Juan J. L. Velázquez and Miguel Escobedo On the theory of Weak Turbulence for the Nonlinear Schrödinger Equation Memoirs of the AMS, 238 2015 http://dx.doi.org/10.1090/memo/1124
Abstract: We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schrödinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior.
2014
Miguel Escobedo and Juan J. L. Velázquez Finite time blow-up and condensation for the bosonic Nordheim equation Inventiones mathematicae, 200(3): 761-847 2014 http://dx.doi.org/10.1007/s00222-014-0539-7
Abstract: The homogeneous bosonic Nordheim equation is a kinetic equation describing the dynamics of the distribution of particles in the space of moments for a homogeneous, weakly interacting, quantum gas of bosons. We show the existence of classical solutions of the homogeneous bosonic Nordheim equation that blow up in finite time. We also prove finite time condensation for a class of weak solutions of the kinetic equation.
Miguel Escobedo and Juan J. L. Velázquez On the Blow Up and Condensation of Supercritical Solutions of the Nordheim Equation for Bosons Communications in Mathematical Physics, 330(1): 331-365 2014 http://dx.doi.org/10.1007/s00220-014-2034-9
Abstract: In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons with bounded initial data blow up in finite time in the L ∞ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contain a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.