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Calmed 3D Navier-Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

Matthew Enlow, Adam Larios, Jiahong Wu

Abstract

We propose a modification to the nonlinear term of the three-dimensional incompressible Navier-Stokes equations (NSE) in either advective or rotational form which "calms" the system in the sense that the algebraic degree of the nonlinearity is effectively reduced. This system, the calmed Navier-Stokes Equations (calmed NSE), utilizes a "calming function" in the nonlinear term to locally constrain large advective velocities. Notably, this approach avoids the direct smoothing or filtering of derivatives, thus we make no modifications to the boundary conditions. Under suitable conditions on the calming function, we are able to prove global well-posedness of calmed NSE and show the convergence of calmed NSE solutions to NSE solutions on the time interval of existence for the latter. In addition, we prove that the dynamical system generated by the calmed NSE in the rotational form possesses both an energy identity and a global attractor. Moreover, we show that strong solutions to the calmed equations converge to strong solutions of the NSE without assuming their existence, providing a new proof of the existence of strong solutions to the 3D Navier-Stokes equations.

Calmed 3D Navier-Stokes Equations: Global Well-Posedness, Energy Identities, Global Attractors, and Convergence

Abstract

We propose a modification to the nonlinear term of the three-dimensional incompressible Navier-Stokes equations (NSE) in either advective or rotational form which "calms" the system in the sense that the algebraic degree of the nonlinearity is effectively reduced. This system, the calmed Navier-Stokes Equations (calmed NSE), utilizes a "calming function" in the nonlinear term to locally constrain large advective velocities. Notably, this approach avoids the direct smoothing or filtering of derivatives, thus we make no modifications to the boundary conditions. Under suitable conditions on the calming function, we are able to prove global well-posedness of calmed NSE and show the convergence of calmed NSE solutions to NSE solutions on the time interval of existence for the latter. In addition, we prove that the dynamical system generated by the calmed NSE in the rotational form possesses both an energy identity and a global attractor. Moreover, we show that strong solutions to the calmed equations converge to strong solutions of the NSE without assuming their existence, providing a new proof of the existence of strong solutions to the 3D Navier-Stokes equations.
Paper Structure (17 sections, 10 theorems, 119 equations)

This paper contains 17 sections, 10 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Existence of Weak Solutions for Calmed NSE
  5. Proof of Theorem \ref{['thm_cNSE_vel_exi']}
  6. Strong solutions
  7. Proof of Theorem \ref{['thm_cNSE_vel_reg']}
  8. Proof of Theorem \ref{['thm_cNSE_vel_uni']}
  9. Convergence to strong solutions of the Navier-Stokes equations
  10. Proof of Theorem \ref{['thm_cNSE_conv']}
  11. Existence of Strong Solutions to 3D Navier-Stokes
  12. Proof of Theorem \ref{['thm_cNSE_Cauchy']}
  13. An Energy Equality for Weak Solutions
  14. Proof of Theorem \ref{['thm_energy_identity']}
  15. A Global Attractor
  16. ...and 2 more sections

Key Result

Proposition 1.4

Consider $\boldsymbol{\zeta}^\epsilon_i$ as described in zeta_choices. For $i = 1,2,4$, $\boldsymbol{\zeta}^\epsilon_i$ satisfies Conditions zeta_cond_Lip-zeta_cond_parll of Definition zeta_def. For $i=3$, $\boldsymbol{\zeta}^\epsilon_i$ satisfies Conditions zeta_cond_Lip, zeta_cond_bdd, and zeta_co

Theorems & Definitions (23)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Proposition 1.4
  • Definition 1.5: Weak solution
  • Definition 1.6: Strong solution
  • Theorem 1.7: Global existence of weak solutions to calmed systems
  • Theorem 1.8: First-order regularity of calmed systems
  • Theorem 1.9: Global well-posedness of strong solutions to calmed systems
  • Theorem 1.10: Convergence
  • ...and 13 more