David M. Ambrose

Well-Posedness of Vortex Sheets with Surface Tension

We study the initial value problem for two-dimensional, periodic vortex sheets with surface tension. We allow the upper and lower fluids 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 o...

Well-posedness of two-phase Hele–Shaw flow without surface tension

We prove short-time well-posedness of a Hele–Shaw system with two fluids and no surface tension (this is also known as the Muskat problem). We restrict our attention here to the stable case of the problem. That is, 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. This sign condition essentially means that the more viscous fluid must displace the less viscous fluid. The proof uses the formulation introduced in the numerical work of Hou, Lowengrub, and Shelley, and relies on energy methods.

Computation of Time-Periodic Solutions of the Benjamin–Ono Equation

We present a spectrally accurate numerical method for finding nontrivial time-periodic solutions of nonlinear 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 in the case of the Benjamin–Ono equation, 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 nontrivial 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. 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 ODEs governing the evolution of solitons using the ansatz suggested by the numerical simulations.

Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension

A numerical method is introduced for the computation of time-periodic vortex sheets with surface tension separating two immiscible, irrotational, two-dimensional ideal fluids of equal density. The approach is based on minimizing a nonlinear functional of the initial conditions and supposed period that is positive unless the solution is periodic, in which case it is zero. An adjoint-based optimal control technique is used to efficiently compute the gradient of this functional. Special care is required to handle singular integrals in the adjoint formulation. Starting with a solution of the linearized problem about the flat rest state, a family of smooth, symmetric breathers is found that, at quarter-period time intervals, alternately pass through a flat state of maximal kinetic energy, and a rest state in which all the energy is stored as potential energy in the interface. In some cases, the interface overturns before returning to the initial, flat configuration. It is found that the bifurcation diagram describing these solutions contains several disjoint curves separated by near-bifurcation events.

Well-posedness of two-phase Darcy flow in 3D

We prove the well-posedness, locally in time, of the motion of two fluids flowing according to Darcys 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.

A small-scale decomposition for 3D boundary integral computations with surface tension ☆

Abstract An efficient, non-stiff boundary integral method for the initial value problem for interfacial Darcy flow (which is a model of porous media flow) in three space dimensions is presented. We consider a ‘doubly-periodic’ interface separating two fluids, with surface tension present at the boundary. Surface tension introduces high order (i.e., high derivative) terms in the governing equation, and this imposes a severe stability constraint on explicit time-integration methods. Furthermore, the high order terms appear in a nonlocal operator, which makes it difficult to design an efficient implicit method. The stiffness is removed by developing a small-scale decomposition in the spirit of prior work in the two-dimensional problem by Hou, Lowengrub, and Shelley. In order to develop this small-scale decomposition, we formulate the problem using a generalized isothermal parameterization of the free surface. An additional difficulty is the efficient calculation of the Birkhoff–Rott integral for the velocity of the interface. We present a new algorithm, based on Ewald summation, to compute this in O ( N log N ) operations, where N is the number of interface grid points. Our non-stiff method is expected to apply widely to problems for doubly-periodic interfacial flow with surface tension which have a boundary integral formulation.

On ill-posedness of truncated series models for water waves

The evolution of surface gravity waves on a large body of water, such as an ocean, is reasonably well approximated by the Euler system for ideal, free-surface flow under the influence of gravity. The well-posedness theory for initial-value problems for these equations, which has a long and distinguished history, reveals that solutions exist, are unique, and depend continuously upon initial data in various function–space contexts. This theory is subtle, and the design of stable, accurate, numerical schemes is likewise challenging. Depending upon the wave regime in question, there are many different approximate models that can be formally derived from the Euler equations. As the Euler system is known to be well-posed, it seems appropriate that associated approximate models should also have this property. This study is directed to this issue. Evidence is presented calling into question the well-posedness of a well-known class of model equations which are widely used in simulations. A simplified version of these models is shown explicitly to be ill-posed and numerical simulations of quadratic- and cubic-order water-wave models, initiated with initial data predicted by the explicit analysis of the simplified model, lends credence to the general contention that these models are ill-posed.

Gravity perturbed Crapper waves

Crapper waves are a family of exact periodic travelling wave solutions of the free-surface irrotational incompressible Euler equations; these are pure capillary waves, meaning that surface tension is accounted for, but gravity is neglected. For certain parameter values, Crapper waves are known to have multi-valued height. Using the implicit function theorem, we prove that any of the Crapper waves can be perturbed by the effect of gravity, yielding the existence of gravity–capillary waves nearby to the Crapper waves. This result implies the existence of travelling gravity–capillary waves with multi-valued height. The solutions we prove to exist include waves with both positive and negative values of the gravity coefficient. We also compute these gravity perturbed Crapper waves by means of a quasi-Newton iterative scheme (again, using both positive and negative values of the gravity coefficient). A phase diagram is generated, which depicts the existence of single-valued and multi-valued travelling waves in the gravity–amplitude plane. A new largest water wave is computed, which is composed of a string of bubbles at the interface.

Preservation of support and positivity for solutions of degenerate evolution equations

We prove that sufficiently smooth solutions of equations of a certain class have two interesting properties. These evolution equations are in a sense degenerate, in that every term on the right-hand side of the evolution equation has either the unknown or its first spatial derivative as a factor. We first find a conserved quantity for the equation: the measure of the set on which the solution is non-zero. Second, we show that solutions which are initially non-negative remain non-negative for all times. These properties rely heavily upon the degeneracy of the leading order term. When the equation is more degenerate, we are able to prove that there are additional conserved quantities: the measure of the set on which the solution is positive and the measure of the set on which the solution is negative. To illustrate these results, we give examples of equations with nonlinear dispersion which have solutions in spaces with sufficient regularity to satisfy the hypotheses of the support and positivity theorems. An important family of equations with nonlinear dispersion are the Rosenau–Hyman compacton equations; there is no existence theory yet for these equations, but the known solutions of the compacton equations are of lower regularity than is needed for the preceding theorems. We prove an additional positivity theorem which applies to solutions of the same family of equations in a function space which includes some solutions of compacton equations.

Fokas integral equations for three dimensional layered-media scattering

Abstract The scattering of acoustic waves by periodic structures is of central importance in a wide range of problems of scientific and technological interest. This paper describes a rapid, high-order numerical algorithm for simulating solutions of Helmholtz equations coupled across irregular (non-trivial) interfaces meant to model acoustic waves incident upon a multiply layered medium. Building upon an interfacial formulation from previous work, we describe an Integral Equation strategy inspired by recent developments of Fokas and collaborators for its numerical approximation. The method requires only the discretization of the layer interfaces (so that the number of unknowns is an order of magnitude smaller than volumetric approaches), while it requires neither specialized quadrature rules nor periodized fundamental solutions characteristic of many popular Boundary Integral/Element Methods. As with previous contributions by the authors on this formulation, this approach is efficient and spectrally accurate for smooth interfaces.