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Burgers equation with a twist: A study on rotational-form equations

Adam Larios

TL;DR

The paper investigates a 3D rotational Burgers-type equation derived from the NSE by removing the Bernoulli pressure, resulting in a model with the same energy balance as NSE but governed solely by the rotational nonlinearity. It proves global well-posedness for the viscous case, shows local well-posedness and finite-time blow-up for the inviscid case, and develops a damped-driven variant with a global attractor. Additionally, it introduces a rotational modification of the 2D Kuramoto–Sivashinsky equation with global well-posedness and presents a 2D rotational numerical scheme and simulations that reveal complex, potentially chaotic dynamics. The work also discusses extensions, numerical approaches, and pedagogical benefits of studying rotational forms as a tractable proxy for NSE and related PDEs. Overall, the rotational framework offers new analytical and computational avenues to understand nonlinear fluid-like dynamics while preserving key energy properties.

Abstract

A new three-dimensional (3D) equation is proposed, which is formed like Burgers' equation by starting with the 3D incompressible Navier-Stokes equations (NSE) and eliminating the pressure and the divergence-free constraint, but instead the Bernoulli pressure is eliminated, leaving only the rotational form of the nonlinearity. This results in a globally well-posed 3D equation which has exactly the same energy balance as the 3D NSE. Moreover, we show in simulations that the system seems to exhibit chaotic dynamics. In the viscous case, we prove the global existence, uniqueness, and higher-order regularity of solutions to this equation with no restriction on the initial data other than smoothness. In the inviscid case, local existence holds, but we give an example of a class of solutions with smooth initial data that develop a singularity in finite time in both 2D and 3D. Moreover, a new numerical algorithm is presented in the 2D case, and simulations are included to illustrate the dynamics. In addition, a rotational-form modification for the 2D Kuramoto-Sivashinsky equations (KSE) is proposed, and global well-posedness is also established. We also discuss several related ``rotational form'' equations, and some pedagogical considerations. Global well-posedness for the original 3D NSE and 2D KSE remains a challenging open problem, but it is hoped that by focusing on the rotational term, new insight may be gained.

Burgers equation with a twist: A study on rotational-form equations

TL;DR

The paper investigates a 3D rotational Burgers-type equation derived from the NSE by removing the Bernoulli pressure, resulting in a model with the same energy balance as NSE but governed solely by the rotational nonlinearity. It proves global well-posedness for the viscous case, shows local well-posedness and finite-time blow-up for the inviscid case, and develops a damped-driven variant with a global attractor. Additionally, it introduces a rotational modification of the 2D Kuramoto–Sivashinsky equation with global well-posedness and presents a 2D rotational numerical scheme and simulations that reveal complex, potentially chaotic dynamics. The work also discusses extensions, numerical approaches, and pedagogical benefits of studying rotational forms as a tractable proxy for NSE and related PDEs. Overall, the rotational framework offers new analytical and computational avenues to understand nonlinear fluid-like dynamics while preserving key energy properties.

Abstract

A new three-dimensional (3D) equation is proposed, which is formed like Burgers' equation by starting with the 3D incompressible Navier-Stokes equations (NSE) and eliminating the pressure and the divergence-free constraint, but instead the Bernoulli pressure is eliminated, leaving only the rotational form of the nonlinearity. This results in a globally well-posed 3D equation which has exactly the same energy balance as the 3D NSE. Moreover, we show in simulations that the system seems to exhibit chaotic dynamics. In the viscous case, we prove the global existence, uniqueness, and higher-order regularity of solutions to this equation with no restriction on the initial data other than smoothness. In the inviscid case, local existence holds, but we give an example of a class of solutions with smooth initial data that develop a singularity in finite time in both 2D and 3D. Moreover, a new numerical algorithm is presented in the 2D case, and simulations are included to illustrate the dynamics. In addition, a rotational-form modification for the 2D Kuramoto-Sivashinsky equations (KSE) is proposed, and global well-posedness is also established. We also discuss several related ``rotational form'' equations, and some pedagogical considerations. Global well-posedness for the original 3D NSE and 2D KSE remains a challenging open problem, but it is hoped that by focusing on the rotational term, new insight may be gained.

Paper Structure

This paper contains 31 sections, 9 theorems, 172 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Relations to Navier-Stokes and other equations
  3. The Bernoulli pressure and the Lamb vector
  4. Related equations and other considerations.
  5. Preliminaries
  6. Global Well-Posedness for a Viscous Rotational Burgers Equation
  7. Higher-order estimates
  8. A Maximum Principle
  9. Global Well-Posedness
  10. The Global Attractor (Damped-Driven Case)
  11. bounds in
  12. bounds
  13. The Inviscid Case
  14. Local well-posedness for the inviscid case
  15. A finite-time singularity
  16. ...and 16 more sections

Key Result

Theorem 4.2

Given $\nu>0$, $T^*>0$, $\mathbf{u}_0\in H^1(\mathbb T^3)$, and $\mathbf{f}\in L^2(0,T^*;\dot{L}^2)$, there exists a $T\in(0,T^*)$ and a unique strong solution to Burgers_twist on $[0,T]$. Moreover, strong solutions depend continuously on the initial data in the sense of $C([0,T];H^1)\cap L^2(0,T;H^

Figures (3)

  • Figure 2.1: (A): Visualization of a randomly-generated smooth 3D vector field $\vec{v}$, along with its curl $\nabla\times \vec{v}$, and the Lamb vector $(\nabla\times \vec{v})\times \vec{v}$. (B): 2D analogs of the fields in (A). Vectors not to scale.
  • Figure 9.1: Snapshots of various quantities at time $T=1000$ for \ref{['Burgers_twist']} (2D). Resolution: $N^2=2048^2$. (a) Energy spectrum $E_k$. Spectrum is resolved to machine precision ($\epsilon=2.22\times10^{-16}$) before dealiasing cutoff ($N/3$) with inertial range roughly $E_k\sim k^{-2}$. (b) Magnitude of the velocity. (c) 2D curl, showing negative and positive fine structures. (d) Divergence, with fine-scale positive structures, and coarse-scale negative structures. Unlabeled axes: (a) $E_k$ vs. wave number $k$; (b), (c), and (d): standard $xy$-axes, $-\pi\leq x, y <\pi$.
  • Figure 9.2: Large-time dynamics as illustrated by various norms of the solution to \ref{['Burgers_twist']}.

Theorems & Definitions (19)

  • Remark 1.1
  • Definition 4.1
  • Theorem 4.2: Local well-posedness: strong solutions
  • proof
  • Remark 4.3
  • Theorem 4.4
  • proof
  • Lemma 4.5: Maximum principle
  • proof
  • Theorem 4.6
  • ...and 9 more