October  2010, 26(4): 1471-1490. doi: 10.3934/dcds.2010.26.1471

On the Lipschitzness of the solution map for the 2 D Navier-Stokes system

1. 

Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523

Received  November 2008 Revised  May 2009 Published  December 2009

We consider the Navier-Stokes system on R2. It is well-known that solutions with $L^2$ data become instantly smooth and persist globally. In this note, we show that the solution map is Lipschitz, when acting in $L^\infty $Hσ (R2) and $L^2_t$Hσ+1 (R2), when $0\leq $ σ<1. This generalizes an earlier result of Gallagher and Planchon [7], who showed the Lipschitzness in $L^2$(R2). The question for the Lipschitzness of the map in Hσ (R2), σ$\geq 1$ remains an interesting open problem, which hinges upon the validity of an endpoint estimate for the trilinear form $(\phi, v, w)\to \int$R2(∂Φ/∂x ∂v/∂y - ∂Φ/∂y ∂v/∂x)wdx.
Citation: Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471
[1]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5421-5448. doi: 10.3934/dcdsb.2020352

[2]

Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631

[3]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647

[4]

Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 279-314. doi: 10.3934/dcdsb.2007.8.279

[5]

Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873

[6]

Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643

[7]

Hongyong Cui, Mirelson M. Freitas, José A. Langa. Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1297-1324. doi: 10.3934/dcdsb.2018152

[8]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[9]

Julia García-Luengo, Pedro Marín-Rubio, José Real. Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603

[10]

Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations & Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191

[11]

Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701

[12]

Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080

[13]

Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047

[14]

Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015

[15]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408

[16]

Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas. Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations. Electronic Research Archive, 2021, 29 (3) : 2223-2247. doi: 10.3934/era.2020113

[17]

Yinnian He, Pengzhan Huang, Jian Li. H2-stability of some second order fully discrete schemes for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2745-2780. doi: 10.3934/dcdsb.2018273

[18]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

[19]

Kerem Uǧurlu. Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D. Communications on Pure & Applied Analysis, 2017, 16 (1) : 189-208. doi: 10.3934/cpaa.2017009

[20]

Adam Larios, Yuan Pei. Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data. Evolution Equations & Control Theory, 2020, 9 (3) : 733-751. doi: 10.3934/eect.2020031

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]