Namkwon Kim

Existence of mixed type solutions in the Chern–Simons gauge theory of rank two in R2

Abstract We consider the self-dual equations arising from the Chern–Simons gauge theory of rank 2 such as the S U ( 3 ) , S O ( 5 ) , and G 2 Chern–Simons model in R 2 . There are three possible types of solutions in these theories, namely, topological, nontopological, and mixed type solutions. The existence of a mixed type solution for an arbitrary configuration of vortex points has been a long-standing open problem. We construct here a family of mixed type solutions ( u 1 , e , u 2 , e ) , e > 0 for an arbitrary configuration of vortex points under a mild non-degeneracy condition. The main new idea of our construction of an approximate solution is to use different scalings for different components. In the process of the finite dimensional reduction, all estimates for this particular construction become rather delicate.

GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A LONG PERIODIC DOMAIN

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a long periodic domain. We show by a simple argument that a strong solution exists globally in time when the initial velocity in and the forcing function in ([0; T);), T > 0, satisfy a certain condition. This condition common appears for the global existence in thin non-periodic domains. Larger and larger initial data and forcing functions satisfy this condition as the thickness of the domain tends to zero.

REGULARITY OF SOLUTIONS OF 3D NAVIER-STOKES EQUATIONS IN A LIPSCHITZ DOMAIN FOR SMALL DATA

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a bounded Lipschitz do- main under Dirichlet boundary condition. We present by a very simple argument that a strong solution exists globally when the product of L2 norms of the initial velocity and the gradient of the initial velocity and Lp,2, p � 4 norm of the forcing function are small enough. Our condition is scale invariant and implies many typical known global existence results for small initial data including the sharp dependence of the bound on the volumn of the domain and viscosity. We also present a similar result in the whole domain with slightly stronger condition for the forcing.