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

Raphaël, Pierre¹ ; Szeftel, Jeremie²

¹ Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
² Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75 005 Paris, France

Journal of the American Mathematical Society, Tome 24 (2011) no. 2, pp. 471-546

Résumé

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 a standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}$ are global in time while a finite time blow-up singularity formation may occur for $\|u\|_{L^2}>M_k$. 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 nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case $k\equiv 1$.

Détail

DOI : 10.1090/S0894-0347-2010-00688-1

Bibliographie

[1] Banica, Valeria Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2004 139 170

[2] Blanchet, Adrien, Carrillo, Jos㩠A., Masmoudi, Nader Infinite time aggregation for the critical Patlak-Keller-Segel model in ℝ² Comm. Pure Appl. Math. 2008 1449 1481

[3] Bourgain, Jean, Wang, W. Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1997

[4] Burq, N., Gã©Rard, P., Tzvetkov, N. Two singular dynamics of the nonlinear Schrödinger equation on a plane domain Geom. Funct. Anal. 2003 1 19

[5] Chang, Shu-Ming, Gustafson, Stephen, Nakanishi, Kenji, Tsai, Tai-Peng Spectra of linearized operators for NLS solitary waves SIAM J. Math. Anal. 2007/08 1070 1111

[6] Duyckaerts, Thomas, Merle, Frank Dynamic of threshold solutions for energy-critical NLS Geom. Funct. Anal. 2009 1787 1840

[7] Fibich, Gadi, Merle, Frank, Raphaã«L, Pierre Proof of a spectral property related to the singularity formation for the 𝐿² critical nonlinear Schrödinger equation Phys. D 2006 1 13

[8] Gidas, B., Ni, Wei Ming, Nirenberg, L. Symmetry and related properties via the maximum principle Comm. Math. Phys. 1979 209 243

[9] Ginibre, J., Velo, G. On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case J. Functional Analysis 1979 1 32

[10] Kenig, Carlos E., Merle, Frank Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case Invent. Math. 2006 645 675

[11] Killip, Rowan, Li, Dong, Visan, Monica, Zhang, Xiaoyi Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS SIAM J. Math. Anal. 2009 219 236

[12] Krieger, Joachim, Lenzmann, Enno, Raphaã«L, Pierre On stability of pseudo-conformal blowup for 𝐿²-critical Hartree NLS Ann. Henri Poincaré 2009 1159 1205

[13] Krieger, Joachim, Martel, Yvan, Raphaã«L, Pierre Two-soliton solutions to the three-dimensional gravitational Hartree equation Comm. Pure Appl. Math. 2009 1501 1550

[14] Krieger, J., Schlag, W. Non-generic blow-up solutions for the critical focusing NLS in 1-D J. Eur. Math. Soc. (JEMS) 2009 1 125

[15] Kwong, Man Kam Uniqueness of positive solutions of Δ𝑢-𝑢+𝑢^{𝑝} Arch. Rational Mech. Anal. 1989 243 266

[16] Lions, P.-L. The concentration-compactness principle in the calculus of variations. The locally compact case. I Ann. Inst. H. Poincaré Anal. Non Linéaire 1984 109 145

[17] Martel, Yvan Asymptotic 𝑁-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations Amer. J. Math. 2005 1103 1140

[18] Martel, Yvan, Merle, Frank Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation Ann. of Math. (2) 2002 235 280

[19] Martel, Yvan, Merle, Frank Nonexistence of blow-up solution with minimal 𝐿²-mass for the critical gKdV equation Duke Math. J. 2002 385 408

[20] Martel, Yvan, Merle, Frank Multi solitary waves for nonlinear Schrödinger equations Ann. Inst. H. Poincaré C Anal. Non Linéaire 2006 849 864

[21] Merle, F. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power Duke Math. J. 1993 427 454

[22] Merle, Frank Construction of solutions with exactly 𝑘 blow-up points for the Schrödinger equation with critical nonlinearity Comm. Math. Phys. 1990 223 240

[23] Merle, Franck Nonexistence of minimal blow-up solutions of equations 𝑖𝑢_{𝑡} Ann. Inst. H. Poincaré Phys. Théor. 1996 33 85

[24] Merle, Frank, Raphael, Pierre The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation Ann. of Math. (2) 2005 157 222

[25] Merle, F., Raphael, P. Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation Geom. Funct. Anal. 2003 591 642

[26] Merle, Frank, Raphael, Pierre On universality of blow-up profile for 𝐿² critical nonlinear Schrödinger equation Invent. Math. 2004 565 672

[27] Merle, Frank, Raphael, Pierre On a sharp lower bound on the blow-up rate for the 𝐿² critical nonlinear Schrödinger equation J. Amer. Math. Soc. 2006 37 90

[28] Planchon, Fabrice, Raphaã«L, Pierre Existence and stability of the log-log blow-up dynamics for the 𝐿²-critical nonlinear Schrödinger equation in a domain Ann. Henri Poincaré 2007 1177 1219

[29] Raphael, Pierre Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation Math. Ann. 2005 577 609

[30] Weinstein, Michael I. Nonlinear Schrödinger equations and sharp interpolation estimates Comm. Math. Phys. 1982/83 567 576

[31] Weinstein, Michael I. Lyapunov stability of ground states of nonlinear dispersive evolution equations Comm. Pure Appl. Math. 1986 51 67