Second order abstract differential equations with singular coefficients

Summary We study a large class of second order linear abstract differential equations, whose coefficients can be singular. In the framework of suitable « weighted » spaces, we prove some existence and uniqueness results for generalized and ordinary solutions of initial value problems for such equations.This work was partially supported by the G.N.A.F.A. and the Istituto di Analisi Numerica of the C.N.R. (Italy).     

A note on the amenable subgroups of PSL (2,R)

In this note we give a characterization of the amenable subgroups of PSL (2,R) in terms of the action on the hyperbolic half-plane.This work was partially supported by G.N.A.F.A. of the C.N.R., Italy.     

On a differential problem coming from cell biology

Summary An initial and boundary value problem for a system of first order linear partial differential equations is studied. This problem arises in cell biology, when the evolution of a homogeneous cell population is modelled. Some existence, uniqueness and regularity results are proven; an estimate for the growth of the cell population is also obtained.This work was in part supported by the Istituto di Analisi Numerica of the C.N.R. in Pavia (Italy), the G.N.A.F.A. of the C.N.R. (Italy) and the Ministero della Pubblica Istruzione (Italy).     

Resonantly Interacting Weakly Nonlinear Hyperbolic Waves in the Presence of Shocks: A Single Space Variable in a Homogeneous,Time Independent Medium

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves for a single space variable in a homogeneous, time independent medium. This theory extends the results previously presented by A. Majda and R. Rosales, under similar hypotheses, to the case where waves break and shocks form. Similarly the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable, is included as a special case. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small-amplitude periodic initial data are prescribed. In the important physical example of the 3 ? 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. These are the same equations derived by Majda and Rosales previously. However, the waves are displaced relative to the positions prescribed by them.     

On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity

We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded as t → ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called as C ^k -type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.     

An equivalence theorem on properties A, B for third order differential equations

Differential equations are often classified according to oscillatory/nonoscillatory properties of their solutions as equations having property A or property B. The aim of the paper is to state an equivalence theorem between property A and property B for third order differential equations. Some applications, to linear as well as to nonlinear equations, are given too. Particularly, we give integral criteria ensuring property A or B for nonlinear equations. Our only assumption on nonlinearity is its superlinearity in neighbourhood of infinity, hence our results apply also to Emden-Fowler type equations.The second author wishes to thank C.N.R. of Italy and Grant Agency of Czech Republic (grant 201/96/0410) which made this research possible.     

On finite element methods for convection dominated phenomena

The aim of this work is to give theoretical justification of several types of finite element approximations to the initial-boundary value problems of first order linear hyperbolic equations. Our approximate scheme is obtained by the piecewise linear continuous finite element method for space variable, x, and the Euler type step by step integration method for time variable, t. An artificial viscosity technique, up-stream type methods are considered within the frame work of L²-theory. The convergence and the error estimate of the approximate solutions to the true one are discussed.     

On construction of weak solutions with higher integrable gradients to linear hyperbolic partial differential systems

Taking linear hyperbolic partial differential equations as an illustration, we attempt to construct weak solutions with higher integrable gradients, in the sense of Gehring, to hyperbolic diffeential equations with initial and boundary conditions. We adopt Rothe's method and follow the calculation which has been expanded by Giaquinta and Struwe in dealing with parabolic equations. To establish the scheme, we evaluate some local estimates for solutions to Rothe's approximations to hyperbolic differential equations. Bibliography: 6 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 30–52.     

On the hyperbolic limit points of groups acting on hyperbolic spaces

We study the hyperbolic limit points of a groupG acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case whereG is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting mapg ⁺↦g ⁻, is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case whereG is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends ofG. Partially supported by a grant from M.U.R.S.T., Italy.     

Resonantly Interacting Weakly Nonlinear Hyperbolic Waves. I. A Single Space Variable

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves in a single space variable. This theory includes as a special case the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small-amplitude periodic initial data are prescribed. In the important physical example of the 3 × 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. (In the general case we give many new conditions guaranteeing nonresonance for a given hyperbolic system with prescribed initial data, as well as other new structural conditions which imply that resonance occurs.) A method for treating resonantly interacting waves in several space variables, together with applications, will be developed by the authors elsewhere.     

