Onsager's Conjecture Research Articles

High dimensionality and h-principle in PDE

In this note we present an analyst's point of view'' on the Nash-Kuiper Theorem and, in particular, highlight the very close connection to turbulence--a paradigm example of a high-dimensional phenomenon. Our aim is to explain recent applications of Nash's ideas in connection with the incompressible Euler equations and Onsager's famous conjecture on the energy dissipation in 3D turbulence.

On the kinetic energy profile of Hölder continuous Euler flows

In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space Lt∞B3,∞1/3 due to low regularity of the energy profile.The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class Ct,x1/5−ϵ. The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Székelyhidi [1].

A heat flow approach to Onsager’s conjecture for the Euler equations on manifolds

We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to $\mathbb{T}^{d}$ or $\mathbb{R}^{d}$, our approach yields an alternative proof of the sharp result of the latter authors. Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.

Anomalous dissipation for 1/5-Hölder Euler flows

Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Holder spaces. The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Holder exponent-albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive e, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Holder class $C^{1/5-\epsilon}$.

2D Homogeneous Solutions to the Euler Equation

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form u = ∇⊥Ψ, Ψ(r, θ) = r λψ(θ), for λ > 0, we show that only trivial solutions exist in the range 0 < λ <1/2, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for λ > 9/2 the number of different non-trivial elliptic solutions is equal to the cardinality of the set . The case λ = 2/3 is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity 1/3.

Dissipative Euler flows and Onsager's conjecture

Building upon the techniques introduced in [15], for any \theta&lt;\frac{1}{10} we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent \theta . A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent \theta&lt;\frac{1}{3} . Our theorem is the first result in this direction.

Lectures on the Onsager conjecture

These lectures give an account of recent results pertaining to the celebrated Onsager conjecture. The conjecture states that the minimal space regularity needed for a weak solution of the Euler equation to conserve energy is $1/3$. Our presentation is based on the Littlewood-Paley method. We start with quasi-local estimates on the energy flux, introduce Onsager criticality, find a positive solution to the conjecture in Besov spaces of smoothness $1/3$. We illuminate important connections with the scaling laws of turbulence. Results for dyadic models and a complete resolution of the Onsager conjecture for those is discussed, as well as recent attempts to construct dissipative solutions for the actual equation.&nbsp;&nbsp;&nbsp;The article is based on a series of four lectures given at the 11th school 'Mathematical Theory in Fluid Mechanics' in Kácov, Czech Republic, May 2009.

Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the loss of smoothness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.

Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood–Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

Regularity of inviscid shell models of turbulence

In this paper we continue the analytical study of the sabra shell model of energy turbulent cascade. We prove the global existence of weak solutions of the inviscid sabra shell model, and show that these solutions are unique for some short interval of time. In addition, we prove that the solutions conserve energy, provided that the components of the solution satisfy /u(n)/ <or= Ck(n)(-1/3)[square root of n log(n+1)](-1) for some positive absolute constant C, which is the analog of the Onsager's conjecture for the Euler's equations. Moreover, we give a Beal-Kato-Majda type criterion for the blow-up of solutions of the inviscid sabra shell model and show the global regularity of the solutions in the "two-dimensional" parameters regime.

THE QUASI-GEOSTROPHIC EQUATION AND ITS TWO REGULARIZATIONS

ABSTRACT We consider the quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with critical or sub-critical indices. The proof[4], [13] of Onsager's conjecture[19] concerning weak solutions of the 3D Euler equations and the notion of dissipative solutions of Duchon and Robert[12] are extended to weak solutions of the quasi-geostrophic equation.

Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations

We study the local equation of energy for weak solutions of three-dimensional incompressible Navier-Stokes and Euler equations. We define a dissipation term D(u) which stems from an eventual lack of smoothness in the solution u. We give in passing a simple proof of Onsager's conjecture on energy conservation for the three-dimensional Euler equation, slightly weakening the assumption of Constantin et al. We suggest calling weak solutions with non-negative D(u) `dissipative'.

Dissipation d'énergie pour des solutions faibles des équations d'euler et navier-stokes incompressibles

We study the local energy equation for weak solutions of 3D incompressible Euler and Navier-Stokes equations. We introduce a dissipation term coming from an eventual lack of regularity in the solution. We use this local energy equation to give a simple proof of Onsager's conjecture, slightly improving the hypothesis given in [ 1]. We propose a new notion of dissipative solution for such weak solutions

Onsager's conjecture on the energy conservation for solutions of Euler's equation

We give a simple proof of a result conjectured by Onsager [1] on energy conservation for weak solutions of Euler's equation.

An alternative interpretation of Onsager's inverted snowball effect of electron solvation

A reasonable interpretation is given for Onsager's conjecture on the inverted snowball effect. It is concluded that his conjecture applies to the solvation of an electron, but not an ion. Although his conjecture has been applied to the solvation of an ion, this is a misapplication.

Microscopic expression for time-dependent solvation energy of ions and dipoles in dense polar liquids

Microscopic expressions for the time dependence of solvation energies of newly created ions and dipoles in a dense dipolar liquid are presented. The dynamics of solvation of an ion differ considerably from that of a dipole, especially that the long wavelength (k = 0) component of solvent response is totally absent for dipoles. The translational modes of the solvent molecules lead to a breakdown of Onsager's conjecture on the distance dependence of solvent polarization relaxation.