The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem...

An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime singularities is naturally related to blow-up phenomena for nonlinear hyperbolic systems. These connections are explained and recent progress in applying the theory of hyperbolic equations in this field is presented. A direction which has turned out to be fruitful is that...

We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{jk}$. Moreover, we assume that the stored energy function is $C^{\infty }$ with respect to x and $e_{jk}$. In our description we assume that Piola-Kirchhoff’s stress tensor $p_{jk}$ depends on the tensor $e_{jk}$. This means...

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal. 178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal.178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...

A new method for computation of the fundamental solution of electrodynamics for general anisotropic nondispersive materials is suggested. It consists of several steps: equations for each column of the fundamental matrix are reduced to a symmetric hyperbolic system; using the Fourier transform with respect to space variables and matrix transformations, formulae for Fourier images of the fundamental matrix columns are obtained; finally, the fundamental solution is computed by the inverse Fourier transform....

A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.

The goal of this talk is to describe the Lamé operator which drives the propagation of linear elastic waves. The main motivation for me is the work I have done in collaboration with Michel Campillo’s group from LGIT (Grenoble) on passive imaging in seismology. From this work, several mathematical problems emerged: equipartition of energy between $S-$ and $P-$waves, high frequency description of surface waves in a stratified medium and related inverse spectral problems.We discuss the following topics:What...

We prove existence and asymptotic behaviour of a weak solutions of a mixed problem for $$\begin{array}{cc}& u^{\text{'}\text{'}}+Au-\Delta u^{\text{'}}+{\left|v\right|}^{\rho +2}{\left|u\right|}^{\rho }u=f_1\hfill \\ & v^{\text{'}\text{'}}+Av-\Delta v^{\text{'}}+{\left|u\right|}^{\rho +2}{\left|v\right|}^{\rho }v=f_2*\hfill \end{array}$$ where $A$ is the pseudo-Laplacian operator.

This talk describes joint work with Chris Sogge and Markus Keel, in which we establish a global existence theorem for null-type quasilinear wave equations in three space dimensions, where we impose Dirichlet conditions on a smooth, compact star-shaped obstacle $𝒦\subset {ℝ}^3$. The key tool, following Christodoulou [1], is to use the Penrose compactification of Minkowski space. In the case under consideration, this reduces matters to a local existence theorem for a singular obstacle problem. Full details will...