On the Riemann matrices of the hyperbolic system dual to the system of equations of one-dimensional gas dynamics

We consider the Cauchy problem for the quasilinear hyperbolic system describing a one-dimensional flow of a gas with the equation of state p = p(ϱ), p′(ϱ) > 0, and with initial data satisfying a monotonicity condition. We suggest an approach to solving it by reduction to the Cauchy problem for the linear hyperbolic system obtained from the original system by the hodograph transformation. These constructions are extended to a system of elasticity equations describing nonlinear vibrations of a one-dimensional medium. The main result is illustrated by two examples.     

Real valued iterative methods for solving complex symmetric linear systems

Complex valued systems of equations with a matrix R + 1S where R and S are real valued arise in many applications. A preconditioned iterative solution method is presented when R and S are symmetric positive semi‐definite and at least one of R, S is positive definite. The condition number of the preconditioned matrix is bounded above by 2, so only very few iterations are required. Applications when solving matrix polynomial equation systems, linear systems of ordinary differential equations, and using time‐stepping integration schemes based on Padé approximation for parabolic and hyperbolic problems are also discussed. Numerical comparisons show that the proposed real valued method is much faster than the iterative complex symmetric QMR method. Copyright © 2000 John Wiley & Sons, Ltd.     

On stability of numerical schemes via frozen coefficients and the magnetic induction equations

We study finite difference discretizations of initial boundary value problems for linear symmetric hyperbolic systems of equations in multiple space dimensions. The goal is to prove stability for SBP-SAT (Summation by Parts—Simultaneous Approximation Term) finite difference schemes for equations with variable coefficients. We show stability by providing a proof for the principle of frozen coefficients, i.e., showing that variable coefficient discretization is stable provided that all corresponding constant coefficient discretizations are stable.     

On the well-posedness of the mixed problem for hyperbolic operators with characteristics of variable multiplicity

The paper is devoted to the study of the well-posedness of mixed problems for hyperbolic equations with constant coefficients and characteristics of variable multiplicity. The authors distinguish a class of higher-order hyperbolic operators with constant coefficients and characteristics of variable multiplicity for which a generalization of the Sakamoto L ₂-well-posedness of the mixed problem is obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 85–98, 2006.     

The Qualitative Analysis and the Critical Hypersurfaces of Elliptic and Hyperbolic PDEs

Many boundary value problems of PDEs of the applied mathematics lead to the solving of equivalent elliptic and hyperbolic quadratic algebraic equations (QAEs) with variable coefficients. The qualitative analysis of elliptic and hyperbolic QAEs is started here by the determination of their behaviors by systematical variation of their free and linear terms, from −∞ to +∞ and by their visualization. It comes out that, for these variations of their coefficients, the elliptic and hyperbolic QAEs have critical hypersurfaces, which are obtained by cancellation of their great determinant as in [1], [2]. The critical hypersurface can be considered as a limit of existence of real solutions of an elliptic QAE. The hyperbolic QAE degenerates jumps and breaks along its critical hypersurface, which is also its asymptote. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identifying unknown terms in hyperbolic and parabolic equations

We recover unknown source terms in nonlinear hyperbolic differential equations and in nonlinear parabolic integro-differential equations in one space variable under the assumption of knowing a first integral (in the hyperbolic case) or the value of the solution at a point inside the domain (in the parabolic case). For this class of problems we prove existence results in classes of smooth solutions. Moreover, for linear hyperbolic and parabolic differential equations in one space variable we recover some characteristic parameters. Conferenza tenuta il giorno 29 Novembre 1999     

Some Remarks on Some Second-order Hyperbolic Differential Equations

We are concerned with the almost automorphic solutions to the second-order hyperbolic differential equations of type ü(s) + 2B ù(s) + Au(s) = f(s) (*), where A, B are densely defined closed linear operators acting in a Hilbert space H, and f : R |—> H is a vector-valued almost automorphic function. Using invariant subspaces, it will be shown that under appropriate assumptions, every solution to (*) is almost automorphic.     

Multipoint problem for hyperbolic equations with variable coefficients

By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2n (n ≥ 1) with coefficients depending onx. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.     

Homogenization of linear and nonlinear transport equations

We study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n-dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two-dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non-local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two-scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.     

A multipoint problem with multiple nodes for linear hyperbolic equations

We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1311–1316, October, 1999.