Global regularity for 2D Burgers with hyper-dissipation

The viscous Burgers equation in two or three dimensions has global smooth solutions because of a maximum principle. Replacing the Laplacian by a higher power of it should, if anything, damp small scales more strongly, yet it destroys the maximum principle, and for large initial data nobody knows whether solutions stay smooth. The question is whether stronger dissipation can ever hurt.

Problem

On the two-dimensional torus $\mathbb{T}^2$, consider the vector field $u(x, t) \in \mathbb{R}^2$ solving

$$\partial_t u + (u \cdot \nabla) u = -\Delta^2 u, \qquad u(\cdot, 0) = u_0,$$

the Burgers equation with the bi-Laplacian as dissipation. Is the equation globally well-posed for large data: for every smooth initial datum $u_0$, does a smooth solution exist for all $t > 0$?

More generally, the same question can be asked for $\partial_t u + \kappa(-\Delta)^\alpha u - \nu \Delta u + (u \cdot \nabla) u = 0$ with $\alpha > 1$, in two or three dimensions.

Background

The problem is listed by the AIM workshop on small scale dynamics in incompressible fluid flows. Larios and Titi raised it as a cautionary example: hyper-viscosity is routinely added to numerical schemes to stabilise them, but for the viscous Burgers equation the only global a priori bound comes from the maximum principle $\|u(t)\|_{L^\infty} \le \|u_0\|_{L^\infty}$, which fails once a fourth-order term is present. Global regularity results would also shed light on the two- and three-dimensional Kuramoto–Sivashinsky equation, which has a similar structure.

What is known

• With ordinary viscosity, $\partial_t u + (u \cdot \nabla) u = \nu \Delta u$ on the torus is globally well-posed in two and three dimensions, thanks to the maximum principle (as summarised by Larios and Titi).
• In one space dimension, global regularity with hyper-viscosity follows from standard energy methods.
• For large data in two dimensions with $\alpha > 1$, Larios and Titi wrote that nothing was known about global regularity.
• For the closely related two-dimensional Kuramoto–Sivashinsky equation, which is also fourth order and lacks a maximum principle, very little is known about global well-posedness; Larios and Yamazaki (2020) proved global well-posedness for an anisotropically reduced model with the same nonlinearity.

What would count as progress

• A proof of global regularity for large data in two dimensions, for instance through a new a priori bound that replaces the maximum principle.
• A proof of finite-time blow-up for some smooth initial datum, or a carefully resolved numerical study pointing to blow-up or regularity, with convergence checks that others can reproduce.
• Results for intermediate models, such as $1 < \alpha < 2$, special symmetric data, or gradient flows $u = \nabla \phi$.

References

1. A. Larios and E. S. Titi, Global regularity versus finite-time singularities: some paradigms on the effect of boundary conditions and certain perturbations, in Recent Progress in the Theory of the Euler and Navier–Stokes Equations, Cambridge University Press (2016), 96–125; arXiv:1401.1534.
2. A. Larios and K. Yamazaki, On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto–Sivashinsky equation, Phys. D 411 (2020), 132560; arXiv:1908.09239.

Source

Imported from AIM Problem Lists smallscalefluid 3, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.