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Aynur Bulut

Publications and source records attributed to Aynur Bulut.

At least 19 recordsLinked to original sources
arXiv2024

Global well-posedness for supercritical SQG with perturbations of radially symmetric data

We study the global well-posedness of the supercritical dissipative surface quasi-geostrophic (SQG) equation, a key model in geophysical fluid dynamics. While local well-posedness is known, achieving global well-posedness for large initial data remains open. Motivated by enhanced decay in radial solutions, we aim to establish global well-posedness for small perturbations of potentially large radial data. Our main result shows that for small perturbations of radial data, the SQG equation admits a unique global solution.

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arXiv2023

Non-uniqueness up to the Onsager threshold for the forced SQG equation

We establish new non-uniqueness results for the forced inviscid surface quasi-geostrophic equation, via an alternating formulation of convex integration techniques. Our results imply non-uniquenesss in the class of weak solutions with $|\nabla|^{-1}θ\in C_tC_x^α$, for any $α<1$.

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arXiv2023

Convex integration above the Onsager exponent for the forced Euler equations

We establish new non-uniqueness results for the Euler equations with external force on $\mathbb{T}^{d}$ $(d\geq3)$. By introducing a novel alternating convex integration scheme, we construct non-unique, almost-everywhere smooth, Hölder-continuous solutions with regularity $\frac{1}{2}-$, which is notably above the Onsager threshold of $\frac{1}{3}$. The solutions we construct differ significantly in nature from those which arise from the recent unstable vortex construction of Vishik; in particular, our solutions are genuinely $d$-dimensional ($d\geq3$), and give non-uniqueness results for any smooth data. To the best of our knowledge, this is the first instance of a convex integration construction above the Onsager exponent.

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arXiv2022

Epochs of regularity for wild Hölder-continuous solutions of the Hypodissipative Navier-Stokes System

We consider the hypodissipative Navier-Stokes equations on $[0,T]\times\mathbb{T}^{d}$ and seek to construct non-unique, Hölder-continuous solutions with epochs of regularity (smooth almost everywhere outside a small singular set in time), using convex integration techniques. In particular, we give quantitative relationships between the power of the fractional Laplacian, the dimension of the singular set, and the regularity of the solution. In addition, we also generalize the usual vector calculus arguments to higher dimensions with Lagrangian coordinates.

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arXiv2021

Non-existence of splash singularities for the two-fluid Euler--Navier-Stokes system

We consider a system of two incompressible fluids separated by a free interface. The first fluid is inviscid, governed by the Euler system, while the second fluid has positive viscosity and is governed by the Navier-Stokes system. We formulate a notion of splash-type self-intersection singularities, and show that this system cannot form such a singularity in finite time. The main obstacle in our analysis is to handle the effect of bulk vorticity induced by the Navier-Stokes flow. To the best of our knowledge this is the first result on preclusion of splash-type singularities in the presence of non-trivial vorticity interior to one of the fluids.

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arXiv2020

A geometric trapping approach to global regularity for 2D Navier-Stokes on manifolds

In this paper, we use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier-Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai [15] which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our arguments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schr\"odinger equation, and ideas from microlocal analysis.

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arXiv2020

Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds

We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schr\"odinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates. As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate.

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arXiv2019

Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.

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arXiv2018

Global well-posedness for the logarithmically energy-supercritical Nonlinear Wave Equation with partial symmetry

We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and extend work of Tao in the radially symmetric setting. The techniques involved include weighted versions of Morawetz and Strichartz estimates, with weights adapted to the partial symmetry assumptions. In an appendix, we establish a corresponding quantitative result for the energy-critical problem.

