Approximate travelling wave solutions to the 2D Euler equation on the torus

Nicolas Crouseilles (1, 2) , Erwan Faou (3, 4)

1 IRMA - Institut de Recherche Mathématique Avancée
2 CALVI - Scientific computation and visualization
3 IPSO - Invariant Preserving SOlvers
4 IRMAR - Institut de Recherche Mathématique de Rennes

We consider the two-dimensional Euler equation with periodic boundary conditions. We construct approximate solutions of this equation made of localized travelling profiles with compact support propagating over a stationary state depending on only one variable. The direction or propagation is orthogonal to this variable, and the support is concentrated around flat points of the stationary state. Under regularity assumptions, we prove that the approximation error can be made exponentially small with respect to the width of the support of the travelling wave. We illustrate this result by numerical simulations.

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Crouseilles, Erwan Faou. Approximate travelling wave solutions to the 2D Euler equation on the torus. Asymptotic Analysis, 2013, 81, pp.31-34. ⟨10.3233/ASY-2012-1117⟩. ⟨hal-00567426⟩