About

Ian Tice is a professor at Carnegie Mellon University in the Department of Mathematical Sciences. His research focuses on the analysis of fluid equations, free boundary problems, and related mathematical topics such as surfactants, fluid droplets, and generalized bending energies. He has supervised numerous Ph.D. and master's students working on topics including the analysis of fluid equations, free boundary problems with generalized bending energies, stability of micropolar fluids, Schauder estimates for elliptic systems, regulated integrals, anisotropic Sobolev spaces, and viscous Faraday wave problems. His postdoctoral collaborators have worked on surfactants and fluid droplets, and his students have gone on to pursue advanced research careers at institutions such as the University of Wisconsin-Madison, University of Illinois at Urbana-Champaign, University of Chicago, Princeton University, and others. Many of his students and researchers have been supported by National Science Foundation grants and fellowships, reflecting the impact and recognition of his research program in mathematical fluid dynamics and analysis.

Research topics

• Mathematical analysis
• Mechanics
• Mathematics
• Classical mechanics
• Computer Science
• Physics
• Geometry
• Thermodynamics

Selected publications

• The Traveling Wave Problem for the Shallow Water Equations: Well-posedness and the Limits of Vanishing Viscosity and Surface Tension

  Communications in Mathematical Physics · 2026-04-04

  preprintOpen accessSenior author
  PublisherOA PDFDOI

• Traveling wave solutions to a general incompressible Navier-Stokes-Fourier system with free boundary

  ArXiv.org · 2026-03-22

  articleOpen accessSenior author

  We study traveling wave solutions to the free boundary problem associated to a generalized Navier-Stokes Fourier system, which models a viscous, incompressible, heat-conducting fluid. The fluid is assumed to occupy a horizontally infinite strip-like domain with flat rigid bottom and moving upper surface. The fluid is acted upon by gravity as well as external sources of bulk force and boundary stress and an external heat source. Additionally, we allow for temperature-dependent viscosity and capillary coefficients, the latter of which gives rise to Marangoni stresses on the free surface. We develop a small data well-posedness theory in Sobolev spaces that shows that if the sources of force, stress, and heat are small, then there exists a unique solution depending continuously on these data.

  PublisherOA PDF

• Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations

  ArXiv.org · 2025-05-30

  preprintOpen accessSenior author

  We give the first mathematical construction of two-dimensional traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations for a single finite depth layer of constant density fluid. Our construction is based on a rigorous justification of the formal shallow water limit, which postulates that in a certain scaling regime the full free boundary traveling Navier-Stokes system of PDEs reduces to a governing system of ODEs. We find heteroclinic orbits solving these ODEs and, through a delicate fixed point argument employing the Stokes problem in thin domains and a nonautonomous orbital perturbation theory, use these ODE solutions as the germs from which we build bore PDE solutions for sufficiently shallow layers.

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• Stationary wave solutions to two dimensional viscous shallow water equations: theory of small and large solutions

  arXiv (Cornell University) · 2025-02-17

  preprintOpen accessSenior author

  We study a system of forced viscous shallow water equations with nontrivial bathymetry in two spatial dimensions. We develop a well-posedness theory for small but arbitrary forcing data, as well as for a fixed data profile but large amplitude. In the latter case, solutions may actually fail to exist for large amplitude, but in this case we prove that one of three physically meaningful breakdown scenarios occurs. Through the use of implicit function theorem techniques and a priori estimates, we construct both spatially periodic and solitary (non-periodic but spatially localized) solutions. The solitary case is substantially more complicated, requiring a delicate analysis in weighted Sobolev spaces. To the best of our knowledge, these results constitute the first general construction of stationary wave solutions, large or otherwise, to the viscous shallow water equations and the first general analysis of large solitary wave solutions to any viscous free boundary fluid model.

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• Traveling wave solutions to the free boundary incompressible Navier-Stokes equations with Navier boundary conditions

  Journal of Differential Equations · 2024-08-14 · 3 citations

  articleOpen accessSenior author
  PublisherDOI

• Traveling Wave Solutions to the One-Phase Muskat Problem: Existence and Stability

  Archive for Rational Mechanics and Analysis · 2024-01-18 · 5 citations

  articleSenior author
  PublisherDOI

• Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

  Journal of Functional Analysis · 2024-08-13 · 4 citations

  articleSenior author
  PublisherDOI

• On a scale of anisotropic Sobolev spaces

  Communications on Pure &amp Applied Analysis · 2024-01-01

  articleOpen accessSenior author

  We introduce a scale of anisotropic Sobolev spaces defined through a three-parameter family of Fourier multipliers and study their functional analytic properties. These spaces arise naturally in PDE when studying traveling wave solutions, and we give some simple applications of the spaces in this direction.

