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Boundary observability for the space semi-discretizations of the 1 – d wave equation

Juan Antonio Infante, Enrique Zuazua (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider space semi-discretizations of the 1-d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a...

Boundary sentinels in cylindrical domains.

J. Saint Jean Paulin, M. Vanninathan (2001)

Revista Matemática Complutense

We study a model describing vibrations of a cylindrical domain with thickness e > 0. A characteristic of this model is that it contains pollution terms in the boundary data and missing terms in the initial data. The method of sentinels'' of J. L. Lions [7] is followed to construct a sentinel using the observed vibrations on the boundary. Such a sentinel, by construction, provides information on pollution terms independent of missing terms. This requires resolution of initial-boundary value...

Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces

Jean-Marc Delort, Jérémie Szeftel (2006)

Annales de l’institut Fourier

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on compact revolution hypersurfaces with non-Hamiltonian nonlinearities, when the data are smooth, small and radial. The method combines normal forms with the fact that the eigenvalues associated to radial eigenfunctions of the Laplacian on such manifolds are simple and satisfy convenient asymptotic expansions.

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Vincent Duchêne (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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...

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Vincent Duchêne (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

Branching random motions, nonlinear hyperbolic systems and travellind waves

Nikita Ratanov (2006)

ESAIM: Probability and Statistics

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independent of random motion, and intensities of reverses are defined by a particle's current direction. A solution of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) has a so-called McKean representation via such processes. Commonly this system possesses travelling-wave solutions. The convergence of solutions with Heaviside terminal...

Breakdown in finite time of solutions to a one-dimensional wave equation.

Mokhtar Kirane, Salim A. Messaoudi (2000)

Revista Matemática Complutense

We consider a special type of a one-dimensional quasilinear wave equation wtt - phi (wt / wx) wxx = 0 in a bounded domain with Dirichlet boundary conditions and show that classical solutions blow up in finite time even for small initial data in some norm.

Breathers for nonlinear wave equations

Michael W. Smiley (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The semilinear differential equation (1), (2), (3), in ℝ × Ω with Ω ∈ ℝ N , (a nonlinear wave equation) is studied. In particular for Ω = ℝ 3 , the existence is shown of a weak solution u ⁢ ( t , x ) , periodic with period T , non-constant with respect to t , and radially symmetric in the spatial variables, that is of the form u ⁢ ( t , x ) = ν ⁢ ( t , | x | ) . The proof is based on a distributional interpretation for a linear equation corresponding to the given problem, on the Paley-Wiener criterion for the Laplace Transform, and on the alternative method of...

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