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Galina Perelman

Publications and source records attributed to Galina Perelman.

12 recordsLinked to original sources
arXiv2020

Global well-posedness for the derivative nonlinear Schrödinger equation

This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in $H^{\frac12}$ with mass strictly less than $4π$ or general initial conditions in the weighted Sobolev space $H^{2, 2}$. In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in $H^{\frac12}$ and that furthermore the $H^{\frac12}$ norm of the solutions remains globally bounded in time. One should recall that for $H^s$, with $s < 1 / 2 $, the associated Cauchy problem is ill-posed in the sense that uniform continuity with respect to the initial data fails. Thus, our result closes the discussion in the setting of the Sobolev spaces $H^s$. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.

math.AP↗
arXiv2019

Blow up dynamics for the hyperbolic vanishing mean curvature flow of surfaces asymptotic to Simons cone

In this article, we establish the existence of a family of hypersurfaces $(Γ(t))_{0< t \leq T}$ which evolve by the vanishing mean curvature flow in Minkowski space and which as $t$ tends to~$0$ blow up towards a hypersurface which behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second order quasilinear wave equation. Our constructive approach consists in proving the existence of finite time blow up solutions of this hyperbolic equation under the form $u(t,x) \sim t^ {ν+1} Q\Big(\frac {x} {t^ {ν+1}} \Big) $, where~$Q$ is a stationary solution and $ν$ an arbitrary large positive irrational number. Our approach roughly follows that of Krieger, Schlag and Tataru. However contrary to these works, the equation to be handled in this article is quasilinear. This induces a number of difficulties to face.

math.AP↗
arXiv2013

Scattering for the critical 2-D NLS with exponential growth

In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schrödinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in \cite{CGT, PV} and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\R^2)$ into the critical Orlicz space ${\cL}(\R^2)$ settled in \cite{BMM}. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.

math.AP↗
arXiv2012

Blow up dynamics for equivariant critical Schrödinger maps

For the Schrödinger map problem from 2+1 dimensions into the 2-sphere we prove the existence of equivariant finite time blow up solutions that are close to a dynamically rescaled lowest energy harmonic map, the scaling parameter being given by $t^{-ν}$ with $ν>3/2$.

math.AP↗
arXiv2012

A solution to the focusing 3d NLS that blows up on a contracting sphere

We rigorously construct radial $H^1$ solutions to the 3d cubic focusing NLS equation $i\partial_t ψ+ Δψ+ 2 |ψ|^2ψ=0$ that blow-up along a contracting sphere. With blow-up time set to $t=0$, the solutions concentrate on a sphere at radius $\sim t^{1/3}$ but focus towards this sphere at the faster rate $\sim t^{2/3}$. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov (1975) and independently later in Holmer-Roudenko (2007), where it was demonstrated to be consistent with all conservation laws of this equation. In the latter paper, it was proposed as a solution that would yield divergence of the $L_x^3$ norm within the "wide" radius $\sim |\nabla u(t)|_{L_x^2}^{-1/2}$ but not within the "tight" radius $\sim |\nabla u(t)|_{L_x^2}^{-2}$, the second being the rate of contraction of self-similar blow-up solutions observed numerically and described in detail in Chapter 7 of Sulem-Sulem (1998).

math.AP↗
arXiv2010

Effective dynamics of double solitons for perturbed mKdV

We show that an interacting double soliton solution to the perturbed mKdV equation is close in $H^2$ to a double soliton following an effective dynamics obtained as Hamilton's equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof.

math.AP↗
arXiv2009

Vey theorem in infinite dimensions and its application to KdV

We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $κ$-smoothing map ($κ\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^\prime: H\to H$, the germ $(F^\prime-id)$ is $κ$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with $κ=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form `identity plus a 1-smoothing analytic map'.

math.DS↗
arXiv2004

Stark-Wannier type operators with purely singular spectrum

We consider the one-dimensional Stark-Wannier type operators with potentials given by a smooth function with a logarithmic growth at infinity plus a periodic function with the Fourier coefficients of the form $(\ln |n|)^{-b}, 0<b<1/2$. We prove that in the case of rational electric field the spectrum of the corresponding operator is purely singular continuous.

math-ph↗