Online North East PDE and Analysis Seminar (ONEPAS)

  • Tuesdays, 3:00-4:00 PM, Eastern Time. Zoom Meeting ID: 586-919-392, Passcode: 889022
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  • Track ONEPAS on researchseminars.org.
  • Watch previous talks on the ONEPAS YouTube channel.
  • ONEPAS is now running in a thematic session format. The thematic sessions focus on a topic in analysis and PDE, broadly construed, with a series of 3-7 weekly one hour talks aimed at non-experts. The goal is to introduce the topic, give insight to its origin and applications, highlight some important achievements, and provide a glimpse of new directions and open problems. We believe that adding some historical perspective and focusing on the long view can add some depth in a way that is complementary to what is gained from regular seminars. In practice, each session is curated by a prominent senior mathematician. Suggestions and spontaneous applications are welcome!

Organizers


Upcoming talks

ONEPAS Thematic Series - Steady water waves: the ebb and flow of the past two centuries

09-15-20, 3:00 PM (ET): Walter Strauss, Brown University
  • Title: Introduction to steady water waves
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract: This is a very basic introduction. No previous knowledge of water waves is required. I will mention the high points of the history of water wave theory. Then the fundamental equations inside the water and on the free boundary will be discussed. Finally, many important directions of current research will be briefly outlined.
  • YouTube link
  • Suggested reading: W. Strauss. Steady water waves. Bull. Amer. Math. Soc. 47 (2010), no. 4, 671-694.
09-22-20, 3:00 PM (ET): John Toland, University of Bath
  • Title: Variational aspects of steady irrotational water wave theory
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract: Among the many modern approaches to abstract nonlinear problems, those based on the implicit function theorem, real-analytic function theory, Nash-Moser theory and topological degree theory have made significant contributions to water-wave theory in recent years. However, the same cannot be said of variational methods (min/max, mountain-pass, Morse index, Lyusternik-Schnirelman genus etc) even though, when the viscosity of water is ignored and the flow is assumed to be irrotational, there are several attractive ways to formulate the equations of wave motion variationally. On the 100th anniversary of the first proof that the equations of motion have non-zero, small-amplitude solutions, this talk will briefly survey these issues and advocate variational methods for analyzing water waves that are \(2\pi\)-periodic in space. In the suggested reading are some publications which, with the references therein, are related to these topics; those marked \(\ast\) are useful places to start.
  • YouTube link
  • Suggested reading: Click here for the pdf.
  • Talk slides.
09-29-20, 3:00 PM (ET): Susanna Haziot, University of Vienna
  • Title: Rotational water waves
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract: One significant difficulty of working with water waves is that the boundary of the fluid domain itself is an unknown. I will begin with a brief presentation of the steady water wave problem for waves with vorticity. Subsequently, I will review some existence results as well as present recent research which involve different methods for transforming the fluid domain into a fixed domain.
  • YouTube link
  • Suggested reading: Adrian Constantin. Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
10-13-20, 3:00 PM (ET): Vera Mikyoung Hur, University of Illinois at Urbana-Champaign
  • Title: Stokes waves in constant vorticity flows
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract: I will discuss recent progress on the numerical computation of Stokes waves in constant vorticity flows. Based on a Babenko-kind equation, our result improves those in the 1980s by Simmen and Saffman, Teles da Silva and Peregrine. Notably, it reveals a plethora of new solutions: Crapper's exact solution (even though there is no surface tension), a fluid disk in rigid body rotation, etc. I will also discuss the effects of vorticity on the extreme wave, particularly, the maximum slope for an almost extreme wave. I will discuss some open problems, both analytical and numerical.
  • YouTube link
  • Suggested reading: S.A. Dyachenko and V.M. Hur. Stokes Waves in a Constant Vorticity Flow. Nonlinear Water Waves. Tutorials, Schools, and Workshops in the Mathematical Sciences. Springer International Publishing, Cham, 2019.
10-20-20, 3:00 PM (ET): Miles Wheeler, University of Bath
  • Title: Solitary waves and fronts
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract: I will give a general introduction to the theory of solitary water waves, that is traveling waves whose surfaces converge to some asymptotic height at infinity. In many respects, the theory for solitary waves is more difficult and more subtle than that for periodic waves. Yet in other ways the problem is much simpler, and indeed many results for solitary waves are stronger than their periodic counterparts.

