Mass-critical NLS Research Articles

The [formula omitted] sequential convergence of a solution to the mass-critical NLS above the ground state

In this paper we generalize a weak sequential result of Fan (2021) to a non-scattering solutions in dimension d≥2. No symmetry assumptions are required for the initial data. We build on a previous result of Dodson (2020) for one dimension.

The $L^{2}$ Sequential Convergence of a Solution to the One-Dimensional, Mass-Critical NLS above the Ground State

In this paper we generalize a weak sequential result of [C. Fan, Int. Math. Res. Not. IMRN, 2021 (2021), pp. 4864--4906] to any nonscattering solutions in one dimension. No symmetry assumptions are...

Refined mass-critical Strichartz estimates for Schrödinger operators

We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article. Using phase space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear restriction estimate for paraboloids to more general Schr\"{o}dinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large data solutions to the $L^2$-critical NLS with a harmonic oscillator potential in dimensions $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.

Low regularity blowup solutions for the mass-critical NLS in higher dimensions

In this paper, we study the Hs-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphaël) for the focusing mass-critical nonlinear Schrödinger equations i∂tu+Δu+|u|4du=0 in Rd with d≥3. We aim to extend the result in Colliander and Raphael (2009) [5] for dimension two to the higher dimensions cases d≥3, where we use the bootstrap argument in the above paper and the commutator estimates in Visan and Zhang (2007) [24].

$L^2$ concentration of blow-up solutions for the mass-critical NLS with inverse-square potential

In this paper, we prove a refined version of a compactness lemma and we use it to establish mass-concentration for the focusing nonlinear Schrodinger equation with an inverse-square potential.

Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

We consider a mass-critical system of nonlinear Sch\"{o}dinger equations \begin{align*} \begin{cases} i\partial_t u +\Delta u =\bar{u}v,\\ i\partial_t v +\kappa \Delta v =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*} where $(u,v)$ is a $\mathbb{C}^2$-valued unknown function and $\kappa >0$ is a constant. If $\kappa =1/2$, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $M(u,v)<M(\phi ,\psi)$, where $M(u,v)$ denotes the mass and $(\phi ,\psi)$ is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson \cite{MR3406535}. Scattering is also obtained without mass-resonance condition under the restriction that $(u,v)$ is radially symmetric.

Existence of multi-solitary waves with logarithmic relative distances for the NLS equation

We construct 2-solitary wave solutions with logarithmic distance to the nonlinear Schrödinger equation,i∂tu+Δu+|u|p−1u=0,t∈R,x∈Rd, in mass-subcritical cases 1<p<1+4d and mass-supercritical cases 1+4d<p<d+2d−2, i.e. solutions u(t) satisfying‖u(t)−eiγ(t)∑k=12Q(⋅−xk(t))‖H1→0 and|x1(t)−x2(t)|∼2log⁡t,ast→+∞, where Q is the ground state. The logarithmic distance is related to strong interactions between solitary waves.In the integrable case (d=1 and p=3), the existence of such solutions is known by inverse scattering (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). The mass-critical case p=1+4d exhibits a specific behavior related to blow-up, previously studied in Y. Martel, P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).

The L2 Weak Sequential Convergence of Radial Focusing Mass Critical NLS Solutions with Mass Above the Ground State

Abstract We study the non-scattering $L^{2}$ solution $u$ to the radial focusing mass-critical nonlinear Schrödinger equation with mass just above the ground state, and show that there exists a time sequence $\{t_{n}\}_{n}$, such that $u(t_{n})$ weakly converges to the ground state $Q$ up to scaling and phase transformation. We also give some partial results on the mass concentration phenomena of the minimal mass blow-up solution.

Scattering for the two dimensional NLS with (full) exponential nonlinearity

We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schrödinger equation \begin{document}$iu_t+Δ u = u(e^{4π |u|^2}-1)$ \end{document} in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mass-critical NLS to employ the techniques of Tao-Visan-Zhang from [25]. This permits us to combine the known spacetime estimates for mass-critical NLS proved by Dodson [12] and the work of [15] and [14] to prove corresponding spacetime estimates which imply scattering.

