## Duke Mathematical Journal

### On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation

#### Abstract

We consider the nonlinear Schrödinger equation $i\partial _{t}u+\Deltau+u|u|^{p-1}=0$ in dimension $N\geq2$ and in the mass supercritical and energy subcritical range $1+\frac{4}{N}\lt p\lt \min\{\frac{N+2}{N-2},5\}$. For initial data $u_{0}\in H^{1}$ with radial symmetry, we prove a universal upper bound on the blow-up speed. We then prove that this bound is sharp and attained on a family of collapsing ring blow-up solutions first formally predicted in Fibich et al.

#### Article information

Source
Duke Math. J., Volume 163, Number 2 (2014), 369-431.

Dates
First available in Project Euclid: 29 January 2014

https://projecteuclid.org/euclid.dmj/1391007590

Digital Object Identifier
doi:10.1215/00127094-2430477

Mathematical Reviews number (MathSciNet)
MR3161317

Zentralblatt MATH identifier
1292.35283

#### Citation

Merle, Frank; Raphaël, Pierre; Szeftel, Jeremie. On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation. Duke Math. J. 163 (2014), no. 2, 369--431. doi:10.1215/00127094-2430477. https://projecteuclid.org/euclid.dmj/1391007590

#### References

• [1] V. Banica, R. Carles, and T. Duyckaerts, Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation, Comm. Partial Differential Equations 36 (2011), 487–531.
• [2] H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 489–492.
• [3] J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 197–215 (1998).
• [4] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Math. 10, New York Univ., Courant Inst. Math. Sci., New York, 2003.
• [5] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549–561.
• [6] S.-M. Chang, S. Gustafson, K. Nakanishi, and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal. 39 (2007), 1070–1111.
• [7] L. Escauriaza, G. A. Seregin, and V. Svverak, $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3–44; translation in Russian Math. Surveys 58 (2003), no. 2, 211–250.
• [8] G. Fibich, N. Gavish, and X.-P. Wang, Singular ring solutions of critical and supercritical nonlinear Schrödinger equations, Phys. D 231 (2007), 55–86.
• [9] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.
• [10] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, I: The Cauchy problem, general case, J. Funct. Anal. 32 (1979), 1–32.
• [11] J. Holmer and S. Roudenko, A class of solutions to the 3d cubic nonlinear Schrödinger equation that blow-up on a circle, Appl. Math. Res. Express AMRX, (2011), 23–94.
• [12] J. Holmer and S. Roudenko, Blow-up solutions on a sphere for the 3d quintic NLS in the energy space, Anal. PDE 5 (2012), 475–512.
• [13] C. E. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, Amer. J. Math. 133 (2011), 1029–1065.
• [14] J. Krieger, E. Lenzman, and P. Raphaël, Nondispersive solutions to the $L^{2}$-critical half-wave equation, Arch. Ration. Mech. Anal. 209 (2013), 61–129.
• [15] J. Krieger, Y. Martel, and P. Raphaël, Two-soliton solutions to the three-dimensional gravitational Hartree equation, Comm. Pure Appl. Math. 62 (2009), 1501–1550.
• [16] J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), 543–615.
• [17] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p}=0$ in ${R}^{n}$, Arch. Rational Mech. Anal. 105 (1989), 243–266.
• [18] Y. Martel, Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, Amer. J. Math. 127 (2005), 1103–1140.
• [19] Y. Martel, F. Merle, and P. Raphaël, Blow up for gKdV I: Dynamics near the ground state, to appear in Acta. Math.
• [20] Y. Martel, F. Merle, and P. Raphaël, Blow up for gKdV II: The minimal mass solution, submitted.
• [21] F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys. 129 (1990), 223–240.
• [22] F. Merle and P. Raphaël, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. Math. 161 (2005), 157–222.
• [23] F. Merle and P. Raphaël, Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), 591–642.
• [24] F. Merle, and P. Raphaël, On universality of blow up profile for $L^{2}$ critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), 565–672.
• [25] F. Merle, and P. Raphaël, Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), 37–90.
• [26] F. Merle, and P. Raphaël, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), 675–704.
• [27] F. Merle, and P. Raphaël, Blow up of the critical norm for some radial $L^{2}$ super critical nonlinear Schrödinger equations, Amer. J. Math. 130 (2008), 945–978.
• [28] F. Merle, P. Raphaël, and I. Rodnianksi, Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map, submitted, arXiv:1106.0912.
• [29] F. Merle, P. Raphaël, and J. Szeftel, Stable self similar blow up dynamics for slightly $L^{2}$ supercritical NLS equations, Geom. Funct. Anal. 20 (2010), 1028–1071.
• [30] F. Merle, P. Raphaël, and J. Szeftel, The instability of Bourgain-Wang solutions for the $L^{2}$ critical NLS, Amer. J. Math. 135 (2013), 567–1017.
• [31] K. Nakanishi and W. Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2011.
• [32] G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré 2 (2001), 605–673.
• [33] G. Perelman, Analysis seminar, Université de Cergy Pontoise, dec 2011 (joint work with J. Holmer and S. Roudenko).
• [34] P. Raphaël, Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation, Math. Ann. 331 (2005), 577–609.
• [35] P. Raphaël, Existence and stability of a solution blowing up on a sphere for a $L^{2}$ supercritical nonlinear Schrödinger equation, Duke Math. J. 134 (2006), 199–258.
• [36] P. Raphaël and I. Rodnianski, Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 1–122.
• [37] P. Raphaël and J. Szeftel, Standing ring blow up solutions to the N-dimensional quintic nonlinear Schrödinger equation, Comm. Math. Phys. 290, 973–996, 2009.
• [38] P. Raphaël and J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc. 24 (2011), 471–546.
• [39] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567–576.
• [40] Y. Zwiers, Standing ring blowup solutions for cubic nonlinear Schrödinger equations, Anal. PDE 4 (2011), 677–727.