Miles H. Wheeler

I am a Lecturer 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

News

• The One World PDE Seminar meets Tuesdays 3:30–4:30pm UK time. Hope to see you there!

• My paper 7 with Vikas Krishnamurthy, Darren Crowdy, and Adrian Constantin is on the cover of the August issue of Proceedings of the Royal Society A! Special thanks to David Lowry-Duda.

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, 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.

Papers and preprints

1

Z Hassainia and MH Wheeler, Multipole vortex patch equilibria for active scalar equations. arxiv (32 pages)

2

MA Johnson, T Truong, and MH Wheeler, Solitary waves in a Whitham equation with small surface tension, submitted. arxiv (34 pages)

3

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. journal (11 pages)

4

VS Krishnamurthy, MH Wheeler, DG Crowdy, and A Constantin, Liouville chains: new hybrid vortex equilibria of the 2D Euler equation, submitted. arxiv (34 pages)

5

T Truong, E Wahlén, and MH Wheeler, Global bifurcation of solitary waves for the Whitham equation, submitted. arxiv (38 pages)

6

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

7(1,2)

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

8

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. journal (15 pages)

9

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

10

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

11

V Kozlov, E Lokharu, and MH Wheeler, Nonexistence of subcritical solitary waves, to appear in Arch. Rational Mech. Anal. arxiv (13 pages)

12

RM Chen, S Walsh, and MH Wheeler, Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics, submitted. arxiv (45 pages)

13

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

14

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. journal, preprint (11 pages)

15

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

16

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 journal, arxiv (39 pages)

17

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

18

MH Wheeler, Integral and asymptotic properties of solitary waves in deep water, Comm. Pure Appl. Math. 71: 1941–1956, 2018. preprint, arxiv, journal (16 pages)

19

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, journal (62 pages)

20

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. journal (5 pages)

21

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, journal (26 pages)

22

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

23

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

24

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

25

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, journal, website (44 pages)

Teaching

• 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 an undergraduates which has appeared in the American Mathematical Monthly.

CV

Last updated February 2021

Some figures (mostly from talks)

A generalized solitary wave solution of the Whitham equation in 2.

An extreme solitary wave solution of the Whitham equation in 5.

An anti-plane shear front in 6, with a graph of its centerline.

A family of point vortex equilibria which “collapses” in 7.

Explicit overturning waves with constant vorticity in 9.

Speed of solitary waves compared to small periodic waves; see 11 and 22.

Bore with a critical layer in 12; see 10 (and 3) for large-amplitude bores but without critical layers.

Streamlines for a hybrid vortex equilibria in 14 and 4; also see 8.

Vortex patch with streamlines from 13 (also see 1). See here for more plots.

Waves with multiple fluid layers in 15 and 17.

An integration by parts argument in 18 and 16

The angle \(θ\) considered in 21

A moving planes argument in 19 (also see 20)

Constructing the global continuum in 24

Waves with localized pressure forcing, see 23

Eigenfunction of the Laplacian on the Vicsek set in 25.