In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an $L^2$-dense set of Hölder continuous initial data in the class of Hölder continuous admissible weak solutions for all exponents below the Onsager-critical $1/3$. This improves previous results on non-uniqueness obtained by Daneri in arXiv:1302.0988 and by Daneri and Szekelyhidi Jr. in arXiv:1603.09714 and generalizes the result obtained by Buckmaster, De Lellis, Szekelyhidi Jr. and Vicol in arXiv:1701.08678.
For any θ<1/10 we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent θ. A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent θ<1/3. Our theorem is the first result in this direction.
The Onsager's conjecture has two parts: conservation of energy, if the exponent is larger than $1/3$ and the possibility of dissipative Euler solutions, if the exponent is less or equal than $1/3$. The paper proves half of the conjecture, the conservation part, in bounded domains.
We establish new non-uniqueness results for the Euler equations with external force on $\mathbb{T}^{d}$ $(d\geq3)$. By introducing a novel alternating convex integration scheme, we construct non-unique, almost-everywhere smooth, Hölder-continuous solutions with regularity $\frac{1}{2}-$, which is notably above the Onsager threshold of $\frac{1}{3}$. The solutions we construct differ significantly in nature from those which arise from the recent unstable vortex construction of Vishik; in particular, our solutions are genuinely $d$-dimensional ($d\geq3$), and give non-uniqueness results for any …
For any $ε>0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L^1_t (C_x^{\frac{1}{3}-ε})$, namely $x\mapsto v (x,t)$ is $(\frac{1}{3}-ε)$-Hölder continuous in space at a.e. time $t$ and the integral $\int [v(\cdot, t)]_{\frac{1}{3}-ε} dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L^\infty_t (C_x^{\frac{1}{3}-ε})$.
This paper addresses the problem of energy conservation for the two- and three-dimensional density-dependent Euler equations. Two types of sufficient conditions on the regularity of solutions are provided to ensure the conservation of total kinetic energy on the entire time interval including the initial time. The first class of data assumes integrability on the spatial gradient of the density, and hence covers the classical result of Constantin-E-Titi for the homogeneous Euler equations. The other type of data imposes extra time Besov regularity on the velocity profile, and the corresponding …
Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of 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 $B^{1/3}_{3,c(\NN)}$. 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 …
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…
The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if $u (\cdot ,t) \in C^{0, θ} (\mathbb{T}^3)$ with $θ> \frac{1}{3}$. In this paper, we prove an analogue of Onsager's conjecture for several subgrid scale $α$-models of turbulence. In particular we find the required Hölder regularity of the solutions that ensures the conservation of energy-like quantities (either the $H^1 (\mathbb{T}^3)$ or $L^2 (\mathbb{T}^3)$ norms) for these models. We establish such results for the Leray-$α$ model, the Euler-$α$ equations (also known as t…
In this note we extend a 2018 result of Bardos and Titi \cite{BT} to a new class of functional spaces $C^{0,α}_λ(\barΩ)$. It is shown that weak solutions $\,u\,$ satisfy the energy equality provided that $u\in L^3((0,T);C^{0,α}_λ(\barΩ))$ with $α\geq\frac{1}{3}$ and $λ>0$. The result is new for $\,α= \,\frac{1}{3}\,.$ Actually, a quite stronger result holds. For convenience we start by a similar extension of a 1994 result of Constantin, E, and Titi, \cite{CET}, in the space periodic case. The proofs follow step by step those of the above authors. For the readers convenience, and completeness, …
The goal of this note is to show that, also in a bounded domain $Ω\subset \mathbb{R}^n$, with $\partial Ω\in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $Ω\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t$, with the impermeability boundary condition: $u\cdot \vec n =0$ on $\partialΩ\times(0,T)$, is of constant energy on the interval $(0,T)$ provided the velocity field $u \in L^3((0,T); C^{0,α}(\overlineΩ))$, with $α>\frac13\,.$
This note addresses the question of energy conservation for the 2D Euler system with an $L^p$-control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if $ω= \nabla \times u \in L^{\frac32}$. An example of a 2D field in the class $ω\in L^{\frac32 - ε}$ for any $ε>0$, and $u\in B^{1/3}_{3,\infty}$ (Onsager critical space) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument. Finally we prove that any solution to the Euler equation produced via a vanishin…
We study stationary homogeneous solutions to the 3D Euler equation. The problem is motivated be recent exclusions of self-similar blowup for Euler and its relation to Onsager conjecture and intermittency. We reveal several new classes of solutions and prove rigidity properties in specific categories of genuinely 3D solutions. In particular, irrotational solutions are characterized by vanishing of the Bernoulli function; and tangential flows are necessarily 2D axisymmetric pure rotations. In several cases solutions are excluded altogether. The arguments reveal geodesic features of the Euler flo…
In this article we study the principle of energy conservation for the Euler-Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler-Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.
