Dongho Chae

Regularity criterion in terms of pressure for the Navier-Stokes equations

In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R. It is known that if a Leray weak solution u belongs to L 2 1−r ( (0, T ) ;L d r ) for some 0 ≤ r ≤ 1, (0.1) then u is regular. It is proved that if the pressure p associated to a Leray weak solution u belongs to L 2 2−r ( (0, T ) ; . M2, dr ( R d )d) , (0.2) where . M2, dr ( R d ) is the critical Morrey-Campanato space (a definition is given in the text) for 0 < r < 1, then the weak solution is actually regular. Since this space . M2, dr is wider than L d r and . Xr, the above regularity criterion (0.2) is an improvement of Zhou’s result.

Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations

In this paper we prove the local existence and uniqueness of C 1+γ solutions of the Boussinesq equations with initial data υ 0 , θ 0 ∈ C 1+γ , ω 0 , ∇θ 0 ∈ L q for 0 q 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C 1+γ solutions of the Boussinesq equations.

Nonexistence of Self-Similar Singularities for the 3D Incompressible Euler Equations

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations if the vorticity decays sufficiently fast near infinity in

Exact controllability for semilinear parabolic equations with Neumann boundary conditions

\mathbb{R}^3

Topological Multivortex Solutions of the Self-Dual Maxwell–Chern–Simons–Higgs System

. By a similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in

Global existence for small initial data in the Born–Infeld equations

\mathbb{R}^n