Local and global solutions to the \(O(3)\)-sigma model with the Maxwell and the Chern-Simons gauges in \(\mathbb{R}^{1 + 1} \)
In this paper the authors study the well-posedness of the \((1 +1)\)-dimensional Maxwell-Chern- Simons gauged \(O(3)\)-sigma model (MCSS) with the Lorenz gauge condition. The Lagrangian density of the \((2+1)\)-dimensional MCSS has the form \[ \mathcal L=-(1/4\rho )F_{\mu\nu }F^{\mu\nu }+ (\kappa /4)\epsilon^{\mu\nu\lambda } F_{\mu\nu }A_{\lambda}+(1/2)D_{\mu }\phi\cdot D^{\mu }\phi + 1/(2\rho ) \partial_{\mu }N\partial^{\mu }N-V(\phi ,N), \] where it is used the Minkowski metric , \(\phi=(\phi_1,\phi_2,\phi_3): \mathbb{R}^{2+1}\to S^2\) is a vector field, \(N:\mathbb{R}^{2+1}\to\mathbb{R}\) is a neutral scalar field and \(A_{\lambda }:\mathbb{R}^{2+1}\to\mathbb{R}\) (\(\lambda =0,1,2\)) is the gauge field, \(F_{\mu\nu }\) is the curvature tensor, and \(D_{\mu }\) is the covariant derivative. One may accept \(\rho =1\). By a standard way it is obtained the Euler-Lagrange system of partial differential equations. There are two possible natural asymptotic conditions for \( (\varphi , N, A_{\mu }, B) \) to make the total energy finite: (i) nontopological boundary condition, and (ii) topological boundary condition. Here the authors consider the nontopological boundary condition. The Euler-Lagrange system is invariant under the gauge transformations. It is accepted the gauge which is fixed by adopting the Lorenz gauge condition \(\partial_0 A_0 - \partial_1 A_1 = 0\). Under the Lorenz gauge condition, the Cauchý problem for the \((1+1)\)-dimensional MCSS system along with initial data can be rewritten in a proper form. In the first part the authors study the global well-posedness of a solution in a high regular space. The main statement is that if the initial data \[ \varphi_0, \ A_{\mu } (0,\cdot ) = a_{0\mu }, \ B(0,\cdot )=b_0, \ N(0,\cdot )=n_0\in H^2, \] \[ \partial_t\phi (0,\cdot )= \varphi_1, \ \partial_tA_{\mu }(0,\cdot )=a_{1\mu }, \ \partial_tB(0,\cdot )= b_1, \ \partial_tN(0,\cdot )=n_1\in H^1, \] and certain constraint hold, then the Cauchý problem for MCSS system has a unique, global in time solution \((\phi , N,A_{\mu },B)\) whose components belong to \(C([0,\infty ),H^2(\mathbb{R}))\cap C^1 ([0,\infty ), H^1(\mathbb{R}))\). Also it is shown that existence of a global solution with corresponding finite total energy. Moreover, the authors discuss the local well-posedness below the energy regularity by exploiting null form and wave-Sobolev space.
Maxwell-Chern-Simons gauge \(O(3)\)-sigma model
scientific article; zbMATH DE number 7315381
scientific article