We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy di…
We prove a version of Onsager's conjecture on the conservation of energy for the incompressible Euler equations in the context of statistical solutions, as introduced recently by Fjordholm et al. As a byproduct, we also obtain a new proof for the conservative direction of Onsager's conjecture for weak solutions. Dedicated to Edriss S. Titi on the occasion of his 60th birthday.
We consider the incompressible Euler equations in a bounded domain in three space dimensions. Recently, the first two authors proved Onsager's conjecture for bounded domains, i.e., that the energy of a solution to these equations is conserved provided the solution is Hölder continuous with exponent greater than 1/3, uniformly up to the boundary. In this contribution we relax this assumption, requiring only interior Hölder regularity and continuity of the normal component of the energy flux near the boundary. The significance of this improvement is given by the fact that our new condition is co…
In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one -- in the viscous case -- with Hölder type assumptions, but without additional ass…
We give a new sufficient criteria to prove the uniqueness of the incompressible Euler equation in dimension $N\geq2$. In their celebrated works by V. Scheffer [18], A. Shnirelman [19], C. De Lellis and L. Székelyhidi Jr. [7] they have obtained the nonuniqeness of weak solutions of incompressible Euler equation. Here we obtain uniqueness criteria for the same equation under some mild regularity condition on weak solutions. Our proof is simple and can be employed to other equations like inhomogeneous incompressible Euler and Euler-Boussinesq equations. One of the key ingredients in our proof is …
Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces (arXiv:1202.1751 and arXiv:1205.3626 (2012)). 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 P. Isett (arXiv:1211.4065) has improved upon our methods, introducing some novel ide…
In this paper we study the uniqueness property of solutions to the steady incompressible Euler equations with perturbations in $\Bbb R^N$. Our perturbations include as special cases the Euler equations with a `single signed' nonlinear term, the self-similar Euler equations, and the steady Navier-Stokes equations. For these equations show that suitable decay assumptions at infinity on the solution or its derivatives, imposed by the $L^q$ conditions imply that the only possible solution is zero.
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is locally integrable with values in the same Lebesgue space.
We show that 1-D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.
We establish a weak-strong uniqueness result for the isentropic compressible Euler equations, that is: As long as a sufficiently regular solution exists, all energy-admissible weak solutions with the same initial data coincide with it. The main novelty in this contribution, compared to previous literature, is that we allow for possible vacuum in the strong solution.
In recent work by Isett (arXiv:1211.4065), and later by Buckmaster, De Lellis, Isett and Székelyhidi Jr. (arXiv:1302.2815), iterative schemes where presented for constructing solutions belonging to the Hölder class $C^{1/5-ε}$ of the 3D incompressible Euler equations which do not conserve energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of $C^{1/3-ε}$ solutions to the Euler equations which dissipate energy. In this note we show how the later scheme can be adapted in order to prove the existence of non-trivial Hölder continuous solu…
In Part II of the paper, we prove linear instability of a certain class of radially symmetric flows of an ideal incompressible fluid in dimension two used in Part I
In [25], Moffatt introduced the concept of helicity in an inviscid fluid and examined the helicity preservation of smooth solution to barotropic compressible flow. In this paper, it is shown that the weak solutions of the above system in Onsager type spaces $\dot{B}^{1/3}_{p,c(\mathbb{N})}$ guarantee the conservation of the helicity. The parallel results of homogeneous incompressible Euler equations and the surface quasi-geostrophic equation are also obtained.
We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonloc…
We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$ boundaries and having corners with arbitrary angles, and can fail without the sign hypothesis when the domain has large angle corners. Hence positive vorticity solutions on such domains are Lagrangian, and we also obtain their uniqueness if the vorticity is initially constant near the boundary.
