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The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with L p initial vorticity, provided that p ≥ 4. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order | log ⁡ ν | − α / 2 with α > 0. [ABSTRACT FROM AUTHOR]

It is shown in Ferrari [Comm. Math. Phys., 155 (1993), pp. 277--294] that if [0,T*) is the maximal time interval of existence of a smooth solution of the incompressible Euler equations in a bounded, simply-connected domain in R³, then ∫T*0∥ω(t,⋅)∥L∞dt=+∞, where ω is the vorticity of the flow. Ferrari's result generalizes the classical Beale-Kato-Majda [Comm. Math. Phys., 94 (1984), pp. 61-66] breakdown criterion in the case of a bounded fluid domain. In this manuscript, we show a breakdown criterion for a smooth solution of the Euler equations describing the motion of an incompressible fluid in a bounded domain in R3 with a free surface boundary. The fluid is under the influence of surface tension. In addition, we show that our breakdown criterion reduces to the one proved by Ferrari [Comm. Math. Phys., 155 (1993), pp. 277-294] when the free surface boundary is fixed. Specifically, the additional control norms on the moving boundary will either become trivial or stop showing up if the kinematic boundary condition on the moving boundary reduces to the slip boundary condition. [ABSTRACT FROM AUTHOR]

Building on the work of Crouseilles and Faou on the 2D case, we construct C ∞ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer N ⩾ 1 we prove that any L q initial stream function can be approximated in L q (strongly when 1 ⩽ q < ∞ and weak-⁎ when q = ∞) by smooth initial data whose solutions are dense on N -dimensional tori. [ABSTRACT FROM AUTHOR]

We show that the log-Hölder coefficients of a solution to the transport equation is preserved in time. [ABSTRACT FROM AUTHOR]

Intermittency phenomena are known to be among the main reasons why Kolmogorov's theory of fully developed Turbulence is not in accordance with several experimental results. This is why some fractal statistical models have been proposed in order to realign the theoretical physical predictions with the empirical experiments. They indicate that energy dissipation, and thus singularities, is not space filling for high Reynolds numbers. This note aims to give a precise mathematical statement on the energy conservation of such fractal models of Turbulence. We prove that for \theta -Hölder continuous weak solutions of the incompressible Euler equations energy conservation holds if the upper Minkowski dimension of the spatial singular set S \subseteq \mathbb {T}^3 (possibly also time-dependent) is small, or more precisely if \overline {\operatorname {dim}}_{\mathcal {M}}(S)<2+3\theta. In particular, the spatial singularities of non-conservative \theta -Hölder continuous weak solutions of Euler are concentrated on a set with dimension lower bound 2+3\theta. This result can be viewed as the fractal counterpart of the celebrated Onsager conjecture and it matches both with the prediction given by the \beta -model introduced by Frisch, Sulem and Nelkin [J. Fluid Mech. 87 (1978), pp. 719–736] and with other mathematical results in the endpoint cases. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with nonlinear desingularization of steady vortex rings in R 3 with a general nonlinearity f. Using the improved vorticity method, we construct a family of steady vortex rings which constitute a desingularization of the classical circular vortex filament in the whole space. The requirements on f are very general, and it may not satisfy the Ambrosettiâ€"Rabinowitz condition. Some qualitative and asymptotic properties are also established. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Rayleigh-Taylor instability of incompressible Euler equations in a horizontal slab domain, which develops the results of Hwang and Guo (2003) in [11] by taking into account the boundary condition. If a steady density profile is non-monotonic, then the smooth steady state is nonlinearly unstable. Moreover, we also give a new proof for the local existence to inhomogeneous incompressible Euler equations in a smooth bounded domain Ω ⊂ R n , with initial data in H s (Ω) (s > n 2 + 1). [ABSTRACT FROM AUTHOR]

Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program based on the Turing completeness of the Euler equations to address the blow-up problem in the Navier-Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of threedimensional fluid flows has remained an elusive open problem since Moore's works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao's approach to the blow-up problem in the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this article we study the limit when α ↓ 0 of solutions to the α-Euler system in the half-plane, with no-slip boundary conditions. We establish the existence of subsequences converging to a weak solution of the 2D incompressible Euler equations, assuming nonnegative initial vorticities in the space of bounded Radon measures in H-1. This result extends the analysis done in [A. V. Busuioc and D. Iftimie, Nonlinearity, 30 (2017), pp. 4534-4557; M. C. Lopes Filho et al., Phys. D, 292-293 (2015), pp. 51-61]. It requires a substantially distinct approach, analogous to that used for Delort's theorem, and a new detailed investigation of the relation between (no-slip) filtered velocity and potential vorticity in the half-plane. [ABSTRACT FROM AUTHOR]

