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The Euler equations for incompressible inviscid fluids may be written as

\displaystyle \partial_t u + (u \cdot \nabla) u = -\nabla p

\displaystyle \nabla \cdot u = 0

where {u: [0,T] \times {\bf R}^n \rightarrow {\bf R}^n} is the velocity field, and {p: [0,T] \times {\bf R}^n \rightarrow {\bf R}} is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that {u} is bounded. We will take the dimension {n} to be at least two, with the three-dimensional case {n=3} being of course especially interesting.

The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when {n=3} is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a noise-tolerant manner (as the viscosity term in the Navier-Stokes equation is to be viewed as perturbative noise).

Perhaps the most prominent obstacles to this route are the conservation laws for the Euler equations, which limit the types of final states that a putative computer could reach from a given initial state. Most famously, we have the conservation of energy

\displaystyle \int_{{\bf R}^n} |u|^2\ dx \ \ \ \ \ (1)

 

(assuming sufficient decay of the velocity field at infinity); thus for instance it would not be possible for a computer to generate a replica of itself which had greater total energy than the initial computer. This by itself is not a fatal obstruction (in this paper of mine, I constructed such a “computer” for an averaged Euler equation that still obeyed energy conservation). However, there are other conservation laws also, for instance in three dimensions one also has conservation of helicity

\displaystyle \int_{{\bf R}^3} u \cdot (\nabla \times u)\ dx \ \ \ \ \ (2)

 

and (formally, at least) one has conservation of momentum

\displaystyle \int_{{\bf R}^3} u\ dx

and angular momentum

\displaystyle \int_{{\bf R}^3} x \times u\ dx

(although, as we shall discuss below, due to the slow decay of {u} at infinity, these integrals have to either be interpreted in a principal value sense, or else replaced with their vorticity-based formulations, namely impulse and moment of impulse). Total vorticity

\displaystyle \int_{{\bf R}^3} \nabla \times u\ dx

is also conserved, although it turns out in three dimensions that this quantity vanishes when one assumes sufficient decay at infinity. Then there are the pointwise conservation laws: the vorticity and the volume form are both transported by the fluid flow, while the velocity field (when viewed as a covector) is transported up to a gradient; among other things, this gives the transport of vortex lines as well as Kelvin’s circulation theorem, and can also be used to deduce the helicity conservation law mentioned above. In my opinion, none of these laws actually prohibits a self-replicating computer from existing within the laws of ideal fluid flow, but they do significantly complicate the task of actually designing such a computer, or of the basic “gates” that such a computer would consist of.

Below the fold I would like to record and derive all the conservation laws mentioned above, which to my knowledge essentially form the complete set of known conserved quantities for the Euler equations. The material here (although not the notation) is drawn from this text of Majda and Bertozzi.

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