Online North East PDE and Analysis Seminar (ONEPAS)

• ONEPAS is currently on hiatus, but you can watch previous talks on the ONEPAS YouTube channel.

Organizers

• Javier Gómez-Serrano
• Benoît Pausader
• Fabio Pusateri
• Ian Tice

Previous talks

ONEPAS Thematic Series - The mathematics of thin structures

• Title: The mathematics of thin structures
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: In this talk we will give a brief survey of the variational approach to the characterization of the energy density of a thin structure, resulting from a 3D elastic material occupying a cylindrical reference configuration whose thickness converges to zero. The asymptotic behavior may fall within different frameworks, including membrane theory, plate theory, and von Kármán theory. The coupling of dimension reducation with other effects, such as fracture, debonding or delamination, micromagnetism, homogenization, pattern formation, and optimal design, will be addressed.
• YouTube link

• Title: Beyond membranes
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: In her talk, Irene Fonseca will give a picture of the current mathematical state of thin membranes. In this talk, I propose to go to higher scalings of the elastic energy as a function of the thickness of the structure. I will provide a quick overview of the hierarchy of models that can be obtained in the footstep of the seminal work of Friesecke-James-Müller. I will also discuss what I view as potential modeling issues that seem inherent to the adopted approach.
• YouTube link

• Title: Geometry and morphogenesis: problems and prospects
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: The shape of a leaf or flower in the garden, or its laboratory analogs built using swelling gels, raise several questions at the interface of biology, physics and mathematics: how might they be described, how can their shapes be predicted, and how can they eventually be controlled for function? We address aspects of this problem from a mathematical perspective, bringing together analysis, geometry, and mechanics to understand the morphogenesis of low-dimensional objects. We first provide a review of the rigorous analytical results on curvature-driven patterning of thin elastic surfaces, especially the asymptotic behaviors of the solutions as the thickness of the surfaces becomes vanishingly small and the local curvatures can become large. We then proceed with linking the classical Nash embedding problem to that of biological morphogenesis and to describe some open problems in this area.
• YouTube link

• Title: Micromagnetics of curved thin films
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: Skyrmions are a class of topologically stable solitons with quasiparticle properties: they behave like particles, but due to their collective nature, they are inherently more complex structures. In micromagnetics, magnetic skyrmions emerge as topological defects in the magnetization texture that carry a specific topological charge referred to as the skyrmion (winding​) number. In the last decade, magnetic systems with the shape of a curved thin film have been subject to extensive experimental and theoretical research (nanotubes, 3d helices, thin spherical shells). Even in anisotropic crystals, curved geometries can induce an effective antisymmetric interaction and, therefore, the formation of magnetic skyrmions. Moreover, the wide range of magnetic properties emerging in curved geometries makes them well-suited for technological applications, from racetrack memory devices to spin-wave filters. An in-depth understanding of their rich structure (e.g., chirality, topological charge, stability) leads to challenging problems in a subject area where geometry and continuum mechanics meet topology and analysis, and this has raised interest in magnetic skyrmions also from a mathematical perspective. In this talk, we will review the existing literature on the micromagnetics of curved thin films and our contributions to the field. Special attention will be given to the analysis of magnetic skyrmions in spherical thin films. This question will lead to a sharp Poincaré-type inequality that allows for a precise characterization of the global minimizers of the micromagnetic energy functional on the 2-sphere.
• References:
  • G. Di Fratta, Micromagnetics of curved thin films. Zeitschrift für angewandte Mathematik und Physik (2020), 71(111).
  • G. Di Fratta, V. Slastikov, A. Zarnescu, On a sharp Poincaré-type inequality on the 2-sphere and its application in micromagnetics. SIAM Journal on Mathematical Analysis (2019), 51(4):3373–3387.
  • G. Di Fratta, C. Muratov, F. Rybakov, V. Slastikov, Variational principles of micromagnetics revisited. SIAM Journal on Mathematical Analysis (2020), 52(4):3580–3599.
  • E. Davoli, G. Di Fratta, D. Praetorius, M. Ruggeri, Micromagnetics of thin films in the presence of Dzyaloshinskii-Moriya interaction. arXiv:2010.15541 (2020).
  • G. Di Fratta, A. Fiorenza, V. Slastikov, On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces. SFB65 Preprint/CVGMT paper 5038 (2021).
• YouTube link

