Euler and Navier–Stokes equations on the hyperbolic plane

Abstract

We show that nonuniqueness of the Leray-Hopf solutions of the Navier-Stokes equation on the hyperbolic plane H 2 observed by Chan and Czubak is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on H n whenever n ≥ 3. We also describe the corresponding general Hamiltonian framework of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.harmonic forms | steady flows | ill-posedness | Dirichlet problem | Dodziuk's theorem C onsider the initial value problem for the Navier-Stokes equations on a complete n-dimensional Riemannian manifold Mvð0; xÞ = v 0 ðxÞ:The symbol ∇ denotes the covariant derivative and L = Δ − 2r, where Δ is the Laplacian on vector fields and r is the Ricci curvature of M. Dropping the linear term Lv from the first equation in Eq. 1 yields the Euler equations of hydrodynamics,[3]Most of the work on well-posedness of the Navier-Stokes equations has focused on the cases where M is either a domain in R n or the flat n-torus T n . In fundamental contributions J. Leray and E. Hopf established existence of an important class of weak solutions described as those divergence-free vector fields v inthat solve the Navier-Stokes equations in the sense of distributions and satisfyfor any 0 ≤ t < ∞ and where Def v = 1 2 ð∇v + ∇v T Þ is the so-called deformation tensor (ref. 2). When n = 2 using interpolation inequalities and energy estimates, it is possible to show that the Leray-Hopf solutions are unique and regular but the problem is in general open for n = 3 (e.g., refs. 3 and 4).There have also been studies on curved spaces, which with few exceptions have been confined to compact manifolds (possibly with boundary) (e.g., ref. 5 and references therein). In a recent paper (1) Chan and Czubak studied the Navier-Stokes equation on the hyperbolic plane H 2 and more general noncompact manifolds of negative curvature. In particular, they showed that in the former case the Cauchy problem (Eq. 1 and 2) admits nonunique Leray-Hopf solutions and in the latter a similar nonuniqueness holds for a modified Navier-Stokes equation using the results of Anderson (6) and Sullivan (7).Our goal in this paper is to provide a direct formulation of the nonuniqueness of the Leray-Hopf solutions on H 2 and explain that it relies on the specific form of the Hodge decomposition for 1-forms (or vector fields) in this case. We also show that no such phenomenon can occur in the hyperbolic space H n with n ≥ 3, thus answering the question raised in ref. 1. As a by-product, we describe the corresponding Hamiltonian setting of the Euler equations on complete Riemannian manifolds (in particular, hyperbolic spaces).We point out that this type of nonuniqueness cannot be found in the Euler equations. Furthermore, it is of a different nature than the examples constructed, e.g., by Shnirelman (8) or De Lellis and Székelyhidi (9). On the other hand, it is similar to nonuniqueness of solutions of the Navier-Stokes equations defined in unbounded domains of the higher-dimensional E...

Remarks on the weak formulation of the Navier–Stokes equations on the 2D hyperbolic space

“…In [17], by means of the result of Dodziuk [7], Khesin and Misio lek showed that our construction can only work in 2D. The main idea is that on H n (−a 2 ) the only L 2 harmonic forms are of degree k = n 2 .…”

“…To see this, we could use the result of [7] to know that dF belongs to L 2 (H 2 (−a 2 )) since it is a harmonic 1-form, and H 2 (−a 2 ) satisfies properties of the manifolds considered in [7]. More directly, as done in [17] and in [26] one can use the conformal equivalence of the Poincaré disk and the Euclidean unit disk together with standard elliptic theory to show that dF is in L 2 . However, to treat more general Riemannian manifolds, we would still need (1.10).…”

“…Zhang [31], Khesin and Misio lek [17], and that of Taylor [26]. In [31] the author shows the non-uniqueness of the weak solution with finite L 2 norm on a connected sum of two copies of R 3 .…”

“…In [31] the author shows the non-uniqueness of the weak solution with finite L 2 norm on a connected sum of two copies of R 3 . In [17] the authors primarily study the stationary Euler equation on the hyperbolic space (but also see Sections 1.2 and 1.3). In [26] the author obtains pointwise estimates for the gradient of harmonic functions on the hyperbolic space (also see remarks after (1.10) and Section 1.3).…”

Turbulence on Hyperbolic Plane: The Fate of Inverse Cascade

“…There is a little catch here [19]. In fact, L 2 (Vect χ ) splits into a direct orthogonal sum of two subspaces…”