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We solve the relativistic Riemann problem in viscous matter using the relativistic Boltzmann equation and the relativistic causal dissipative fluid-dynamical approach of Israel and Stewart. Comparisons between these two approaches clarify…
We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We…
For general hyperbolic systems of conservation laws we show that dissipative weak solutions belonging to an appropriate Besov space $B^{\alpha,\infty}_q$ and satisfying a one-sided bound condition are unique within the class of dissipative…
In this article, a perturbation theory of the compressible Navier-Stokes equations in $\mathbb{R}^n$ $(n \geq 3)$ is studied to investigate decay estimate of solutions around a non-constant state. As a concrete problem, stability is…
We investigate the Riemann Problem for a shallow water model with porosity and terrain data. Based on recent results on the local existence, we build the solution in the large settings (the magnitude of the jump in the initial data is not…
Hamiltonian perturbations of the simplest hyperbolic equation $u_t + a(u) u_x=0$ are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be…
A quadratic interaction potential $t \mapsto \Upsilon(t)$ for hyperbolic systems of conservation laws is constructed, whose value $\Upsilon(\bar t)$ at time $\bar t$ depends only on the present and the future profiles of the solution and…
We establish new sufficient conditions for the existence of classical hyperbolic quasiperiodic solutions for natural Lagrangian system on Riemannian manifold with time-quasiperiodic force function
We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically…
Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic--parabolic system of conservation laws possesses a translation-invariant center…
In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow…
We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce here an augmented formulation which allows for the modeling of the dynamics of interfaces between…
In the first part of this article we present some exact solutions for special hyperbolic-parabolic systems with sustained oscillations induced by the initial data, most notably the compressible Navier-Stokes system with non-monotone…
We consider a p-system of conservation laws that emerges in one dimensional elasticity theory. Such system is determined by a function $W$, called strain-energy function. We consider four forms of $W$ which are known in the literature.…
We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. The semi-discrete version of the proposed method is based on a technique…
In this article, we discuss about the resolution of the Riemann problem for a 2x2 system in nonconservative form exhibiting parabolic degeneracy. The system can be perceived as the limiting equation (depending on a parameter tending to 0)…
The Riemann solutions to Chaplygin Euler equations with a scaled pressure are considered. When the pressure vanishes, there are three cases. The Riemann solution containing two shock waves converges to the delta shock wave solution of the…
We consider the Cauchy problem for a strictly hyperbolic, $n\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation. We show that the solutions of the viscous approximations…
Hall-MHD is a mixed hyperbolic-parabolic partial differential equation that describes the dynamics of an ideal two fluid plasma with massless electrons. We study the only shock wave family that exists in this system (the other…
Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only H\"older continuous…