J. Math. Fluid Mech. 20 (2018), 161–187
2017 Springer International Publishing
Journal of Mathematical
Traveling Gravity Water Waves with Critical Layers
Ailo Aasen and Kristoﬀer Varholm
Communicated by A. Constantin
Abstract. We establish the existence of small-amplitude uni- and bimodal steady periodic gravity waves with an aﬃne
vorticity distribution, using a bifurcation argument that diﬀers slightly from earlier theory. The solutions describe waves
with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis
is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions.
Finally, we investigate the asymptotic behavior of the bifurcating solutions.
Mathematics Subject Classiﬁcation. Primary 35Q31; Secondary 35B32, 35C07, 76B15.
Up until fairly recently, most authors working with steady water waves have made the assumption that
of the velocity ﬁeld (u, v) vanishes identically. Such waves are known as irrotational, as opposed to
rotational waves where ω is allowed to be nonzero. Rotational waves can exhibit more exotic behavior
than irrotational ones, including interior stagnation points and critical layers of closed streamlines .
Stagnation points correspond to ﬂuid particles that are stationary with respect to the wave, and for
irrotational ﬂows this can only occur at a sharp crest .
Irrotational waves are mathematically simpler to work with than rotational ones, due to the existence
of the velocity potential. The velocity potential is the harmonic conjugate of the stream function, thus
enabling the use of tools such as complex analysis, which are typically not available with nonzero vorticity.
The survey  treats the theory of Stokes waves—an important class of irrotational waves—and the
results on the so-called Stokes conjecture for such waves. This conjecture was not fully settled until the
appearance of the paper .
Although rotational waves were considered intractable for mathematical analysis, they have long been
important in more applied ﬁelds because rotational waves are not uncommon in nature: There are many
physical eﬀects that can induce rotation in waves, such as wind and thermal or salinity gradients ,
and rotational waves are also important in wave-current interactions .
The ﬁrst, and still the only known, explicit example of a nontrivial traveling gravity water wave solution
to the Euler equations was given in  (see also  for a more modern treatment) and is rotational; a fact
which was only later pointed out by Stokes. Much later came the ﬁrst existence result for small-amplitude
waves with general vorticity distributions . It was not, however, before the pioneering article  that
large-amplitude waves were constructed, using an extension of the global bifurcation theory of Rabinowitz
[16,28], leading to renewed interest in rotational waves. A corresponding result on deep water, where the
lack of compactness is an obstacle, was established in .
A. Aasen was supported by a Grant from the I. K. Lykke Fund. The authors also acknowledge the support of the project
Nonlinear Water Waves by the Research Council of Norway (Grant No. 231668).