Decay of solutions to a water wave model with a nonlocal viscous term

Serge Dumont (1, 2) , Olivier Goubet (3) , Imen Manoubi (4)

1 UNIMES - Université de Nîmes
2 IMAG - Institut Montpelliérain Alexander Grothendieck
3 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
4 Département de Mathématiques [Monastir]

We update here the results on the decay of solutions to a nonlocal water wave equation that reads $$u_t+u_x+\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t}\int_0^t\frac{u(s)}{\sqrt{t−s}}ds+uu_x=u_{xx}$$ where $$\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t}\int_0^t\frac{u(s)}{\sqrt{t−s}}ds$$ is the Riemann–Liouville half-order derivative.

Mots clés: Nonlocal water wave equation; Decay rate of solutions

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Serge Dumont, Olivier Goubet, Imen Manoubi. Decay of solutions to a water wave model with a nonlocal viscous term. Afrika Matematika, 2020, 31 (1), pp.115-127. ⟨10.1007/s13370-019-00748-2⟩. ⟨hal-02068336⟩