1,721,149 research outputs found
The Cauchy problem for weakly hyperbolic systems
No full textWe consider the well-posedness of the Cauchy problem in Gevrey spaces for first order weakly hyperbolic systems. The question is to know wether the general results of M.D.Bron stein cite{Br} and K.Kajitani cite{Ka2} can be improved when the coefficients depend only on time and are smooth,
as it has been done for the scalar wave equation in cite{CJS}. The anwser is no for general systems, and yes when the system is uniformly diagonalizable: in this case we show that the Cauchy problem is well posed in all Gevrey classes when the coefficients are . Moreover, for systems and some other
special cases, we prove that the Cauchy problem is well posed in for s < 1+k when the coefficients are , which is sharp following the counterexamples of S.Tarama cite{Tar}. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, , the coefficients of which are polynomials of and the coefficients of and
Numerical Analysis of Very Weakly Well Posed Hyperbolic Cauchy Problems
Full text linkThis paper analyses the approximate solution of very weakly hyperbolic Cauchy problems. These problems have very sensitive dependence on initial data. We treat a single family of such problems showing that in spite of the sensitive dependence, approximate solutions with desired precision \eps can be computed in finite precision arithmetic with cost growing polynomially in 1/\eps. The sensitive dependence requires high finite precision. The analysis required a new Gevrey stability estimate for the leap frog scheme. The latter depends on a new discrete Glaeser inequality. The cost of calculating solutions with features on scale grows as
Sur l'unicité de la solution du problème de Cauchy pour des opérateurs elliptiques dégénérés du second ordre
Full text linkOn the regularity of solutions of hyperbolic equations with discontinuous coefficients variable in time
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Un teorema di regolarità alla frontiera per soluzioni di sistemi ellittici quasi lineari
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Energy estimates at infinity for hyperbolic equations with oscillating coefficients
No full textAbstractWe study the behaviour, for t→∞, of the energy of the solutions to the Cauchy problem for some strictly hyperbolic second order equations with coefficients very rapidly oscillating
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