CPAA
On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$
Keyan Wang
In this paper we prove the global regularity of classical solutions to the incompressible Navier-Stokes equations in $\mathbf R^3$ for a family of large initial data with finite energy.
keywords: Incompressible Navier-Stokes equations global regularity large data.
CPAA
Global well-posedness for a transport equation with non-local velocity and critical diffusion
Keyan Wang
This paper is concerned with a one dimensional transport equation with a non-local velocity and critical diffusion. In [Ann. of Math. 162 (2005), 1377--1389], A. Córdoba, D. Córdoba and Marco A. Fontelos showed the finite time singularities for a family of initial data without diffusion, global existence with subcritical diffusion and the global existence for small initial data with critical diffusion. We prove the global existence with critical diffusion, removing the smallness constraint on the initial data.
keywords: non-local velocity global existence Critical diffusion transport equation.
DCDS-B
Stability of the two dimensional magnetohydrodynamic flows in $\mathbb{R}^3$
Keyan Wang Yi Du
We prove that two dimensional incompressible magnetohydrodynamic flows are stable in $\mathbb{R}^3$. As a corollary, we show the global existence of classical solutions to the three dimensional incompressible magnetohydrodynamic equations with small initial data. Furthermore, our smallness assumption of the perturbed initial data $(u_0, B_0)$ from that of the two dimensional case is only imposed on the scaling invariant quantity $\|u_0\|_{L^2}\|(\xi\cdot\nabla)u_0\|_{L^2} + \|B_0\|_{L^2}\|(\xi\cdot\nabla)B_0\|_{L^2}$ for one direction $\xi$, while $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2} + \|B_0\|_{L^2}\|\nabla B_0\|_{L^2}$ may be arbitrarily large.
keywords: large data Magnetohydrodynamic equations global existence. stability
DCDS
Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation
Jingrui Wang Keyan Wang

In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $\dot{H}^{-α}(\mathbb{R}^{3})$ or $\dot{H}^{-α}(\mathbb{T}^{3})$ with $0<α≤ 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $t≥0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlović and Staffilani on (SIMA) in which $α$ is restricted to $0<α<\frac{1}{4}$.

keywords: Navier-Stokes equation random initial data

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