Hyungjin Huh

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

We prove global existence of a classical solution for small initial in the Cauchy problem of the Born–Infeld system describing nonlinear electromagnetism. For the proof we crucially use the null form structure of the the nonlinear terms under the Lorentz gauge condition.

Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field

We prove the existence of infinitely many radially symmetric standing-wave solutions of the Chern-Simons-Schrodinger system. Our result is established by applying the mountain-pass theorem to the functional, which is obtained by representing gauge fields Aμ in terms of a scalar field ϕ.

Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge

ABSTRACT In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of Xs, b spaces, we obtain local well-posedness of (CSD) for initial data . Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data , , which improves the previous result of Selberg and Tesfahun.

Low regularity solutions of the Chern?Simons?Higgs equations

We study low regularity solutions of the Chern–Simons–Higgs equations. Under the Lorentz gauge condition they are formulated in the second order hyperbolic equations with null form. The various Strichartz and null form estimates will be used to lower the regularity condition. With the temporal gauge condition divergence–curl decomposition and elliptic estimates will be used.

Energy Solution to the Chern-Simons-Schrödinger Equations

We prove that the Chern-Simons-Schrodinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space . The Coulomb gauge provides elliptic features for gauge fields . The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wentes inequalities.

Remarks on the repulsive Wigner-Poisson system

We present an estimate for the space-time integral of classical solutions to the repulsive Wigner-Poisson system. We use Markowich’s formalism between Wigner-Possion and Schrodinger-Poisson systems. Through this formalism and Morawetz interaction potentials, we derive the same a priori estimate given by Chae and Ha for the repulsive Vlasov-Poisson system.

The equivalence of the Chern-Simons-Schrödinger equations and its self-dual system

In this paper, we discuss the equivalence of the second order Chern-Simons-Schrodinger equations and its first order self-dual system.

Blow-Up Solutions of Modified Schrödinger Maps

We study blow-up solutions of modified Schrödinger maps. We observe the pseudo-conformal invariance by which explicit blow-up solutions can be constructed.

Nonexistence Results of Semilinear Elliptic Equations Coupled with the Chern-Simons Gauge Field

We discuss the nonexistence of nontrivial solutions for the Chern-Simons-Higgs and Chern-Simons-Schrodinger equations. The Derrick-Pohozaev type identities are derived to prove it.

Global Strong Solutions to Some Nonlinear Dirac Equations inSuper-Critical Space

We study the initial value problem of some nonlinear Dirac equations which are critical. Corresponding to the structure of nonlinear terms, global strong solutions can be obtained in different Lebesgue spaces by using solution representation formula. The uniqueness of weak solutions is proved for the solution .