Current research topics


Blowup phenomena in hydrodynamics, mechanism of developed turbulence, spontaneously stochastic solutions, shell models

We discuss a theoretical framework to define an optimal sub-grid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of Kolmogorov formulated for similar quantities for the original three dimensional Navier-Stokes turbulence. 

L. Biferale, A.A. Mailybaev and G. Parisi, An optimal subgrid scheme for shell models of turbulence, Phys. Rev. E 95 (2017), 043108. [Link] [PDF]

We present a theoretical argument showing that a classical deterministic solution before a finite-time blowup must be continued as a stochastic process after the blowup, representing a unique physically relevant description in the inviscid limit.

A.A. Mailybaev, Spontaneously stochastic solutions in one-dimensional inviscid systems, Nonlinearity 29 (2016), 2238-2252. [Link] [PDF]

Incompressible 3D Euler equations develop high vorticity in very thin pancake-like regions from generic large-scale initial conditions. In this work we propose an exact solution of the Euler equations for the asymptotic pancake evolution. 

D.S. Agafontsev, E.A. Kuznetsov and A.A. Mailybaev, Asymptotic solution for high vorticity regions in incompressible 3D Euler equations, Journal of Fluid Mechanics 813, R1 (2017). [Link] [PDF]

Singularities in Mathematical Physics

Exceptional points in non-Hermitian systems, applications in optics and quantum physics

Physical systems with loss or gain have resonant modes that decay or grow exponentially with time. Whenever two such modes coalesce both in their resonant frequency and their rate of decay or growth, an ‘exceptional point’ occurs.

We present experimental results from a waveguide structure that steers incoming waves around an exceptional point during the transmission process. In this way, mode transitions are induced that transform this device into a robust and asymmetric switch between different waveguide modes. 

J. Doppler, A.A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T.J. Milburn, P. Rabl, N. Moiseyev, S. Rotter, Dynamically encircling an exceptional point for asymmetric mode switching, Nature (2016) 537:7618, 76-79.

 [Link]    [Online version]   Supplementary material [PDF]

Selected for: Nature Physics News & Views.

Multiscale dynamics of a breaking wave

Singularities, instabilities, droplets formation

We disclose a new multiscale mechanism in which small-scale perturbations (ripples) on water surface are controlled by the large-scale wave dynamics in a way analogous to the adiabatic transport in quantum mechanics. Unlike classical hydrodynamic instabilities, this results in a super-exponential steepness amplification for small-scale ripples. We propose that this explosive increase of ripple steepness explains the breakdown of a smooth water surface into a spray with small droplets, commonly observed in ocean waves prior to overturning even in the absence of wind. The developed quantitative theory is shown to be in excellent agreement with numerical simulations.

(photo courtesy of Keahi de Aboitiz)

A.A. Mailybaev and A. Nachbin, Breakdown along an ocean wave surface prior to overturning, 2017. ArXiv:1707.07516.

[PDF] [Supplementary video]

Reactive Flows in Porous Media

Singular traveling waves in conservation laws, in-situ combustion, enhanced oil recovery (EOR)

This paper combines analytical and numerical studies of light oil recovery by air injection. Our solution shows that between regimes of total and partial oxygen consumption there is a change in the oxidation wave, which may have negative implications for oxygen breakthrough.

F.P. Santos, A.A. Mailybaev and D. Marchesin, Oxidation wave structure and oxygen breakthrough for air injection into light oil reservoirs, Computational Geosciences (2016) 20, 1095-1107. [Link] [PDF]

We study a nonlinear wave for a system of balance laws, which describes combustion for two-phase (gas and liquid) flow in a porous medium. The problem is formulated for a general N-component liquid for modeling multicomponent effects. Despite the immense complexity of the model, the problem allows an analytic solution. The clue to this solution is a special form of a folding singularity at an internal point of the wave profile. 

M.A. Endo Kokubun and A.A. Mailybaev, Singularity of a combustion wave profile: a clue to the multicomponent theory for liquid-gas filtration, SIAM Journal on Applied Mathematics 77 (2017), 1375–1396. [Link] [PDF]