    Beginning with the linear theory, or perhaps more accurately the lack of a linear theory, I will explain how small-amplitude waves can be rigorously constructed via a center manifold reduction. Next I will collect a series of results which together guarantee that any solitary wave, regardless of amplitude, is symmetric and decreasing about a central crest and travels at a “supercritical” speed \(c>\sqrt{gd}\). Finally, I will explain how the significant obstacles to applying global bifurcation techniques can be overcome by taking advantage of the above properties together with the nonexistence of front-type solutions. This approach is surprisingly robust, and has recently been generalized to apply to front-type solutions in addition to solitary waves.
  • Suggested reading:
10-27-20, 3:00 PM (ET): Samuel Walsh, University of Missouri
  • Title: Water waves with density stratification or localized vorticity
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract: This talk will serve as a general introduction to two areas of active research in water waves. In the ocean, the presence of salt and temperature gradients can lead to substantial stratification of the density. This phenomenon is well-known to have significant physical implications. Indeed, it makes possible enormous ``internal waves'' that can dwarf even the largest waves seen on the surface. We will present an overview of the mathematical work on this subject, focusing primarily on recent results regarding the existence of large-amplitude solitary stratified waves.

    The second part of the talk will discuss waves with localized distributions of vorticity. These include water waves with submerged point vortices, dipoles, vortex patches, and those exhibiting a vortex spike.

    This is joint work with Robin Ming Chen, Mats Ehrnström, Jalal Shatah, Kristoffer Varholm, Erik Wahlén, Miles H. Wheeler, and Chongchun Zeng.
11-03-20, 3:00 PM (ET): Erik Wahlén, Lund University
  • Title:
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract:

ONEPAS Thematic Series - Dynamical systems

11-10-20, 3:00 PM (ET): Rafael de la Llave, Georgia Tech
  • Title:
  • Zoom Meeting ID: 586-919-392 , Passcode: 889022
  • Abstract:

