Sufficient Conditions For Hyperbolicity Research Articles

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.

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.

Sufficient conditions for hyperbolicity and consistency of the dynamic equations for fluid-saturated solids

Previous studies showed that the dynamic equations for a porous fluid-saturated solid may lose hyperbolicity and thus render the boundary-value problem ill-posed while the equations for the same but dry solid remain hyperbolic. This paper presents sufficient conditions for hyperbolicity in both dry and saturated states. Fluid-saturated solids are described by two different systems of equations depending on whether the permeability is zero or nonzero (locally undrained and drained conditions, respectively). The paper also introduces a notion of wave speed consistency between the two systems as a necessary condition which must be satisfied in order for the solution in the locally drained case to tend to the undrained solution as the permeability tends to zero. It is shown that the symmetry and positive definiteness of the acoustic tensor of the skeleton guarantee both hyperbolicity and the wave speed consistency of the equations.

Stability Of Shear Shallow Water Flows with Free Surface

Stability of inviscid shear shallow water flows with free surface is studied in the framework of the Benney equations. This is done by investigating the generalized hyperbolicity of the integrodifferential Benney system of equations. It is shown that all shear flows having monotonic convex velocity profiles are stable. The hydrodynamic approximations of the model corresponding to the classes of flows with piecewise linear continuous and discontinuous velocity profiles are derived and studied. It is shown that these approximations possess Hamiltonian structure and a complete system of Riemann invariants, which are found in an explicit form. Sufficient conditions for hyperbolicity of the governing equations for such multilayer flows are formulated. The generalization of the above results to the case of stratified fluid is less obvious, however, it is established that vorticity has a stabilizing effect.

A thermodynamically compatible splitting procedure in hyperelasticity

A material is hyperelastic if the stress tensor is obtained by variation of the stored energy function. The corresponding 3D mathematical model of hyperelasticity written in the Eulerian coordinates represents a system of 14 conservative partial differential equations submitted to stationary differential constraints. A classical approach for numerical solving of such a 3D system is a geometrical splitting: the 3D system is split into three 1D systems along each spatial direction and solved then by using a Godunov type scheme. Each 1D system has 7 independent eigenfields corresponding to contact discontinuity, longitudinal waves and shear waves. The construction of the corresponding Riemann solvers is not an easy task even in the case of isotropic solids. Indeed, for a given specific energy it is extremely difficult, if not impossible, to check its rank-one convexity which is a necessary and sufficient condition for hyperbolicity of the governing equations.In this paper, we consider a particular case where the specific energy is a sum of two terms. The first term is the hydrodynamic energy depending only on the density and the entropy, and the second term is the shear energy which is unaffected by the volume change. In this case a very simple criterion of hyperbolicity can be formulated. We propose then a new splitting procedure which allows us to find a numerical solution of each 1D system by solving successively three 1D sub-systems. Each sub-system is hyperbolic, if the full system is hyperbolic. Moreover, each sub-system has only three waves instead of seven, and the velocities of these waves are given in explicit form. The last property allows us to construct reliable Riemann solvers. Numerical 1D tests confirm the advantage of the new approach. A multi-dimensional extension of the splitting procedure is also proposed.

Gaussian Thermostats as Geodesic Flows of Nonsymmetric Linear Connections

We establish that Gaussian thermostats are geodesic flows of special metric connections. We give sufficient conditions for hyperbolicity of geodesic flows of metric connections in terms of their curvature and torsion.

Relative hyperbolicity and Artin groups

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a two-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for hyperbolicity of the Deligne complex which applies to a large class of these groups for which the underlying Coxeter group is hyperbolic. The key tool in the proof is an extension of the Milnor-Svarc Lemma which states that if a group G admits a discontinuous, co-compact action by isometries on a Gromov hyperbolic metric space, then G is weakly hyperbolic relative to the isotropy subgroups of the action.

A hyperbolicity criterion for periodic solutions of functional-differential equations: The case of rational periods

We establish necessary and sufficient conditions for hyperbolicity of periodic solutions of nonlinear functional-differential equations.

Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of the billiard dynamics, and give lower estimation for the Lyapunov exponent. This extends our recent results for non-magnetic billiards on surfaces of constant curvature. Using these conditions, we construct large classes of magnetic billiard tables with positive Lyapunov exponents on the plane, on the sphere and on the hyperbolic plane.

