Brown PDE Seminar

Fridays 3:00pm- 4:00pm EDT

Meeting ID: 927 6053 7508 (password protected)

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Organizers

• Ryan Creedon
• Yan Guo

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• Upcoming Talks (Fall 2025)

• Global solutions to the one-phase Muskat problem
  
  Hyunwoo Kwon, Brown
  12-09-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• We consider the one-phase Muskat problem, which models the motion of a fluid interface in porous media governed by Darcy’s law. With surface tension, we establish the global well-posedness of small periodic initial data and show that solutions decay to zero as t→∞ . For the case of a flat bottom with finite depth D, we prove global well-posedness for small initial data in the natural critical space. This talk is based on ongoing joint projects with Hongjie Dong (surface tension case) and Benoit Pausader (finite depth case). 

• Numerical approximations to nonlinear dispersive equations, from short to long times
  
  Yvonne Alama Bronsard, MIT
  19-09-2025 - 3:00PM (EST), in-person at 182 George St, room 110 (no Zoom available)
  
• Abstract
• The first part of this talk deals with the numerical approximation to nonlinear dispersive equations, such as the prototypical nonlinear Schrödinger or Korteweg-de Vries equations. We introduce integration techniques allowing for the construction of schemes which perform well both in smooth and non-smooth settings. Higher order extensions will be presented, following techniques based on decorated trees series inspired by singular stochastic PDEs via the theory of regularity structures. In the second part, we introduce a new approach for designing and analyzing schemes for some nonlinear and nonlocal integrable PDEs, including the Benjamin-Ono equation. This work is based upon recent theoretical breakthroughs lead by Patrick Gérard and his collaborators, on explicit formulas for nonlinear integrable equations. It opens the way to numerical approximations which are far more accurate and efficient for simulating these integrable PDEs, from short up to long times.

• Stability of geophysical fluids
  
  Haram Ko, Brown
  26-09-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
• Abstract
• Although Euler and Navier-Stokes equations are the two most popular choices for modeling the dynamics of fluids, geophysical fluids tend to be influenced by additional structures. The rotation of the planet and the stratification of the fluid are considered the two most important factors among geophysicists, which often comes with the dispersion mechanism. I will discuss several different scenarios where we can use this mechanism to prove stability of the system, with the emphasis on the role of natural parameters of the systems. In particular, while giving a detailed linear analysis, I will present a version of stationary phase estimate that is general enough to be adapted to other dispersive PDEs. The talk will include joint works with Catalina Jurja, Benoit Pausader, Takada Ryo, and Klaus Widmayer.

• Non-uniqueness for the Navier-Stokes equations from critical data
  
  Stan Palasek, IAS
  03-10-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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  Zhongtian Hu, Princeton
  10-10-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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  Ian Tice, CMU
  17-10-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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• Higher order hamiltonians: The biharmonic NLS equations
  
  Shijun Zheng, Georgia Southern University
  24-10-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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  Catalina Jurga, UZH
  31-10-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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• Spatially quasi-periodic water waves
  
  Xinyu Zhao, NJIT
  07-11-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
• Abstract
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  Joe Kraisler, Amherst College
  14-11-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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  Gonzalo Cao-Labora, Courant
  21-11-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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  Martin Dindos, University of Edinburgh
  21-11-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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  Bekarys Bekmaganbetov, Brown
  12-12-2025 - 3:00PM (EST), in-person at 179 Hope St, room 108, and via Zoom
  
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• TBA

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• Past Talks (Spring 2025)

• The Modulational Instability Spectrum of Periodic Traveling-Waves
  
  Ryan Creedon, Brown
  25-04-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• The modulational instability is among the most recognized of instabilities that can occur in periodic traveling waves of dispersive Hamiltonian systems. One of the hallmark features of this instability is a figure-eight curve of eigenvalues centered at the origin of the stability spectrum of the periodic waves. In this talk, we illustrate a perturbation method that extracts important asymptotic information from these unstable eigenvalues, including an approximate parameterization for the figure-eight curve as well as the leading-order behavior of the most unstable eigenvalue. The method applies to a broad class of PDEs, encompassing, for instance, the Kawahara, Whitham, and Intermediate Long-Wave equations. As a result, a general index can be derived to detect modulational instability, and leading-order asymptotic approximations for the resulting growth rates can be obtained. Both the index and growth rates are expressed only in terms of the linear dispersion relation of these models, making results of this method both general and easy to use. Under appropriate restrictions, the method can also be made rigorous.
  