Remark on the periodic mass critical nonlinear Schrödinger equation

We consider the mass critical NLS onT\mathbb {T}andT2\mathbb {T} ^2. In theRd\mathbb {R}^dcase the Strichartz estimates enable us to show well-posedness of the IVP inL2L^2(at least for small data) via the Picard iteration method. However, counterexamples to theL6L^6Strichartz onT\mathbb {T}and theL4L^4Strichartz onT2\mathbb {T}^2were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP onT\mathbb {T}andT2\mathbb {T}^2cannot have a smooth data-to-solution map inL2L^2even for small initial data. The same results are also obtained for most of the two dimensional irrational tori.

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

For 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 solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for Hx1 initial data in dimensions d = 2, 3 with spherical symmetry and d ⩾ 4 with certain splitting-spherically symmetric initial data.

Maximizers for the Strichartz norm for small solutions of mass-critical NLS

Consider the mass-critical nonlinear Schrodinger equations in both focusing and defocusing cases for initial data in L2 in space dimension N . By Strichartz inequality, solutions to the corresponding linear problem belong to a global L p space in the time and space variables, where p = 2 + 4 N . In 1D and 2D, the best constant for the Strichartz inequality was computed by D. Foschi who has also shown that the maximizers are the solutions with Gaussian initial data. Solutions to the nonlinear problem with small initial data in L2 are globally defined and belong to the same global L p space. In this work we show that the maximum of the L p norm is attained for a given small mass. In addition, in 1D and 2D, we show that the maximizer is unique and obtain a precise estimate of the maximum. In order to prove this we show that the maximum for the linear problem in 1D and 2D is nondegenerated. Mathematics Subject Classification (2010): 35Q55 (primary); 35P25, 35B50, 35B45 (secondary).

Global Well-posedness and Scattering for the Defocusing Cubic nonlinear Schrödinger equation in Four Dimensions

In this short note, we present a new proof of the global well-posedness and scattering result for the defocusing energy-critical nonlinear Schrödinger equation (NLS) in four space dimensions obtained previously by Ryckman and Visan [“Global well-posedness and scattering for the defocusing energycritical nonlinear Schrödinger equation in ⁠.” American Journal of Mathematics 129 (2007): 1–60. MR2288737]. The argument is inspired by the recent work of Dodson [“Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d≥3.” (2009): preprint arXiv:0912.2467.] on the mass-critical NLS.

Global well-posedness and scattering for the mass-critical NLS

Global well-posedness and scattering for the mass-critical NLS

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 $H^1$ solutions with $\|u\|_{L^2} M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow up elements at a non degenerate point, hence extending the pioneering work by Merle who treated the pseudo conformal invariant case $k\equiv 1$.

Regularity 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 e > 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 scattering conjecture for mass critical NLS without reducing to three enemies [17], [18]. As another important application, we establish a Liouville type result for Lx initial data with ground state mass. We prove that if a radial Lx solution to focusing mass critical problem has the ground state mass and does not scatter in both time directions, then it must be global and coincide with the solitary wave up to symmetries. Here the ground state is the unique, positive, radial solution to elliptic equation ∆Q − Q + Q 4 d = 0. This is the first rigidity type result in scale invariant space Lx.

Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

We consider the focusing mass-critical NLS iut+�u = −|u| 4/d u in high dimensions d ≥ 4, with initial data u(0) = u0 hav- ing finite mass M(u0) = R Rd |u0(x)| 2 dx < ∞. It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class C 0 t,loc L 2 ∩ L 2loc L 2d/(d 2) x , and also admits global (but not unique) weak solutions in L 1 L 2. In this paper we introduce an intermediate class of solution, which we call a semi-Strichartz class solution, for which one does have global ex- istence and uniqueness in dimensions d ≥ 4. In dimensions d ≥ 5 and assuming spherical symmetry, we also show the equivalence of the Strichartz class and the strong solution class (and also of the semi-Strichartz class and the semi-strong solution class), thus establishing unconditional uniqueness results in the strong and semi-strong classes. With these assumptions we also characterise these solutions in terms of the continuity properties of the mass function t 7→ M(u(t)).

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

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 minimal-mass nonscattering solutions are, up to phase rotation and scaling, the pseudoconformal ground state and the ground state solitary wave.