Linear symmetric hyperbolic systems with characteristic boundary

The author studies the mixed problem for the linear symmetric hyperbolic systems with maximally non-negative and characteristic boundary condition. Existence of a unique solution is proved inside a suitable class of functions of weighted Sobolev type which takes account of the loss of regularity in the normal direction to the characteristic boundary.     

Non-characteristic mixed problems for non-linear symmetric hyperbolic systems

In this paper the existence and uniqueness of solutions of the following initial boundary value problem for non-linear symmetric hyperbolic equations of the first order are shown, where M = I ⁺ ? S , has the same from as the Kreiss' condition, but S must be sufficiently small ( I ⁺ is the unit matrix in the space generated by eigenvectors of the matrix ? A · n? , corresponding to positive eigenvalues) and n? is a unit outward vector normal to the boundary. The main result of the paper is obtaining an a priori estimate for non-linear equations. This estimate is obtained for sufficiently small time and norms of given data functions. The existence of solutions is proved by the method of successive approximations, which can be used because at each step such properties as symmetry of matrices and the numbers of positive and negative eigenvalues of the matrix ? A · n? are assured. This can be done because we restrict our attention to such systems of equations for which these properties are satisfied for solutions from some neighbourhood of initial data u ₀. Therefore, using the fact that solutions in the class of continuous functions are sought, these properties can be satisfied for sufficiently small time. Moreover, some examples of initial boundary value problems for equations of hydrodynamics and magnetohydrodynamics are considered.     

Dissipative structure for symmetric hyperbolic systems with memory

We study symmetric hyperbolic systems with memory-type dissipation and investigate their dissipative structures. We treat two cases: memory-type diffusion and memory-type relaxation, and observe that the dissipative structures of these two cases are essentially different. Namely, we show that the dissipative structure of the system with memory-type diffusion is of the standard type, while that of the system with memory-type relaxation is of the regularity-loss type.     

Asymptotic properties of eigenfunctions— the hyperbolic plane

Dedicated to Professor Shmuel Agmon     

The propagation of weak discontinuities in quasi-linear symmetric hyperbolic systems

Zusammenfassung Die vorliegende Abhandlung untersucht die Fortpflanzung kleiner Unstetigkeiten in Systemen von nichtlinearen hyperbolischen Differentialgleichungen. Ein Ausdruck wird abgeleitet, der die Änderung in der Intensität der Unstetigkeit angibt, wenn diese sich entlang eines Strahls des hyperbolischen Gleichungssystems fortbewegt. Schliesslich wird als Beispiel mit Hilfe des angegebenen Verfahrens die Fortpflanzung von Schallwellen behandelt.     

Explicit finite element schemes for first order symmetric hyperbolic systems

Summary Finite element methods of up to fourth order accuracy admitting explicit discrete equations are constructed for linear symmetric first order hyperbolic equations having sufficiently smooth solutions. Lumping of the mass matrix at the forward time level is achieved by the addition of a differential operator, which for smooth spline spaces is dissipative and strongly enhances the stability properties of the resulting scheme.     

Global wellposedness for quasi-linear symmetric hyperbolic systems with regularized coefficients

Following the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi-linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local-in-time convergence to the classical solution of the non-regularized problem is proved.     

Simple digital quantum algorithm for symmetric first-order linear hyperbolic systems

Numerical Algorithms - This paper is devoted to the derivation of a digital quantum algorithm for the Cauchy problem for symmetric first-order linear hyperbolic systems, thanks to the reservoir...     

Asymptotic properties of symmetric stable distributions with small index

The family of symmetric stable distributions is considered. A local limit theorem is established which takes account of large deviations as the characteristic exponent α tends to zero. On the basis of this theorem the asymptotics of Fisher information with respect to α is obtained. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. Based on the existence results of global classical solutions, we prove that when t tends to infinity, the solution approaches a combination of C¹ travelling wave solutions, provided that L¹ ∩ L^∞ norm of the initial data as well as its derivative are bounded. Application is given for the time‐like extremal surface in Minkowski space. Copyright © 2006 John Wiley & Sons, Ltd.     

On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (= width) _ε0${\epsilon }_0$, the same happens for the solution u(t,⋅)$u(t,\cdot )$ for a certain radius ε(t)$\epsilon (t)$, as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t)$\epsilon (t)$ as t grows.     

Some algebraic properties of the hyperbolic systems

Abstract The technique of quasi-symmetrizer has been applied to the well-posedness of the Cauchy problem for scalar operators [10], [13] and linear systems [5], [15], [4], and to the propagation of analitycity for solutions to semi-linear systems [6]. In all these works, it is assumed that the principal symbol depends only on the time variable. In this note we illustrate, in some special cases, a new property of the quasisymmetrizer which allows us to generalize the result in [6] to semi-linear systems with coefficients depending also on the space variables [21]. Such a property is closely connected with some interesting inequalities on the eigenvalues of a hyperbolic matrix. We expect that this technique applies also to other problems. Keywords: First order hyperbolic systems, Quasi-symmetrizer, Glaeser inequality