4,084 research outputs found
Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS
We consider the 2-dimensional focusing mass critical NLS with an
inhomogeneous nonlinearity: . From
standard argument, there exists a threshold such that solutions
with are global in time while a finite time blow up
singularity formation may occur for . In this paper, we
consider the dynamics at threshold and give a necessary and
sufficient condition on to ensure the existence of critical mass finite
time blow up elements. Moreover, we give a complete classification in the
energy class of the minimal finite time blow up elements at a non degenerate
point, hence extending the pioneering work by Merle who treated the pseudo
conformal invariant case
On melting and freezing for the 2d radial Stefan problem
We consider the two dimensional free boundary Stefan problem describing the
evolution of a spherically symmetric ice ball . We
revisit the pioneering analysis of [20] and prove the existence in the radial
class of finite time melting regimes which respectively
correspond to the fundamental stable melting rate, and a sequence of
codimension excited regimes. Our analysis fully revisits a
related construction for the harmonic heat flow in [42] by introducing a new
and canonical functional framework for the study of type II (i.e. non self
similar) blow up. We also show a deep duality between the construction of the
melting regimes and the derivation of a discrete sequence of global-in-time
freezing regimes which correspond
respectively to the fundamental stable freezing rate, and excited regimes which
are codimension stable.Comment: 70 pages, a few references added and typos correcte
On the stability of critical chemotactic aggregation
We consider the two dimensional parabolic-elliptic Patlak-Keller-Segel model
of chemotactic aggregation for radially symmetric initial data. We show the
existence of a stable mechanism of singularity formation and obtain a complete
description of the associated aggregation process.Comment: 80 page
Blow up for the critical gKdV equation II: minimal mass dynamics
We fully revisit the near soliton dynamics for the mass critical (gKdV)
equation.
In Part I, for a class of initial data close to the soliton, we prove that
only three scenario can occur:
(BLOW UP) the solution blows up in finite time in a universal regime with
speed ;
(SOLITON) the solution is global and converges to a soliton in large time;
(EXIT) the solution leaves any small neighborhood of the modulated family of
solitons in the scale invariant norm.
Regimes (BLOW UP) and (EXIT) are proved to be stable. We also show in this
class that any nonpositive energy initial data (except solitons) yields finite
time blow up, thus obtaining the classification of the solitary wave at zero
energy.
In Part II, we classify minimal mass blow up by proving existence and
uniqueness (up to invariances of the equation) of a minimal mass blow up
solution . We also completely describe the blow up behavior of .
Second, we prove that is the universal attractor in the (EXIT) case,
i.e. any solution as above in the (EXIT) case is close to (up to
invariances) in at the exit time. In particular, assuming scattering for
(in large positive time), we obtain that any solution in the (EXIT)
scenario also scatters, thus achieving the description of the near soliton
dynamics
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