Samuel Walsh is an Assistant Professor in the Department of Mathematics at the University of Missouri. He received a Ph.D. in Applied Mathematics from Brown University in 2010, under the direction of Walter Strauss. Prior to coming to MU, he held the postdoctoral position of Courant Instructor at New York University.
- 2010 Ph.D., Applied Mathematics, Brown University
- 2005 B.S., Mathematical Sciences, Carnegie Mellon University
- MATH 4100. Differential Equations
- MATH 8445. Partial Differential Equations I
Professor Walsh's research is in the area of nonlinear partial differential equations, particularly those pertaining to water waves. A large part of these efforts have been devoted to investigations of steady waves with vorticity: proving their existence in various regimes, diagnosing their stability properties, and determining their qualitative features. The overarching goal of this program is to take deep ideas from PDEs, analysis, and dynamical systems, and bring them to bear on physically important problems in fluid mechanics. For example, recent projects address: the wind-driven generation of ocean waves, traveling waves with compactly supported vorticity, and the reconstruction of waves with density stratification from deep sea pressure measurements.
Walsh is 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.
Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity, (with R. M. Chen and M. H. Wheeler),
Arch. Rational Mech. Anal., vol. 234(2) (2019), pp. 595--633.
Solitary water waves with discontinuous vorticity , (with A. Akers),
J. Math. Pures Appl., vol. 124 (2019), pp. 220--272.
Existence and qualitative theory for stratified solitary water waves, (with R. M. Chen and M. H. Wheeler),
Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 25(2) (2018), pp. 517--576.
Reconstruction of stratified steady water waves from pressure readings on the ocean bed, (with R. M. Chen),
J. Differential Equations, vol. 264(1) (2018), pp. 115--133.
On the wind generation of water waves, (with O. Bühler, J. Shatah, and C. Zeng),
Arch. Rational Mech. Anal., vol 222(2) (2016), pp. 827--878.
Continuous dependence on the density for stratified steady water waves, (with R. M. Chen),
Arch. Rational Mech. Anal., vol. 219(2) (2016).
Nonlinear resonances with a potential: multilinear estimates and an application to NLS (with P. Germain and Z. Hani),
Internat. Math. Res. Notices, vol. 2015(18) (2015).
Steady stratified periodic gravity waves with surface tension II: Global bifurcation,
Discrete Cont. Dyn. Syst. Ser. A, no. 8 (2014), pp. 3241--3285.
Steady stratified periodic gravity waves with surface tension I: Local bifurcation,
Discrete Cont. Dyn. Syst. Ser. A, no. 8 (2014), pp. 3287--3315.
Travelling water waves with compactly supported vorticity (with J. Shatah and C. Zeng),
Nonlinearity, 26 (2013), pp. 1529--1564.
Steady water waves in the presence of wind (with O. Bühler and J. Shatah),
SIAM J. Math. Anal. 45 (2013), pp.2182--2227.
Stratified steady periodic water waves,
SIAM J. Math. Anal., 41 (2009), pp. 1054--1105.