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Nikolay Tzvetkov

Publications and source records attributed to Nikolay Tzvetkov.

At least 19 recordsLinked to original sources
arXiv2026

Numerical study of probabilistic well-posedness of one dimensional fractional nonlinear wave equations

The three dimensional cubic defocusing nonlinear wave equation is known to be ill-posed for general low regularity initial data. Well-posedness can however be recovered globally in time on a probabilistic level, provided the Fourier coefficients of the random initial data follow a sub-Gaussian law, an assumption on which the available proofs rely heavily. In this article we perform numerical simulations of the one dimensional fractional cubic defocusing wave equation in a periodic setting, with low regularity random initial data drawn either from a Gaussian or from a heavy tailed Student law. Besides illustrating probabilistic well-posedness and norm inflation in both the energy subcritical and supercritical regimes, our simulations display no qualitative difference between the heavy tailed and the sub-Gaussian cases. This leads us to conjecture that probabilistic well-posedness holds under the sole assumption that the Fourier coefficients are centered with finite variance.

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arXiv2026

On probabilistic ill-posedness

In this note, we introduce an enhanced notion of probabilistic well-posedness for dispersive PDEs with random initial data by imposing stability at the origin as the amplitude of randomization tends to $0$. We then use this notion to re-interpret recent works on "beyond variance blowup" for dispersive PDEs, by the authors with their collaborators (2025, 2026), as probabilistic ill-posedness results. By drawing an analogy to the failure of $C^k$-smoothness of a solution map in the deterministic setting, we interpret variance blowup results as mild probabilistic ill-posedness.

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arXiv2026

Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the $3d$ ball

We construct probabilistic strong solutions to the cubic Schrödinger equation on the three-dimensional ball with radial initial data, which is a significant improvement of a result by Bourgain--Bulut. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng--Nahmod--Yue. We achieve this result through gauge transformations that do not modify the equation, combined with a refined modulation analysis using random averaging operators.

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arXiv2026

Gibbs measure dynamics for the fractional NLS

We construct global solutions on a full measure set with respect to the Gibbs measure for the one dimensional cubic fractional nonlinear Schrödinger equation (FNLS) with weak dispersion $(-\partial_x^2)^{α/2}$, $α<2$ by quite different methods, depending on the value of $α$. We show that if $α>\frac{6}{5}$, the sequence of smooth solutions for FNLS with truncated initial data converges almost surely, and the obtained limit has recurrence properties as the time goes to infinity. The analysis requires to go beyond the available deterministic theory of the equation. When $1<α\leq \frac{6}{5}$, we are not able so far to get the recurrence properties but we succeeded to use a method of Bourgain-Bulut to prove the convergence of the solutions of the FNLS equation with regularized both data and nonlinearity. Finally, if $\frac{7}{8}<α\leq 1$ we can construct global solutions in a much weaker sense by a classical compactness argument.

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arXiv2026

Hyperbolic nonlinear Schrödinger equations on $\mathbb{R}\times \mathbb{T}$

In this paper, we consider the hyperbolic nonlinear Schrödinger equations (HNLS) on $\mathbb{R}\times\mathbb{T}$. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for higher odd nonlinearities. Moreover, when the initial data is small, we prove the global existence and scattering for the solutions to HNLS with higher nonlinearities (except the cubic one) in critical Sobolev spaces. The main ingredient of the proof is the sharp up to the endpoint local/global-in-time Strichartz estimates.

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arXiv2026

Large torus limit of global dynamics of the two-dimensional dispersive Anderson model

We continue the study of the two-dimensional dispersive Anderson model (DAM), i.e. the nonlinear Schrödinger equation with multiplicative spatial white noise. For this model, global well-posedness on the periodic domain was established by Visciglia and the second author (2023), and global well-posedness on the full space was established by Debussche, Visciglia, and the authors (2024). We show that, under suitable initial conditions and suitable periodization procedure of the noise, the periodic global dynamics of the DAM converges in spaces of local domains to that of the DAM on the full space as the period goes to infinity. In order to control the growth of the noise and obtain a priori bounds for solutions independent of the periodicity, we introduce periodic weights and construct weighted function spaces on periodic domains. In Appendix, we also discuss the same problem for the parabolic Anderson model.

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arXiv2026

The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$

We extend Bourgain's $L^2$-wellposedness result for the KP-II equation on $\mathbb{T}^2$ to initial data with negative Sobolev regularity. The key ingredient is a new linear $L^4$-Strichartz estimate which is effective on frequency-dependent time scales. The $L^4$-Strichartz estimates follow from combining an $\ell^2$-decoupling inequality recently proved by Guth--Maldague--Oh with semiclassical Strichartz estimates. Moreover, we rely on a variant of Bourgain's bilinear Strichartz estimate on frequency-dependent times, which is proved via the Córdoba--Fefferman square function estimate.

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arXiv2025

Almost sure global nonlinear smoothing for the 2D NLS

In this article, we prove an almost-sure global in time nonlinear smoothing effect for NLS on the two-dimensional torus. For deterministic data, this phenomenon was proved for the NLS on the circle by Erdoğan--Tzirakis, which remains unknown on multidimensional torus. Our argument is based on a quantitative quasi-invariance of Gaussian measures with covariance operator $(1-Δ)^{-s}$ for $s>2$.

