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

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.