Displaying similar documents to “Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data”

Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Silvia Cingolani, Louis Jeanjean, Simone Secchi (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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In this work we consider the magnetic NLS equation ( ℏ i ∇ - A ( x ) ) 2 u + V ( x ) u - f ( | u | 2 ) u = 0 in ℝ N ( 0 . 1 ) where N ≥ 3 , A : ℝ N → ℝ N is a magnetic potential, possibly unbounded, V : ℝ N → ℝ is a multi-well electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : ℝ N → ℂ to (0.1), under conditions on the nonlinearity which are nearly optimal.

Regularity and uniqueness in quasilinear parabolic systems

Pavel Krejčí, Lucia Panizzi (2011)

Applications of Mathematics

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Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.