8. (with Jon Wilkening) Time-periodic solutions of the Benjamin-Ono equation,
Submitted.
Abstract: We present a spectrally accurate numerical method for finding
non-trivial time-periodic solutions of non-linear partial
differential equations. The method is based on minimizing a
functional (of the initial condition and the period) that is
positive unless the solution is periodic, in which case it is zero.
We solve an adjoint PDE to compute the gradient of this functional
with respect to the initial condition. We include additional terms
in the functional to specify the free parameters, which are
the mean, a spatial phase, a temporal phase and the real part of
one of the Fourier modes at t=0.
We use our method to study global paths of non-trivial time-periodic
solutions connecting stationary and traveling waves of the
Benjamin-Ono equation. As a starting guess for each path, we
compute periodic solutions of the linearized problem by solving an
infinite dimensional eigenvalue problem in closed form. We then use
our numerical method to continue these solutions beyond the realm of
linear theory until another traveling wave is reached (or until the
solution blows up). By experimentation with data fitting, we
identify the analytical form of the solutions on the path connecting
the one-hump stationary solution to the two-hump traveling wave. We
then derive exact formulas for these solutions by explicitly solving
the system of ODE's governing the evolution of solitons using the
ansatz suggested by the numerical simulations.
7. Singularity formation in a model for the vortex sheet with surface tension, Accepted,
Math. Comput. Simulation, (2008).
Abstract: In a recent analytical study, the author has proved well-posedness of the vortex sheet with surface
tension. This work included using a formulation of the problem introduced by Hou, Lowengrub, and
Shelley for a numerical study of the same problem. The analytical study required identification of a term
in the evolution equations which can be viewed as being responsible for the Kelvin-Helmholtz instability;
this term is of lower order than the surface tension term. In the present work, the author introduces
a simple model for the vortex sheet with surface tension which maintains the same dispersion relation
and the same destabilizing force as in the vortex sheet with surface tension. For the model problem, it
is found that finite-time singularities can form when the initial data is taken from a certain class. For
the vortex sheet with surface tension, the only observed singularities thus far in numerical work have
coincided with self-intersection of the fluid interface. There is no analogue of self-intersection in the model
problem, and thus the singularities observed in the present work may well be related to a previously
unobserved singularity for the full vortex sheet problem.
6. (with Nader Masmoudi) The zero surface tension limit of three-dimensional water waves,
Accepted, Indiana U. Math. J., (2007).
Abstract: We provide a new proof of existence of
irrotational water waves in three space dimensions. We do this
by establishing that the limit of the water wave with surface tension, as surface tension vanishes,
is the water wave without surface tension.
The main tool is an energy estimate which is uniform in the surface tension parameter.
Before establishing estimates, we reformulate the problem using suitable variables and an
isothermal parameterization. With these variables and parameterizations, estimates for the
water wave with or without surface tension are straightforward, and the existence proof is more
elementary and shorter than other proofs.
5. (with Nader Masmoudi) Well-posedness
of 3D vortex sheets with surface tension,
Comm. Math. Sci., (2007) 5:391-430.
Abstract: We prove well-posedness for the initial value
problem for a vortex sheet in 3D fluids, in the presence of surface tension. We
first reformulate the problem by making a favorable choice of variables and
parameterizations. We then perform energy estimates for the evolution equations.
It is important to note that the Kelvin-Helmholtz instability is present for the
vortex sheet in the absence of surface tension. Accordingly, we must construct
the energy functional carefully with an eye toward the regularization of this
instability. Well-posedness follows from the estimates.
4.
Well-posedness of two-phase Darcy flows in 3D, Quart. Appl. Math., (2007) 65:189-203.
Abstract: We prove the well-posedness, locally in time, of the
motion of two fluids flowing according to Darcy's law, separated by a sharp
interface in the absence of surface tension. We first reformulate the problem
using favorable variables and coordinates. This results in a quasilinear
parabolic system. Energy estimates are performed, and these estimates imply that
the motion is well-posed for a short time with data in a Sobolev space, as long
as a condition is satisfied. This condition essentially says that the more
viscous fluid must displace the less viscous fluid. It should be true that small
solutions exist for all time; however, this question is not addressed in the
present work.
3. (with Nader Masmoudi) The
zero surface tension limit of two-dimensional water waves, Comm. Pure Appl.
Math., (2005) 58:1287-1315.