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arXiv2025

On low regularity well-posedness of the binormal flow

We focus on a class of solutions of the binormal flow, model of the evolution of vortex filaments, that generate several corner singularities in finite time. This phenomenon has been studied earlier in the regular case, which in this context is in terms of the summability of the angles of the corners generated. Our goal here is to investigate the lower regularity case, using further the Hasimoto approach that allows to use the 1D cubic nonlinear Schrödinger to study the binormal flow. We first obtain a deterministic result by proving an existence result for general binormal flow solutions at low regularity. Then we obtain improved results on the above class of solutions by a suitable randomization of the curvature and torsion of the vortex filament. To do so, we prove a scattering result for a quasi-invariance measure associated with a suitable 1D cubic nonlinear Schrödinger equation that we consider of independent interest. An interesting feature of this result is that we are able to identify a limit measure, which is usually not possible when working on quasi-invariant Gaussian measures for Hamiltonian PDEs on bounded domains.

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arXiv2025

Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation

We investigate a possible extension of probabilistic well-posedness theory of nonlinear dispersive PDEs with random initial data beyond variance blowup. As a model equation, we study the Benjamin-Bona-Mahony equation (BBM) with Gaussian random initial data. By introducing a suitable vanishing multiplicative renormalization constant on the initial data, we show that solutions to BBM with the renormalized Gaussian random initial data beyond variance blowup converge in law to a solution to the stochastic BBM forced by the derivative of a spatial white noise. By considering alternative renormalization, we show that solutions to the renormalized BBM with the frequency-truncated Gaussian initial data converges in law to a solution to the linear stochastic BBM with the full Gaussian initial data, forced by the derivative of a spatial white noise. This latter result holds for the Gaussian random initial data of arbitrarily low regularity. We also establish analogous results for the stochastic BBM forced by a fractional derivative of a space-time white noise.

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arXiv2025

Probabilistic well-posedeness for the nonlinear Schrödinger equation on the $2d$ sphere I: positive regularities

We establish the probabilistic well-posedness of the nonlinear Schrödinger equation on the $2d$ sphere $\mathbb{S}^{2}$. The initial data are distributed according to Gaussian measures with typical regularity $H^{s}(\mathbb{S}^{2})$, for $s>0$. This level of regularity goes significantly beyond existing deterministic results, in a regime where the flow map cannot be extended uniformly continuously.

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arXiv2025

Quasi-invariance of Gaussian measures for the $3d$ energy critical nonlinear Schr\" odinger equation

We consider the $3d$ energy critical nonlinear Schr\" odinger equation with data distributed according to the Gaussian measure with covariance operator $(1-Δ)^{-s}$, where $Δ$ is the Laplace operator and $s$ is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from $1d$ to higher dimensions.

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arXiv2025

Global results for weakly dispersive KP-II equations on the cylinder

We consider the dispersion-generalized KP-II equation on a partially periodic domain in the weakly dispersive regime. We use Fourier decoupling techniques to derive essentially sharp Strichartz estimates. With these at hand, we show global well-posedness of the quasilinear Cauchy problem in $L^2(\mathbb{R} \times \mathbb{T})$. Finally, we prove a long-time decay property of solutions with small mass by using the Kato smoothing effect in the fractional case.

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arXiv2025

New bounds on the high Sobolev norms of the 1d NLS solutions

We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the $1$D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.

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arXiv2024

Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic $Φ^4_3$-model

We study the fractional $Φ^4_3$-measure (with order $α> 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity, also called the fractional hyperbolic $Φ^4_3$-model. We first construct the fractional $Φ^4_3$-measure via the variational approach by Barashkov-Gubinelli (2020). When $α\leq \frac{9}{8}$, this fractional $Φ^4_3$-measure turns out to be singular with respect to the base Gaussian measure. We then prove almost sure global well-posedness of the fractional hyperbolic $Φ^4_3$-model and invariance of the fractional $Φ^4_3$-measure for all $α> 1$ by further developing the globalization framework due to Oh-Okamoto-Tolomeo (2024) on the hyperbolic $Φ^3_3$-model. Furthermore, when $α> \frac{9}{8}$ , we prove weak universality of the fractional hyperbolic $Φ^4_3$-model by utilizing the convergence of Gibbs measures.

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arXiv2024

Local well-posedness for the periodic Boltzmann equation with constant collision kernel

We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain $\mathbb{T}^d$, $d\geq 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in $L^{2,r}_vH^s_x$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. To reach the result, the main tool we establish is the $L^4$ Strichartz estimate for solutions to the corresponding linear equation.

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arXiv2024

Almost surely smoothed scattering for cubic NLS

We consider cubic NLS in dimensions 2, 3, 4 and we prove that almost surely solutions with randomized initial data at low regularity scatter. Moreover, we establish some smoothing properties of the associated scattering operator and precise the rate of convergence.

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