Miles H. Wheeler

I am a Lecturer (Assistant Professor) in Analysis in the Department of Mathematical Sciences at the University of Bath. Before coming to Bath, I was a University Assistant at the Faculty of Mathematics at the University of Vienna, and before that I was a postdoc at the Courant Institute of Mathematical Sciences supported by an NSF fellowship. I am interested in partial differential equations coming from fluid mechanics, especially “large-amplitude” solutions that cannot be captured using perturbative techniques.

Email:

mw2319@bath.ac.uk

Office:

4W 1.12

Office hours:

By appointment

Papers and preprints

[1]

RM Chen, L Fan, S Walsh, and MH Wheeler, Rigidity of three-dimensional internal waves with constant vorticity, submitted. arxiv (12 pages)

[2]

SV Haziot and MH Wheeler, Large-amplitude steady solitary water waves with constant vorticity, submitted. arxiv (35 pages)

[3] (1,2)

SV Haziot, VM Hur, WA Strauss, JF Toland, E Wahlén, S Walsh, and MH Wheeler, Traveling water waves — the ebb and flow of two centuries, Quart. Appl. Math. 80:317, 2022. arxiv (85 pages)

[4]

VM Hur and MH Wheeler, Overhanging and touching waves in constant vorticity flows, J. Differential Equations 338:572, 2022. arxiv (19 pages)

[5]

Z Hassainia and MH Wheeler, Multipole vortex patch equilibria for active scalar equations, to appear in SIAM J. Math. Anal., arxiv (37 pages)

[6]

MA Johnson, T Truong, and MH Wheeler, Solitary waves in a Whitham equation with small surface tension, Stud. Appl. Math. 148(2):773, 2022. arxiv (40 pages)

[7]

RM Chen, S Walsh, and MH Wheeler, Large-amplitude internal fronts in two-fluid systems, C. R. Math. Acad. Sci. Paris 358(9-10):1073, 2020. (11 pages)

[8]

VS Krishnamurthy, MH Wheeler, DG Crowdy, and A Constantin, Liouville chains: new hybrid vortex equilibria of the 2D Euler equation, J. Fluid Mech. 921:A1, 2021. arxiv (35 pages)

[9]

T Truong, E Wahlén, and MH Wheeler, Global bifurcation of solitary waves for the Whitham equation, Math. Ann. 383:1521, 2022. arxiv (45 pages)

[10]

RM Chen, S Walsh, and MH Wheeler, Global bifurcation of anti-plane shear fronts, J. Nonlinear Sci. 31:28, 2021. arxiv (31 pages)

[11]

VS Krishnamurthy, MH Wheeler, DG Crowdy, and A Constantin, A transformation between stationary point vortex equilibria, Proc. R. Soc. A. 476:20200310, 2020. postprint, (21 pages)

[12]

A Constantin, DG Crowdy, VS Krishnamurthy, and MH Wheeler, Stuart-type polar vortices on a rotating sphere, Discrete Contin. Dyn. Syst. 41(1):201, 2021. (15 pages)

[13]

VM Hur and MH Wheeler, Exact free surfaces in constant vorticity flows, J. Fluid Mech. (Rapids) 896:R1, 2020. postprint (10 pages)

[14]

RM Chen, S Walsh, and MH Wheeler, Global bifurcation for monotone fronts of elliptic equations, submitted. arxiv (60 pages)

[15]

V Kozlov, E Lokharu, and MH Wheeler, Nonexistence of subcritical solitary waves, Arch. Rational Mech. Anal. 241:535, 2021. arxiv (18 pages)

[16]

RM Chen, S Walsh, and MH Wheeler, Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics, Nonlinearity 35(4):1927, 2022. arxiv (59 pages)

[17]

Z Hassainia, N Masmoudi, and MH Wheeler, Global bifurcation of rotating vortex patches, Comm. Pure Appl. Math. 73(9):1933, 2020. arxiv (48 pages)

[18]

VS Krishnamurthy, MH Wheeler, DG Crowdy, and A Constantin, Steady point vortex pair in a field of Stuart-type vorticity, J. Fluid Mech. (Rapids) 874:R1, 2019. preprint (11 pages)

[19]

MH Wheeler, On stratified water waves with critical layers and Coriolis forces, Discrete Contin. Dyn. Syst. 39(8):4747, 2019. preprint (24 pages)

[20]

RM Chen, S Walsh, and MH Wheeler, Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity, Arch. Rational Mech. Anal., 234(2):595, 2019 arxiv (39 pages)

[21]

MH Wheeler, Simplified Models for Equatorial Waves with Vertical Structure, Oceanography 31(3):36, 2018. (6 pages)