Abstract: We consider two-dimensional water waves
of infinite depth, periodic in the horizontal direction. It has been proved by
Wu (in the slightly different non-periodic setting) that solutions to this
initial value problem exist in the absence of surface tension. Recently, Ambrose
has proved that solutions exist when surface tension is taken into account. In
this paper, we provide a shorter, more elementary proof of existence of
solutions to the water wave initial value problem both with and without surface
tension. Our proof requires estimating the growth of geometric quantities using
an arclength parameterization of the free surface and using physical quantities
related to the tangential velocity of the free surface. Using this formulation,
we find that as surface tension goes to zero the water wave without surface
tension is the limit of the water wave with surface tension. Far from being a
simple adaptation of previous works, our method requires a very original choice
of variables; these variables turn out to be physical and well adapted to
both cases.
2. Well-posedness
of two-phase Hele-Shaw flow without surface tension, European J. Appl.
Math., (2004) 15:597-607.
Abstract: We prove short-time well-posedness of a
Hele-Shaw system with two fluids and no surface tension. In order for the motion
to be well-posed, the initial data must satisfy a sign condition which is a
generalization of a condition of Saffman and Taylor. The proof uses the
formulation introduced in the numerical work of Hou, Lowengrub, and Shelley, and
relies on energy methods.
1. Well-posedness
of vortex sheets with surface tension, SIAM J. Math. Anal., (2003)
35:211-244.
Abstract: We study the initial value problem for
two-dimensional, periodic vortex sheets with surface tension. We allow the upper
and lower fluid to have different densities. Without surface tension, the vortex
sheet is ill-posed: it exhibits the well-known Kelvin-Helmholtz instability. In
the linearized equations of motion, surface tension removes the instability. It
has been conjectured that surface tension also makes the full problem
well-posed. We prove that this conjecture is correct using energy methods. In
particular, for the initial value problem for vortex sheets with surface tension
with sufficiently smooth data, it is proved that solutions exist locally in
time, are unique, and depend continuously on the initial data. The analysis uses
two important ideas from the numerical work of Hou, Lowengrub, and Shelley.
First, the tangent angle and arclength of the vortex sheet are used rather than
Cartesian variables. Second, instead of a purely Lagrangian formulation, a
special tangential velocity is used in order to simplify the evolution
equations. A special case of the result is well-posedness of water waves with
surface tension; this is the first proof (with surface tension) which allows the
wave to overturn.
3. Short-time well-posedness of free-surface problems in irrotational 3D
fluids. Hyperbolic Problems: Theory, Numerics, and Applications.
Proceedings of the Eleventh International Conference, Lyon (July, 2006).
Springer-Verlag (2008), pages 307-314.
We discuss the proof of short-time well-posedness for free-surface problems in
irrotational three-dimensional fluids. We consider three situations: the vortex sheet with
surface tension, the water wave, and Darcy flow. A common framework is described for treating
each of these problems. In this framework, we choose convenient parameterizations and variables. In each case, we arrive at a system of evolution equations which is amenable to
use of the energy method. The work on the vortex sheet and the water wave is joint with Nader Masmoudi.
2. Regularization
of the Kelvin-Helmholtz instability by surface tension.
Phil. Trans. R. Soc. A, (2007) 365:2253-2266. Proceedings of the
Semester on Wave Motion; Institut Mittag-Leffler, Djursholm, Sweden (Fall,
2005).
Abstract: The Kelvin-Helmholtz instabilty is present in the motion of
a vortex sheet without surface tension. This can be seen from the linearization
of the equations of motion, and there have also been proofs of ill-posedness for
the full, nonlinear equations. In the presence of surface tension, the
linearized equations no longer exhibit an instability, and it has been believed
that the full equations should then be well-posed. In this paper, we sketch a
proof that the vortex sheet with surface tension is well-posed in the case of
both two-dimensional and three-dimensional fluids. The proof in the case of
three-dimensional fluids is joint work with Nader Masmoudi. The method is to
first reformulate the problem using suitable variables and parameterizations,
and then to perform energy estimates. The choice of variables and
parameterizations in the two-dimensional case is the same as that of Hou,
Lowengrub, and Shelley in prior numerical work.
1. Short-time
well-posedness of free-surface problems in 2D fluids, Hyperbolic Problems: Theory, Numerics, and Applications.
Proceedings of the Tenth International Conference, Osaka (September, 2004).
Yokohama Publishers (2006) vol. 1, pages 247-254.
Abstract: We
describe recent progress in the analysis of free-surface problems in 2D fluids.
This includes the solution of two long-standing open problems: the
well-posedness of the vortex sheet with surface tension and the well-posedness
of the Muskat problem. We also describe a new proof of well-posedness of 2D
water waves. The method of proof is similar for all of these problems: the
evolution equations are first reformulated using geometric variables and an
arclength parameterization, and then energy methods are employed.