Global solutions of Maxwell-Higgs on Minkowski space (Q1279942)

The article concerns the initial value problem for the Euler equations \[ D^jD_j\Phi= -U_{\overline\Phi},\quad \partial_kF^{jk}= i(\overline\Phi D^j\Phi- \overline{D^j\Phi}\Phi) \] of the action functional \[ \int_{\mathbb{R}} (\overline{D^j\Phi} D_j\Phi- U(\Phi)- \textstyle{{1\over 4}} F^{jk} F_{jk})d^3x, \] where \(D^j= \partial^j+ iA^j\), \(F^{jk}= \partial^j A^k- \partial^kA^j\), \(U(\Phi)= \lambda(|\Phi|^2- 1)^2\). This section involves the field \(\Phi(x)\) and the vector potential \(A(x)\) which are smooth functions in the Minkowski space \(\mathbb{R}\) with coordinates \(x^j\) \((0\leq j\leq 2,\;x^0= t)\) and metrices \((g_{jk})= \text{diag}(1, -1,-1)\). The author proves the existence of a unique global solution in a Sobolev space depending continuously on the initial data provided the necessary condition \[ \partial_t\text{ div }A(x, 0)- i(\partial_t\overline\Phi(x, 0) \Phi(x, 0)-\partial_t \Phi(x, 0)\overline\Phi(x, 0))= 0 \] be satisfied.