Zhen Lei

TL;DR: In this paper, the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near-equilibrium initial data was proved.

TL;DR: The Oldroyd model describing fluids with viscoelastic properties with small initial displacements with main difficulty is the lack of the damping mechanism on the deformation tensor.

TL;DR: In this article, the authors established a global well-posedness of mild solutions to the Navier-Stokes equations if the initial data are in the space X-1 defined by X -1 = {f E D'(R 3 ): ∫ ℝ 3 |ξ| -1 |f|dξ < ∞}.

Abstract: In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.

TL;DR: In this paper, the authors derived a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in the Sobolev space.