We present a formal, approximate model for singularity formation in classical fluid dynamics in three dimensions. The construction utilizes an approximation of local two-dimensionality to study an anti-parallel hairpin vortex structure with a cross-section equivalent to the 2D Chaplygin-Lamb dipole vortex. The model exhibits a finite time Euler singularity at an isolated point, with only finite local stretching of vortex lines. The model also suggests an associated Navier-Stokes problem, which exhibits a finite-time point singularity, provided that a Reynolds number is sufficiently large. The singularities are compatible with both the BKM [1] and CF[2] conditions. The vorticity support is infinite in volume but the singularity forms as a result of local processes requiring only finite energy input.
The present paper is motivated by recent mathematical work on the incompressible Euler and Navier-Stokes equations, partly having physically problematic results and unrealistic expectations. The Euler and Navier-Stokes equations are rederived here from the roots, starting at the kinetic equation for the distribution function in phase space. The derivation shows that the Euler and Navier-Stokes equations are valid only if the fluid under consideration is an ideal gas, and if deviations from equilibrium are small in a defined sense, thereby excluding fully nonlinear solutions. Furthermore, the d…
Stretching and folding dynamics in the incompressible, stratified 3D Euler and Navier-Stokes equations are reviewed in the context of the vector $\bdB = \nabla q\times\nablaθ$ where $q=\bom\cdot\nablaθ$. The variable $θ$ is the temperature and $\bdB$ satisfies $\partial_{t}\bdB = \mbox{curl}\,(\bu\times\bdB)$. These ideas are then discussed in the context of the full compressible Navier-Stokes equations where $q$ takes the two forms $q = \bom\cdot\nablaρ$ and $q = \bom\cdot\nabla(\lnρ)$.
This is a rather comprehensive study on the dynamics of Navier-Stokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a). zero viscosity limit of the spectra of linear Navier-Stokes operator, (b). heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Besides Navier-Stokes and Euler equations, we also study two models of them.
In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. This distribution is labeled as the physical solution in the remainder of the article. The dynamics is governed by an assumption of a Lagrangian formulation, with the velocity time derivatives as the momenta conjugate to the velocity configurations. From these definitions and assumptions, we show mathematically that a maximum entropy production principle selects the physical measure from among alternate solutions of the Navi…
We study the asymptotic behavior of the forced linear Euler and nonlinear Navier-Stokes equations close to Couette flow in a periodic channel. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $H^{s}, s>-1$. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier-Stokes equations. Hence, these results provide an indication that nonli…
In this paper we introduce (I,J) similar method for incompressible two and three dimensional Euler equations and Navier-Stokes equations, obtain a series of explicit (I,J) similar solutions to the incompressible two dimensional Euler equations, they include all of twin wave solutions, some new singularity solutions, and some global smooth solutions with finite energy. We also discover that twin wave solution and affine solution to two dimensional incompressible Euler equations are respectively plane wave and constant vector. Finally, we supply some explicit piecewise smooth solutions to incomp…
We propose a two-dimensional generalization of Constantin-Lax-Majda model [2]. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), we present some further model equations. They possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. We also make some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations.
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray $\cite{jL34}$ showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer and Shnirelman obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about the behavior of solutio…
We propose in this work new systems of equations which we call $p$-Euler equations and $p$-Navier-Stokes equations. $p$-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-$p$ distances, with incompressibility constraint. $p$-Euler equations have similar structures with the usual Euler equations but the `momentum' is the signed ($p-1$)-th power of the velocity. In the 2D case, the $p$-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the $p$-Laplacian of the stream…
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Hamiltonian setting of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.
We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable and, furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with re…
Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given through the lens of infinite-dimensional dynamical systems, and various open problems are listed and discussed.
A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of $L^{2m}$-norms of vorticity $Ω_{m}$ for $m\geq 1$. These are scaled to form the dimensionless sequence $D_{m}= (\varpi_{0}^{-1}Ω_{m})^{α_{m}}$ where $\varpi_{0}$ is a constant frequency and $α_{m} = 2m/(4m-3)$. A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of $D_{m}$ and $D_{m+1}$ while another is furnished by imposing a critical lower bound on $\int_{0}^{t}D_{m}\,dτ$. The behaviour of the $D_{…
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^α$ vorticities not necessarily vanishing on the boundary with any $0<α<1$, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of $\mathbb{R}^3$ via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in $\…
The search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by such that the flow contracts more rapidly into one direction than into the other, it can be shown that the dynamics of an axially symmetric flow with swirl may be approximated to a simpler hyperbolic system. By using the method of characteristics, it can be shown that generically the velocity flow exhibits multi-valued solutions appearing on a rim at a finite d…
Based on Makino's solutions with radially symmetry, we extend the corresponding ones with elliptic symmetry for the compressible Euler and Navier-Stokes equations in R^{N} (N\geq2). By the separation method, we reduce the Euler and Navier-Stokes equations into 1+N differential functional equations. In detail, the velocity is constructed by the novel Emden dynamical system: {<K1.1/>| <K1.1 ilk="MATRIX" > a_{i}(t)=(ξ/(a_{i}(t)(Πa_{k}(t))^{γ-1})), for i=1,2,....,N a_{i}(0)=a_{i0}>0, a_{i}(0)=a_{i1} </K1.1> with arbitrary constants ξ, a_{i0} and a_{i1}. Some blowup phenomena or global existences o…
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 study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional torus $\mathbb{T}^2$. We assume the system to be controlled by a degenerate forcing applied to fixed number of modes. In our previous work \cite{ASpb,AS43,ASDAN} we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity ($ν>0$). Methods of differential geometric/Lie algebraic control theory have been used for tha…
In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the E…
We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus ${\mathbb T}^d$, assuming that the solutions have norms for Besov space $B^{σ,\infty}_3({\mathbb T}^d),$ $σ\in (0,1],$ that are bounded in the $L^3$-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form $O(ν^{(3σ-1)/(σ+1)}),$ vanishing as $ν\to0$ if $σ>1/3.$ A consequence is that Onsager-type "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation vanishes in the limit $ν\to 0$, as lon…
We formulate a stochastic least-action principle for solutions of the incompressible Navier-Stokes equation, which formally reduces to Hamilton's principle for the incompressible Euler solutions in the case of zero viscosity. We use this principle to give a new derivation of a stochastic Kelvin Theorem for the Navier-Stokes equation, recently established by Constantin and Iyer, which shows that this stochastic conservation law arises from particle-relabelling symmetry of the action. We discuss issues of irreversibility, energy dissipation, and the inviscid limit of Navier-Stokes solutions in t…
We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are describ…
We consider the incompressible Euler or Navier-Stokes (NS) equations on a torus T^d in the functional setting of the Sobolev spaces H^n(T^d) of divergence free, zero mean vector fields on T^d, for n > d/2+1. We present a general theory of approximate solutions for the Euler/NS Cauchy problem; this allows to infer a lower bound T_c on the time of existence of the exact solution u analyzing a posteriori any approximate solution u_a, and also to construct a function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for all t in [0,T_c). Both T_c and R_n are determined solving suitable "control inequa…
We estimate the inertial force field in the 3d Euler/Navier-Stokes equation in function of |u|_\infty and |\nabla u |_\infty and provide an application to the well-posed problem
The aim of this note is to present the recent results in [Buckmaster, Cao-Labora, Gómez-Serrano, arXiv:2208.09445, 2022], concerning the existence of "imploding singularities" for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski, and Szeftel [Merle, Raphaël, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] and proves the existence of self-similar profiles for all adiabatic exponents $γ>1$ in the case of Euler; as well as …