• Title: The energy of a Möbius strip
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: In 1930 Sadowsky considered the problem of finding the equilibrium configurations of an unstretchable Möbius strip. To tackle this problem he formally derived the bending energy for a strip whose width is much smaller than the length. This energy depends on the curvature and torsion of the centerline of the band and it is singular at points with zero curvature. In this talk we will re-examine the derivation of the Sadowsky energy using Gamma-convergence and we will discuss how this relates to the Kirchhoff energy for plates and to 3d nonlinear elasticity.
• YouTube link

• Title: Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: This talk is devoted to highlighting the interplay between fracture and delamination in thin films. The usual scaling law on the elasticity parameters and the toughness of the medium with respect to its thickness gives rise to traditional cracks which are invariant in the transverse direction. We will show that, upon playing on this scaling law, it is also possible to observe debonding effects (delamination as well as decohesion) through the appearance of cracks which are orthogonal to the thin direction. Starting from a three-dimensional brittle elastic thin film, we will first present how both phenomena can be recovered independently through a Gamma-convergence analysis as the thickness tends to zero. Then, working on a “toy model" for scalar anti-plane displacements, we will show how both phenomena can be obtained at the same time. Some partial results in the full three-dimensional case will be presented. These are joint works with Blaise Bourdin, Duvan Henao, Andrès Leon Baldelli. and Corrado Maurini.
• YouTube link

• Title: One-dimensional domain walls in thin film ferromagnets: an overview
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: Ferromagnetic materials offer a prime example of systems exhibiting a rich variety of spatial patterns driven by minimization of energy. A competition between the short-range exchange and the long-range magnetostatic interaction in ferromagnets often gives rise to the emergence of magnetic domains, in which the magnetization remains nearly constant and aligned along particular preferred directions in extended regions of space. These regions are punctuated by sharp transition layers referred to as domain wall, in which the magnetization abruptly rotates between different preferred orientations. In this talk, I will give an overview of the current state of the art in the understanding of the domain wall solutions in thin ferromagnetic films in which the magnetization is constrained to lie in the film plane. This setting leads to a number of challenging problems in the analysis of nonlinear PDEs involving fractional Laplacian.
• YouTube link

ONEPAS Thematic Series - The partial differential equations of quantum mechanics

• Title: Overview
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: In this talk I will describe key equations of quantum mechanics. I will touch upon results and open problems.
• YouTube link

• Title: Solitons and dynamics of Landau-Lifshitz equations in 2D
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: The Landau-Lifshitz equation is the basic dynamical equation in a micromagnetic description of a ferromagnet. It is naturally viewed as geometric evolution PDE of dispersive, Hamiltonian type ("Schrodinger map"), which scales critically with respect to the physical energy in two space dimensions. We discuss some results and open questions around well-posedness and topological soliton solutions.
• YouTube link

• Title: Evolution of NLS with bounded data
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: We study the nonlinear Schrodinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm (arXiv:2003.08849; J.Stat.Phys, 2020). The case of initial data which is smooth and L^p , p less than infinity, is proved for cubic NLS in one dimension (arXiv:2012.14355). These are joint works with Ben Dodson and Tom Spencer.

• Title: Symmetry and symmetry breaking in functional inequalities
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: A functional that is invariant under a symmetry group has in general minimizers that are not invariant under the group. There are notable exceptions but it is often quite delicate to prove that the minimizers are indeed symmetric. We shall explore this issue in a number of examples chief among them the Caffarelli-Kohn Nirenberg inequalities. Applications to quantum mechanical systems with magnetic fields will be also discussed. Most of these results were obtained in collaboration with Jean Dolbeault and Maria Esteban. If time permits, some new results obtained in collaboration with Rupert Frank will be presented.
• YouTube link

• Title: Novel solutions in Ginzburg–Landau theory
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: Ginzburg–Landau theory is one of the oldest models in classical field theory for spontaneous symmetry breaking through the Higgs mechanism. Minimizers of the Ginzburg–Landau free energy, called vortices, are well-understood, especially in 2 dimensions. Much less is known about nonminimal solutions. On the flat plane no such solution exists, but this is not true on compact domains. Motivated by (and building on) works of Sigal et al. and Taubes, I give conditions for the non/existence of nonminimal solutions. Moreover, I compute approximate forms of such solutions. Part of this work is joint with Gonçalo Oliveira.
• YouTube link

• Title: Eigenvalue statistics for some random Schrödinger operators and random band matrices
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: This review talk will present some recent results concerning local eigenvalue statistics (LES) for a family of one-dimensional random Schrödinger operators (RSO) and random band matrices by the presenter and others. The motivation for studying these two models is the conjecture that RSO in three or more dimensions are expected to exhibit a localization-delocalization transition. It is anticipated that the LES in the localized phase is given by a Poisson point process, whereas in the delocalized phase the LES is the same as the Gaussian orthogonal ensemble in random matrix theory. Two simple one-dimensional models, a scaled-disorder RSO and a RBM, for which a LES transition has been proved to exist, will be described.
• YouTube link