Previous talks

Spring and Summer 2020

08-04-20, 3:30 PM (ET): Stanley Snelson, Florida Tech
  • Title: Low-regularity well-posedness for the Landau equation with large initial data
  • Abstract: The Landau equation is a kinetic evolution model in plasma physics. Mathematically, it features the interaction of transport with nonlocal diffusion. For initial data that is close to an equilibrium state, solutions are known to exist for all time and converge to equilibrium, but global existence for general initial data is a challenging open problem. In this talk, we consider the local existence question, and present a recent result that establishes a short-time solution for any bounded, measurable initial data with polynomial decay of order 5 in the velocity variable. We will also discuss why uniqueness is a delicate issue in this regime--in fact, our proof of uniqueness requires the additional assumption that the initial data be Holder continuous. This talk is based on joint work with Christopher Henderson (Arizona) and Andrei Tarfulea (Louisiana State).
07-28-20, 3:30 PM (ET): Marius Beceanu, SUNY Albany
  • Title: Evolution equations with random time-dependent potential
  • Abstract: We present some results about the Schroedinger equation with time-dependent random potential and nonlinearity. This work is in collaboration with Avy Soffer.
07-21-20, 3:30 PM (ET): Stephen Cameron, Courant
  • Title: Global Existence for the 3D Muskat problem
  • Abstract: The Muskat problem studies the evolution of the interface between two incompressible, immiscible fluids in a porous media. In the case that the fluids have equal viscosity and the interface is graphical, this system reduces to a single nonlinear, nonlocal parabolic equation for the parametrization. Even in this stable regime, wave turning can occur leading to finite time blowup for the slope of the interface. Before that blowup though, we prove that an imperfect comparison principle still holds. Using this, we are able to show that solutions exist for all time so long as either the initial slope is not too large, or the slope stays bounded for a sufficiently long time.
07-14-20, 3:30 PM (ET): Jaemin Park, Georgia Tech
  • Title: On radial symmetry of stationary solutions to active scalar equations
  • Abstract: In this talk, we will study radial symmetry of stationary/ uniformly rotating solutions for 2D Euler equation under the assumption that the vorticity is compactly supported. Our main results are the following: (1) On the one hand, we are able to prove that if the vorticity non-negative, then it has to be radially symmetric up to a translation. (2) On the other hand, we can construct a non-radial stationary solution by allowing the vorticity to change sign. We have also obtained some symmetry results for uniformly-rotating solutions for 2D Euler equation, as well as stationary/rotating solutions for the SQG equation. The symmetry results are mainly obtained by calculus of variations and elliptic equation techniques and the construction of non-radial solution is obtained from the bifurcation theory. This is a joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.
07-07-20, 3:30 PM (ET): Casey Rodriguez, MIT
  • Title: The radiative uniqueness conjecture for bubbling wave maps
  • Abstract: We will discuss the finite time breakdown of solutions to a canonical example of a geometric wave equation, energy critical wave maps. Breakthrough works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz and Raphaël-Rodnianski produced examples of wave maps that develop singularities in finite time. These solutions break down by concentrating energy at a point in space (via bubbling a harmonic map) but have a regular limit, away from the singular point, as time approaches the final time of existence. The regular limit is referred to as the radiation. This mechanism of breakdown occurs in many other PDE including energy critical wave equations, Schrödinger maps and Yang-Mills equations. A basic question is the following: can we give a precise description of all bubbling singularities for wave maps with the goal of finding the natural unique continuation of such solutions past the singularity? In this talk, we will discuss recent work (joint with J. Jendrej and A. Lawrie) which is the first to directly and explicitly connect the radiative component to the bubbling dynamics by constructing and classifying bubbling solutions with a simple form of prescribed radiation. Our results serve as an important first step in formulating and proving the following Radiative Uniqueness Conjecture for a large class of wave maps: every bubbling solution is uniquely characterized by it’s radiation, and thus, every bubbling solution can be uniquely continued past blow-up time while conserving energy.
06-30-20, 3:30 PM (ET): Evan Miller, McMaster University
  • Title: The self-amplification of strain as a possible mechanism for finite-time blowup for the Navier-Stokes equation
