On Vanishing dissipative-dispersive perturbations of hyperbolic conservation laws

In presence of linear diffusion and non-positive dispersion, we prove well-posedness of the nonlinear conservation equation u_t+f(u)_x=\eps u_xx -\del((u_xx)^2)_x. Then, as the right-hand perturbations vanish, we prove convergence of the previous solutions to the entropy weak solution of the hyperbolic conservation law u_t+f(u)_x=0

Comparison of solutions of Boussinesq systems

International audienceWe compare the solution of the generalized Boussinesq systems, for various values of a,b,c,d, \begin{eqnarray}\nonumber \eta_t +u_x +\varepsilon ((\eta u)_x +au_{xxx}-b\eta_{xxt}) &=& 0 \\\nonumber u_t +\eta_x +\varepsilon (uu_x +c\eta_{xxx} -du_{xxt}) &=&0.\end{eqnarray}These systems describe the two-way propagation of small amplitude long waves in shallow water. We prove, using an energy method introduced by Bona, Pritchard and Scott, that respective solutions of Boussinesq systems, starting from the same initial datum, remain close on a time interval inversely proportional to the wave amplitude

Numerical Study of Solutions of the 3D Generalized Kadomtsev-Petviashvili Equations for Long Times

International audienceFrom a spectral method combined with a predictor-corrector scheme, we numerically study the behavior in time of solutions of the three-dimensional generalized Kadomtsev-Petviashvili equations. In a systematic way, the dispersion, the blow-up in finite time, the solitonic behavior and the transverse instabilities are numerically inspected

Unique continuation property for Boussinesq-type systems

We prove, that if the solution of the Cauchy problem for a regularized version of the Boussinesq systems, has a compact support for all time, then this solution vanishes identically

ÉQUATIONS DISPERSIVES ET DISSIPATIVES ET QUELQUES APPLICATIONS EN ÉCOLOGIE

This report presents my main results in three separate chapters.The first one is about nonlinear dispersive partial differential equations. The Cauchy problem is studied through a method of normal form and dispersion inequalities. Two unique continuation properties are established via a Carleman inequality and an estimate of Bourgain.The consideration of the viscosity and viscoelasticity is the subject of the second chapter. Described mathematically by a nonlocal operator, one shows the dissipation and the stability of the energy. The convergence to the entropy solution of the hyperbolic conversion law is also proved.In the third chapter, completely independent of the first two, we focus on the modeling of plants and their environment. The interactions between plants, pathogens and pests are drawn up

On the decay in time of solutions of the generalized regularized Boussinesq system

We are interested in dispersive properties of the Boussinesq system for small initial data. We prove that, for a nonlinearity of sufficiently high order, the solution of the Boussinesq system exists for all time and tends to zero when the time tends to infinity