Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS

@article{Killip2009CharacterizationOM,
  title={Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS},
  author={Rowan Killip and Dong Li and Monica Visan and Xiaoyi Zhang},
  journal={SIAM J. Math. Anal.},
  year={2009},
  volume={41},
  pages={219-236}
}
Let $d\geq4$ and let u be a global solution to the focusing mass-critical nonlinear Schrodinger equation $iu_t+\Delta u=-|u|^{\frac{4}{d}}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state Q. We prove that if u does not scatter, then, up to phase rotation and scaling, u is the solitary wave $e^{it}Q$. Combining this result with that of Merle [Duke Math. J., 69 (1993), pp. 427–453], we obtain that in dimensions $d\geq4$, the only spherically symmetric… 

ON THE RIGIDITY OF MINIMAL MASS SOLUTIONS TO THE FOCUSING MASS-CRITICAL NLS FOR ROUGH INITIAL DATA

For the focusing mass-critical nonlinear Schrodinger equation iut+ u = | u| 4/d u, an important problem is to establish Liouville type results for solutions with ground state mass. Here the ground

Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+\Delta u+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that

The instability of Bourgain-Wang solutions for the L2 critical NLS

We consider the two dimensional $L^2$ critical nonlinear Schr\"odinger equation $i\partial_tu+\Delta u+uu^2=0$. In their pioneering 1997 work, Bourgain and Wang have constructed smooth solutions

On the blow up phenomenon for the mass critical focusing Hartree equation with inverse-square potential.

In this paper, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties

Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications

In this paper, we consider the $L_x^2$ solution $u$ to mass critical NLS $iu_t+\Delta u=\pm |u|^{\frac 4d} u$. We prove that in dimensions $d\ge 4$, if the solution is spherically symmetric and is

On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d⩾2

AbstractFor the focusing mass-critical NLS $$iu_t + \Delta u = - \left| u \right|^{\tfrac{4} {d}} u$$, it is conjectured that the only global non-scattering solution with ground state mass must be a

On the focusing mass critical problem in six dimensions with splitting spherically symmetric initial data

In this paper, we consider the six-dimensional focusing mass criti- cal NLS: iut+�u = | u| 2 3 u with splitting-spherical initial data u0(x1,··· x6) = u0( q x 2 + x 2 + x 2, q x 2 + x 2 + x 2). We

Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schr\"odinger equation

We consider the focusing mass-critical NLS $iu_t + \Delta u = - |u|^{4/d} u$ in high dimensions $d \geq 4$, with initial data $u(0) = u_0$ having finite mass $M(u_0) = \int_{\R^d} |u_0(x)|^2 dx <

Minimal mass non-scattering solutions of the focusing $L^2$-critical Hartree equations with radial data

We prove that for the Cauchy problem of focusing \begin{document}$L^2$\end{document} -critical Hartree equations with spherically symmetric \begin{document}$H^1$\end{document} data in dimensions

References

SHOWING 1-10 OF 19 REFERENCES

Minimal-mass blowup solutions of the mass-critical NLS

Abstract We consider the minimal mass m 0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|4/d u to blow up. If m 0 is finite, we show that there exists

The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d)

Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation

In this paper, we show that any solution of the nonlinear Schr{o}dinger equation $iu_t+\Delta u\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the

The cubic nonlinear Schr\"odinger equation in two dimensions with radial data

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2)

On uniqueness and continuation properties after blow‐up time of self‐similar solutions of nonlinear schrödinger equation with critical exponent and critical mass

We consider the nonlinear Schrodinger equation with critical power where u: (0, T) × ℝN C and o ϕ H1 ∪ {o;|x|o ϵ L2}. With this nonlinear term, the equation (1)-(1′) has a conformal

On the Defect of Compactness for the Strichartz Estimates of the Schrödinger Equations

Abstract In this paper, we prove that every sequence of solutions to the linear Schrodinger equation, with bounded data in H1( R d), d⩾3, can be written, up to a subsequence, as an almost orthogonal

Nonlinear Schrödinger equations and sharp interpolation estimates

AbstractA sharp sufficient condition for global existence is obtained for the nonlinear Schrödinger equation $$\begin{array}{*{20}c} {(NLS)} & {2i\phi _t + \Delta \phi + \left| \phi \right|^{2\sigma

On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations

Solutions to the Cauchy problem for the equation iu t =Δu+F(|u|  2 )u (x∈ n , t>0), u(x,0)=φ(x), are considered. Conditions on φ and F are given so that, for solutions with nonpositive energy, the

Uniqueness of positive solutions of Δu−u+up=0 in Rn

We establish the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n≧1, with the Neumann

Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations

A simple duality argument shows these two problems are completely equivalent ifp and q are dual indices, (]/) + (I/q) ]. ]nteresl in Problem A when S is a sphere stems from the work of C. Fefferman