Cauchy Problem

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The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform

Feng Jiao - One of the best experts on this subject based on the ideXlab platform.

Guy Métivier - One of the best experts on this subject based on the ideXlab platform.

  • The Cauchy Problem for weakly hyperbolic systems
    2017
    Co-Authors: F Colombini, Guy Métivier
    Abstract:

    We consider the well-posedness of the Cauchy Problem in Gevrey spaces for N × N first order weakly hyperbolic systems. The question is to know wether the general results of M.D.Bronštein [Br] and K.Kajitani [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 [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 G s when the coefficients are C ∞. Moreover, for 2 × 2 systems and some other special cases, we prove that the Cauchy Problem is well posed in G s for s < 1 + k when the coefficients are C k , which is sharp following the counterexamples of S.Tarama [Ta1]. The main new ingredient is the construction, for all hyperbolic matrix A, of a family of approximate symmetrizers, S ε , the coefficients of which are polynomials of ε and the coefficients of A and A *. MSC Classification : 35 L 50, 35 L 45, 35 L 40.

  • Counterexamples to the well posedness of the Cauchy Problem for hyperbolic systems
    Analysis & PDEs, 2015
    Co-Authors: F Colombini, Guy Métivier
    Abstract:

    This paper is concerned with the well posedness of the Cauchy Problem for first order symmetric hyperbolic systems in the sense of Friedrichs. The classical theory says that if the coefficients of the system and if the coefficients of the symmetrizer are Lipschitz continuous, then the Cauchy Problem is well posed in L 2. When the symmetrizer is Log-Lipschtiz or when the coefficients are analytic or quasi-analytic, the Cauchy Problem is well posed C ∞. In this paper we give counterexamples which show that these results are sharp. We discuss both the smoothness of the symmetrizer and of the coefficients.

  • The Cauchy Problem for Wave Equations with non
    2008
    Co-Authors: Lipschitz Coefficients, Ferruccio Colombini, Guy Métivier
    Abstract:

    In this paper we study the Cauchy Problem for second order strictly hyperbolic operators of the form when the coefficients of the principal part are not Lipschitz c but only "Log-Lipschitz" with respect to all the variables. This class of equation is invariant under changes of variables and therefore suit- able for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy Problem.

  • The Cauchy Problem for Wave Equations with NonLipschitz Coefficients
    Annales Scientifiques de l'École Normale Supérieure, 2008
    Co-Authors: Ferruccio Colombini, Guy Métivier
    Abstract:

    In this paper we study the Cauchy Problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy Problem.

P M Santini - One of the best experts on this subject based on the ideXlab platform.

  • the Cauchy Problem on the plane for the dispersionless kadomtsev petviashvili equation
    arXiv: Exactly Solvable and Integrable Systems, 2006
    Co-Authors: S V Manakov, P M Santini
    Abstract:

    We construct the formal solution of the Cauchy Problem for the dispersionless Kadomtsev - Petviashvili equation as application of the Inverse Scattering Transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy Problem for the Kadomtsev - Petviashvili equation, associated with the Inverse Scattering Transform of the time dependent Schroedinger operator for a quantum particle in a time-dependent potential.

  • the Cauchy Problem on the plane for the dispersionless kadomtsev petviashvili equation
    Jetp Letters, 2006
    Co-Authors: S V Manakov, P M Santini
    Abstract:

    We construct the formal solution to the Cauchy Problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy Problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrodinger operator for a quantum particle in a time-dependent potential.

Simon Garruto - One of the best experts on this subject based on the ideXlab platform.

Berikbol T Torebek - One of the best experts on this subject based on the ideXlab platform.

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