Local and global strong solutions to the stochastic incompressible Navier-Stokes equations in critical Besov space

Abstract

Considering the stochastic Navier-Stokes system in R d forced by a multiplicative white noise, we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical spaceḂ d p −1 p,r (R d ). The proof is based on the contraction mapping principle, stopping time and stochastic estimates. Then we prove the global existence of strong solutions in probability if the initial data are sufficiently small, which contain a class of highly oscillating "large" data.

Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations

“…J. U. Kim [27] obtained the local and global existence of the strong solution for the three-dimensional stochastic Navier-Stokes equations when the initial data are in H α+1/2 (R 3 ), 0 < α < 1 2 . Recently, we obtain a similar result in the space [15]. On the other hand, many authors considered the martingale solutions to the stochastic Navier-Stokes equations, see e.g.…”

Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

“…where α ∈ (0, 1]. Following [6], we introduce the definition of local and global strong solutions for the equation (1.1).…”

“…Example 1 Here, we give an example to show that the noise coefficient in the above theorems is non-empty. Motivated by [6], let M > 0 be arbitrary, we take…”

“…In particular. Du and Zhang [6] obtained local and global existence of strong solutions for the SNS in the critical Besov space Ḃ d p −1 p,r . Chang and Yang [3] studied the initial-boundary value problem of the SNS in the half space.…”

Initial-boundary value problem of the Navier–Stokes equations in the half space with nonhomogeneous data

“…Given a positive number ε, there exists a positive number δ such that if (2) The standard Besov spaces are used throughout the paper (cf. [12]).…”