On the regularity of solutions of hyperbolic nonlinear equations

Summary Some regularity results for the solutions of certain hyperbolic nonlinear equations are given. This paper was written while the author was visiting Department of Mathematics, Sir GeorgeWilliams University, Montreal (Canada). Entrata in Redazione il 22 dicembre 1971.     

Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity

In this paper, we study the regularity of the eigenvalues and eigenvectors and the existence of normalized coordinates for quasilinear hyperbolic systems with characteristic fields of constant multiplicity. We prove that the eigenvalues and eigenvectors of the system have the same regularity as the coefficients of the system. On the other hand, we show that, for the quasilinear hyperbolic system of conservation laws with characteristic fields of constant multiplicity, the normalized coordinates exist on the domain under consideration.     

Boundary-value problems for hyperbolic equations with constant coefficients

By using a metric approach, we study the problem of well-posedness of boundary-value problems for hyperbolic equations of ordern (n2) with constant coefficients in a cylindrical domain. Conditions of existence and uniqueness of solutions are formulated in number-theoretic terms. We prove a metric theorem on lower estimates of small denominators that appear when constructing solutions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 795–802, July, 1994.     

Decay and regularity for dispersive equations with constant coefficients

This article discusses some smoothing estimates of the initial value problem for dispersive equations with constant coefficients. In particular, it is shown that a certain condition for the principal part of the symbol (see the assumption (1.3) below, which is equivalent to the one “of principal type” in the paper by Ben-Artzi and Devinatz [2]) is necessary and sufficient for the maximal smoothing in space-time. Dedicated to Professor Norio Shimakura The author was supported in part by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 13640187).     

The Levi-Lax condition for partial differential equations with real characteristics of constant multiplicity

The object of this paper is to show that the Levi-Lax condition is necessary and sufficient for the Cauchy problem to be well-posed.     

Symmetric hyperbolic equations in the nonlinear elasticity theory

The basic results are overviewed concerning the formulation of nonlinear elasticity equations in the form of symmetric hyperbolic systems in long-time studies performed under the direction of the first author. The underlying principles developed in those studies are stated, and some inaccuracies and errors are corrected. Procedures are described for computing the coefficient matrices of the equations from a given equation of state.     

A note on weakly hyperbolic equations with analytic principal part

In this note we show how to include low order terms in the C^∞$C^{\infty }$ well-posedness results for weakly hyperbolic equations with analytic time-dependent coefficients. This is achieved by doing a different reduction to a system from the previously used one. We find the Levi conditions such that the C^∞$C^{\infty }$ well-posedness continues to hold.     

Time regularity of the solutions to second order hyperbolic equations

We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class g^s₀\gamma^{s_{0}} and the Cauchy data belong to g^s₁\gamma^{s_{1}}, then the Cauchy problem has a solution in  g^s₀([0,T^*];g^s₁(\mathbbR))\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R})) for some T ^*>0, provided 1≤s ₁≤2−1/s ₀. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤s ₁≤s ₀.