Benjamin Melinand

About me

I am currently an associate professor (maître de conférences) at Paris Dauphine University.

I did my PhD (From September 2013 to August 2016) under the supervision of David Lannes (defense : June 2016) at Bordeaux University. I am working on partial differential equations and more precisely on the water waves equations. I want to understand the effect of pressure disturbances, moving bottoms and the Coriolis force on water waves. My work deals with several physical phenomena : Meteotsunamis, landslide-tsunamis, Proudman resonance, Coriolis forcing, Poincaré waves.

I was a Zorn Postdoctoral Fellow (From August 2016 to May 2018) under the mentorship of Kevin Zumbrun (Indiana University) where I worked on the stability of steady states of viscous equations (viscous conservation laws, the isentropic Navier-Stokes equations,...).

Mon CV en français.

My Phd thesis

Météotsunamis, résonance de Proudman et effet Coriolis pour les équations des vagues.

Research

Conservation laws

Existence and behavior of steady solutions solutions on a interval for general hyperbolic-parabolic systems of conservation laws. (with K. Zumbrun). In this paper, we study the existence, spectral stability and the structure of steady solutions of general hyperbolic-parabolic systems conservations laws on a bounded interval.

Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier-Stokes equations. (with B.Barker and K. Zumbrun), To appear in Confluentes Mathematici (2023). In this paper, we study the existence and the spectral stability of steady solutions of viscous conservations laws on a bounded interval with noncharacteristic boundary conditions. We study in particular the full Navier-Stokes equations.

Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions (with K. Zumbrun), Physica D. (2019), 394:16-25. In this paper, we study existence and stability of steady solutions of the isentropic compressible Navier-Stokes equations on a Finite interval with noncharacteristic boundary conditions.

Water waves

Dispersive estimates for nonhomogeneous radial phases: An application to weakly dispersive equations and water wave models, (2023), submitted. In this paper, we provide linear dispersive estimates for weakly dispersive systems. We then apply our estimates to different water wave models.

Rectification of a deep water model for surface gravity waves. (with V. Duchêne), (2022), submitted. In this paper, we discuss a deep irrotational water waves model that is weakly non linear. We show we can rectify it to recover wellposedness without sacrifying the Hamiltonian structure and the cubic accurary. We illustrate our study with detailed numerical simulations.

The KP approximation under a weak Coriolis forcing, Journal of mathematical fluid mechanics (2018), Volume 20, Issue 3, 1229–1247. In this paper, I rigorously justify the Rotation-modified KP equation (also called Grimshaw-Melville equation) under a weak Coriolis forcing. I also fully justify the KP equation when the Coriolis forcing is very weak.

A splitting method for deep water with bathymetry (with A. Bouharguane), IMA Journal of Numerical Analysis (2017), Volume 38, Issue 3, 1324–1350. In this paper, we propose and justify a numerical scheme to study water waves in deep water. Then, we give different applications. For instance, we study the homogenization effect of rapidly varying topographies on water waves.

Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation, to appear in Proceedings of the Royal Society of Edinburgh Section A : Mathematics (2018), volume 148, Issue 6, 1201-1237. In this paper, I propose a generalization of the Boussinesq equations under a Coriolis forcing and I study different asymptotic models in this framework as the Ostrovsky equation or the KdV equation. I also justify the Poincaré waves (or Sverdrup waves).

Coriolis effect on water waves , ESAIM: M2AN (2017), volume 51(5), p.1957-1985. In this paper, I prove a local wellposedness result for the water waves equations with vorticity and a Coriolis forcing. Then, I justify the derivation of the nonlinear rotating shallow water equations from the water waves equations.

A mathematical study of meteo and landslides tsunamis : The Proudman resonance, Nonlinearity (2015), volume 28, p. 4037-4080. In this paper, I prove a local wellposedness result for the water waves equations with a non constant pressure at the surface and a moving bottom. Then, I study the influence of the pressure and the moving bottom in different linear asymptotic models and extend the so called Proudman resonance in deep waters.

Proceeding

Parametric study of the accuracy of an approximate solution for the mild slope equation, (with E. Audusse, O. Lafitte, A. Leroy, C.T. Pham and P. Quemar) SYNACSC 2017. In this proceeding, we develop an approximate analytical solution of the mild slope equation and we perform a parametric study of this approximate solution in various norm comparisons.