Dynamics of Ideal Fluid Flows

TL;DR

The work investigates the dynamics of the incompressible Euler equations for ideal fluids, emphasizing nonlocality from volume preservation, least-action derivations, and the vorticity formulation. It develops a variational/Weber framework for deriving Euler flows, analyzes steady-state structures (including Arnold’s structure theorem and 2D reductions), and examines asymptotic behavior such as inviscid damping and phase mixing. It then addresses solvability, including local/global existence results and Beale–Kato–Majda-type blow-up criteria, before detailing singularity formation through model equations, blow-up scenarios, self-similar ansatz, and a perturbative scheme to obtain true self-similar solutions from approximate ones. The synthesis highlights the rich interplay between variational principles, dynamical stability, and nonlinear, nonlocal mechanisms driving long-time behavior and potential finite-time singularities in both 2D and 3D settings, with implications for understanding realistic fluid flows and their mathematical limits.

Abstract

We will discuss various aspects of the incompressible Euler equation. We will discuss, in particular, problems related to the least action principle, the existence of special solutions, the problem of solvability, singularity formation, and asymptotic behavior.

Figures (4)

• Figure 1: Caricature of the long-time dynamics for passive scalars (left) and the 2d Euler equation (right). The set in the middle, in each case, is the set $\mathcal{A}_\infty$ that attracts all solutions. In the passive scalar case it is an infinite dimensional linear space that consists of steady states, time-periodic solutions, etc, though generically it only consists of steady states. For the 2d Euler equation, it is conjectured to consist of those solutions lying on compact orbits (including steady states, periodic solutions, etc.). While in the passive scalar case the limit set is linear, the limit set in the 2d Euler case is certainly nonlinear and may be singular, as discussed above.
• Figure 2: Caricature of the singularity reported by Luo and Hou in LuoHou. The horizontal axis, in the picture, is a solid boundary. The red regions should be seen as regions of fluid with high density outside of which is fluid with low density. Since gravity is pointing down, the light fluid in the center is ejected upwards and the high density bumps try to settle. The singularity reported in LuoHou predicts that a type of collision occurs at the origin in the picture (as on the right). The intensity of the "collision" at the origin should depend on the degree of vanishing of flatness of the density at $x=0$, as in singularities in the Burgers equation CGM. This is also the geometric scenario considered in ZlatosIPM, with a corner in EJBEJ3dE, and the numerical work Wang. See also CKY1CKY2CHH for works on models of this scenario.
• Figure 3: Caricature of the singularity constructed in E_Classical. A caricature of the data is on the left, while the solution just prior to the singularity is on the right. Particles flow down and out symmetrically along each axial plane. The initial vorticity is axi-symmetric without swirl and vanishes only very weakly on the symmetry axis. This allows for particles at the axis to come down much faster than those away from the axis. At the final time, the vorticity becomes unbounded at the origin and the velocity develops a cusp discontinuity at $x=0$ while remaining smooth away from the symmetry axis.
• Figure 4: Caricature of the singularity constructed in EP. The doted lines indicate level sets of the density while the circular $+$ and $-$ figures represent vorticity of positive and negative sign, respectively. The left figure represents the initial configuration with unstably stratified density plus a small perturbing vorticity. The right figure represents the short time evolution of the level sets of density by the perturbing vorticity. The change in the shape of the level sets of the density leads to the creation of more vorticity, which is represented by the $+$ sign in the upper right corner. The main barrier to singularity formation in this scenario is the flattening of the level sets of the density due to incompressibility due to stretching in the horizontal direction. The singularity constructed in EP overcomes this issue in the setting of $C^{1,\alpha}$ solutions that are smooth in the angular direction at $x=0.$ It may very well be that truly smooth solutions develop a singularity in this setting.

Theorems & Definitions (25)

• Corollary 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 3.1
• Theorem 3.3
• Theorem 4.1
• Remark 4.2
• Theorem 4.3
• Definition 4.4
• Conjecture 4.5