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William J. Trenberth

Publications and source records attributed to William J. Trenberth.

3 recordsLinked to original sources
arXiv2021

Quasi-invariance of fractional Gaussian fields nonlinear wave equation with polynomial nonlinearity

We prove quasi-invariance of Gaussian measures $μ_s$ with Cameron-Martin space $H^s$ under the flow of the defocusing nonlinear wave equation with polynomial nonlinearities of any order for all $s>5/2$, including fractional $s$. This extends work of Oh-Tzvetkov [15] and Gunaratnam-Oh-Tzvetkov-Weber [10] who proved this result this result for a cubic nonlinearity and $s$ an even integer. The main contributions are a modified construction of a weighted adapted to higher order nonlinearity, and an energy estimate for the derivative of the energy replacing the integration by parts argument introduced in previous works. We also address the question of (non) quasi-invariance for the dispersionless model raised in the introductions to [15,10].

math.AP↗
arXiv2019

Global well-posedness for the two-dimensional stochastic complex Ginzburg-Landau equation

We study the stochastic complex Ginzburg-Landau equation (SCGL) with an additive space-time white noise forcing on the two-dimensional torus. This equation is singular and thus we need to renormalize the nonlinearity in order to give proper meaning to the equation. Unlike the real-valued stochastic quantization equation, SCGL is complex valued and hence we are forced to work with the generalized Laguerre polynomials for the sake of renormalization. In handling nonlinearities of arbitrary degree, we derive a useful algebraic identity on the renormalization in the complex-valued setting and prove that the renormalized SCGL is locally well-posed. We prove global well-posedness using an energy estimate and almost sure global well-posedness under different conditions using an invariant measure argument.

math.AP↗