具常重特征的非齐次拟线性双曲组的整体弱间断解

This paper considers the Cauchy problem with a kind of non-smooth initial data for general inhomogeneous quasilinear hyperbolic systems with characteristics with constant multiplicity. Under the matching condition, based on the refined fomulas on the decomposition of waves, we obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution to the Cauchy problem.     

Initial Value Problem for General Quasilinear Hyperbolic Systems with Characteristics with Constant Multiplicity

The authors consider the global existence and the blow-up phenomenon of classical solutions with small amplitude to the Cauchy problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity and given some applications.     

Short-wave asymptotics of solutions of the Cauchy problem for hyperbolic systems with constant coefficients and with characteristics of variable multiplicity

We construct asymptotic expansions of solutions of the Cauchy problem with rapidly oscillating initial data for hyperbolic systems with constant coefficients and with characteristics of a variable multiplicity. By way of example, we consider the system of Maxwell equations.     

CAUCHY PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS WITH CHARACTERISTICS WITH CONSTANT MULTIPLICITY

For quasilinear hyperbolic systems with characteristics of constant multiplicity, suppose that characteristics of constant multiplicity(> 1) are linearly degenerate, by means of generalized normalized coordinates we get the global existence and the blow-up phenomenon of the C^1 solution to the Cauchy problem under an additional hypothesis.     

Sur la monodromie du problème de Cauchy ramifié

In this paper, we study the monodromy of the ramified Cauchy problem for operators with multiple characteristics of constant multiplicity. More precisely, we give an estimation of the eigenvalues of the solution's monodromy, first with the assumptions of the theorem of Hamada–Leray–Wagschal, then with the assumptions of the theorem of Leichtnam.     

NONLINEAR HYPERBOLIC CAUCHY PROBLEMS IN GEVREY CLASSES

91. IntroductionAny linear hyperbolic operator P with Cx coefficients with respect to time, 0 < X < 1, hasa well posed Cauchy problem in suitable Gevrey classes G6 whereas this is not true in Coo.Focusing on operators with characteristics of constaat multip1icity we have the followingbounds for the Gevrey index:1a < he, if P is strictly hyperbolicl','l'5'], (0.1)ta < min{1 X, b}, r 2 2 the largest multip1icity[']. (0.2)Nom (0.1) and (0.2) we have well posedness forra < --3 r 2 1r -- xand…     

Nonstandard characteristics and localized asymptotic solutions of a linearized magnetohydrodynamic system with small viscosity and drag

We describe asymptotic solutions of the Cauchy problem for a linearized system of magnetohydrodynamics with initial conditions localized in a small neighborhood of a curve or a two-dimensional surface. We investigate how a change of the multiplicity of characteristics affects such solutions and prove a uniform estimate of the residual.     

A degenerate hyperbolic equation under Levi conditions

Abstract We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order. Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C^∞ well posedness of the Cauchy problem. Keywords: Cauchy problem, Hyperbolic equations, Levi conditions     

Monodromie du problème de Cauchy ramifiéMonodromy of the ramified Cauchy problem

In this Note, we study the monodromy of the ramified Cauchy problem for operators with multiple characteristics of constant multiplicity. More precisely, we give an estimation of the eigenvalues of the solution's monodromy, first with the assumptions of the theorem of Hamada–Leray–Wagschal, then with the assumptions of the theorem of Leichtnam. To cite this article: R. Camalès, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 639–642.     

On the well-posedness of the cauchy problem and the mixed problem for some class of hyperbolic systems and equations with constant coefficients and characteristics of variable multiplicity

We study the well-posedness of the mixed problem for hyperbolic equations with constant coefficients and with characteristics of variable multiplicity. We single out a class of higher-order hyperbolic operators with constant coefficients and with characteristics of variable multiplicity, for which we obtain a generalization of the Sakamoto conditions for the well-posedness of the mixed problem in L ₂.     

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.     

Local analytic continuation of holomorphic solutions of partial differential equations

Summary We ask. When is it possible to continue analytically holomorphic solutions of partial differential equations. Using objects called cones of analytic continuation we get sufficient conditions generalizing results by J.-M. Bony and P. Schapira and by Y. Tsuno. The results are counterparts of earlier results by the author on local uniqueness in the Cauchy problem. We also give a necessary condition by constructing solutions with singularities. We think that the technique used here should also have other applications. Anyhow this result is a generalization of a result by Y. Tsuno related to simple characteristic hypersurfaces. It corresponds to the existence of local null solution when the initial hypersurface has constant multiplicity in the Cauchy problem. Entrata in Redazione il 24 dicembre 1975.     

Wave front set at infinity for tempered ultradistributions and hyperbolic equations

Abstract We study the propagation of singularities and the microlocal behaviour at infinity for the solution of the Cauchy problem associated to an SG-hyperbolic operator with one characteristic of constant multiplicity. We perform our analysis in the framework of tempered ultradistributions, cf. Introduction, using an appropriate notion of wave front set. Keywords: Wave front set at infinity, Tempered ultradistributions, Hyperbolic equations     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

Equivalent forms of Levi condition

In this paper we consider hyperbolic differential operators with characteristic roots of constant multiplicity and we prove the equivalence of some conditions, called Levi conditions, for the correctness of the Cauchy problem inC ^∞ and in Gevrey classes.

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Sommario In questo articolo prendiamo in considerazione operatori differenziali iperbolici con caratteristiche di molteplicità costante e dimostriamo l’equivalenza di alcune condizioni, note come condizioni di Levi, necessarie e sufficienti per la buona positura del problema di Cauchy nelle classiC ^∞ e Gevrey.

     

Conditions of hyperbolicity of systems with constant multiplicity

We define the Levi conditions for an N × N linear first order system of differential operators. We prove that these conditions are sufficient for the well-posedness of the Cauchy problem and are necessary up to the multiplicity five.     

Necessary conditions for hyperbolic systems

We consider the Cauchy problem for first order hyperbolic systems that have characteristic points of higher multiplicity. This means that the determinant of the principal symbol has multiple characteristic points. In the case where, on a multiple characteristic point, the principal symbol has corank 2, we give necessary conditions for the well posedness of the Cauchy problem. These conditions involve a suitably defined noncommutative determinant of the full symbol of the system.     

Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficients. In doing so, we extend earlier results in two articles of Balser and Lutz, Miyake, and Schäfke for the complex heat equation, as well as in a paper of Balser and Miyake, who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE where the formal solution of the Cauchy problem is not k-summable for whatever value of k, but instead is multisummable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now.     

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.     

Exponential Function of Pseudo-Differential Operators in Gevrey Spaces

The article is devoted to the construction of the exponential function of the matrix pseudo-differential operator, which does not satisfy conditions of any known theorem (see, e.g. Sec. 8 Ch. VIII and Sec. 2 Ch. XI of Treves in Introduction to the theory of pseudodifferential and Fourier integral operator, vols. 1 & 2, Plenum Press, New York, 1980). An application of the exponential function to the fundamental solution of the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity is given.