On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge
Jianjun Yuan
Discrete & Continuous Dynamical Systems - A 2014, 34(5): 2389-2403 doi: 10.3934/dcds.2014.34.2389
In this paper, we investigate the well-posedness of the Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. In particular, we prove that the system is globally wellposed in the energy space. As an application, we prove that the solution of the Maxwell-Chern-Simons-Higgs system converges to that of Maxwell-Higgs system in $H^s\times H^{s-1}$($s\geq1$) as the Chern-Simons coupling constant $\kappa\rightarrow0$.
keywords: Lorenz gauge. Maxwell-Higgs Maxwell-Chern-Simons-Higgs
Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$
Jianjun Yuan
Communications on Pure & Applied Analysis 2015, 14(5): 1941-1960 doi: 10.3934/cpaa.2015.14.1941
In this paper, we investigate the dynamics of a boson gas with three-body interactions in $T^2$. We prove that when the particle number $N$ tends to infinity, the BBGKY hierarchy of $k$-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized $N$-body initial datum, we show that this $N\rightarrow\infty$ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.
keywords: BBGKY hierarchy Gross-Pitaevskii hierarchy Schrödinger equation.
Global solutions of two coupled Maxwell systems in the temporal gauge
Jianjun Yuan
Discrete & Continuous Dynamical Systems - A 2016, 36(3): 1709-1719 doi: 10.3934/dcds.2016.36.1709
In this paper, we consider the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge. By using the fact that when the spatial gauge potentials are in the Coulomb gauge, their $\dot{H}^1$ norms can be controlled by the energy of the corresponding system and their $L^2$ norms, and the gauge invariance of the systems, we show that finite energy solutions of these two systems exist globally in this gauge.
keywords: Maxwell-Chern-Simons-Higgs temporal gauge Coulomb gauge. Maxwell-Klein-Gordon

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