Department of Mathematics

Mathematical Physics and Gauge Theory

  •  Jonas Lührmann
  •  Probabilistic scattering for the 4D energy-critical defocusing nonlinear wave equation
  •  01/25/2018
  •  11:00 AM - 12:00 PM
  •  C304 Wells Hall

We consider the Cauchy problem for the energy-critical defocusing nonlinear wave equation in four space dimensions. It is known that for initial data at energy regularity, the solutions exist globally in time and scatter to free waves. However, the problem is ill-posed for initial data at super-critical regularity, i.e. for regularities below the energy regularity. In this talk we study the super-critical data regime for this Cauchy problem from a probabilistic point of view, using a randomization procedure that is based on a unit-scale decomposition of frequency space. We will present an almost sure global existence and scattering result for randomized radially symmetric initial data of super-critical regularity. This is the first almost sure scattering result for an energy-critical dispersive or hyperbolic equation for scaling super-critical initial data. The main novelties of our proof are the introduction of an approximate Morawetz estimate to the random data setting and new large deviation estimates for the free wave evolution of randomized radially symmetric data. This is joint work with Ben Dodson and Dana Mendelson.



Department of Mathematics
Michigan State University
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C212 Wells Hall
East Lansing, MI 48824

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