Parametrising the attractor of the two-dimensional Navier-Stokes equations with a finite number of nodal values

Abstract We consider the solutions lying on the global attractor of the two-dimensional Navier–Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg–Landau equation, the Kuramoto–Sivashinsky equation, and reaction–diffusion equations with analytic nonlinearities.

[1]  R. Kraichnan.  Inertial Ranges in Two‐Dimensional Turbulence , 1967 .

[2]  Keith Promislow,et al.  Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations , 1991 .

[3]  J. Eckmann,et al.  A global attracting set for the Kuramoto-Sivashinsky equation , 1993 .

[4]  S. Krantz.  Fractal geometry , 1989 .

[5]  R. Temam,et al.  Attractors Representing Turbulent Flows , 1985 .

[6]  Bernardo Cockburn,et al.  Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems , 1997, Math. Comput..

[7]  I. Kukavica.  Hausdorff length of level sets for solutions of the Ginzburg-Landau equation , 1995 .

[8]  I. Kukavica.  Oscillations of solutions of the Kuramoto-Sivashinsky equation , 1994 .

[9]  E. Titi,et al.  Global Gevrey Regularity for the Bénard Convection in a Porous Medium with Zero Darcy-Prandtl Number , 1999 .

[10]  Y. Pesin.  On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions , 1993 .

[11]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[12]  W. Rudin.  Principles of mathematical analysis , 1964 .

[13]  A. Eden,et al.  Exponential Attractors for Dissipative Evolution Equations , 1995 .

[14]  Dean A. Jones,et al.  On the number of determining nodes for the 2D Navier-Stokes equations , 1992 .

[15]  James C. Robinson.  A rigorous treatment of ‘experimental’ observations for the two–dimensional Navier–Stokes equations† , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  Jack K. Hale,et al.  Infinite dimensional dynamical systems , 1983 .

[17]  Charles R. Doering,et al.  On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation , 1990 .

[18]  Igor Kukavica,et al.  On the number of determining nodes for the Ginzburg-Landau equation , 1992 .

[19]  M. Yamazato.  Absolute continuity of operator-self-decomposable distributions on Rd , 1983 .

[20]  I. Kukavica.  An upper bound for the winding number for solutions of the Ginzburg-Landau equation , 1992 .

[21]  Edriss S. Titi,et al.  GEVREY REGULARITY FOR NONLINEAR ANALYTIC PARABOLIC EQUATIONS , 1998 .

[22]  C. Doering,et al.  Applied analysis of the Navier-Stokes equations: Index , 1995 .

[23]  E. Olson,et al.  Finite fractal dimension and Holder-Lipshitz parametrization , 1996 .

[24]  R. Temam,et al.  Asymptotic analysis of the navier-stokes equations , 1983 .

[25]  I. Kukavica,et al.  Space Analyticity for the Nonlinear Heat Equation in a Bounded Domain , 1999 .

[26]  R. Temam,et al.  Determination of the solutions of the Navier-Stokes equations by a set of nodal values , 1984 .

[27]  I. Kukavica,et al.  Space Analyticity for the NavierStokes and Related Equations with Initial Data in L p , 1998 .

[28]  B. Hunt.  Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces , 1992, math/9210220.

[29]  R. Mañé,et al.  On the dimension of the compact invariant sets of certain non-linear maps , 1981 .

[30]  Edriss S. Titi,et al.  Determining nodes, finite difference schemes and inertial manifolds , 1991 .

[31]  R. Temam,et al.  On the dimension of the attractors in two-dimensional turbulence , 1988 .

[32]  Donald A. Jones,et al.  Determining finite volume elements for the 2D Navier-Stokes equations , 1992 .

[33]  John D. Gibbon,et al.  Attractor dimension and small length scale estimates for the three-dimensional Navier - Stokes equations , 1997 .

[34]  W. Rudin.  Real and complex analysis , 1968 .

[35]  James C. Robinson,et al.  Smooth Attractors Have Zero “Thickness”☆ , 1999 .

[36]  Xiaosong Liu.  Gevrey class regularity and approximate inertial manifolds for the Kuramoto-Sivashinsky equation , 1991 .

[37]  Roger Temam,et al.  Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attr , 1985 .

[38]  Igor Kukavica,et al.  Space Analyticity for the Navier–Stokes and Related Equations with Initial Data inLp , 1998 .

[39]  C. Foias,et al.  Determining nodes for the Kuramoto-Sivashinsky equation , 1995 .

[40]  Brian R. Hunt,et al.  Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces , 1999 .

[41]  Spatial Analyticity on the Global Attractor for the Kuramoto–Sivashinsky Equation , 2000 .

[42]  M. Marion,et al.  Approximate inertial manifolds for reaction-diffusion equations in high space dimension , 1989 .

[43]  E. Titi,et al.  ANALYTICITY OF THE ATTRACTOR AND THE NUMBER OF DETERMINING NODES FOR A WEAKLY DAMPED DRIVEN NONLINEAR SCHRODINGER EQUATION , 1998 .

[44]  Pierre Collet,et al.  Analyticity for the Kuramoto-Sivashinsky equation , 1993 .

[45]  B. Nicolaenko,et al.  Exponential attractors and their relevance to fluid dynamics systems , 1993 .

[46]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[47]  Edriss S. Titi,et al.  Regularity of solutions and the convergence of the galerkin method in the ginzburg-landau equation , 1993 .

[48]  James A. Yorke,et al.  Prevalence: an addendum , 1993 .

[49]  Richard E. Mortensen,et al.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam) , 1991, SIAM Rev..

[50]  Edriss S. Titi,et al.  Gevrey Regularity for Nonlinear Analytic Parabolic Equations on the Sphere , 2000 .

[51]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[52]  K. Falconer.  The geometry of fractal sets: Contents , 1985 .

[53]  Darryl D. Holm,et al.  Low-dimensional behaviour in the complex Ginzburg-Landau equation , 1988 .

[54]  M. Marion.  Attractors for reaction-diffusion equations: existence and estimate of their dimension , 1987 .

[55]  R. Range.  Holomorphic Functions and Integral Representations in Several Complex Variables , 1998 .

[56]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[57]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[58]  Charles R. Doering,et al.  Applied analysis of the Navier-Stokes equations: Index , 1995 .

[59]  S. Hofmann.  Weighted norm inequalities and vector valued inequalities for certain rough operators , 1993 .

[60]  P. Collet,et al.  Determining nodes for extended dissipative systems , 1996 .

[61]  R. Temam,et al.  Gevrey class regularity for the solutions of the Navier-Stokes equations , 1989 .