4 research outputs found
On the two-dimensional singular stochastic viscous nonlinear wave equations
We study the stochastic viscous nonlinear wave equations (SvNLW) on , forced by a fractional derivative of the space-time white noise . In
particular, we consider SvNLW with the singular additive forcing
such that solutions are expected to be merely distributions.
By introducing an appropriate renormalization, we prove local well-posedness of
SvNLW. By establishing an energy bound via a Yudovich-type argument, we also
prove global well-posedness of the defocusing cubic SvNLW. Lastly, in the
defocusing case, we prove almost sure global well-posedness of SvNLW with
respect to certain Gaussian random initial data.Comment: 20 page
On the two-dimensional singular stochastic viscous nonlinear wave equations
We study the stochastic viscous nonlinear wave equations (SvNLW) on , forced by a fractional derivative of the space-time white noise . In particular, we consider SvNLW with the singular additive forcing such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove pathwise global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data
Norm inflation for the viscous nonlinear wave equation
In this article, we study the ill-posedness of the viscous nonlinear wave
equation for any polynomial nonlinearity in negative Sobolev spaces. In
particular, we prove a norm inflation result above the scaling critical
regularity in some cases. We also show failure of -continuity, for the
power of the nonlinearity, up to some regularity threshold.Comment: 33 page