arXiv subjects

Miles H. Wheeler

Publications and source records attributed to Miles H. Wheeler.

At least 19 recordsLinked to original sources

arXiv2026

On the vorticity threshold for steady water waves

One of the most striking features of two-dimensional steady water waves with adverse vorticity is the emergence of stagnation points in the flow, even along branches of solutions that are initially unidirectional. In this paper we show that, for constant adverse vorticity above a certain explicit threshold, the first stagnation point must appear on the bed directly below the crest. This is in contrast to waves with favourable or even small adverse vorticity, where it is known that bottom stagnation cannot occur, leading to extreme waves exhibiting surface singularities. We also establish several related inequalities for unidirectional waves with strong adverse vorticity, including a new upper bound on the amplitude.

math.AP↗

arXiv2026

Global bifurcation for steady viscous roll waves on an incline

We construct a branch of travelling periodic `roll wave' solutions to the free-boundary incompressible Navier--Stokes equations on an inclined plane in two dimensions. These solutions bifurcate from a parallel shear flow, under natural assumptions on the related Orr--Sommerfeld equation. Using techniques from analytic global bifurcation theory, we extend the local branch to a global curve of solutions. A key step of the proof is reformulating the problem, including the unknown free boundary, as an elliptic system in the sense of Agmon--Douglis--Nirenberg. Finally, we verify the hypotheses on the Orr--Sommerfeld equation for two regimes: small wavenumber and low Reynolds number.

math.AP↗

arXiv2025

Extreme internal waves: gravity currents and overturning fronts

Hydrodynamic bores are front-type traveling wave solutions to the two-layer free boundary Euler equations in two dimensions. The velocity field in each layer is assumed to be incompressible and irrotational, and it limits to distinct laminar flows upstream and downstream. Rigid horizontal boundaries confine the fluids from above and below. A constant gravitational force acts on the waves, but surface tension is neglected. It was recently shown by the authors that there exist two large-amplitude families of hydrodynamic bores: a curve of depression bores and a curve of elevation bores. We now prove that in the limit along the elevation bore family, the solutions must overturn: the interface separating the layers develops a vertical tangent. This type of behavior was first observed over 45 years ago in numerical computations of internal gravity waves and gravity water waves with vorticity. Despite considerable progress over the past decade in constructing families of water waves that potentially overturn, a proof that overturning definitively occurs has been stubbornly elusive. We further show that in the limit along the depression bore family, either overturning occurs or the solutions converge to a gravity current: the free boundary contacts the upper wall and the relative velocity in the upper fluid is stagnant. We also determine the contact angle between the interface and the rigid barrier for the limiting gravity current, giving the first rigorous confirmation of a conjecture of von Kármán. The resolutions of these questions in the specific case of hydrodynamic bores is accomplished through the use of novel geometric analysis techniques, including bounds on the decay of the velocity field near a hypothetical double stagnation point. These ideas may have broader applications to bifurcation theoretic studies of large-amplitude waves.

math.AP↗

arXiv2025

Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane

In this note we construct smooth bounded domains $Ω\subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} Δu + λu &= 0 &\qquad& \text{ in } Ω, \newline u &= b &\qquad& \text{ on } \partial Ω, \newline \frac{\partial u}{\partial n} &= c &\qquad& \text{ on } \partial Ω \end{alignedat} \right. $$ has a solution for some constants $λ,b,c \ne 0$. These appear to be the first counterexamples to a conjecture of Willms and Gladwell [WG94].

math.AP↗

arXiv2025

Large-amplitude periodic solutions to the steady Euler equations with piecewise constant vorticity

We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation theory, we rigorously construct curves of solutions that terminate either with stagnation on the interface or when the conformal equivalence between one of the layers and a strip breaks down in a $C^1$ sense. We give numerical evidence that, depending on parameters, these occur either as a corner forming on the interface or as one of the layers developing regions of arbitrarily thin width. Our proof relies on a novel formulation of the problem as an elliptic system for the velocity components in each layer, conformal mappings for each layer, and a horizontal distortion which makes these mappings agree on the interface. This appears to be the first local formulation for a multi-layer problem which allows for both overhanging wave profiles and stagnation points.

math.AP↗

arXiv2025

Finite-time self-similar implosion of hollow vortices

In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves $Γ_1, \ldots, Γ_M$, and such that the flow is irrotational and incompressible, but with a nonzero circulation around each boundary component. The region bounded by $Γ_k$ is a ``vortex core'', modeled as a bubble of ideal gas: the pressure is constant in space and inversely proportional to the area of the vortex. This can be thought of as the isobaric approximation assuming isothermal flow. Our results come in two parts. There exist explicit families of purely circular rotating and imploding hollow vortices. Implosion means more precisely that the vortex core shrinks to the origin in finite time, while the absolute value of the pressure simultaneously diverges to infinity. We prove that for any $m \geq 2$, there exist near-circular $m$-fold symmetric rotating hollow vortices. By contrast, for all $m \geq 2$, the purely circular imploding vortices are locally unique among all collapsing vortices with uniform velocity at infinity. The second part concerns configurations of multiple hollow vortices. The existence of configurations of point vortices that collapse into a common point in finite time is classical. We prove that generically, these can be desingularized to yield families of hollow vortex configurations exhibiting self-similar finite-time implosion. Specific examples of an imploding trio and quartet of hollow vortices are given.