• An action approach to nodal and least energy normalized solutions for nonlinear Schrödinger equations
  
  Damien Galant, UMONS/UPHF
  18-04-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• In this talk, I will present two notions of stationary state solutions to the nonlinear Schrödinger equation: those with a fixed frequency, corresponding to critical points of the action functional, and those with fixed mass (normalized solutions), corresponding to critical points of the energy functional constrained on a L²-sphere. In general, it is somewhat easier to treat the problem with a fixed frequency. In particular, in this case one is able to find least action solutions and least action nodal solutions for all Sobolev-subcritical exponents in the nonlinearity. Regarding the fixed mass solutions, a new critical exponent appears. Finding normalized solutions in the "mass-supercritical" regime is usually a difficult problem, explored since pioneering work by Jeanjean in the late 1990s and often imposing geometrical conditions on the domain on which the equation is set. We will present a new method which allows to characterize the masses of the least action solutions and the least action normalized solutions, therefore building a bridge between the study of solutions having a fixed frequency and those having a fixed mass. As we will see, we will do so in a new "variational" fashion since one does not expect in general to have continuous branches of solutions for all values of the frequency. This is joint work with Colette De Coster (CERAMATHS/DMATHS, UPHF and INSA HdF, Valenciennes, France), Simone Dovetta (Politecnico di Torino, Italy) and Enrico Serra (Politecnico di Torino).
  
• On Vortices of Ginzburg-Landau evolutions
  
  Fabio Pusateri, University of Toronto
  11-04-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• We present some recent results on the (linear) stability of vortices in relativistic Ginzburg-Landau equations.
  
• Evolution of singular coherent structures in incompressible fluids
  
  Francisco Gancedo, University of Seville
  04-04-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• In this talk, we discuss two new results on the evolution of singular coherent structures in incompressible fluids. For the 3D Euler equations, we establish the existence of weak dissipative solutions with initial vorticity given by a vortex filament concentrated on a circle. For the 2D one-phase Muskat problem with contact points on vertical impermeable walls, we derive global-in-time energy estimates for initial data close to the stationary state.
  
• Exact boundary controllability for the ideal magneto-hydrodynamic equations
  
  Igor Kukavica, USC
  21-03-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• We consider the three-dimensional ideal MHD system on a domain in with a controllable part of the boundary where we prescribe the boundary data. The basic question of boundary controllability is whether, given two states, one can by means of the control on the boundary drive one state to another. We will review the existing literature on this problem and provide a positive result for domains with only Sobolev regularity. The results are based on works with Matthew Novack, Wojciech Ozanski, and Vlad Vicol.
  
• Global stability of Minkowski spacetime with minimal and borderline decay
  
  Dawei Shen, Columbia
  14-03-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• The global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou-Klainerman in 1993. In 2007, Bieri extended the result of Christodoulou-Klainerman under lower decay and regularity assumptions on the initial data. In this talk, I will introduce my recent work, which extends the result of Bieri to minimal decay assumptions. Moreover, I will also discuss another recent work, which proves that the exterior stability of Minkowski holds with decay which is borderline.
  
• From instability to singularity formation in incompressible fluids
  
  Federico Pasqualotto, UCSD
  07-03-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• In this talk, I will first review the singularity formation problem in incompressible fluid dynamics, describing how particle transport poses the main challenge in constructing blow-up solutions for incompressible fluids. I will then outline a new mechanism that allows us to overcome the effects of particle transport, leveraging the instability seen in the classical Taylor–Couette experiment. Using this mechanism, we construct the first swirl-driven singularity for the incompressible Euler equations in R^3. I will describe the core ideas of our proof, based on the analysis of the Boussinesq system. This is joint work with Tarek Elgindi (Duke University).
  