  • Abstract: In this talk, I will discuss the role of the strain matrix in the Navier-Stokes regularity problem. First, I will go over an identity for enstrophy growth in terms of the determinant of strain, and a regularity criterion in terms of \(\lambda_2^+\), the positive part of the middle eigenvalue of the strain matrix. Then I will prove the existence of finite-time blowup solutions for a toy model of the the strain equation. By treating the full Navier-Stokes equation as a perturbation of this model equation, I will show how we can obtain a new conditional blowup result for solutions of the full Navier-Stokes equation.
06-23-20, 3:30 PM (ET): Yilun Wu, University of Oklahoma
  • Title: Complete integrability of the Intermediate Long Wave equation
  • Abstract: The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. It models a regime intermediate between the KdV equation and the Benjamin-Ono equation. Kodama, Satsuma and Ablowitz discovered the formal complete integrability of ILW and formulated the direct and inverse scattering transform solution. However, it is unclear whether the formal scattering problems actually admit well-behaved solutions. No rigorous analysis of this method has ever been done. In this talk, I present some recent progress on the direct problem with small data. The solution is related to the theory of analytic functions on a strip. This is joint work with Peter Perry and Joel Klipfel.
06-16-20, 3:30 PM (ET): Alex Blumenthal, University of Maryland
  • Title: Chaos, scalar mixing, and passive scalar turbulence for models in fluid mechanics
  • Abstract: In models of fluid mechanics, Lagrangian flow \(\phi^t\) on the fluid domain describes the motion of a passive particle advected by the fluid. It is anticipated that typically, Lagrangian flow \(\phi^t\) is chaotic in the sense of (1) sensitivity with respect to initial conditions and (2) fast mixing of passive scalars (equivalently \(H^{-1}\) decay for passive scalars). I will present joint work with Jacob Bedrossian (U Maryland) and Sam Punshon-Smith (Brown U) in which we rigorously verify these chaotic properties for various incompressible and stochastically forced fluid models on the periodic box, including stochastic 2D Navier-Stokes and hyperviscous 3D Navier-Stokes. I will also present our recent application of these result to the study of passive scalar turbulence in the Batchelor regime, i.e., the steady state of passive scalars in a fluid (at fixed viscosity) attained as molecular diffusivity goes to 0. In this setting, we are able to prove Batchelor's inverse power law for the power spectrum, the passive scalar analogue of Kolmogorov's \(-4/3\) law for the power spectrum in the inertial range of a turbulent 3D fluid.
06-09-20, 3:30 PM (ET): Zhiyuan Zhang, Courant
  • Title: Linear stability and magnetic confinement of the relativistic Vlasov-Maxwell system
  • Abstract: The talk consists of two parts. In the first part, we consider the relativistic Vlasov-Maxwell system on a general axisymmetric spatial domain with perfect conducting boundary which reflects particles specularly, and look at a certain class of equilibria, assuming axisymmetry in the problem. We prove a sharp criterion of spectral stability under these settings and then provide several explicit families of stable/unstable equilibria. In the second part, we verify, for the 1.5D relativistic Vlasov-Maxwell system on an interval \((0, 1)\), that for a plasma in a spatial domain with a boundary, the specular reflection effect of the boundary can be approximated by a large magnetic confinement field in the near-boundary region.
06-02-20, 3:30 PM (ET): Thomas Beck, University of North Carolina, Chapel Hill
  • Title: Two-phase free boundary problems and the Friedland-Hayman inequality
  • Abstract: The classical Friedland-Hayman inequality provides a lower bound on the first Dirichlet eigenvalues of complementary subsets of the round sphere. In this talk, we will describe a variant of this inequality to geodesically convex subsets of the sphere with Neumann boundary conditions. As an application, we establish the Lipschitz continuity up to the boundary of the minimizers of a two-phase free boundary problem for a class of convex domains. This is joint work with David Jerison and Sarah Raynor.
05-26-20, 3:30 PM (ET): Antoine Remond-Tiedrez, University of Wisconsin, Madison
  • Title: Instability of an anisotropic micropolar fluid
  • Abstract: Many aerosols and suspensions, and more broadly fluids containing a non-trivial structure at a microscopic scale, can be described by the theory of micropolar fluids. The resulting equations couple two elements: (1) the Navier-Stokes equations which describe the macroscopic motion of the fluid and (2) evolution equations for the angular momentum and the moment of inertia associated with the microscopic structure. In this talk we will discuss the case of viscous incompressible three-dimensional micropolar fluids. We will discuss how, when subject to a fixed torque acting at the microscopic scale, the nonlinear stability of the unique equilibrium of this system depends on the shape of the microstructure.
