Partial differential equations and dynamical systems

This thesis falls within the broad framework of partial differential equations and dynamical systems, and focuses more specifically on two independent topics.

The first one is the study of the discrete coagulation-fragmentation equations with diffusion. Using duality lemma we establish new $L^p$ estimates for polynomial moments of the solutions, under an assumption of convergence of the diffusion coefficients. These moment estimates are then used to obtain new results of smoothness and to prove that strong enough fragmentation can prevent gelation even in the diffusive case.

The second topic is the one of computer-assisted proofs for dynamical systems. We improve and apply a method enabling to a posteriori validate numerical solutions, which is based on Banach's fixed point theorem. More precisely, we extend the range of applicability of the method to include operators with a dominant linear tridiagonal part, we improve an existing technique allowing to compute and validate invariant manifolds, and we introduce an new technique that significantly improves the usage of polynomial interpolation for a posteriori validation methods. 

Then, we apply those techniques to prove the existence of traveling waves for the suspended bridge equation, and to study inhomogeneous steady states of a cross-diffusion system.