We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasilinear hyperbolic equations and they describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV eq...

We replace a Fourier type law by a Cattaneo type law in the derivation of the fundamental equations of fluid mechanics. This leads to hyperbolicly perturbed quasilinear Navier-Stokes equations. For this problem the standard approach by means of quasilinear symmetric hyperbolic systems seems to fail by the fact that finite propagation speed might not be expected. Therefore a somewhat different a...

Hydrodynamic type systems are systems of quasilinear equations of the first order. They naturally arise in continuum mechanics but also occur as a result of semi-classical approximations of nonlinear dispersive waves. The mathematical theory of one-dimensional hyperbolic quasilinear equations initiated by B. Riemann in XIX century has been developed into a rich and diverse area of applied mathe...

We establish existence and uniqueness results for the singular initial value problem associated with a class of quasilinear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an extension of earlier work by the authors for the same problem in one space dimension.

We consider a hyperbolicly perturbed Navier-Stokes initial value problem in R, n = 2, 3, arising from using a Cattaneo type relation instead of a Fourier type one in the constitutive equations. The resulting system is a hyperbolic one with quasilinear nonlinearities. The global existence of smooth solutions for small data is proved, and relations to the classical Navier-Stokes systems are discu...