Ground State Mass Concentration for Nls
نویسندگان
چکیده
We consider finite time blowup solutions of the L 2-critical cubic focusing nonlinear Schrödinger equation on R 2. Such functions, when in H 1 , are known to concentrate a fixed L 2-mass (the mass of the ground state) at the point of blowup. Blowup solutions from initial data that is only in L 2 are known to concentrate at least a small amount of mass. In this paper we consider the intermediate case of blowup solutions from initial data in H s , with 1 > s > sQ, where sQ ≤ 1 5 + 1 5 √ 11. Our main result is that such solutions, when radially symmetric, concentrate at least the mass of the ground state at the origin at blowup time.
منابع مشابه
Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger Equation in 1d
We consider the L-critical quintic focusing nonlinear Schrödinger equation (NLS) on R. It is well known that H solutions of the aforementioned equation blow-up in finite time. In higher dimensions, for H spherically symmetric blow-up solutions of the L-critical focusing NLS, there is a minimal amount of concentration of the L-norm (the mass of the ground state) at the origin. In this paper we p...
متن کاملRegularity of Almost Periodic modulo Scaling Solutions for Mass-critical Nls and Applications
In this paper, we consider the Lx solution u to mass critical NLS iut + ∆u = ±|u| 4 d u. We prove that in dimensions d ≥ 4, if the solution is spherically symmetric and is almost periodic modulo scaling, then it must lie in H x for some ε > 0. Moreover, the kinetic energy of the solution is localized uniformly in time. One important application of the theorem is a simplified proof of the scatte...
متن کامل. A P ] 1 0 A pr 2 00 9 ON THE RIGIDITY OF SOLITARY WAVES FOR THE FOCUSING MASS - CRITICAL NLS IN DIMENSIONS d ≥ 2
For the focusing mass-critical NLS iut + ∆u = −|u| 4 d u, it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for H x initial data in dimensions d = 2, 3 with spherical symmetry and d ≥ 4 with certain splittingspherically symmetric initial data.
متن کاملAn Invariant Set in Energy Space for Supercritical Nls in 1d
We consider radial solutions of a mass supercritical monic NLS and we prove the existence of a set, which looks like a hypersurface, in the space of finite energy functions, invariant for the flow and formed by solutions which converge to ground states. §
متن کامل0 90 2 . 08 02 v 1 [ m at h . A P ] 4 F eb 2 00 9 ON THE RIGIDITY OF SOLITARY WAVES FOR THE FOCUSING MASS - CRITICAL NLS IN DIMENSIONS d ≥ 2
For the focusing mass-critical NLS iut + ∆u = −|u| 4 d u, it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for H x initial data in dimensions d = 2, 3 with spherical symmetry and d ≥ 4 with certain splittingspherically symmetric initial data. As an off-shoot of ...
متن کامل