Axisymmetric Vortex Fluid Flow in a Long Elastic Tube

A mathematical model for axisymmetric eddy motion of a perfect incompressible fluid in a long tube with thin elastic walls is proposed. Necessary and sufficient conditions for hyperbolicity of the system of equations of motion for flows with monotonic radial velocity profiles are formulated. The propagation velocities of the characteristics of the system under study and the characteristic shape of this system are calculated. The existence of simple waves continuously attached to a given steady shear flow is proved. The group of transformations admitted by the system is found, and submodels that determine invariant solutions are given. By integrating factor‐systems, new classes of exact solutions of equations of motion are found.

Vortex motions of liquid in a narrow channel

Quasi-linear integrodifferential equations that describe vortex flows of an ideal incomparessible liquid in a narrow curved channel in the Eulerian-Lagrangian coordinate system are considered. The necessary and sufficient conditions for hyperbolicity of the system of equations of motion are obtained for flows with a monotonic velocity depth profile. The propagation velocities of the characteristics and the characteristic form of the system are calculated. A particular solution is given in which the system of integrodifferential equations changes type with time. The solution of the Cauchy problem is given for linearized equations. An example of initial data for which the Cauchy problem is ill-posed is constructed.

Subgroups of Word Hyperbolic Groups in Dimension 2

If G is a word hyperbolic group of cohomological dimension 2, then every subgroup of G of type FP2 is also word hyperbolic. Isoperimetric inequalities are defined for groups of type FP2 and it is shown that the linear isoperimetric inequality in this generalized context is equivalent to word hyperbolicity. A sufficient condition for hyperbolicity of a general graph is given along with an application to ‘relative hyperbolicity’. Finitely presented subgroups of Lyndon's small cancellation groups of hyperbolic type are word hyperbolic. Finitely presented subgroups of hyperbolic 1-relator groups are hyperbolic. Finitely presented subgroups of free Burnside groups are finite in the stable range.

Non-linear equations of the dynamics of an elastic micropolar medium

The non-linear equations for a continuous elastic medium with three additional degrees of freedom associated with local rotation, are considered. Such an elastic medium is called micropolar /1/. The existence of an elastic potential is proposed for it; thermal effects are neglected. The purpose of this paper is to study certain qualtiative properties of the equations that are closely associated with the concept of hyperbolicity. The complete set of equations is represented as a system of local conservation laws, closed by finite relationships yielding the rheology of the material. The possibility of such a representation is based /2, 3/ on the fact that the gradients of the particle displacment and angle of rotation are used as a measure of the deformation. Local conservation laws for the compatibility of the strain and velocity fields of fairly simple structure are formulated. The velocities of propgation of characteristic surfaces are studied for the dynamic equations for the general case of the material under consideration. The existence of real velocities, the necessary condition for hyperbolicity, results in a constraint on the form of the elastic potential function, which is an analog of the SE-inequality /4/ in the classical theory of non-linearly elastic media. The system of non-linear equations being studied is reduced to symmetric form by replacing the vector of the solution. The necessary condition for such a transformation /5/ is the existence of an additional energy conservation law that follows from the system under consideration. The symmetric form of the equations enables us to formulate the sufficient condition for hyperbolicity — the condition of convexity of the elastic potential in its arguments. An estimate is obtained for the growth of the solutions of the Cauchy problem and the ensuing uniqueness theorem. The presence of the symmetric form of the system enables a general form to be obtained for the transport equation that governs the rate of change of a weak discontinuity along a bicharacteristic.

A Sufficient Condition for Hyperbolicity of Partial Differential Operators With Constant Coefficient Principal Part

Let $P$ be a differential operator with principal part ${P_m}$, and suppose that ${P_m}$ has constant coefficients and is hyperbolic. It is shown that the condition for hyperbolicity of $P$ when $P$ has constant coefficients, namely, that $P$ is weaker than ${P_m}$ is also a sufficient condition for hyperbolicity in the case where $P$ does not have constant coefficients. Some generalizations are also made to the case where $P$ is a square matrix of differential operators.

A sufficient condition for hyperbolicity of partial differential operators with constant coefficient principal part

Let P P be a differential operator with principal part P m {P_m} , and suppose that P m {P_m} has constant coefficients and is hyperbolic. It is shown that the condition for hyperbolicity of P P when P P has constant coefficients, namely, that P P is weaker than P m {P_m} is also a sufficient condition for hyperbolicity in the case where P P does not have constant coefficients. Some generalizations are also made to the case where P P is a square matrix of differential operators.