Existence and Uniqueness of Energy Solutions to the Stochastic Diffusive\n Surface Quasi-Geostrophic Equation with Additive Noise

2020/06/30 by Nathan Totz, Totz, Nathan
Mathematics · Computer Science · #Navier-Stokes equation solutions #Stochastic processes and statistical mechanics #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2007.00061

Abstract

We continue our study of the dynamics of a nearly inviscid periodic surface\nquasi-geostrophic equation. Here we consider a slightly diffusive stochastic\nSQG equation of the form \
begincases d
theta_t +\n|D|^2
delta
theta_t
,dx + (u_t
cdot
nabla)
theta_t
,dx + |D|
^deltadW_t =\n0

u_t =
nabla^
perp|D|^-1
theta_t.
endcases We\nconstruct global energy solutions as introduced by P. Goncalves and M. Jara\n(2014) for any \δ > 0, so that any small amount of diffusion permits us\nto construct solutions. We show moreover that pathwise uniqueness of these\nenergy solutions holds in the presence of sufficiently high diffusion \δ >\n frac32.\n

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