The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation

By F. Merle, Pierre Raphael. 2005

373 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bfe1f87cd9bab9ac1b4cc1ee0b56ce0543240e30

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In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

On Blow-up Solutions to the 3D Cubic Nonlinear Schrödinger Equation (2010, result): “Raphaël’s result [21] draws upon a large body of breakthrough work by Merle-Raphaël [12]-[17] [21] (see also Fibich-Merle-Raphaël [6]) on the blow-up problem for the L critical NLS.”
Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS (2010, methodology): “Uniqueness then follow by integrating the flow close to uc backwards from infinity using a dynamical system type of argument and the a priori 7which is useless until we know the blow up speed 8as opposed to the works [20], [26], [9]”
On universality of blow-up profile for L2 critical nonlinear Schrödinger equation (2004, methodology): “Remark 1 Note that this property has been proved in [ 22 ] for dimension N = 1 and constant 2 − = 9 .”
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems (2009, background): “[28℄ Merle, F.; Raphaël, P., Sharp lower bound on the blow up rate for riti al nonlinearS hrödinger equation, J. Amer.”
On growth rate near the blowup surface for semilinear wave equations (2005, background): “, Giga and Kohn [5, 6, 7], and Merle and Zaag [15] for the semilinear heat equation, and see Merle and Raphael [12, 13, 14] for the critical nonlinear Schrödinger equation (NLS)).”
Kink dynamics in the $\phi^4$ model: asymptotic stability for odd perturbations in the energy space (2015, methodology): “Using the arguments of the proof of Lemma 13 in [27], (4.”
On collapsing ring blow up solutions to the mass supercritical NLS (2012, background): “The situation is better understood in the mass critical case sc = 0 since the series of works [33], [23], [24], [25], [26], [27] where a stable blow up regime of "log-log" type is exhibited in dimension N ≤ 5 with a complete description of the associated bubble of concentration.”
On the formation of singularities in the critical $O(3)$ $\sigma$-model (2006, neutral): “Merle and Raphael in their work [ 28 ] on the blow-up analysis for the critical focussing non-linear Schr¨odinger equation.”
Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation (2008, methodology): “For a similar approach of the blow-up problem, see Martel and Merle [11], [10] for the critical KdV equation; see also Merle and Raphaël [12], [13] for the NLS equation.”
The cubic nonlinear Schr\"odinger equation in two dimensions with radial data (2007, background): “In a series of papers [41, 42, 43], Merle and Raphael gave a more or less complete description of blowup behaviour at masses slightly above that of the ground state (and finite energy).”
Uniqueness of Radial Solutions for the Fractional Laplacian (2013, neutral): “[37, 32]).”
Minimal-mass blowup solutions of the mass-critical NLS (2006, neutral): “[2], [4], [6], [7], [8], [17], [20], [21], [22], [23], [24], [25], [26], [27], [39].”
Proof of a Spectral Property related to the singularity formation for the L2 critical nonlinear Schrödinger equation (2006, methodology): “Using this new orthogonality condition and Lemma 8 in [16] in the analysis of the series of papers [14] [17], we can prove Theorem 1.”
Determination of the blow-up rate for a critical semilinear wave equation (2005, background): “The wave equation has no new blow-up rate at the critical case, unlike NLS where new blow-up rates appear at the critical case (with respect to the conformal invariance), see Merle and Raphaël [13], [12], [14], or the semilinear heat equation where we have new blow-up rates at the critical case (wit”
The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities (2005, background): “For results on this problem and a more detailed list of references see [17, 18].”
Existence and Stability of the log–log Blow-up Dynamics for the L2-Critical Nonlinear Schrödinger Equation in a Domain (2007, methodology): “For N = 1, this property was proved in [22], using the explicit formula for the ground state Q.”
Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension (2007, methodology): “See Martel and Merle [13] for the case of the L2 critical Korteweg de Vries equation, and Merle and Raphaël [14] for the case of the L2 critical nonlinear Schrödinger equation.”