Remarks on Global a Priori Estimates for the Nonlinear Schrödinger Equation
نویسندگان
چکیده
We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schrödinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in two dimensions. We bound the restricted L4tL 4 γ Strichartz norm of the solution on any curve γ in R . For the specific case of a straight line we upgrade this estimate to a weighted Strichartz estimate valid in the full plane.
منابع مشابه
On a generalized nonlinear equation of Schrödinger type
Here is established the global existence of smooth solutions to a generalized nonlinear equation of Schrödinger type in the usual Sobolev spaces H and certain weighted Sobolev spaces by using Leray-Schauder fixed point theorem and delicate a priori estimates.
متن کاملar X iv : 0 90 8 . 06 44 v 1 [ m at h . A P ] 5 A ug 2 00 9 REMARKS ON GLOBAL A PRIORI ESTIMATES FOR THE NONLINEAR SCHRÖDINGER EQUATION
We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schrödinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in t...
متن کاملOptimal order finite element approximation for a hyperbolic integro-differential equation
Semidiscrete finite element approximation of a hyperbolic type integro-differential equation is studied. The model problem is treated as the wave equation which is perturbed with a memory term. Stability estimates are obtained for a slightly more general problem. These, based on energy method, are used to prove optimal order a priori error estimates.
متن کاملGlobal existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds
We consider the nonlinear Schrödinger equation with a pure power repulsive nonlinearity on Schwarzschild manifolds. Equations of this type arise when a nonlinear wave equation on a Schwarzschild manifold is written in Hamiltonian form, cf. [2], [10]. For radial solutions with sufficiently localized initial data, we obtain global existence, L estimates, and the existence and asymptotic completen...
متن کاملImproved almost Morawetz estimates for the cubic nonlinear Schrödinger equation
We prove global well-posedness for the cubic, defocusing, nonlinear Schrödinger equation on R2 with data u0 ∈ H s(R2), s > 1/4. We accomplish this by improving the almost Morawetz estimates in [9].
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009