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arXiv2015

Negative energy blowup results for the focusing Hartree hierarchy via identities of virial and localized virial type

We establish virial and localized virial identities for solutions to the Hartree hierarchy, an infinite system of partial differential equations which arises in mathematical modeling of many body quantum systems. As an application, we use arguments originally developed in the study of the nonlinear Schr\"odinger equation (see work of Zakharov, Glassey, and Ogawa--Tsutsumi) to show that certain classes of negative energy solutions must blow up in finite time. The most delicate case of this analysis is the proof of negative energy blowup without the assumption of finite variance; in this case, we make use of the localized virial estimates, combined with the quantum de Finetti theorem of Hudson and Moody and several algebraic identities adapted to our particular setting. Application of a carefully chosen truncation lemma then allows for the additional terms produced in the localization argument to be controlled.

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arXiv2015

The defocusing energy-supercritical cubic nonlinear wave equation in dimension five

We consider the energy-supercritical nonlinear wave equation $u_{tt}-Δu+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time {\it a priori} bound on the critical norm implies that solutions exist globally in time and scatter at infinity in both time directions. Together with our earlier works in dimensions $d\geq 6$ with general data and dimension $d=5$ with radial data, the present work completes the study of global well-posedness and scattering in the energy-supercritical regime for the cubic nonlinearity under the assumption of uniform-in-time control over the critical norm.

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arXiv2014

An optimal decay estimate for the linearized water wave equation in 2D

We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-\Delta)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-\Delta)^{1/2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $|t|^{-1/2}$ for solutions corresponding to data $u(0)=\varphi$, assuming only bounds on $\lVert \varphi\rVert_{H_x^1(\mathbb{R})}$ and $\lVert x\partial_x\varphi\rVert_{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $iu_t-(-\Delta)^{\alpha/2}u=0$ for a wider range of $\alpha$.

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arXiv2013

Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball I: the 2D case

Our first purpose is to extend the results from \cite{T} on the radial defocusing NLS on the disc in $\mathbb{R}^2$ to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs measure (which is the natural extension of the classical flow for smooth data). We follow a similar approach as in \cite{BB-1} exploiting certain additional a priori space-time bounds that are provided by the invariance of the Gibbs measure. Next, we consider the radial focusing equation with cubic nonlinearity (the mass-subcritical case was studied in \cite{T2}) where the Gibbs measure is subject to an $L^2$-norm restriction. A phase transition is established, of the same nature as studied in the work of Lebowitz-Rose-Speer \cite{LRS} on the torus. For sufficiently small $L^2$-norm, the Gibbs measure is absolutely continuous with respect to the free measure, and moreover we have a well-defined dynamics.

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arXiv2013

Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3D ball

We establish new global well-posedness results along Gibbs measure evolution for the nonlinear wave equation posed on the unit ball in $\mathbb{R}^3$ via two distinct approaches. The first approach invokes the method established in the works \cite{B1,B2,B3} based on a contraction-mapping principle and applies to a certain range of nonlinearities. The second approach allows to cover the full range of nonlinearities admissible to treatment by Gibbs measure, working instead with a delicate analysis of convergence properties of solutions. The method of the second approach is quite general, and we shall give applications to the nonlinear Schrödinger equation on the unit ball in subsequent works \cite{BB1,BB2}.

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arXiv2013

Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3D case

We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in R^d to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in R^3. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation, and results are obtained almost surely with respect to this probability measure. The key tools used include a class of probabilistic a priori bounds for finite-dimensional projections of the equation and a delicate trilinear estimate on the nonlinearity, which - when combined with the invariance of the Gibbs measure - enables the a priori bounds to be enhanced to obtain convergence of the sequence of approximate solutions.

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arXiv2011

The radial defocusing energy-supercritical cubic nonlinear wave equation in R^{1+5}

In this work, we consider the energy-supercritical defocusing cubic nonlinear wave equation in dimension d=5 for radially symmetric initial data. We prove that an a priori bound in the critical space implies global well-posedness and scattering. The main tool that we use is a frequency localized version of the classical Morawetz inequality, inspired by recent developments in the study of the mass and energy critical nonlinear Schrödinger equation.

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