We consider extensions of excluded volume interactions for complex corpora that generalize simple rod-like particles. The Onsager equation can be defined for quite general configuration spaces, and the dimension reduction of the phase space in the limit of highly intense interaction can be shown. The formalism describes both freely articulated and interacting N-rods and the example of interacting 2-rods is given in detail.
Non degeneracy of blow-up solutions of non-quantized singular Liouville-type equations and the convexity of the mean field entropy of the Onsager vortex model with singular sources
Onsager's conjecture for the 3D Navier-Stokes equations concerns the validity of energy equality of weak solutions with regards to their smoothness. In this note we establish energy equality for weak solutions in a large class of function spaces. These conditions are weak-in-time with optimal space regularity and therefore weaker than all previous classical results. Heuristics using intermittency argument suggests the possible sharpness of our results.
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.
The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non degeneracy assumptions on the associated $m$-vortex Hamiltonian, the $m$-point bubbling solutions of the mean field equation are non degenerate as well. Then we deduce that the Onsager mean field equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind.
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 …
We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $L_t^3 C_x^{1/3 - \varepsilon}$, for any fixed $\varepsilon >0$. Our construction combines ideas of [BDL22] and [CCS22].
On strictly starshaped domains of second kind we find natural sufficient conditions which allow the solution of two long standing open problems closely related to the mean field equation $\prl$ below. On one side we catch the global behaviour of the Entropy for the mean field Microcanonical Variational Principle ((MVP) for short) arising in the Onsager description of two-dimensional turbulence. This is the completion of well known results first established in Caglioti et al. Comm.Math.Phys. (1995). Among other things we find a full unbounded interval of strict convexity of the Entropy. On the …
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.
We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This questio…
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…
This note serves to provide additional details for the proof of Lemma 3.6 in our paper [Liu, Zhang and Zhang, Comm. Math. Sci., 3(2005), pp.201-218]. Moreover, we will also present an alternative, yet simpler, proof based on arguments in [Wang, Zhang and Zhang, CPAM, 68(2015), no. 8, 1326-1398].
In this paper we consider the minimizing sequence for some energy functional of an elliptic equation associated with the mean field limit of the point vortex distribution one-sided Borel probability measure. If such a sequence blows up, we derive some estimate which is related to the behavior of solution near the blow-up point. Moreover, we study the two intensities case to consider the sufficient condition for this estimate. Our main results are new for the standard mean field equation as well.
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 …
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…
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…
Properties of an infinite system of nonlinearly coupled ordinary differential equations are discussed. This system models some properties present in the equations of motion for an inviscid fluid such as the skew symmetry and the 3-dimensional scaling of the quadratic nonlinearity. It is proved that the system with forcing has a unique equilibrium and that every solution blows up in finite time in $H^{5/6}$-norm. Onsager's conjecture is confirmed for the model system.
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 $α>\frac13\,.$
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…
We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals without hydrodynamics. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position $x\in \mathbb{R}^d(d=2,3)$ and orientation vector $m\in\mathbb{S}^2$ (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat f…
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 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, …
We consider a vanishing viscosity sequence of weak solutions of the three-dimensional Navier--Stokes equations on a bounded domain. In a seminal paper [25] Kato showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato's criterion holds for Hölder continuous solutions with the regularity index arbitrarily close to the Onsager's critical exponent through a new boundary layer foliation and a global mollification.
We compare two mean field equations describing hydrodynamic turbulence in equilibrium, which are derived under a deterministic vs.\ stochastic assumption on the variable vortex intensity distribution. Mathematically, such equations correspond to non-local Liouville type problems, and the critical temperature corresponds to the optimal Moser-Trudinger constant. We consider the radial case and we assume that the inverse temperature is near its critical value. Under these assumptions we show that, unlike previously existing results, the qualitative properties of the solution set in the determinis…
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 $L^\infty_t (C_x^{\frac{1}{3}-ε})$.
We are motivated by the study of the Microcanonical Variational Principle within the Onsager's description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and "thin" enough domains in the supercritical (with respect to the Moser-Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known \un{only} on multiply connected domains. T…
In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{α,p}$, we determine Onsager type exponents $α$ that guarantee the conservation of all entropies. In particular, the Onsager exponent $α$ is smaller than $α= 1/3$ established for fluid models. Entropies conservation is equivalent to the renormalization property, which have been introduced by DiPerna--Lions for studying well-posedness of passive transport equatio…
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 the bifurcation diagram of the Onsager free-energy func- tional for liquid crystals with orientation parameter on the sphere in three dimensions. In particular, we concentrate on a very general class of two- body interaction potentials including the Onsager kernel. The problem is reformulated as a non-linear eigenvalue problem for the kernel operator, and a general method to find the corresponding eigenvalues and eigenfunctions is presented. Our main tools for this analysis are spherical harmonics and a special algorithm for computing expansions of products of spherical harmon- ics in…
Nonlocal Onsager Operators and Entropy Dissipation for Finite-State Schrödinger Bridges
Higher Institute of Computer Science and Multimedia of Sfax, University of Sfax（萨法克斯多媒体与计算机科学高等学院，萨法克斯大学）；Qassim University（卡西姆大学）；Sfax Preparatory Engineering Institute, University of Sfax（萨法克斯预备工程学院，萨法克斯大学）
We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain $Ω\subset \mathbb{R}^d$, $d\ge 2$, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity $u\in L^3(0,T;B_{3}^{1/3, c_0})$. On an arbitrary thin neighborhood of the boundary, we assume boundedness of velocity and pressure and, at the boundary, we assume continuity of wall-normal velocity. We also prove two theorems which establish that the global viscous dissipation vanishes in the inviscid limit for Leray--Hopf solutions $u^ν$ of the Navier-Stokes equa…
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.
Starting from Doi-Onsager equation for the liquid crystal, we first derive the Q-tensor equation by the Bingham closure. Then we derive the Ericksen-Leslie equation from the Q-tensor equation by taking the small Deborah number limit.
We consider the 2D quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with subcritical or critical indices. The proof of Onsager's conjecture concerning weak solutions of the 3D Euler equations and the notion of dissipative solutions of Duchon and Robert are extended to weak solutions of the quasi-geostrophic equation.
