We study an infinite system of nonlinear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3D Euler and Navier-Stokes equations in a rough approximation of wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case
In this paper we prove that the lack of uniqueness for solutions of the tree
dyadic model of turbulence is overcome with the introduction of a suitable
noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial
conditions and is proven using a technique relying on the properties of the
$q$-matrix associated to a continuous time Markov chain
We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation.As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation.As an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift-Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. Moreover we use energy estimates to show that solutions of the Ginzburg-Landau equation are Hölder continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.