Yu Deng
The 4 dimensional Anderson model
We establish Gaussian scaling limit, for the (elliptic) Anderson model in 4 dimensions, with small coupling constant. This is a simplest example of a critical model in SPDEs, a theme which has been a major challenge of the field. Our theorem provides the first example where infinitely many renormalization constants are present and controlled. This is joint work with Hao Shen.
Dallas Albritton
Instability, self-similarity, and non-uniqueness for the Euler equations
We will survey aspects of the (in)stability theory for the Euler equations, to which Susan Friedlander has contributed in a fundamental way. We will conclude with an explanation of how the Vishik non-unique solutions can arise in the inviscid limit.
Alexis Vasseur
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Helena Nussenzveig -Lopes
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Nathan Glatt-Holtz
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Jacob Bedrossian
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Gigliola Staffilani
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Peter Constantin
Ideal Magnetic Reconnection
Magnetic fields do not change topology during smooth dynamics of ideal MHD. But topology change does occur in nature. Magnetic resistivity and near singularities have been suggested as a possible explanation. In this talk I will focus on a different explanation, also suggested in the physics literature: reconnection due to magneto-hydrodynamic inertia. I will describe 2D models that exhibit rigorous ideal topology change. The models are Hamiltonian, have global smooth solutions and also global unique Yudovich class solutions. The reconnection is obtained from merger of a pair of active scalars transported by incompressible velocities they create. Finite time merger without singularities and without resistivity or viscosity is proved rigorously. Limited smoothness of patch boundaries of one class of models persists through merger. Numerics suggest a very rich phase space. This is based on joint work with Lydia Boubendir, Hezekiah Grayer II and Zhongtian (Kevin) Hu.
Tarek Elgindi
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Misha Vishik
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Roman Shvydkoy
Large data relaxation for solutions of kinetic models arising in collective dynamics.
In this talk we will present a new estimate on the spectral gap of a general alignment force defined by a symmetric communication kernel. Such forces arise in various models of collective dynamics including the kinetic Cucker-Smale and symmetrized Motsch-Tadmor models. The novelty of the estimate is that it demonstrates dependence of the spectral gap only on the thickness of the flock, and not on the upper bound on the density as in previously known attempts. As an application, we will prove relaxation of solutions to the Fokker-Planck-alignment models starting from large data.
Alexander Kiselev
Singularity suppression by fluid flow
Transport by fluid flow can provide one of the less understood regularization mechanisms in PDE. In this talk, I will focus on the 2D Keller-Segel equation for chemotaxis set on a general domain and coupled via buoyancy with the fluid obeying Darcy's law - a much studied model of the incompressible fluid flow in porous media. It is well known that solutions to the 2D Keller-Segel equation can form singularities in finite time if the mass of the initial data is larger than critical. It turns out that if the equation is coupled with fluid flow obeying Darcy's law via buoyancy, this completely regularizes the system, leading to globally regular solutions for arbitrarily large initial data. One of the key ingredients in the proof is a new generalized Nash inequality, which employs anisotropic norm that is natural in the context of the incompressible porous media flow. This talk is based on works joint with Kevin Hu, Naji Sarsam, and Yao Yao.
Anna Mazzucato
Global existence for the 2D Kuramoto-Sivashinsky equation
I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation in 2 space dimensions with and without advection by an incompressible flow. The KSE is a model of flame-front propagation in combustion and more generally a model of long-wave instability in dissipative systems.
Tom Hou
Nonuniqueness of Leray–Hopf Solutions for the Unforced 3D Incompressible Navier–Stokes Equations
The nonuniqueness of Leray–Hopf solutions to the unforced 3D incompressible Navier–Stokes equations is a central open problem in mathematical fluid dynamics. In joint work with Yixuan Wang and Changhe Yang, we develop the first rigorous computer-assisted proof of such nonuniqueness. Inspired by earlier work in this area, we construct a Leray–Hopf solution in a self-similar setting and then prove the existence of a second solution by studying the linearized operator around this profile and showing that it admits an unstable perturbation. Our approach combines a novel high-precision numerical method for computing candidate solutions with a rigorous framework for establishing exact solutions in their neighborhood. A key ingredient is the decomposition of the linearized operator into a coercive part and a compact perturbation, followed by a finite-rank approximation of the compact part up to a small error. We then rigorously verify, by computer-assisted proof, the invertibility of the linearized operator restricted to the image of this finite-rank approximation. This yields a certified unstable eigenpair and, consequently, a second solution—indeed, infinitely many Leray–Hopf solutions.
Alexey Cheskidov
When viscosity fails to select: three non-uniqueness mechanisms for the Navier-Stokes equations
Recent work shows that non-uniqueness for fluid equations is not confined to inviscid regimes or rough weak solutions. In this talk, I will discuss three recent non-uniqueness mechanisms that require viscosity and occur for solutions that can enjoy critical or even subcritical regularity. The first produces non-unique dissipative solutions for data that admit a local smooth solution, showing that Lions’ weak-strong uniqueness principle for the Euler equations breaks down in the viscous setting. Second, there exist singular stationary solutions in Besov spaces of arbitrarily negative smoothness, showing that non-uniqueness can occur even in subcritical spaces: a perturbative mild solution can arise from an initial datum that is itself a stationary solution. Third, for any smooth (or even zero) initial data, there are infinitely many weak solutions belonging to the Koch-Tataru path space and enjoying the same regularity properties as Koch-Tataru solutions. These non-unique solutions are created by a complete energy cascade from infinite wavenumber, leading to instantaneous blow-up. Again, viscosity is essential. The unifying theme is that viscosity not only regularizes, but also opens new mechanisms for non-uniqueness.
Mimi Dai
Blowup phenomena for fluid equations
We will discuss some constructions which lead to instantaneous blowup for fluid equations in the viscous case, and both instantaneous and finite time blowup in the inviscid dyadic case.
Jerry Bona
Bore propagation with dynamic boundary conditions
The lecture will sketch the derivation and analysis of a model for bore propagation that includes cross-river variation and dynamical boundary conditions.
Anthony Suen
Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime
We address a generalised three-dimensional $\alpha$-Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when $\alpha\in[0,1)$ and also prove global-in-time existence for strong solutions when $\alpha\in[0,\frac{1}{2})$ with initial data controlled by explicit constants. We obtain maximum principles for the $L^{\infty}$-norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these $L^{\infty}$-norms. Finally, we give some convergence results for strong solutions as $\alpha\to0^+$. This is a joint work with Qasim Khan and Bao Quoc Tang.
Hongqiu Chen
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