  PublisherDOI

• Anisotropic micropolar fluids subject to a uniform microtorque: the stable case

  Analysis & PDE · 2024-02-05 · 1 citations

  articleOpen accessSenior author

  We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain.Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium.We prove that when the microstructure is inertially oblate (i.e., pancake-like) this equilibrium is nonlinearly asymptotically stable.Our proof employs a nonlinear energy method built from the natural energy dissipation structure of the problem.Numerous difficulties arise due to the dissipative-conservative structure of the problem.Indeed, the dissipation fails to be coercive over the energy, which itself is weakly coupled in the sense that, while it provides estimates for the fluid velocity and microstructure angular velocity, it only provides control of two of the six components of the microinertia tensor.To overcome these problems, our method relies on a delicate combination of two distinct tiers of energy-dissipation estimates, together with transport-like advection-rotation estimates for the microinertia.When combined with a quantitative rigidity result for the microinertia, these allow us to deduce the existence of global-in-time decaying solutions near equilibrium.1. Introduction 42 2. Strategy and difficulties 49 3. Notation 61 4. A priori estimates 63 5. Local well-posedness 93 6. Continuation argument 110 7. Global well-posedness and decay 119 Appendix A. Identities involving the microinertia 126 Appendix B. Analytical results 128 References 130This paper, together with the companion paper [Remond-Tiedrez and Tice 2021], provides a sharp nonlinear stability criterion for an anisotropic micropolar fluid subject to a uniform microtorque.The companion paper is concerned with the unstable regime; we tackle the stable regime here.Note to the reader: The introduction of Section 1 serves as a "shortest path" to the main result recorded in Theorem 1.2, providing the necessary physical and mathematical background to appropriately state the main result.For a more detailed discussion of the problem and the strategy employed to prove nonlinear stability, we direct the reader's attention to Section 2.

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• Global Well-Posedness of Contact Lines: 2D Navier-Stokes Flow

  arXiv (Cornell University) · 2024-07-25

  preprintOpen access

  Based on the global a priori estimates in [Guo-Tice, J. Eur. Math. Soc. (2024)], we establish the well-posedness of a viscous fluid model satisfying the dynamic law for the contact line \begin{equation*} \mathscr{W}(\p_tζ(\pm\ell,t))=[\![γ]\!]\mpσ\frac{\p_1ζ}{(1+|\p_1ζ|^2)^{1/2}}(\pm\ell,t) \end{equation*} in 2D domain, where $ζ(x_1,t)$ is a free surface with two contact points $ζ(\pm\ell,t)$, $[\![γ]\!]$ and $σ$ are constants characterizing the solid-fluid-gas free energy, and the increasing $\mathscr{W}$ is the contact point velocity response function. Motivated by the energy-dissipation structure, our construction relies on the construction of a pressureless weak solution for the coupled velocity and free interface for the linearized problems, via a Galerkin approximation with a time-dependent basis and an artificial regularization for the capillary operator.

  PublisherOA PDFDOI

Recent grants

• Analysis of Free Boundaries: Contact Lines and Viscous Traveling Waves

  NSF · $369k · 2022–2026

• CAREER: Analysis of Partial Differential Equations in Moving Interface Problems

  NSF · $420k · 2017–2023

Frequent coauthors

• Yan Guo

  24 shared

• Sylvia Serfaty

  11 shared

• Juhi Jang

  University of Southern California

  11 shared

• Stéphane Serfaty

  CY Cergy Paris Université

  10 shared

• Yanjin Wang

  Henan University of Economic and Law

  9 shared

• Noah Stevenson

  8 shared

• Dongfen Bian

  8 shared

• Antoine Remond-Tiedrez

  University of Wisconsin–Madison

  6 shared

Labs

• Ian Tice's LabPI

  Research in fluid equations, free boundary problems, and related topics.

Education

• Ph.D., Mathematics

  Courant Institute, New York University

Awards & honors

• Julius Ashkin Teaching Award, 2019
• NSF CAREER Award