Quansen Jiu

The 2D magnetohydrodynamic equations with magnetic diffusion

This paper examines the initial-value problem for the two-dimensional magnetohydrodynamic equation with only magnetic diffusion (without velocity dissipation). Whether or not its classical solutions develop finite time singularities is a difficult problem and remains open. This paper establishes two main results. The first result features a regularity criterion in terms of the magnetic field. This criterion comes naturally from our approach to obtain a global bound for the vorticity. Due to the lack of velocity dissipation, it is difficult to conclude the boundedness of the vorticity from the vorticity equation itself. Instead we derive and involve a new equation for the combined quantity of the vorticity and a singular integral operator on the tensor product of the magnetic field. This criterion may be verifiable. Our second main result is a weaker version of the small data global existence result, which is shown by the bootstrap argument.

Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation

This paper studies solutions of the two-dimensional incompressible Boussinesq equations with fractional dissipation. The spatial domain is a periodic box. The Boussinesq equations concerned here govern the coupled evolution of the fluid velocity and the temperature and have applications in fluid mechanics and geophysics. When the dissipation is in the supercritical regime (the sum of the fractional powers of the Laplacians in the velocity and the temperature equations is less than 1), the classical solutions of the Boussinesq equations are not known to be global in time. Leray–Hopf type weak solutions do exist. This paper proves that such weak solutions become eventually regular (smooth after some time

Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions

T>0

Global classical solution of the Cauchy problem to 1D compressible Navier–Stokes equations with large initial data

T>0) when the fractional Laplacian powers are in a suitable supercritical range. This eventual regularity is established by exploiting the regularity of a combined quantity of the vorticity and the temperature as well as the eventual regularity of a generalized supercritical surface quasi-geostrophic equation.

Remarks on blow-up of smooth solutions to the compressible fluid with constant and degenerate viscosities

This paper focuses on the initial- and boundary-value problem for the two-dimensional micropolar equations with only angular velocity dissipation in a smooth bounded domain. The aim here is to establish the global existence and uniqueness of solutions by imposing natural boundary conditions and minimal regularity assumptions on the initial data. Besides, the global solution is shown to possess higher regularity when the initial datum is more regular. To obtain these results, we overcome two main difficulties: one due to the lack of full dissipation and one due to the boundary conditions. In addition to the global regularity problem, we also examine the large time behavior of solutions and obtain explicit decay rates.

A remark on free boundary problem of 1‐D compressible Navier–Stokes equations with density‐dependent viscosity

This paper concerns the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space Lq with 2 < q ≤ ∞, we show that the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In a special case when the vorticity is in L∞, an explicit convergence rate is obtained.