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

Frank Merle; Pierre Raphael

Displaying similar documents to “Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation”

Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain

Valeria Banica (2003)

Journées équations aux dérivées partielles

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We concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound from below the blow-up rate for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than ${(T - t)}^{- 1}$ , the expected one. Moreover, we state that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.

Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle (2011)

Journal of the European Mathematical Society

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Consider the energy-critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let $W$ be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially...

Energy Critical nonlinear Schrödinger equations in the presence of periodic geodesics

Sebastian Herr (2010)

Journées Équations aux dérivées partielles

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This is a report on recent progress concerning the global well-posedness problem for energy-critical nonlinear Schrödinger equations posed on specific Riemannian manifolds $M$ with small initial data in $H^{1} (M)$ . The results include small data GWP for the quintic NLS in the case of the $3 d$ flat rational torus $M = 𝕋^{3}$ and small data GWP for the corresponding cubic NLS in the cases $M = ℝ^{2} \times 𝕋^{2}$ and $M = ℝ^{3} \times 𝕋$ . The main ingredients are bi-linear and tri-linear refinements of Strichartz estimates which obey the critical scaling,...

Blow-up of solutions for the non-Newtonian polytropic filtration equation with a generalized source

Jun Zhou (2016)

Annales Polonici Mathematici

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This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation $u_{t} - d i v (| \nabla u^{m} |^{p - 2} \nabla u^{m}) = f (u)$ with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.

Existence globale et diffusion pour l’équation de Schrödinger non linéaire répulsive cubique sur $m a t h b b R^{3}$ en dessous l’espace d’énergie

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao (2002)

Journées équations aux dérivées partielles

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We sketch a proof of global existence and scattering for the defocusing cubic nonlinear Schrödinger equation in $H^{s} (ℝ^{3})$ for $s > \frac{4}{5}$ . The proof uses a new estimate of Morawetz type.

Changing blow-up time in nonlinear Schrödinger equations

Rémi Carles (2003)

Journées équations aux dérivées partielles

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Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^{2}$ -critical....

Global well-posedness and scattering for the mass-critical NLS

Benjamin Dodson (2011)

Journées Équations aux dérivées partielles

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Blow-up on the boundary: a survey

Marek Fila, Ján Filo (1996)

Banach Center Publications

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Non-generic blow-up solutions for the critical focusing NLS in 1-D

Joachim Krieger, Wilhelm Schlag (2009)

Journal of the European Mathematical Society

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We consider the $L^{2}$ -critical focusing non-linear Schrödinger equation in $1 + 1$ -d. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow-up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a codimension one stable blow-up manifold in the measurable category.

Blow up and near soliton dynamics for the $L^{2}$ critical gKdV equation

Yvan Martel, Frank Merle, Pierre Raphaël (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

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These notes present the main results of [, , ] concerning the mass critical (gKdV) equation $u_{t} + {(u_{x x} + u^{5})}_{x} = 0$ for initial data in $H^{1}$ close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in $H^{1}$ , construction of various exotic blow up rates in $H^{1}$ , including grow up in infinite time. ...