The h-principle and Onsager’s conjecture

The h-principle is a concept introduced by Gromov which pertains to various problems in differential geometry, where one expects high flexibility of the moduli spaces of solutions due to the high-dimensionality (or underdetermined nature) of the problem. Interestingly, in some cases a form of the h-principle holds even for systems of partial differential relations which are, formally, not underdetermined. Perhaps the most famous instance is the Nash-Kuiper theorem on C1 isometric Euclidean embeddings of n-dimensional Riemannian manifolds. In the classical situation of embedding (two-dimensional) surfaces in three-space, the resulting maps comprise three unknown functions that must satisfy a system of three independent partial differential equations. This is a determined system and, indeed, sufficiently regular solutions satisfy additional constraints (C2, i.e. continuous second order derivatives, suffices). The oldest example of such a constraint is the Theorema Egregium of Gauss: the determinant of the differential of the Gauss map (a-priori an “extrinsic” quantity) equals a function which can be computed directly from the metric, i.e. the intrinsic Gauss curvature of the original surface. At a global level there are much more restrictive consequences: for instance any (C2) isometric embedding u of the standard 2-sphere S2 in R3 can be extend in a unique way to an isometry of R3 and must therefore map S2 affinely onto the boundary of a unitary ball. In other words C2 isometric embeddings of the standard 2-sphere in R3 are rigid; in fact the same holds for any metric on the 2-sphere which has positive Gauss curvature. Nevertheless, the outcome of the Nash-Kuiper theorem is that C1 solutions are very flexible and all forms of the aforementioned rigidity are lost. In a sense, in this situation low regularity serves as a replacement for high-dimensionality. A similar phenomenon has been found recently for solutions of a very classical system of partial differential equations in mathematical physics: the Euler equations for ideal incompressible fluids. Regular (C1) solutions of this system are determined by the boundary and initial data, whereas continuous solutions are not unique and might even violate the law of conservation of kinetic energy. Although at a rigorous mathematical level this was proved only recently, the latter phenomenon was predicted in 1949 by Lars Onsager in his famous note [41] about statistical hydrodynamics. Onsager conjectured a threshold regularity for the conservation of the kinetic energy. The conjecture is still open and the threshold has deep connections with the Kolmogorov’s theory of fully developed turbulence. In this brief note we will first review the isometric embedding problem, emphasizing the h-principle aspects. We will then turn to some “h-principle-type statements” in the theory of differential inclusions, proved in the last three decades by several authors. These results were developed independently of Gromov’s work, but a fruitful relation was pointed out in a groundbreaking paper by Müller and Šverak fifteen years ago, see [38]. There is however a fundamental difference: in differential geometry the h-principle results are in the “C0 category”, whereas the corresponding statements in the theory of differential inclusions hold in the “L∞ category”. Indeed “L∞ h-principle statements” in differential geometry are usually trivial, whereas “C0 h-principle statements” in the theory of differential inclusions are usually false. Surprisingly both aspects are present and nontrivial when dealing with solutions of the incompressible Euler equations: the last two sections of this note will be devoted to them.