05-19-20, 3:30 PM (ET): Yu Deng, University of Southern California
  • Title: Invariant Gibbs measures and global strong solutions for 2D nonlinear Schrödinger equations
  • Abstract: We solve the long-standing problem of proving almost sure global well-posedness (i.e. existence with uniqueness) for the nonlinear Schrödinger equation (NLS) on \(\mathbb{T}^2\) on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is done by the new method of random averaging operator, which precisely captures the implicit randomness structure of the high-low interactions. This is joint work with Andrea R. Nahmod (UMass Amherst) and Haitian Yue (USC).
05-12-20, 3:30 PM (ET): Logan Stokols, University of Texas at Austin
  • Title: SQG on bounded domains
  • Abstract: The surface quasi-geostrophic (SQG) equation on \(\mathbb{R}^2\) was shown in the late `00s to be well posed with smooth solutions. Recently, Constantin and Ignatova proposed a model for SQG on bounded open subsets of \(\mathbb{R}^2\), defined in terms of the Dirichlet Laplacian. This model is particularly complex because it involves a nonlocal operator on a bounded domain. We will discuss this model, including physical motivation, existence, and regularity.
05-05-20, 3:30 PM (ET): Jessica Lin, McGill University
  • Title: Homogenization of an Allen-Cahn equation in periodic media
  • Abstract: I will discuss periodic homogenization for the Allen-Cahn equation with Neumann boundary conditions. It is well-known that rescaled solutions of the (homogeneous) Allen-Cahn equation converge to generalized solutions of mean-curvature flow. Using a variational approach, I will show that under suitable hypotheses, a similar result holds true for Allen-Cahn equations with periodic reaction terms. This talk is based on joint work with Rustum Choksi, Irene Fonseca, and Raghavendra Venkatraman.
04-28-20, 3:30 PM (ET): Huy Nguyen, Brown University
  • Title: On well-posedness of the Muskat problem
  • Abstract: The Muskat problem concerns the evolution of the interface between two fluids in porous media. The fluids have different densities and different viscosities in the general case. The interface is driven by gravity and surface tension. We will present well-posedness results for the Muskat problem with and without surface tension in all subcritical Sobolev spaces, which control Holder norms of the slope of the interface. At the same level of regularity, we prove that solutions of the problem with surface tension converge to solution of the problem without surface tension in the vanishing surface tension limit. In particular, this low regularity result allows for interfaces that have unbounded curvature. If time permits, we will discuss an ongoing work on the one-fluid Muskat problem with Lipschitz surface. Joint work with F. Gancedo, P. Flynn and B. Pausader.
04-21-20, 3:30 PM (ET): Sohrab Shahshahani, University of Massachusetts Amherst
  • Title: Well-posedness for the hard phase model with free boundary
  • Abstract: The hard phase model is a relativistic fluid model in which the fluid is assumed to be irrotational and the sound speed is assumed to be one. During gravitational collapse of the degenerate core of a massive star, the mass-energy density exceeds the nuclear saturation density and the sound speed approaches the speed of light. The hard phase is an idealized model for this physical situation. In this talk, I will discuss recent work on the well-posedness of the hard phase model with vacuum free boundary in the Minkowski background, and extensions to the more general case of barotropic fluids with free boundary. This is joint work with Shuang Miao and Sijue Wu.
04-14-20, 3:30 PM (ET): Zaher Hani, University of Michigan
  • Title: On the rigorous derivation of the wave kinetic equations
  • Abstract: Wave turbulence theory conjectures that the behavior of “generic" solutions of nonlinear dispersive equations is governed (at least over certain long timescales) by the so-called wave kinetic equation (WKE). This approximation is supposed to hold in the limit when the size L of the domain goes to infinity, and the strength \(\alpha\) of the nonlinearity goes to 0. We will discuss some recent progress towards settling this conjecture, focusing on a recent joint work with Yu Deng (USC), in which we show that the answer seems to depend on the “scaling law” with which the limit is taken. More precisely, we identify two favorable scaling laws for which we justify rigorously this kinetic picture for very large times that are arbitrarily close to the kinetic time scale (i.e. within \(L^\epsilon\) for arbitrarily small \(\epsilon\)). These two scaling laws are similar to how the Boltzmann-Grad scaling law is imposed in the derivation of Boltzmann's equation. We also give counterexamples showing certain divergences for other scaling laws.