Tatsuo Nishitani

Hyperbolicity of two by two systems with two independent variables

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 hyperbolic systems which includes a fortiori symmetrizable systems.

Microlocal Energy Estimates for Hyperbolic Operators with Double Characteristics

Publisher Summary This chapter discusses microlocal energy estimates for two different types of operators with double characteristics. One operator is effectively hyperbolic and the other is non-effectively hyperbolic operator whose fundamental matrix has a Jordan block of size four. The coefficient matrix of the Hamiltonian system with the quadratic Hamiltonian Pp is called the fundamental or Hamilton matrix of P. The chapter discusses the calculus of pseudodifferential operators and presents a derivation of microlocal energy estimates for the localized operator. In the most simple case, the decomposability is equivalent to that there is no bicharacteristic issuing from a simple characteristic point that has a limit point in the double characteristic set.

Necessary conditions for the well-posedness of the Cauchy problem for hyperbolic systems

In the present paper we study necessary conditions for the we ll-posedness of the Cauchy problem for hyperbolic systems of arbitrary order wi th multiple characteristics. In the scalar case the seminal paper of Ivrii and Petkov [6] ha s shown that the correctness of the Cauchy problem implies that for a given hy perbolic differential operator, near a multiple characteristic point, a set of vanis hi g conditions on the homogeneous parts of the lower order terms must be satisfied. Evidently in the case of hyperbolic differential systems th e situation is more complex, the vector structure playing a relevant role. As a cons equence the above mentioned result may not be true any more as we shall see in some ex amples in the second section of the paper. Before stating our main result we would like to recall, as a mo tivation, some results in the case of hyperbolic systems already existing in t he literature. Let us introduce the notation and the definition of well-pose dness for a differential equation (system of differential equations). We work i n an open subset of R +1, with coordinates = (0 1 . . . ) = ( 0 ′), and assume that the origin belongs to ; let = { ∈ | 0 < }, = { ∈ | 0 > }.

Necessary Conditions for Hyperbolic Systems

In this article we study the Cauchy problem for a first order system

Regularity of solutions to characteristic boundary value problem for symmetric systems

L(x,D) = {D_0} + \sum\limits_{j = 1}^n {{A_j}(x){D_j} + B(x) = {L_1}(x,D) + {L_0}(x)}

ON THE CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS: THE GEVREY 3 WELL-POSEDNESS

where Ai(x) andB(x)arerxrsmooth matrices and

On the Cauchy Problem for D t 2-D xa(t,x)D x in the Gevrey Class of Order s > 2

{L_1}(x,D) = {D_0} = \sum\limits_{J = 1}^n {{A_j}(x){D_j},{L_0}(x) = B} (x)