Global well-posedness and scattering for the energy-critical Schrödinger equation in \(\mathbb R^{3}\)
The subject of the work is the nonlinear Schrödinger equation in the three-dimensional space, with the self-defocusing quintic nonlinearity: \[ iu_t + \Delta u = |u|^4u, \] where \(\Delta\) is the three-dimensional Laplacian. This equation is considered to be of the ``energy-critical'' type because its Hamiltonian remains invariant under the scaling transformation, \(u\to\lambda^{1/2}u\), \(x\to \lambda^{-1}x\), \(t\to \lambda^{-2}t\). In previous works, it has been proven that the initial problem for this equation is a globally well-posed one for small values of the energy (Hamiltonian), or locally well-posed for arbitrary values of the energy. It has also been demonstrated that the initial-value problem is well-posed assuming the spherical symmetry of solutions. The main result of the present work is a rigorous proof of the global existence of a single solution to the initial-value problem, with arbitrary finite energy \(E\). The solution obeys a constraint, \[ \int_{-\infty}^{+\infty}\int_{R^{3}} |u(t,{\mathbf x})|^{10} dtd{\mathbf x}\leq C(E), \] with constant \(C\) that depends only on the energy. The spherical symmetry of the solutions is not assumed. As a consequence, the scattering property of the solutions is also proved, i.e., the fact that, at \(t\to\infty\), a general solution of the nonlinear equation asymptotically approaches a solution of the linear three-dimensional Schrödinger equation.
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