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  1. ; ; (, Annals of PDE)
    Free, publicly-accessible full text available June 1, 2026
  2. ; ; (, EMS Surveys in Mathematical Sciences)
    WS: it is difficult to write about Thomas Kappeler in the past tense. He was a brilliant mathematician, but more importantly he was a wonderfully open, generous, and friendly person. I was fortunate that we had many opportunities to spend time together and discuss mathematics. I greatly miss him.A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. Even Stokes waves of very small amplitude are unstable when subjected to various perturbations. We present a brief survey of this phenomenon with emphasis on transverse perturbations. 
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    Free, publicly-accessible full text available May 16, 2026
  3. ; (, Archive for Rational Mechanics and Analysis)
    We study the Muskat problem for one fluid in an arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that, for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem. 
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