No blow-up by nonlinear Itô noise for the Euler equations
Marco Bagnara, Mario Maurelli, Fanhui Xu
TL;DR
This work addresses the blow-up problem for the stochastic Euler equations by introducing a radial, nonlinear Itô noise with superlinear growth that stabilizes the dynamics. The authors develop a general variational framework for monotone SPDEs, employing Galerkin approximations and a Lyapunov function $V(x)=\log\|x\|_{E_1}$ to obtain uniform bounds, tightness, and convergence to a global weak solution, along with pathwise uniqueness under suitable conditions. They then apply the theory to 2D and 3D stochastic Euler equations with diffusion $\sigma(u)=c(1+\|u\|_{W^{1,\infty}}^2)^{\beta/2}u$, proving global existence and uniqueness of strong solutions in $\mathcal{H}^s(D)$ for $s>d/2+2$, provided $\beta>1/2$ (or $\beta=1/2$ with large $c$). A brief discussion of the Stratonovich case shows that the no-explosion effect may fail under Stratonovich integration with a single radial noise, while multiplicity of noise or different formulations could restore non-explosion. This work highlights a mechanism by which stochastic forcing can prevent finite-time blow-ups in fluid models and provides a general framework for regularization by noise in hyperbolic-type SPDEs.
Abstract
By employing a suitable multiplicative Itô noise with radial structure and with more than linear growth, we show the existence of a unique, global-in-time, strong solution for the stochastic Euler equations in two and three dimensions. More generally, we consider a class of stochastic partial differential equations (SPDEs) with a superlinear growth drift and suitable nonlinear, multiplicative Itô noise, with the stochastic Euler equations as a special case within this class. We prove that the addition of such a noise effectively prevents blow-ups in the solution of these SPDEs.