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1. Gradient Estimates for Stokes and Navier--Stokes Systems with Piecewise DMO Coefficients

   https://doi.org/10.1137/21M1423518  

   Choi, Jongkeun ; Dong, Hongjie ; Xu, Longjuan ( June 2022 , SIAM Journal on Mathematical Analysis)
2. Analyticity up to the boundary for the Stokes and the Navier-Stokes systems

   https://doi.org/10.1090/tran/7990  

   Camliyurt, Guher ; Kukavica, Igor ; Vicol, Vlad ( May 2020 , Transactions of the American Mathematical Society)
3. Connection Between Grad-Div Stabilized Stokes Finite Elements and Divergence-Free Stokes Finite Elements

   Neilan, Michael ; Zytoon, Ahmed ( October 2020 , International journal of numerical analysis and modeling)

   null (Ed.)
4. Stokes drift and its discontents

   https://doi.org/10.1098/rsta.2021.0032  

   Vanneste, Jacques ; Young, William R. ( June 2022 , Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   The Stokes velocity u S , defined approximately by Stokes (1847, Trans. Camb. Philos. Soc. , 8 , 441–455.), and exactly via the Generalized Lagrangian Mean, is divergent even in an incompressible fluid. We show that the Stokes velocity can be naturally decomposed into a solenoidal component, u sol S , and a remainder that is small for waves with slowly varying amplitudes. We further show that u sol S arises as the sole Stokes velocity when the Lagrangian mean flow is suitably redefined to ensure its exact incompressibility. The construction is an application of Soward & Roberts’s glm theory (2010, J. Fluid Mech. , 661 , 45–72. ( doi:10.1017/S0022112010002867 )) which we specialize to surface gravity waves and implement effectively using a Lie series expansion. We further show that the corresponding Lagrangian-mean momentum equation is formally identical to the Craik–Leibovich (CL) equation with u sol S replacing u S , and we discuss the form of the Stokes pumping associated with both u S and u sol S . This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’. 

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5. Superharmonic instability of stokes waves

   https://doi.org/10.1111/sapm.12535  

   Korotkevich, A. O. ; Lushnikov, P. M. ; Semenova, A. ; Dyachenko, S. A. ( October 2022 , Studies in Applied Mathematics)

   A stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically in approximation of an infinite depth. Investigation of the stability properties can give one an insight into the evolution of the Stokes wave. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical equations we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behavior of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.orgwebsite.

    

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