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Hamilton-Jacobi functional differential equations with unbounded delay

Adam Nadolski (2003)

Annales Polonici Mathematici

The Cauchy problem for nonlinear functional differential equations on the Haar pyramid is considered. The phase space for generalized solutions is constructed. An existence theorem is proved by using the method of successive approximations. The theory of characteristics and integral inequalities are used. Examples of phase spaces are given.

Hyperbolicity of two by two systems with two independent variables

Tatsuo Nishitani (1998)

Journées équations aux dérivées partielles

We study the simplest system of partial differential equations: that is, two equations of first order partial differential equation with two independent variables with real analytic coefficients. We describe a necessary and sufficient condition for the Cauchy problem to the system to be C infinity well posed. The condition will be expressed by inclusion relations of the Newton polygons of some scalar functions attached to the system. In particular, we can give a characterization of the strongly...

Involutivity and Symple Waves in R^2

Kolev, Dimitar (1997)

Serdica Mathematical Journal

A strictly hyperbolic quasi-linear 2×2 system in two independent variables with C2 coefficients is considered. The existence of a simple wave solution in the sense that the solution is a 2-dimensional vector-valued function of the so called Riemann invariant is discussed. It is shown, through a purely geometrical approach, that there always exists simple wave solution for the general system when the coefficients are arbitrary C^2 functions depending on both, dependent and independent variables.

L 2 well-posed Cauchy problems and symmetrizability of first order systems

Guy Métivier (2014)

Journal de l’École polytechnique — Mathématiques

The Cauchy problem for first order system L ( t , x , ∂ t , ∂ x ) is known to be well-posed in L 2 when it admits a microlocal symmetrizer S ( t , x , ξ ) which is smooth in ξ and Lipschitz continuous in ( t , x ) . This paper contains three main results. First we show that a Lipschitz smoothness globally in ( t , x , ξ ) is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point...

L p - L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

Jerzy Gawinecki (1991)

Annales Polonici Mathematici

We prove the L p - L q -time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the L p - L q -time decay estimates.

L2 stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods

Yingjie Liu, Chi-Wang Shu, Eitan Tadmor, Mengping Zhang (2008)

ESAIM: Mathematical Modelling and Numerical Analysis


We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central discontinuous Galerkin method and the regular discontinuous Galerkin method in this context is also made. Numerical experiments are provided to validate the quantitative conclusions from the analysis.

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