Numerical Analysis of Very Weakly Well Posed Hyperbolic Cauchy Problems

This 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 $\ell \ll 1$ grows as $e^{C\ell^{-1/2} }$

Incoming and disappearing solutions for Maxwell's equations

We prove that in contrast to the free wave equation in $\R^3$ there are no incoming solutions of Maxwell's equations in the form of spherical or modulated spherical waves. We construct solutions which are corrected by lower order incoming waves. With their aid, we construct dissipative boundary conditions and solutions to Maxwell's equations in the exterior of a sphere which decay exponentially as $t \to +\infty$. They are asymptotically disappearing. Disappearing solutions which are identically zero for $t \geq T > 0$ are constructed which satisfy maximal dissipative boundary conditions which depend on time $t$. Both types are invisible in scattering theory

A Simple Example of Localized Parametric Resonance for the Wave Equation

2000 Mathematics Subject Classification: 35L05, 35P25, 47A40.The problem studied here was suggested to us by V. Petkov. Since the beginning of our careers, we have benefitted from his insights in partial differential equations and mathematical physics. In his writings and many discussions, the conjuction of deep analysis and specially interesting problems has been a source inspiration for us.The research of J. Rauch is partially supported by the U.S. National Science Foundation under grant NSF-DMS-010409

A Note on Non-homogeneous Hyperbolic Operators with Low Regularity Coefficients

In this paper we obtain an energy estimate for a complete strictly hyperbolic operator with second order coefficients satisfying a log-Zygmund-continuity condition with respect to $t$, uniformly with respect to $x$, and a log-Lipschitz-continuity condition with respect to $x$,uniformly with respect to $t$

Counterexamples to the well posedness of the Cauchy problem for hyperbolic systems

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 $C^\infty$. 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