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Oana Pocovnicu

Publications and source records attributed to Oana Pocovnicu.

At least 19 recordsLinked to original sources
arXiv2025

Invariant Gibbs dynamics for two-dimensional fractional wave equations in negative Sobolev spaces

We consider a fractional nonlinear wave equations (fNLW) with a general power-type nonlinearity, on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for a renormalized fNLW. We first construct the Gibbs measure associated with this equation by using the variational approach of Barashkov and Gubinelli. We then prove almost sure local well-posedness with respect to Gibbsian initial data, by exploiting the second order expansion. Finally, we extend solutions globally in time using Bourgain's invariant measure argument.

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arXiv2024

A remark on randomization of a general function of negative regularity

In the study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, we typically decompose a classically ill-defined solution map into two steps, where, in the first step, we use stochastic analysis to construct various stochastic objects. The simplest kind of such stochastic objects is the Wick powers of a basic stochastic term (namely a random linear solution, a stochastic convolution, or their sum). In the case of randomized initial data of a general function of negative regularity for studying nonlinear wave equations (NLW), we show necessity of imposing additional Fourier-Lebesgue regularity for constructing Wick powers by exhibiting examples of functions slightly outside $L^2(\mathbb T^d)$ such that the associated Wick powers do not exist. This shows that probabilistic well-posedness theory for NLW with general randomized initial data fails in negative Sobolev spaces (even with renormalization). Similar examples also apply to stochastic NLW and stochastic nonlinear heat equations with general white-in-time stochastic forcing, showing necessity of appropriate Fourier-Lebesgue $γ$-radonifying regularity in the construction of the Wick powers of the associated stochastic convolution.

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arXiv2020

Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces

We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local well-posedness of the renormalized NLW in negative Sobolev spaces. We also prove a new instability result for the defocusing cubic NLW without renormalization in negative Sobolev spaces, which is in the spirit of the so-called triviality in the study of stochastic partial differential equations. More precisely, by studying (un-renormalized) NLW with given smooth deterministic initial data plus a certain truncated random initial data, we show that, as the truncation is removed, the solutions converge to $0$ in the distributional sense for any deterministic initial data.

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arXiv2018

Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^3$

We consider the cubic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work [2]. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.

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arXiv2017

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $\mathbb{R}^d$, $d=5,6$, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $\mathbb{R}^d$, $d = 5, 6$, in the defocusing case, and (ii) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

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arXiv2016

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-wave Equation on The Real Line

We consider the focusing cubic half-wave equation on the real line $$i \partial_t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(ξ)=|ξ|\hat{u}(ξ), \ \ (t,x)\in \Bbb R_+\times \Bbb R.$$ We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small $L^2$-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its $H^1$-norm on a finite time interval, followed by (ii) a saturation regime in which the $H^1$-norm remains stationary large forever in time.

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arXiv2016

Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line

In this paper, we study ill-posedness of cubic fractional nonlinear Schrödinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on $\mathbb R$. In particular, we prove the following ill-posedness results: (i) failure of local uniform continuity of the solution map in $H^s(\mathbb R)$ for $s\in (0,\frac 12)$, and also for $s=0$ in the focusing case; (ii) failure of $C^3$-smoothness of the solution map in $L^2(\mathbb R)$; (iii) norm inflation and, in particular, failure of continuity of the solution map in $H^s(\mathbb R)$, $s<0$. By a similar argument, we also prove norm inflation in negative Sobolev spaces for the cubic fractional NLS. Surprisingly, we obtain norm inflation above the scaling critical regularity in the case of dispersion $|D|^β$ with $β>2$.

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arXiv2015

On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb R^d$, $d \geq 3$

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on $\mathbb R^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_\text{crit} = \frac{d-2}{2}$. More precisely, given a function on $\mathbb R^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove `conditional' almost sure global well-posedness for $d = 4$ in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when $d \ne 4$, we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

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arXiv2015

Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d=4$ and $5$

We consider the energy-critical defocusing nonlinear wave equation (NLW) on $\mathbb{R}^d$, $d=4$ and $5$. We prove almost sure global existence and uniqueness for NLW with rough random initial data in $H^s(\mathbb{R}^d)\times H^{s-1}(\mathbb{R}^d)$, with $0< s\leq 1$ if $d=4$, and $0\leq s\leq 1$ if $d=5$. The randomization we consider is naturally associated with the Wiener decomposition and with modulation spaces. The proof is based on a probabilistic perturbation theory. Under some additional assumptions, for $d=4$, we also prove the probabilistic continuous dependence of the flow with respect to the initial data (in the sense proposed by Burq and Tzvetkov).

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arXiv2014

Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on $\mathbb{R}^3$

We consider the cubic-quintic nonlinear Schrödinger equation: \[ i\partial_t u = -Δu - |u|^2u + |u|^4u. \] In the first part of the paper, we analyze the one-parameter family of ground-state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo--Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region $\mathcal{R}$ of the mass/energy plane where the virial is positive. We show this region is partially bounded by solitons but also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.

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arXiv2014

Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on $\mathbb R^4$ is almost surely locally well-posed with respect to randomized initial data below the energy space.

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arXiv2011

Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions

We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schrödinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.

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arXiv2011

First and second order approximations for a nonlinear wave equation

We consider the nonlinear wave equation (NLW) iv_t-|D|v=|v|^2v on the real line. We show that in a certain regime, the resonant dynamics is given by a completely integrable nonlinear equation called the Szego equation. Moreover, the Szego equation provides a first order approximation for NLW for a large time. The proof is based on the renormalization group method of Chen, Goldenfeld, and Oono. As a corollary, we give an example of solution of NLW whose high Sobolev norms exhibit relative growth. An analogous result of approximation was proved by Gerard and Grellier on the torus using the theory of Birkhoff normal forms. We improve this result by finding the second order approximation on the torus with the help of an averaging method introduced by Temam and Wirosoetisno. We show that the effective dynamics will no longer be given by the Szego equation.

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arXiv2011

Soliton interaction with small Toeplitz potentials for the Szego equation on the real line

We consider the cubic Szego equation perturbed by a small Toeplitz potential (natural generalization of a multiplicative potential) and having as initial condition a soliton of the unperturbed equation. We show that the solution preserves the soliton shape for a long time, and we determine the effective ODEs satisfied by the modulation parameters of the soliton.

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