• Title: Tight-binding limits in strong magnetic fields and the integer quantum Hall effect
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: Tight-binding models of non-interacting electrons in solids are commonly used when studying the integer quantum Hall effect or more generally topological insulators. However, up until recently there was no proof that the topological properties of these discrete Schrodinger operators agree with those of the continuum models of which they are the tight-binding limit. Before tending to this question, we first tackle the basic issue of the double-well eigenvalue splitting in strong perpendicular constant magnetic fields in 2D. Once this is understood, we set up norm-resolvent convergence of a scaled continuum Schrodinger operator on \(L^2(\mathbb{R}^2)\) (a magnetic Laplacian plus a lattice potential) to its tight-binding limit and finally show why the Chern numbers of these two models, discrete and continuum respectively, must agree. This talk is based on joint collaborations with C. L. Fefferman and M. I. Weinstein.
• YouTube link

ONEPAS Thematic Series - Constructive methods for the long term analysis of dynamics

• Title: Overview
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: The long term behavior of dynamical systems is an important practical problem in technology and a deep challenge for mathematics. In this set of lectures, we plan to present some ideas of a concrete methodology that can be used to study the long term dynamics of some given systems. The key idea is to:
  • Identify some geometric structures that imply interesting long term behaviors.
  • Develop rigorous methods that, with a finite computations can establish the existence of these objects in concrete systems.
  We can only cover some generalities and some vignettes that illustrate these program both in finite dimensional problems and in infinite dimensional systems. This first lecture will present some of the main tools from analysis and topology.
• YouTube link

• Title: Hard implicit function theorems
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: The main idea is the Newton method with iteration. We will discuss the conjugacy of circle maps, following Moser's Pisa Lectures. A proof of the theorem will be given along with numerical implementation.
• YouTube link

• Title: Computer-assisted applications of KAM theory
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: Stability of Hamiltonian systems is relevant to problems from celestial mechanics, particle accelerators, plasma confinement, quasigeostrophic flows, etc. Motivated by such applications, one is interested in detecting mechanisms of stability in concrete models, and in providing quantitative information on the stable trajectories. KAM theory concerns the existence of quasi-periodic solutions, that are geometrically described as orbits lying inside invariant tori.
  
  In this lecture we will overview a methodology for rigorously detecting quasi-periodic orbits in Hamiltonian systems. The methodology involves analytical, geometrical and computational methods and covers from pen and paper rigorous results to computer-assisted rigorous results, passing through algorithms (and the study of their convergence) and implementations. We will present some ideas for performing computer assisted proofs in this context. In particular, we will see FFT-methods for representing rigorously real-analytic periodic functions, that are used to parameterize tori in phase space. We will see some applications in this context. We will finish the lecture with some other applications and further topics.
• YouTube link

• Title: Topological methods and Hamiltonian instability
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: Instability of Hamiltonian systems is relevant to problems from celestial mechanics, particle accelerators, plasma confinement, quasigeostrophic flows, etc. Motivated by such applications, one is interested in detecting mechanisms of instability in concrete models, and in providing quantitative information on the unstable trajectories.
  
  We will describe a topological method based on `correctly aligned windows', which can be used to derive properties concerning the long-term behavior of dynamical systems. In particular, this method enables one to detect topological horseshoes. The method can be implemented in computer assisted proofs, via validated numerical computations. We will show application of this method to Hamiltonian instability. Concrete examples will include mechanical systems consisting of rotators and penduli, and the three-body problem in celestial mechanics.
• YouTube link

• Title: Invariant objects of infinite dimensional systems
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract: To find invariant objects with specified dynamics (e.g. periodic, quasi-periodic, etc.), we consider the space of functions in the corresponding class and impose that they are solutions. This leads to solving functional equations. One advantage of this method is that it does not need to study the evolution, which in some infinite dimensional problems is problematic. We will present some examples of solutions of special kinds in state dependent delay equations, for which even the natural phase space is debatable.
• YouTube link

