Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm

We study the long-time behaviour of the focusing cubic NLS on $\mathbf R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein [27], [26]. In the sequel to this paper we generalize this result to other nonlinear Schrodinger equations. Our arguments are based on the "$I$-method" from earlier papers [9]-[15] which pushes down from the energy norm, as well as an "upside-down $I$-method" which pushes up from the $L^2$ norm.

[1]  Terence Tao,et al.  Sharp global well-posedness for KdV and modified KdV on ℝ and , 2003 .

[2]  F. Merle,et al.  Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation , 2004, math/0405229.

[3]  H. Takaoka,et al.  Almost conservation laws and global rough solutions to a Nonlinear Schr , 2002, math/0203218.

[4]  Terence Tao,et al.  A Refined Global Well-Posedness Result for Schrödinger Equations with Derivative , 2001, SIAM J. Math. Anal..

[5]  T. Tao,et al.  Multilinear estimates for periodic KdV equations, and applications , 2001, math/0110049.

[6]  C. Kenig,et al.  Bilinear estimates and applications to 2d NLS , 2001 .

[7]  Luis Vega,et al.  On the ill-posedness of some canonical dispersive equations , 2001 .

[8]  Terence Tao,et al.  Global Well-Posedness for Schrödinger Equations with Derivative , 2001, SIAM J. Math. Anal..

[9]  T. Tao,et al.  Global well-posedness for KdV in Sobolev spaces of negative index , 2001, math/0101261.

[10]  K. Nakanishi.  Energy Scattering for Nonlinear Klein–Gordon and Schrödinger Equations in Spatial Dimensions 1 and 2☆ , 1999 .

[11]  J. Colliander,et al.  Global wellposedness for KdV below ${L^2}$ , 1999 .

[12]  J. Bourgain.  Scattering in the energy space and below for 3D NLS , 1998 .

[13]  T. Tao,et al.  Endpoint Strichartz estimates , 1998 .

[14]  T. Tao,et al.  Local and global well-posedness of wave maps on $\R^{1+1}$ for rough data , 1998, math/9807171.

[15]  J. Bourgain,et al.  Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity , 1998 .

[16]  G. Staffilani.  On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations , 1997 .

[17]  J. Bourgain.  On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE , 1996 .

[18]  Jean Ginibre,et al.  Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace , 1995 .

[19]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[20]  Thierry Cazenave,et al.  The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s , 1990 .

[21]  Y. Tsutsumi.  L$^2$-Solutions for Nonlinear Schrodinger Equations and Nonlinear Groups , 1985 .

[22]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[23]  C. V. Coffman.  Uniqueness of the ground state solution for Δu−u+u3=0 and a variational characterization of other solutions , 1972 .