For a large class of non smooth bounded domains, existence of a global weak solution of the 2D Euler equations, with bounded vorticity, was established by Gérard-Varet and Lacave. In the case of sharp domains, the question of uniqueness for such weak solutions is more involved due to the bad behavior of $Δ^{-1}$ close to the boundary. In the present work, we show uniqueness for any bounded and simply connected domain with a finite number of corners of angles smaller than $π/2$. Our strategy relies on a log-Lipschitz type regularity for the velocity field.
We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.
In [Isett,13], 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 $L_t^\infty B_{3,\infty}^{1/3}$ due to low regularity of the energy profile. In this paper, we establish two theorems that may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölde…
We consider the 2D incompressible Euler equation on a corner domain $Ω$ with angle $νπ$ with $\frac{1}{2}<ν<1$. We prove that if the initial vorticity $ω_0 \in L^{1}(Ω)\cap L^{\infty}(Ω)$ and if $ω_0$ is non-negative and supported on one side of the angle bisector of the domain, then the weak solutions are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is nontrivial around the boundary.
Uniqueness and stability of steady vortex rings for 3D incompressible Euler equation
Institute of Applied Mathematics, Chinese Academy of Sciences（中国科学院应用数学研究所）；University of Chinese Academy of Sciences（中国科学院大学）；School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）；Department of Mathematics, Sichuan University（四川大学数学系）
We consider the Cauchy problem for the isentropic compressible Euler equations in a three-dimensional periodic domain under general pressure laws. For any smooth initial density away from the vacuum, we construct infinitely many entropy solutions with no presence of shock. In particular, the constructed density is smooth and the momentum is $α$-Hölder continuous for $α<1/7$. Also, we provide a continuous entropy solution satisfying the entropy inequality strictly.
We show that for any $\al<\frac 17$ there exist $\al$-Hölder continuous weak solutions of the three-dimensional incompressible Euler equation, which satisfy the local energy inequality and strictly dissipate the total kinetic energy. The proof relies on the convex integration scheme and the main building blocks of the solution are various Mikado flows with disjoint supports in space and time.
The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures associated to a suitable measure--valued solution of the problem. We show that dissipative solutions coincide with weak solutions starting from the same initial data on condition that: {\bf (i)} the weak solution enjoys certain Besov regularity; {\bf (ii)} the symmetric velocity gradient of the weak solution satisfies a one--sided Lipschitz bound.
We study the Riemann problem for the multidimensional compressible isentropic Euler equations. Using the framework developed by Chiodaroli, De Lellis, Kreml and based on the techniques of De Lellis and Székelyhidi, we extend our previous results and prove that whenever the initial Riemann data give rise to a self-similar solution consisting of one admissible shock and one rarefaction wave and are not too far from lying on a simple shock wave, the problem admits also infinitely many admissible weak solutions.
In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the po…
Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure-valued solutions. Motivated from [Feireisl, Ghoshal and Jana, Commun. Partial Differ. Equ., 2019], we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space $B^{α,\infty}_{p}$ with $α>1/2$. We prove that the Besov solution satisfying the above-mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that …
We prove the stability of contact discontinuities without shear, a family of special discontinuous solutions for the three-dimensional full Euler systems, in the class of vanishing dissipation limits of the corresponding Navier-Stokes-Fourier system. We also show that solutions of the Navier-Stokes-Fourier system converge to the contact discontinuity when the initial datum converges to the contact discontinuity itself. This implies the uniqueness of the contact discontinuity in the class that we are considering. Our results give an answer to the open question, whether the contact discontinuity…
The Euler-$α$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $α$. We show that there exists $β>1$ such that weak solutions to the two and three dimensional Euler-$α$ equations in the class $C^0_t H^β_x$ are not unique and may not conserve the Hamiltonian of the system, thus demonstrating flexibility in this regularity class. The construction utilizes a Nash-style intermittent convex integration scheme. We also formulate an appropriate version of the Onsager conjecture for Euler-$α$, postulating that the threshold between rig…
In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. Based on estimates developed in [Buckmaster-De Lellis-Isett-Székelyhidi], we prove that the set of Hölder $1\slash 5-\eps$ wild initial data is dense in $L^2$, where we call an initial datum wild if it admits infinitely many admissible Hölder $1\slash 5-\eps$ weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows to recover arbitrary Reynolds stresses.
