We consider the equation $u_t\mathrm{-}i\mathrm{\Lambda }u=0$, where $\mathrm{\Lambda }=\lambda \left(D_x\right)$ is a first order pseudo-differential operator with real symbol $\lambda \left(\xi \right)$. Under a suitable convexity assumption on $\lambda$ we find the decay properties for $u\left(t,x\right)$. These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem $u_t\mathrm{-}i\mathrm{\Lambda }u=f\left(u\right)$, $u\left(0,x\right)=g\left(x\right)$. If $f\left(u\right)$ is a smooth function which satisfies $f\left(u\right)\simeq {\left|u\right|}^p$ near $u=0$, and $g$ is small in suitably Sobolev norm, we prove global existence theorems provided $p$ is greater than a critical exponent.

In this paper the exact formula for the critical time of generating discontinuity (shock wave) in a solution of a $2\times 2$ quasilinear hyperbolic system is derived. The applicability of the formula in the engineering praxis is shown on one-dimensional equations of isentropic non-viscous compressible fluid flow.

We obtain non-constant periodic solutions for a class of second-order autonomous dynamic systems whose potential is subquadratic at infinity. We give a theorem on conjugate points for convex potentials.

A mathematical analysis of poroacoustic traveling wave phenomena is presented. Assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy's law, exact traveling wave solutions (TWS)s, as well as asymptotic/approximate expressions, are derived and examined. In particular, stability issues are addressed, shock and acceleration waves are shown to arise, and special/limiting cases are noted. Lastly, connections to other fields are...

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing...

The aim of this work is to establish, from a mathematical point of view, the limit α → +∞ in the system $i{\partial }_{t}E+\nabla (\nabla .E)-{\alpha }^2\nabla \times \nabla \times E=-{\left|E\right|}^{2\sigma }E,$ where $E:{ℝ}^3\to {ℂ}^3$. This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the $H^1\left({ℝ}^3\right)$ norm.

Semilinear hyperbolic problems with source terms piecewise smooth and discontinuous across characteristic surfaces yield similarly piecewise smooth solutions. If the discontinuous source is replaced with a smooth transition layer, the discontinuity of the solution is replaced by a smooth internal layer. In this paper we describe how the layer structure of the solution can be computed from the layer structure of the source in the limit of thin layers. The key idea is to use a transmission problem...

We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to...

In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme...

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in...

On s’intéresse à des systèmes symétriques hyperboliques multidimensionnels en présence d’une semilinéarité. Il est bien connu que ces systèmes admettent des solutions discontinues, régulières de part et d’autre d’une hypersurface lisse caractéristique de multiplicité constante. Une telle solution $u^0$ étant donnée, on montre que $u^0$ est limite quand $\epsilon \to 0$ de solutions ${\left(u^{\epsilon }\right)}_{\epsilon \in ]0,1]}$ du système perturbé par une viscosité de taille $\epsilon$. La preuve utilise un problème mixte parabolique et des développements de couches limites....