Log Log Blowup Regime

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This page outlines a series of lectures to be given on the loglog blowup regime for L2-critical NLS.

Contents

Lecture 1: Overview (2 October 2008)

Lecture 2: Q

  • Variational Characterization / sharp Gagliardo-Nirenberg
  • Orbital Stability

Sources: [Raph 08, Section 2.5, p24-25] - solely a statement in concentration compactness. This introduces the notation/ideas of profile decomposition, and proves the only part of [Merl 04, Theorem 2] we can without limiting measures or the spectral machinery. Raphaël’s proof is based on [Gera 98], whose proof is outlined [Raph 08, Proposition 4, p8-12].


Lecture 3: Self-Similar

  • More on Orbitary Stability

That weakly converging profiles are stable under converging initial data. [Merl 04, Lemma 3, p589]. In particular, maximal lifetime is (essentially) lower semicontinuous.

  • No Blowups Occur with Scaling Lower Bound Speed

[Merl 04, Corollary 2, p607-608]

  • Self-Similar Profiles
    • Asymptotic behaviour of Qb (not in L2 ) is shown in [John 93].
    • Existence of Qb , convergence to Q, and decay/degeneracy are first proven [Merl 03, p603-614].
    • Convergence is proven again [Merl 04, Proposition 8, p609], but with more precise control on where Qb is truncated.
    • Radiation ζb and its properties are first introduced [Merl 04, Lemma 15, p638]. To involve ζ

in the refined virial estimate, [Merl 06, lines (2.18) and (2.20)] introduces two new properties.

Lecture 4: Energy, Momentum and Orthogonal Decomposition

  • Linearized Hamiltonian and Weinstein

A proper motivation/derivation of the Spectral Hypothesis using Weinstein’s algebraic properties of the linearized Hamiltonian would not be out of place. The only real discussion in the MR papers is [Merl 05c, p171-172].

  • Bootstrap/Dynamic Nature

These terms are really interchangeable. We can't over-emphasize that \nabla \epsilon controls the dynamics of parameters that control orthogonal relationships that control \nabla \epsilon. Reduce Reuse Recycle.

  • Momentum

To avoid global momentum estimates, [Merl 05b] and [Plan 07] decompose about {\tilde{Q}}_{b, \beta} = e^{i \beta \cdot y} Q_b . It may be worth incorporating this. Otherwise, I suggest the exposition of [Raph 05]. (Incidentally, the same approach as [Merl 05a]).

Lecture 5: Blowup-speed Alternative

  • Log-log Upper Bound

By following the approach of [Raph 05], the virial estimate can also be used to prove a nice compliment to Lecture 2,

  • No Blowups Between Log-Log and t − 1.

Lecture 6: Sharp Lower Bound

Technical. For a succinct overview of the original [Merl 06], please refer to [Merl 05b, p942-949].

Lecture 7: Omitted Blockbusters MIA

  • [Merl 04, Theorem 5].

There is only one blowup solution that converges to zero in Σ; for all other negative energy solutions the classic virial is wide of the mark.

  • [Merl 05a]

Excluding the blowup profile, fast blowups converge strongly in H^1_x at blowup time. Log-log solutions do not, and leave behind a characteristic distribution of mass. Proving the distribution of mass at log-log blowup time makes use of the control on \| u \|_{L^1_t H^1_x}.

References

  • [Gera 98] P. G´erard. “Description du D´efaut de Compacit´e de L’Injection de Sobolev”. ESAIM Control Optom. Calc. Var., Vol. 3, pp. 213–233, 1998.
  • [John 93] R. Johnson and X. Pan. “On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation”. Proc. Roy. Soc. Edin, Vol. 123A, pp. 763–782, 1993.
  • [Merl 03] F. Merle and P. Rapha¨el. “Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation”. Geom. Funct. Anal., Vol. 13, No. 3, pp. 591–642, 2003.
  • [Merl 04] F. Merle and P. Rapha¨el. “On universality of blow-up profile for L2 critical nonlinear Schrödinger equation”. Invent. Math., Vol. 156, No. 3, pp. 565–672, 2004.
  • [Merl 05a] F. Merle and P. Rapha¨e. “Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation”. Commun. Math. Phys., Vol. 253, pp. 675–704, 2005.
  • [Merl 05b] F. Merle and P. Rapha¨el. “On one blow up point solutions to the critical nonlinear Schrödinger equation”. J. Hyperbolic Diff. Eqn., Vol. 2, No. 4, pp. 919–962, 2005.
  • [Merl 05c] F. Merle and P. Rapha¨el. “The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation”. Ann. Math., Vol. 161, pp. 157–222, 2005.

Despite date of publication, content precedes that of [Merl 03].

  • [Merl 06] F. Merle and P. Rapha¨el. “On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation”. J. Amer. Math. Soc., Vol. 19, No. 1, pp. 37–90, 2006.

Despite date of publication, content directly succeeds [Merl 04].

  • [Plan 07] F. Planchon and P. Rapha¨el. “Existence and Stability of the log-log Blow-up Dynamics for the L2 -Critical Nonlinear Schrödinger Equation in a Domain”. Ann. Henri Poincar´e, Vol. 8, pp. 1177–1219, 2007.
  • [Raph 05] P. Rapha¨el. “Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation”. Math. Ann., Vol. 331, pp. 577–609, 2005.
  • [Raph 08] P. Rapha¨el. “Stability and blow up for the non linear Schrödinger equation”. 2008. Lecture notes from CMI Summer School at Zürich.
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