Rigorous validations of periodic orbits in the Kuramoto-Sivashinsky PDE

Rigorous validations of periodic orbits in the Kuramoto-Sivashinsky PDE

-
Jordi-Lluis Figueras, Uppsala University
Fine Hall 322

We will give a-posteriori results suitable for proving the existence of periodic orbits for some PDEs. Among these we could have parabolic PDEs or ill-posed equations like Boussinesq equation.  We will show how to implement these a-posteriori results as Computer-Assisted Proofs in the 1D Kuramoto-Sivashinsky PDE: \[ \partial_t u = \partial_{xxxx} u+\alpha \partial_{xx} u+\partial_x (u^2), \] where $\alpha > 0$ and $u: \mathbb R\times [0, 1] \rightarrow \mathbb R$ is odd-periodic ($u(t,x)=u(t, x+1)$, $u(t,0) =0$. The validations are based on proving that certain smooth functionals have an isolated zero. For doing so, a combination of functional analysis and rigorous estimates using computers are needed.  We will finish the talk showing some examples.  Joint work with Rafael de la Llave (Georgia Institute of Technology), and Marcio Gameiro (Universidade de Sao Pablo a Sao Carlos) and Jean-Philippe Lessard (Laval University).