Almost Sure Existence of Global Weak Solutions for Supercritical Navier--Stokes Equations

Andrea R. Nahmod, Nataša Pavlović, Gigliola Staffilani

Type: Article

Publication Date: 2013-01-01

Citations: 61

DOI: https://doi.org/10.1137/120882184

Abstract

In this paper we show that after suitable data randomization there exists a large set of supercritical periodic initial data, in $H^{-\alpha}(\boldsymbol{T}^d)$ for some $\alpha(d) > 0$, for both two- and three-dimensional Navier--Stokes equations for which global energy bounds hold. As a consequence, we obtain almost sure large data supercritical global weak solutions. We also show that in two dimensions these global weak solutions are unique.

Locations

arXiv (Cornell University) - View - PDF
DSpace@MIT (Massachusetts Institute of Technology) - View - PDF
SIAM Journal on Mathematical Analysis - View

Similar Works

Action	Title	Year	Authors
+	Almost Sure Existence of Global Weak Solutions for Supercritical Navier--Stokes Equations	2013	Andrea R. Nahmod Nataša Pavlović Gigliola Staffilani
+	Almost sure existence of global weak solutions for super-critical Navier-Stokes equations	2012	Andrea R. Nahmod Nataša Pavlović Gigliola Staffilani
+	Almost sure existence of global weak solutions for super-critical Navier-Stokes equations	2012	Andrea R. Nahmod Nataša Pavlović Gigliola Staffilani
+	Almost sure existence of Navier-Stokes Equations with randomized data in the whole space	2013	Robin Ming Chen Dehua Wang Song Yao Cheng Yu
+ PDF Chat	Almost sure existence of global weak solutions for supercritical electron MHD	2023	Mimi Dai
+	Almost sure existence of global weak solutions for supercritical electron MHD	2022	Mimi Dai
+ PDF Chat	Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation	2017	Wang Jingrui Keyan Wang
+	Almost Sure Existence of Global Weak Solutions to the 3D Incompressible Navier-Stokes Equation	2016	Wang Jingrui Keyan Wang
+	Almost sure existence of global solutions for supercritical semilinear wave equations	2020	Mickaël Latocca
+	Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space	2020	Zachary Bradshaw Igor Kukavica Wojciech S. Ożański
+ PDF Chat	Global Weak Solutions of the Navier–Stokes Equations for Intermittent Initial Data in Half-Space	2022	Zachary Bradshaw Igor Kukavica Wojciech S. Ożański
+	Existence and Blowup Behavior of Global Strong Solutions to the Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data	2012	Xiangdi Huang Jing Li
+	Almost sure existence of global weak solutions for incompressible MHD equations in negative-order Sobolev space	2017	Lihuai Du Ting Zhang
+	Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier–Stokes system with vacuum and large initial data	2016	Xiangdi Huang Jing Li
+	Almost sure global well-posedness for the energy supercritical NLS on the unit ball of $\mathbb{R}^3$	2020	Mouhamadou Sy Xueying Yu
+ PDF Chat	Randomization improved Strichartz estimates and global well-posedness for supercritical data	2021	Nicolas Burq Joachim Krieger
+ PDF Chat	Existence of Suitable Weak Solutions to the Navier–Stokes Equations for Intermittent Data	2019	Zachary Bradshaw Igor Kukavica
+	Solutions of the Navier-Stokes equations for large oscillatory data	2013	Igor Kukavica Walter Rusin Mohammed Ziane
+ PDF Chat	Random data Cauchy theory for supercritical wave equations II: a global existence result	2008	Nicolas Burq Nikolay Tzvetkov
+	Randomization improved Strichartz estimates and global well-posedness for supercritical data	2019	Nicolas Burq Joachim Krieger

Works That Cite This (60)

Action	Title	Year	Authors
+ PDF Chat	A PDE Construction of the Euclidean $$\Phi ^4_3$$ Quantum Field Theory	2021	Massimiliano Gubinelli Martina Hofmanová
+ PDF Chat	Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS	2015	Árpád Bényi Tadahiro Oh Oana Pocovnicu
+ PDF Chat	Randomization and the Gross–Pitaevskii Hierarchy	2015	Vedran Sohinger Gigliola Staffilani
+	Random data Cauchy theory for nonlinear wave equations of power-type on $\mathbb{R}^3$	2013	Jonas Lührmann Dana Mendelson
+	Probabilistic pointwise convergence problem of some dispersive equations	2020	Wei Yan Jinqiao Duan Yongsheng Li Meihua Yang
+	Almost sure existence of global solutions for supercritical semilinear wave equations	2020	Mickaël Latocca
+	Ergodic attractors and almost-everywhere asymptotics of scalar semilinear parabolic differential equations	2018	Siniša Slijepčević
+	Random Data Cauchy Theory for Nonlinear Wave Equations of Power-Type on ℝ<sup>3</sup>	2014	Jonas Lührmann Dana Mendelson
+	Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation	2019	Benjamin Dodson Jonas Lührmann Dana Mendelson
+ PDF Chat	Almost sure global well-posedness for the energy supercritical Schrödinger equations	2021	Mouhamadou Sy

Works Cited by This (20)

Action	Title	Year	Authors
+	An alternative approach to regularity for the Navier–Stokes equations in critical spaces	2010	Carlos E. Kenig Gabriel S. Koch
+ PDF Chat	Unicité dans $L^3 (\mathbb R^3)$ et d'autres espaces fonctionnels limites pour Navier-Stokes	2000	Giulia Furioli Pierre Gilles Lemarié–Rieusset Elide Terraneo
+	StrongL p -solutions of the Navier-Stokes equation inR m , with applications to weak solutions	1984	Tosio Kato
+	Partial regularity of suitable weak solutions of the navier‐stokes equations	1982	Luis Caffarelli Robert V. Kohn Louis Nirenberg
+	Well-posedness for the Navier–Stokes Equations	2001	Herbert Koch Daniel Tataru
+	<i>L</i><sub>3,∞</sub>-solutions of the Navier-Stokes equations and backward uniqueness	2003	Luis Escauriaza G. Serëgin Vladimír Šverák
+	On the Navier-Stokes initial value problem. I	1964	Hiroshi Fujita Tosio Kato
+	A new proof of partial regularity of solutions to Navier-Stokes equations	2007	Alexis Vasseur
+	Random-data Cauchy problem for the Navier–Stokes equations on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi mathvariant="double-struck">T</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math>	2011	Chao Deng Shangbin Cui
+	Random data Cauchy theory for the incompressible three dimensional Navier–Stokes equations	2011	Ting Zhang Daoyuan Fang