Tuesday, October 24, 2023 - 14:15 in V5-148
Wave maps with values in the sphere defined on the future light cone with random data on the boundary
A talk in the Stochastic Numerics Seminar series by
Zdzislaw Brzezniak
| Abstract: |
We study wave maps with values in $\mathbb{S}^d$, defined on the
future light cone $\{|x| \leq t\} \subset \mathbb{R}^{1+1}$,
with prescribed data at the boundary $\{|x| = t\}$.
Based on the work of Keel and Tao, we prove that the problem is
well-posed for locally absolutely continuous boundary data.
We design a discrete version of the problem and prove that for every
absolutely continuous boundary data,
the sequence of solutions of the discretised problem converges to the
corresponding continuous wave map
as the mesh size tends to $0$.
Next, we consider the boundary data given by the $\mathbb{S}^d$-valued Brownian motion.
We prove that the sequence of solutions of the discretised problems
has an accumulation point
for the topology of locally uniform convergence.
We argue that the resulting random field can be interpreted as the
wave-map evolution
corresponding to the initial data given by the Gibbs distribution.
This talk is based on a joint work with Jacek Jendrej (Université
Sorbonne Paris Nord)
Within the CRC this talk is associated to the project(s): B3 |
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