Samuel Walsh

Associate Professor

Mathematics Department

University of Missouri

Associate Professor

Mathematics Department

University of Missouri

I am an Associate Professor in the math department at the University of Missouri. I received my PhD in Applied Mathematics from Brown University in 2010; my thesis advisor was Walter Strauss. My undergraduate degree is a BS in Mathematical Sciences from Carnegie Mellon University. Prior to coming to MU, I was a Courant Instructor at NYU.

A summary of my teaching and research experience can be found in my CV [pdf].

Email: walshsa "at" missouri.edu

Office: 307 Math Sciences Building

Phone: (573) 882-4426

My office hours for Spring 2024 are Wednesdays and Thursdays 3–4PM, and by appointment.

My research is in the area of *nonlinear partial differential equations*, particularly those pertaining to *water waves*. A large part of my work has been devoted to the study of *steady waves*. Steady waves are a special class of solution to a time-dependent PDE which, when viewed in an appropriately chosen moving reference frame, become time-independent. Steady waves have been an object of fascination for hundreds of years (Cauchy wrote one of the original treatises on the subject), but fundamental questions about them remain. For instance, the existence of large-amplitude traveling waves, potentially even overhanging waves, is not yet established in a number of physically important regimes. Still less is understood about the qualitative features of steady waves, or even whether or not they are stable in many instances.

I am also interested in the broader topic of *dispersive nonlinear PDEs*. A dispersive PDE is one for which a solution that is localized in frequency will tend to propagate in space with a speed and direction determined by that frequency. Water waves are one example of this phenomenon, but it is found in many physical settings, e.g., quantum mechanics and nonlinear optics.

My work has been supported in part by the National Science Foundation through DMS-1514950, DMS-1812436, and DMS-2306243, as well as the Simons Foundation through award 960210.

I serve as the organizer for the Differential Equations Seminar at MU. The list of upcoming talks can be found here.

*Vortex-carrying solitary gravity waves of large amplitude*, (with R. M. Chen, K. Varholm, and M. H. Wheeler),

submitted [arXiv].*Desingularization and global continuation for hollow vortices*, (with R. M. Chen and M. H. Wheeler),

submitted [arXiv].*Global bifurcation for monotone fronts of elliptic equations*, (with R. M. Chen and M. H. Wheeler),

to appear in J. Eur. Math. Soc. [arXiv].*Rigidity of three-dimensional internal waves with constant vorticity*, (with R. M. Chen, L. Fan, and M. H. Wheeler),

J. Math. Fluid Mech.,**25**:71 (2023) [article, arXiv].*Smooth stationary water waves with exponentially localized vorticity*, (with M. Ehrnström and C. Zeng),

J. Eur. Math. Soc.,**25**(3) (2023), 1045–1090 [article, arXiv].*Orbital stability of internal waves*, (with R. M. Chen),

Comm. Math. Phys.,**391**(2022), 1091–1141 [article, arXiv].*Traveling water waves — The ebb and flow of two centuries*, (with S. V. Haziot, V. M. Hur, W. Strauss, J. F. Toland, E. Wahlén, and M. H. Wheeler),

Q. Appl. Math,**80**(2) (2022), 317–401 [article, arXiv].*Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics*, (with R. M. Chen and M. H. Wheeler),

Nonlinearity,**35**(4) (2022), 1927–1985 [article, arXiv].*Global bifurcation of anti-plane shear fronts*, (with R. M. Chen and M. H. Wheeler),

J. Nonlinear Sci.,**31**(28) (2021) [article, arXiv].*Large-amplitude internal fronts in two-fluid systems*, (with R. M. Chen and M. H. Wheeler),

Comptes Rendus. Mathématique,**358**(9–10) (2020), 1073–1083 [article, arXiv].*On the stability of solitary water waves with a point vortex*, (with K. Varholm and E. Wahlén),

Comm. Pure Appl. Math.,**73**(12) (2020), 2634–2684. [article, arXiv].*Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity*, (with R. M. Chen and M. H. Wheeler),

Arch. Rational Mech. Anal.,**234**(2) (2019), 595–633 [article, arXiv].*Solitary water waves with discontinuous vorticity*, (with A. Akers),

J. Math. Pures Appl.,**124**(2019), 220–272 [article, arXiv].*Existence and qualitative theory for stratified solitary water waves*, (with R. M. Chen and M. H. Wheeler),

Ann. Inst. H. Poincaré Anal. Non Linéaire,**25**(2) (2018), 517–576 [article, arXiv].*Unique determination of stratified steady water waves from pressure*, (with R. M. Chen),

J. Differential Equations,**264**(1) (2018), 115–133 [article, arXiv].*Pressure transfer functions for interfacial fluid problems*, (with R. M. Chen and V. M. Hur),

J. Math. Fluid Mech.,**19**(1) (2017), 59–76 [article, arXiv].*On the wind generation of water waves*, (with O. Bühler, J. Shatah, and C. Zeng),

Arch. Rational Mech. Anal.,**222**(2) (2016), 827–878 [article, arXiv].*On the existence and qualitative theory for stratified solitary water waves*, (with R. M. Chen and M. H. Wheeler),

C. R. Acad. Sci. Paris, Ser. I,**354**(6) (2016), 601–605 [article].*Continuous dependence on the density for stratified steady water waves*, (with R. M. Chen),

Arch. Rational Mech. Anal.,**219**(2) (2016), 741–792 [article, arXiv].*Nonlinear resonances with a potential: multilinear estimates and an application to NLS*(with P. Germain and Z. Hani),

Internat. Math. Res. Notices,**2015**(18) (2015), 8484–8544 [article, arXiv].*Steady stratified periodic gravity waves with surface tension I: Local bifurcation*,

Discrete Cont. Dyn. Syst. Ser. A,**8**(2014), 3287–3315 [article].*Steady stratified periodic gravity waves with surface tension II: Global bifurcation*,

Discrete Cont. Dyn. Syst. Ser. A,**8**(2014), 3241–3285 [article].*Travelling water waves with compactly supported vorticity*, (with J. Shatah and C. Zeng),

Nonlinearity,**26**(2013), 1529–1564 [article, arXiv].*Steady water waves in the presence of wind*, (with O. Bühler and J. Shatah),

SIAM J. Math. Anal.,**45**(2013), 2182–2227 [article, arXiv].*Some criteria for the symmetry of stratified water waves*,

Wave Motion,**46**(2009), 350–362 [article, arXiv].*Stratified steady periodic water waves*,

SIAM J. Math. Anal.,**41**(2009), 1054–1105 [article, arXiv].

In Spring 2024, I am teaching MATH 3000W: Introduction to Advanced Mathematics and MATH 8446: Partial Differential Equations II. My office hours are WTh 3-4PM and by appointment.

I am currently looking to recruit a PhD and/or MS student to work on projects related to meteotsunamis and the existence/stability of hollow vortices.

My past students are:

- Thomas Hogancamp (PhD, 2023)
- Daniel Sinambela (PhD, 2022)
- Sebastian Henderson (MS, 2022)
- Hung Le (PhD, 2019)
- Max Highsmith (MS, 2018)
- Adelaide Akers (PhD, 2017)
- Jessie Bleile (MS, 2016)
- Evan Datz (MS, 2016).

I am the Math Competition Advisor at MU. If you are an undergraduate at MU and are interested in taking the Putnam exam, please contact me.

I was a faculty mentor to three students participating in the 2012 Summer Undergraduate Research Experience (S.U.R.E.) program at Courant. You can see their report here.