Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation

By Yvan Martel, F. Merle. 2004

162 citations recorded by the source corpus.

The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H 1 (L 2 norm and energy). We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions (and thus global existence). From [15], there do exist for this equation solutions u(t) such that |u(t)| H1 → +∞ as t ↑ T, where T < +∞ (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs. For solutions with L 2 mass close to the minimal mass allowing blow up and with decay in L 2 at the right, we prove after rescaling and translation which leave invariant the L 2 norm that the solution converges to a universal profile locally in space at the blow-up time T. From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e660c9753de32c7b2f8237b0adffee8ca99f9cdb

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Citations in the corpus

The role of self-similarity in singularities of partial differential equations (2008, background): “In [104], [103] it was shown that in the critical case p = 5 solutions may blow-up both in finite and in infinite time.”
Local well-posedness and blow up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations (2010, methodology): “Remark finally that in addition to [25] and [30], two further works ([26] and [27]) provide refined information about the blow up phenomenon for the critical (gKdV) equation close to the soliton family.”
On universality of blow-up profile for L2 critical nonlinear Schrödinger equation (2004, background): “3 Stability of weak convergence: proof of Lemma 3 The proof is similar to the one exhibited in [18] and we shall briefly sketch arguments.”
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation (2005, methodology): “It has been proved in the papers [13], [14], [15], [16], [17] that for α0 small enough, if E(u0) < 0 and ∫ |u0|(2) < ∫ Q2 + α0, then: (i) Blow-up occurs in finite or infinite time, i.”
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]”
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”
Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation (2005, unknown): “For example, for the generalized critical KdV equation, the blow up picture is similar in some sense to the one of (1), and indeed universality of Q as a blow up profile has been proved in [8].”
Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation (2009, methodology): “by classification of compact solutions of the GKdV equation: see Martel Merle [MM00, MM01, MM02 ]; - for the mass-critical NLS equation”
Classification of radial solutions of the focusing, energy-critical wave equation (2012, methodology): “Results of this type for other equations in the global case include those for data close to the soliton for subcritical nonlinearities: see the works of Martel and Merle [27] in the case of generalized KdV equations, of Buslaev and Perelman [4, 3] in the case of one dimensional NLS with specific non”
Blow up for the critical gKdV equation III: exotic regimes (2012, background): “It is proved in [44, 33] that initial data u0 such that (1.”
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.”
Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension (2007, methodology): “Proof of Proposition 2.1: See Antonini and Merle [4] and Appendix A.”