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Forthcoming events in this series
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Long-time existence for Yang-Mills flow
Abstract
I'll discuss the problem of controlling energy concentration in YM flow over a four-manifold. Based on a study of the rotationally symmetric case, it was conjectured in 1997 that bubbling can only occur at infinite time. My thesis contained some strong elementary results on this problem, which I've now solved in full generality by a more involved method.
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Boundary regularity for strong local minimizers and Weierstrass problem
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Minimal hypersurfaces with bounded index
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Ancient solutions of Geometric Flows
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In this talk we will discuss Uniqueness Theorems for ancient solutions to geometric partial differential equations such as the Mean curvature flow, the Ricci flow and the Yamabe flow. We will also discuss the construction of new ancient solutions from the parabolic gluing of one or more solitons.
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Regularity Theory for Symmetric-Convex Functionals of Linear Growth
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A counterexample concerning regularity properties for systems of conservation laws
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Stochastic Conservation Laws
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Quantization of time-like energy for wave maps into spheres
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Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
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Prandtl equations in Sobolev Spaces
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Sharp decay estimates for waves on black holes and Price's law
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Some regularity results for classes of elliptic systems with "structure"
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Concentration Compactness for the Critical Maxwell-Klein-Gordon Equation
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Time-Periodic Einstein-Klein-Gordon Bifurcations Of Kerr
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For a positive measure set of Klein-Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein-Klein-Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein-Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein-Klein-Gordon equations. This is joint work with Otis Chodosh.
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Blow up by bubbling in critical parabolic problems
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Regularity of level sets and flow lines
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Meaning of infinities in singular SPDEs
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Obstacle problems of Signorini type, and for non-local operators
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Weak solutions to the Navier-Stokes initial boundary value problem in exterior domains with initial data in L(3,∞)
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We consider the Navier-Stokes initial boundary value problem (NS-IBVP) in a smooth exterior domain. We are interested in establishing existence of weak solutions (we mean weak solutions as synonym of solutions global in time) with an initial data in L(3,∞)
Non-orientable line defects in the Landau-de Gennes theory of nematic liquid crystals
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Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvement
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energy but involving a conservation statement for topological charge of the line defect field for its evolution will be shown to succeed. This is joint work with Chiqun Zhang, graduate student at CMU.
Energy decay in a 1D coupled heat-wave system
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We study a simple one-dimensional coupled heat wave system, obtaining a sharp estimate for the rate of energy decay of classical solutions. Our approach is based on the asymptotic theory of $C_0$-semigroups and in particular on a result due to Borichev and Tomilov (2010), which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the semigroup generator. This technique not only leads to an optimal result, it is also simpler than the methods used by other authors in similar situations and moreover extends to problems on higher-dimensional domains. Joint work with C.J.K. Batty (Oxford) and L. Paunonen (Tampere).
Ancient Solutions to Navier-Stokes Equations in Half Space
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The relationship between the so-called ancient (backwards) solutions to the Navier-Stokes equations in the space or in a half space and the global well-posedness of initial boundary value problems for these equations will be explained. If time permits I will sketch details of an equivalence theorem and a proof of smoothness properties of mild bounded ancient solutions in the half space, which is a joint work with Gregory Seregin
Quantitative flatness results for nonlocal minimal surfaces in low dimensions
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