In this lecture, I will briefly discuss the difficulty of the inviscid limit problem of Navier-Stokes. As it will be clear in the text, the issue is not due to the fact that the million-dollar regularity problem remains unsolved, but rather nature of the singular perturbation problem. Unless otherwise noted, throughout the course the solutions of both Euler and Navier-Stokes are assumed to be sufficiently smooth as one wishes (for instance, one works in the two-dimensional case or with local-in-time solutions, with which smooth solutions are known to exist).

Precisely, consider the incompressible NS equations in or (periodic setting), with ,

with small viscosity constant . Unknowns in the equation are velocity field and the pressure . (note here that since , one gets , with denoting the vector field ). Assume that the initial data and the corresponding solutions are sufficiently smooth. The most basic quantity associated with (1) is the total energy:

Let us calculate the rate of change of the total energy with respect to time:

We now apply the integration by parts (that is, using the divergence theorem and noting that there is no boundary contribution in our case as ). The middle term vanishes, since . The last term is computed as follows:

This calculation (we shall refer to it as the standard energy estimate!) yields the energy balance for NS solutions:

That is, smooth solutions of NS equations dissipate energy, and in particular, smooth solutions of Euler () conserve energy (i.e., constant in time).

**A side remark:** this latter fact turns out to be false for low-regularity Euler solutions. Precisely, Onsager (1949) conjectured that the Euler weak solutions conserve energy when , and in fact dissipate energy when . The first part of the conjecture was proved in 1994 by Eyink, and then by Constantin-E-Titi. The second part is essentially proved by the recent breakthrough of De Lellis and Székelyhidi Jr., and also Isett and Buckmaster (see, for instance this paper, or Isett’s PhD thesis, and the references therein), using convex integration techniques (introduced by Nash in his famous isometric embedding theorem, and later developed by Gromov in his study of h-principle). This power is closely related to the theory of turbulence of Kolmogorov in 1941. I plan to expand this side remark on another blog post in the near future!

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