math.AP↗

arXiv2024

Overhanging solitary water waves

We provide the first construction of overhanging gravity water waves having the approximate form of a disk joined to a strip by a thin neck. The waves are solitary with constant vorticity, and exist when an appropriate dimensionless gravitational constant $g>0$ is sufficiently small. Our construction involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, Hélein, and Pacard \cite{hauswirth-helein-pacard}. The method developed here is related to the construction of constant mean curvature surfaces through gluing.

math.AP↗

arXiv2024

Vortex-carrying solitary gravity waves of large amplitude

In this paper, we study two-dimensional traveling waves in finite-depth water that are acted upon solely by gravity. We prove that, for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family $\mathcal{C}$ of solitary waves in equilibrium with a submerged point vortex. This family bifurcates from an irrotational uniform flow, and, at least for large Froude numbers, extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. We also provide a numerical bifurcation study of traveling periodic gravity waves with submerged point vortices, which strongly suggests that some of these waves indeed overturn. Finally, we prove that at generic solutions on $\mathcal{C}$ $\unicode{x2013}$ including those that are large amplitude or even overhanging $\unicode{x2013}$ the point vortex can be desingularized to obtain solitary waves with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air.

math.AP↗

arXiv2023

Solitary solutions to the steady Euler equations with piecewise constant vorticity in a channel

We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or difference in density between the two layers, and the flow is inviscid. Unlike in previous studies, we consider solutions which are localised perturbations rather than periodic or quasi-periodic perturbations of a background shear flow. We rigorously construct a curve of exact solutions and give the leading order terms in an asymptotic expansion. We also give a thorough qualitative description of the fluid particle paths, which can include stagnation points, critical layers, and streamlines which meet the boundary.

math.AP↗

arXiv2023

Global bifurcation for monotone fronts of elliptic equations

In this paper, we present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer and Buffoni--Toland, but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs. As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.

math.AP↗

arXiv2023

Desingularization and global continuation for hollow vortices

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; it can therefore be interpreted as a spinning bubble of air in water. This paper gives a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. Our machinery simultaneously treats the translating, rotating, and stationary regimes. Through global bifurcation theory, we further obtain maximal curves of solutions that continue until the onset of a singularity. As specific examples, we give the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington's classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve.

math.AP↗

arXiv2022

Rigidity of three-dimensional internal waves with constant vorticity

This paper studies the structural implications of constant vorticity for steady three-dimensional internal water waves. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two dimensional. The situation is more subtle for internal waves that traveling along the interface between two immiscible fluids. When the layers have the same density, there is a large class of explicit steady waves with constant vorticity that are three-dimensional in that the velocity field and pressure depend on one horizontal variable while the interface is an arbitrary function of the other. We prove the following rigidity result: every three-dimensional traveling internal wave with bounded velocity for which the vorticities in the upper and lower layers are nonzero, constant, and parallel must belong to this family. If the densities in each layer are distinct, then in fact the flow is fully two dimensional.

math.AP↗

arXiv2022

Multipole vortex patch equilibria for active scalar equations

We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu \cite{Hmidi-Mateu} for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point-vortex limit. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem.

math.AP↗

arXiv2022

Overhanging and touching waves in constant vorticity flows

We show the existence of periodic traveling waves at the free surface of a two dimensional, infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper's celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity.

math.AP↗

arXiv2021

Solitary waves in a Whitham equation with small surface tension

Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator is seen to undergo either a reversible $0^{2+}(i k_0)$ bifurcation or a reversible $(i s)^2$ bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second-order or third-order terms. These truncated systems relate directly to systems obtained in the study of the full gravity-capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity-capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity-capillary Whitham equation and the full two-dimensional gravity-capillary water wave problem.

math.AP↗

arXiv2021

Traveling water waves -- the ebb and flow of two centuries

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.

math.AP↗

arXiv2021

Global bifurcation of solitary waves for the Whitham equation

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnström and Wahlén. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.

math.AP↗

arXiv2020

Liouville chains: new hybrid vortex equilibria of the 2D Euler equation

A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a "pure" point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter $A$ in the hybrid solutions approaches the limits $A\to 0,\infty$. While $A\to 0$ reproduces the input equilibrium, $A\to\infty$ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by $A$ as a "Liouville link". In some cases, the emergent point vortex equilibrium as $A\to\infty$ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a "Liouville chain". We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (2003) and by Krishnamurthy et al. (2019) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (2020) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the 2D Euler equation.

physics.flu-dyn↗