[22]

MH Wheeler, Integral and asymptotic properties of solitary waves in deep water, Comm. Pure Appl. Math. 71: 1941, 2018. preprint, arxiv (16 pages)

[23]

RM Chen, S Walsh, and MH Wheeler, Existence and qualitative theory for stratified solitary water waves, Ann. Inst. H. Poincaré Anal. Non Linéaire 35(2):517, 2017. arxiv (62 pages)

[24]

RM Chen, S Walsh, and MH Wheeler, On the existence and qualitative theory for stratified solitary water waves, C. R. Math. Acad. Sci. Paris 354(6):601, 2016. (5 pages)

[25]

WA Strauss and MH Wheeler, Bound on the slope of steady water waves with favorable vorticity, Arch. Rational Mech. Anal. 222:1555, 2016. preprint, arxiv (26 pages)

[26]

MH Wheeler, The Froude number for solitary water waves with vorticity, J. Fluid Mech. 768:91, 2015. arxiv (22 pages)

[27]

MH Wheeler, Solitary water waves of large amplitude generated by surface pressure, Arch. Rational Mech. Anal. 218(2):1131, 2015. preprint (57 pages)

[28]

MH Wheeler, Large-amplitude solitary water waves with vorticity. SIAM J. Math. Anal. 45(5):2937, 2013. preprint (58 pages)

[29]

S Constantin, RS Strichartz, and M Wheeler, Analysis of the Laplacian and spectral operators on the Vicsek set Commun. Pure Appl. Anal. 10(1):1, 2011. arxiv, website (44 pages)

Expository notes and talks

For a more accessible introduction to the sort of work I do, see this expository talk on solitary waves and fronts (and Section 5 of [3]), or this short introduction to local and global bifurcation theory. The talk is from a series on steady water waves in the ONEPAS seminar, and the notes are from a 2019 lecture to a group of masters students in mathematics and physics.

Teaching

• Spring 2022, University of Bath: Theory of Partial Differential Equations

• Fall 2021, University of Bath: Advanced Real Analysis

• Spring 2021, University of Bath: Theory of Partial Differential Equations

• Fall 2020, University of Bath: Advanced Real Analysis

• Spring 2020, University of Bath: Theory of Partial Differential Equations

• Summer 2019, University of Vienna: Tutorial on ordinary differential equations.

• Winter 2018, University of Vienna: Topics in analysis: fluid mechanics

• Summer 2018, University of Vienna: Tutorial on ordinary differential equations.

• Winter 2017, University of Vienna: Tutorial on partial differential equations.

• Fall 2016, NYU: Instructor for Math-UA 325, Analysis I (Section 3).

• Spring 2015, NYU: Instructor for Math-UA 121, Calculus I (Section 6).

• Fall 2014, NYU: Instructor for Math-UA 122, Calculus II (Section 3).

• Fall 2012, Brown University: Instructor (TF) for Math 200, Intermediate (Multivariable, Physics/Engineering) Calculus.

• Spring 2012, Brown University: Instructor (TF) for Math 200, Intermediate (Multivariable, Physics/Engineering) Calculus.

• Spring 2012, Brown University: TA for Math 100 (Calculus II).

• Fall 2010, Brown University: TA for Math 100 (Calculus II).

David Lowry-Duda and I also wrote an expository paper aimed at undergraduates which appeared in the American Mathematical Monthly and won an award from the MAA.

CV

Last updated August 2022

Some figures (mostly from talks)

An overturning solitary wave in [2].

A generalized solitary wave solution of the Whitham equation in [6].

Simple mechanical analogy for solitary waves and bores in the survey [3].

Speed of solitary waves compared to small periodic waves; see [15] and [26].

Explicit overturning waves with constant vorticity in [13]; also see [4].

An extreme solitary wave solution of the Whitham equation in [9].

An anti-plane shear front in [10], with a graph of its centerline.

Streamlines for a hybrid vortex equilibria in [18] and [8]; also see [12].

A family of point vortex equilibria which “collapses” in [11]. Also see the August 2020 cover of Proceedings of the Royal Society A!

Bore with a critical layer in [16]; see [14] (and [7]) for large-amplitude bores but without critical layers.

An integration by parts argument in [22] and [20]

A moving planes argument in [23] (also see [24]).

The angle \(θ\) considered in [25]

Waves with localized pressure forcing, see [27].

Vortex patch with streamlines from [17] (also see [5]). See here for more plots.

Constructing the global continuum in [28].

Waves with multiple fluid layers in [19], [21], and [1].

Eigenfunction of the Laplacian on the Vicsek set in [29].