Blow up in finite time and dynamics of blow up solutions for the L^2-critical generalized KdV equation

By Yvan Martel, F. Merle. 2002

160 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/35ea6137bfb6ff175978bcc4044400bddc1a00ca

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gKdV Equations (2001, background): “– In the critical case p = 5, any solution with negative energy initially close to the soliton blows up in finite or infinite time in H 1 (Merle [20]), and actually blows up in finite time if the initial data satisfies in addition a polynomial decay condition on the right in space (Martel and Merle ”
Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications (2008, background): “Finally, in Section 6 we present a general argument (in the spirit of [24, 25]), which shows that the general “rigidity theorem” is a consequence of the special cases proved in Section 5.”
On universality of blow-up profile for L2 critical nonlinear Schrödinger equation (2004, result): “This last property is based on a surprising algebraic cancellation similar to the one exhibited in [19].”
Blow up for the critical gKdV equation II: minimal mass dynamics (2012, background): “In the series of works [26, 27, 38, 28, 29], Martel and Merle obtain the first qualitative information on the flow for small super critical mass initial data ‖Q‖L2 < ‖u0‖L2 < ‖Q‖L2 < α∗, 0 < α∗ ≪ 1, in particular the existence of finite time blow up solutions for E0 < 0 and the classification of Q a”
Multi-Soliton Solutions for the Supercritical gKdV Equations (2010, unknown): “In the case p = 5, the existence of finite time blow up solutions was proved by Merle [17] and Martel and Merle [12].”
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems (2009, background): “To derive the precise law for b we examine 4Such an upper bound is already sufficient to conclude the finite time blow up and establish a lower bound on the concentration scale λ(t), see [23], [22] for related considerations.”
On the formation of singularities in the critical $O(3)$ $\sigma$-model (2006, unknown): “See, however, the work of Martel/Merle [27] on the critical KdV problem for an example of a constructive finite time blowup mechanism.”
Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation (2005, background): “On the other hand, for p = 4 there is a finite-time blow-up [19].”
On the Cauchy Problem for the Generalized Korteweg-de Vries Equation (2003, background): “On the other hand it has been recently proved in Martel and Merle (2002), Merle (2001) that blow up in finite time can occur in the critical case k ¼ 4.”