Special Session 9: Recent Progress in Mathematical Theory of Stability and Instability in Fluid Dynamics

Model for bore propagation with dynamic boundary conditions

Jerry Bona University of Illinois at Chicago USA Co-Author(s):    Jerry L. Bona and Hongqiu Chen

Abstract: A model for the propagation of bores that allows for cross-river variation is put forward and analysed.

The long way of a viscous vortex dipole

Michele Dolce EPFL Switzerland Co-Author(s):    Thierry Gallay

Abstract: The Helmholtz Kirchhoff system governs the evolution of two counter rotating point vortices in a 2D inviscid fluid, resulting in constant translation at a constant speed. However, at large but finite Reynolds numbers, the size of the vortex cores grows because of diffusion. This means that the point vortex approximation is not valid over long times for the resulting viscous dipole. This talk aims to systematically define an asymptotic expansion accounting for streamlines deformation from vortex interactions and to understand the finite size effects on the dipole`s translation speed. We then prove that the exact solution remains close to our approximation over a very long time interval, extending unboundedly as the Reynolds number approaches infinity. The proof relies on energy estimates inspired by Arnold`s variational characterization of the steady states of the 2D Euler equation, as recently revised by Gallay and Sverak for viscous fluids. This work is a collaboration with T. Gallay.

Traveling waves near shear flows for 2D Euler

Daniel Lear Universidad de Cantabria Spain Co-Author(s):    Angel Castro

Abstract: In this talk we will consider the existence of traveling waves arbitrarily close to shear flows for the 2D incompressible Euler equations. In particular we shall present some results concerning the existence of such solutions near the Couete, Taylor-Couete and the Poiseuille flows. In the first part of the talk we will introduce the problem and review some well known results on this topic. In the second one some of the ideas behind the construction of our traveling waves will be sketched.

Viscosity driven instability of shear flows without boundaries

Hui Li New York University Abu Dhabi United Arab Emirates Co-Author(s):    Hui Li and Weiren Zhao

Abstract: In this talk, we discuss the instability caused by viscous dissipation in a domain without boundaries. We construct a shear flow that is initially spectrally stable but evolves into a spectrally unstable state due to the influence of viscous dissipation.

Nonlinear Inviscid damping for 2-D inhomogeneous incompressible Euler equations

Chen Qi Zhejiang University School of mathematical sciences Peoples Rep of China Co-Author(s):    Dongyi Wei, Ping Zhang, Zhifei Zhang

Abstract: We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on TxR. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as t tends to infinity.

THE TRANSITION TO INSTABILITY FOR STABLE SHEAR FLOWS IN INVISCID FLUIDS

Daniel Sinambela NYUAD United Arab Emirates Co-Author(s):    Weiren Zhao

Abstract: This talk focuses on the generation of eigenvalues of a stable monotonic shear flow under perturbations in $C^s$ with $s

Small-amplitude finite-depth Stokes waves are transversally unstable

Zhao Yang Academy of Mathematics and Systems Science Peoples Rep of China Co-Author(s):    Ziang Jiao, L. Miguel Rodrigues, Changzhen Sun, Zhao Yang

Abstract: We prove that all irrotational planar periodic traveling waves of sufficiently small-amplitude are spectrally unstable as solutions to three-dimensional inviscid finite-depth gravity water-waves equations.

Asymptotic stability of Couette flow for the Stokes-transport equations

Ruizhao Zi Central China Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we investigate the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equation. We observe that a similar lift-up effect to the three-dimensional Navier-Stokes equation near Couette flow destabilizes the system, while the inviscid damping type decay due to the Couette flow (Y,0,0) together with the damping structure caused by the decreasing background density $\varrho_{\rm s}(Y)$ stabilizes the system. This is based on a joint work with Weiren Zhao and Daniel Sinambela.

On Resonances in Dissipative Magnetohydrodynamics

Christian Zillinger Karlsruhe Institute of Technology Germany Co-Author(s):    Niklas Knobel

Abstract: We consider the stability and long time behavior of the inviscid magnetohydrodynamics equations with magnetic dissipation near a combination of a shear flow and a constant magnetic field. While the linearized equations around this stationary solution are stable in Sobolev regularity, we show that in any small analytic neighborhood there exist non-trivial low frequency solutions of the nonlinear problem, which are unstable. More precisely, we show that the critical regularity of the corresponding linearized problem is given by a Gevrey class. This talk is based on joint work with Niklas Knobel.