• Zoom Meeting ID: 942-0776-5949 , Passcode: 889022
• YouTube link
• Speaker: Patrick Flynn , Brown University
• Title: Asymptotic behavior for Vlasov-Poisson via the pseudo-conformal transformation
• Abstract: We study the Vlasov-Poisson system in the limit as time goes to infinity, showing modified scattering of small initial data and the existence of wave operators. In contrast to previous results in this direction, we use the classical analogue of the pseudo-conformal transformation to invert time, allowing us to recast the problem as local existence of Cauchy problem of a transport equation. Although this equation is singular in time, this singularity does not appear in the continuity equation. We leverage this fact, along with leading order approximations for the characteristics near time zero, so as to attain convergence of the electric field under weak assumptions on the data in the case of modified scattering, and attain local well-posedness via standard fixed-point methods in the case of the wave operators.
• Speaker: Milen Ivanov , Brown University Division of Applied Mathematics
• Title: Reaction-diffusion systems: defects in patterns
• Abstract: Reaction-diffusion systems exhibit spiral wave patterns, which may have defects. We use spatial dynamics to interpret such a defect as a homoclinic orbit in a Hilbert space and then construct a finite-dimensional model of the problem. In that model we use normal hyperbolicity to prove existence of said patterns and then we comment on spectral stability. It turns out the patterns are spectrally stable with periodic boundary conditions, and spectrally unstable with Neumann boundary conditions.
• Speaker: Stefano Pasquali, Lund University
• Title: Chaotic–like transfers of energy in Hamiltonian PDEs
• Abstract: A fundamental problem in nonlinear Hamiltonian PDEs on compact manifolds is understanding how solutions can exchange energy among Fourier modes. After discussing results addressing the problem of transfer of energy and growth of Sobolev norms, I will present a recent result which shows chaotic-like transfers of energy for some PDEs on the 2-dimensional torus by combining techniques from dynamical systems and PDEs . This is a joint work with M. Guardia, F. Giuliani and P. Martin (UPC, Barcelona).

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

• 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.

• 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.

• 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.

• 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.

• 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.
• YouTube link
• Suggested reading:
  • V. Kozlov, E. Lokharu, and M.H. Wheeler. Nonexistence of subcritical solitary waves. Preprint, 2020.
  • M.H. Wheeler. The Froude number for solitary water waves with vorticity. J. Fluid Mech. 768:91, 2015. arXiv link.
  • R.M. Chen, S. Walsh, and M.H. Wheeler. Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics. Preprint, 2019.
  • R.M. Chen, S. Walsh, and M.H. Wheeler. Global bifurcation for monotone fronts of elliptic equations. Preprint, 2020.

• 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.
• YouTube link

• Title:Three-dimensional water waves
• Zoom Meeting ID: 586-919-392 , Passcode: 889022
• Abstract:The other talks in the series have concentrated on two-dimensional water waves. In my talk I will give an overview of the considerably younger three-dimensional theory. In the irrotational case, there is by now a rich existence theory for small-amplitude solutions. These can have different behaviours in different horizontal directions, e.g. periodic or solitary. In the talk I will mainly focus on waves which are "doubly periodic", that is, periodic in two different horizontal directions, or "fully localised", that is, solitary in all horizontal directions. As we will see, surface tension plays a much more crucial role than in 2D. I will also discuss some recent work on 3D water waves with vorticity.
• YouTube link
• Suggested reading:
  • B. Buffoni, M. D. Groves, E. Wahlén. A variational reduction and the existence of a fully localised solitary wave for the three-dimensional water-wave problem with weak surface tension. Arch. Ration. Mech. Anal. 228 (2018), 773–820.
  • B. Buffoni, M. D. Groves, S. M. Sun, E. Wahlén. Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves. J. Differential Equations 254 (2013), 1006–1096.
  • W. Craig, D. P. Nicholls. Travelling two and three dimensional capillary gravity water waves. SIAM J. Math. Anal. 32 (2000), 323–359.
  • M. D. Groves. Three-dimensional travelling gravity-capillary water waves. GAMM-Mitt. 30 (2007), 8–43.
  • M. D. Groves, S.-M. Sun. Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem. Arch. Ration. Mech. Anal. 188 (2008), 1–91.
  • G. Iooss, P. I. Plotnikov. Small divisor problem in the theory of three-dimensional water gravity waves. Mem. Amer. Math. Soc. 200 (2009), no. 940.
  • E. Lokharu, D. S. Seth, E. Wahlén. An existence theory for small-amplitude doubly periodic water waves with vorticity. Arch. Ration. Mech. Anal. 238 (2020), 607–637.
  • E. Wahlén. Non-existence of three-dimensional travelling water waves with constant non-zero vorticity. J. Fluid Mech. 746 (2014), R2.

Spring and Summer 2020

• 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).

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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).

• 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.

• 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.

• 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.

• 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.

• 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.

• iantice@andrew.cmu.edu
• Carnegie Mellon University
  Department of Mathematical Sciences
  Wean Hall 6206
  Pittsburgh, PA 15213