This paper deals with the non-conservative coupling of two one-dimensional barotropic Euler systems at an interface at x = 0. The closure pressure laws differ in the domains x < 0 and x > 0, and a Dirac source term concentrated at x = 0 models singular pressure losses. We propose two numerical methods. The first one relies on ghost state reconstructions at the interface while the second is based on a suitable relaxation framework. Both methods satisfy a well-balanced property for stationary...

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and are separated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupied by several fluid components require a “mixed-cell” EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell...

We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension 3 with linear multiplicative spatially homogeneous gaussian noise that is white in time. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison,...

We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations...

We discuss invariants and conservation laws for a nonlinear system of Klein-Gordon equations with Hamiltonian structure ⎧$u_{tt}-\Delta u+m²u=-F₁\left(\right|u|²,|v\left|²\right)u$, ⎨ ⎩$v_{tt}-\Delta v+m²v=-F₂\left(\right|u|²,|v\left|²\right)v$ for which there exists a function F(λ,μ) such that ∂F(λ,μ)/∂λ = F₁(λ,μ), ∂F(λ,μ)/∂μ = F₂(λ,μ). Based on Morawetz-type identity, we prove that solutions to the above system decay to zero in local L²-norm, and local energy also decays to zero if the initial energy satisfies $E(u,v,ℝⁿ,0)=1/2{\int }_{ℝⁿ}\left(\right|\nabla u\left(0\right)|²+|u_t\left(0\right)|²+m²|u\left(0\right)|²+|\nabla v\left(0\right)|²+|v_t\left(0\right)|²+m²|v\left(0\right)|²+F(\left|u\left(0\right)\right|²,\left|v\left(0\right)\right|²\left)\right)dx<\infty$, and F₁(|u|²,|v|²)|u|² + F₂(|u|²,|v|²)|v|² - F(|u|²,|v|²) ≥ aF(|u|²,|v|²) ≥ 0, a > 0.

We consider the inverse problem of determining how the physiological structure of a harvested population evolves in time, and of finding the time-dependent effort to be expended in harvesting, so that the weighted integral of the density, which may be, for example, the total number of individuals or the total biomass, has prescribed dynamics. We give conditions for the existence of a unique, global, weak solution to the problem. Our investigation is carried out using the method of characteristics...

Inertial manifold with delay (IMD) for dissipative systems of second order in time is constructed. This result is applied to the study of different asymptotic properties of solutions. Using IMD, we construct approximate inertial manifolds containing all the stationary solutions and give a new characterization of the K-invariant manifold.

A strictly hyperbolic quasi-linear 2×2 system in two independent variables with C2 coefficients is considered. The existence of a simple wave solution in the sense that the solution is a 2-dimensional vector-valued function of the so called Riemann invariant is discussed. It is shown, through a purely geometrical approach, that there always exists simple wave solution for the general system when the coefficients are arbitrary C^2 functions depending on both, dependent and independent variables.