to design a new family of approximate criteria, valid for any flow condition. Numerical investigations demonstrate the reliability of this approach, which can be easily implemented in numerical schemes for preserving the hyperbolicity.

Topological and geometric hyperbolicity criteria for polynomial automorphisms of

AbstractWe prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$ . Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

Hyperbolicity and causality of Einstein-Gauss-Bonnet gravity in warped product spacetimes

In Einstein-Gauss-Bonnet gravity, for a group of warped product spacetimes, we get a generalized master equation for the perturbation of tensor type. We show that the "effective metric" or "acoustic metric" for the tensor perturbation equation can be defined even without a static condition. Since this master equation does not depend on the mode expansion, the hyperbolicity and causality of the tensor perturbation equation can be investigated for every mode of the perturbation. Based on the master equation, we study the hyperbolicity and causality for all relavent vacuum solutions of this theory. For each solution, we give the exact hyperbolic condition of the tensor perturbation equations. Our approach can also applied to dynamical spacetimes, and Vaidya spacetime have been investigated as an example.