We develop, analyze, and test an approximate, global data assimilation/synchronization
algorithm based on purely local observations for the two-dimensional Navier--Stokes …

There is little analytical theory for the behavior of solutions of the Kuramoto–Sivashinsky
equation in two spatial dimensions over long times. We study the case in which the spatial …

Abstract We consider the Kuramoto–Sivashinsky equation in two space dimensions. We
establish the first proof of global existence of solutions in the presence of a linearly growing …

Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work
by the second and third named authors showed that with quantifiable high probability …

In this paper, we first investigate the instantaneous radius of space analyticity for the
solutions of 3D Navier–Stokes system with initial data in the Besov spaces B˙ p, qs (R 3) for …

This paper considers a nudging-based scheme for data assimilation for the two-dimensional
(2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the …

We introduce a new method for establishing local analyticity and estimating the local
analyticity radius of a solutions to the 3D Navier–Stokes equations at interior points. The …

We provide sufficient conditions for mathematically rigorous proofs of the third order
universal laws capturing the energy flux to large scales and enstrophy flux to small scales for …

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof
of a global mild solution of the three-dimensional Navier-Stokes equations in function …

We consider the incompressible 3D Navier–Stokes equations subject to a shear induced by
noisy movement of part of the boundary. The effect of the noise is quantified by upper …