• Singularities in General Relativity and BKL bounces
  
  Warren Li, Princeton
  28-02-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• In work spanning the late 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general ansatz for spacetimes solving the Einstein vacuum equations near singularities. They suggest that the spacetime dynamics at different spatial points on the singularity decouple and are well-approximated by a system of autonomous nonlinear ODEs, whose orbits are governed by a chaotic cascade of "BKL bounces”. In this talk, we present recent work verifying BKL's heuristics in a large class of symmetric, but spatially inhomogeneous, spacetimes, i.e. we prove decoupling even in the presence of (up to one) BKL bounce.
  
• Long time existence of space periodic water waves
  
  Massimiliano Berti, SISSA
  21-02-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• This talk surveys recent results on long time existence of space-periodic solutions for water wave equations with constant vorticity, subject to gravity and possibly capillary forces. The governing equations inherit from the underlining Euler equations a Hamiltonian and reversible structure. These algebraic properties -which in finite dimensional dynamical systems theory play a crucial role, e.g. for normal forms, KAM and Nekhoroshev theorems- had never been effectively used since recent times for the water waves equations mainly for the difficulty of the quasi-linearity of the vector field and its non local nature. On the other hand for space periodic water waves no dispersive effects of the flow are available and the Hamiltonian and reversible properties govern in a fundamental role the long time water waves dynamics. A key challenge lies in the fact that many classical techniques for proving local well-posedness disrupt this crucial Hamiltonian structure.
  
• The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients
  
  Seick Kim, Yonsei University
  14-02-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• In this talk, we discuss the Dirichlet problem for a non-divergence form elliptic operator L in a bounded domain. Under certain conditions on the coefficients of L, we establish the existence of a unique Green’s function in a ball and derive two-sided pointwise estimates for it. Using these results, we show that regular points for L coincide with those for the Laplace operator, as characterized by the Wiener test. This equivalence ensures the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Additionally, we construct the Green’s function for L in regular domains and establish pointwise bounds for it.
  
• Negative regularity mixing of passive scalars in stochastic fluid mechanics
  
  Patrick Flynn, UCLA
  07-02-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• Consider a passive scalar advected by a random vector field on a compact manifold, such as the solution to the 2D stochastic Navier-Stokes equation on a periodic box. In this talk, I discuss my work with J. Bedrossian and S. Punshon-Smith, where we show that if the passive scalar is initially in some negative regularity Sobolev space, then it will decay exponentially in the same space (in expectation). We prove this result using techniques from dynamical systems theory and semiclassical analysis. Going forward, we hope to apply this result to the problem of turbulence.
  
• Global axisymmetric stability of rotation, uniform in viscosity, of 3d incompressible Navier-Stokes equation
  
  Haram Ko, Brown
  31-01-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• Although global solutions to Navier-Stokes-Coriolis equations, Navier-Stokes equation in the rotating background, have been obtained, previous methods were not able to achieve the same for low viscosities compared to the speed of rotation. I will present how to resolve this for small axisymmetric data, and uniformly in viscosity for fixed speed of rotation, by adapting the methods of [Guo, Pausader, Widmayer, '23] used to prove global existence for Euler-Coriolis equation which builds on the well-developed theory for small data nonlinear dispersive equations.
  
• Global Bifurcation of Surface Capillary Waves on a 2D Droplet
  
  Yilun (Allen) Wu, University of Oklahoma
  24-01-2025 - 3:00PM (EST), in-person at 170 Hope St, room 108, and via Zoom
  
• Abstract
• The construction of steady traveling waves is a classical problem in water wave theory. The well-known Stokes waves, among many other generalizations, form a global continuum of traveling waves that approach limiting singular solutions. Recently, Dyachenko et al. obtained rotational traveling waves bifurcating from a circular droplet in 2D via numerics, as attempts to model spray and white capping. In this talk, I will show a rigorous global bifurcation result constructing a curve of such solutions. The obtained solutions are steady surface capillary waves in 2D, and have $m$-fold rotational symmetry. This is joint work with Gary Moon.