In this paper, we consider the combined quasi‐neutral and inviscid limits of the two‐fluid compressible Navier–Stokes–Poisson system in the unbounded domain R2×T with the ill‐prepared initial data. We prove that the weak solutions of the compressible Navier–Stokes–Poisson system converge to the strong solution of the incompressible Euler equation as long as the latter exists. Moreover, the convergence rates are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Euler equations have a smooth solution. It is important to remark that when the incompressible Euler equations have a global smooth solution, the existence time of the solution for the compressible Euler–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Euler–Korteweg equations towards those for the incompressible Euler equations with a convergence rate. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Rayleigh–Taylor instability of three-dimensional inhomogeneous incompressible Euler equations with damping in a horizontal slab. We show that if the steady density profile is non-monotonous along the height, then the Euler system with damping is nonlinearly unstable around the given steady state. In this article, we develop a new variational structure to construct the growing mode solution, and overcome the difficulty in proving the sharp exponential growth rate by exploiting the structures in linearized Euler equations. Then combined with error estimates and a standard bootstrapping argument, we finish the nonlinear instability. [ABSTRACT FROM AUTHOR]

In this paper, we present port-Hamiltonian formulations of the incompressible Euler equations with a free surface governed by surface tension and gravity forces, modelling e.g. capillary and gravity waves and the evolution of droplets in air. Three sets of variables are considered, namely (v , Σ) , (η , ϕ ∂ , Σ) and (ω , ϕ ∂ , Σ) , with v the velocity, η the solenoidal velocity, ϕ ∂ a potential, ω the vorticity, and Σ the free surface, resulting in the incompressible Euler equations in primitive variables and the vorticity equation. First, the Hamiltonian formulation for the incompressible Euler equations in a domain with a free surface combined with a fixed boundary surface with a homogeneous boundary condition will be derived in the proper Sobolev spaces of differential forms. Next, these results will be extended to port-Hamiltonian formulations allowing inhomogeneous boundary conditions and a non-zero energy flow through the boundaries. Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. Finally, it is proven that the state dependent Dirac structures are related to Poisson brackets that are linear, skew-symmetric and satisfy the Jacobi identity. [ABSTRACT FROM AUTHOR]

We prove that free boundary incompressible Euler equations are locally well posed in a class of solutions in which the interfaces can exhibit corners and cusps. Contrary to what happens in all the previously known non- C 1 water waves, the angle of these crests can change in time. [ABSTRACT FROM AUTHOR]

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. (2017). As a byproduct, we also obtain an alternative proof for the conservative direction of Onsager’s conjecture for weak solutions, under a weaker Besov-type regularity assumption than previously known. [ABSTRACT FROM AUTHOR]

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D. [ABSTRACT FROM AUTHOR]

We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical limit as c → ∞. As our main result, we prove that the fluid pressure of solutions of these incompressible „relativistic" Euler equations satisfies an elliptic equation on each of the surfaces orthogonal to the fluid four-velocity, which indicates infinite speed of propagation. [ABSTRACT FROM AUTHOR]

This papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the square root of the time. We study the confinement properties of a blob of vorticity initially located around the first point vortex and moving in the velocity field produced by itself and by the other point vortices. We find a sufficient condition on the point vortices such that the vorticity stays confined around the first point vortex at a rate better than the square root of the time. The relevance to the large time behavior of the Euler equations is discussed. [ABSTRACT FROM AUTHOR]

The combined quasi-neutral and inviscid limit of the Navier-Stokes-Poisson-Korteweg system with density-dependent viscosity and cold pressure in the torus T3 is studied. It is shown that, for the well-prepared initial data, the global weak solution of the Navier-Stokes-Poisson-Korteweg system converges strongly to the strong solution of the incompressible Euler equations when the Debye length and the viscosity coefficient go to zero simultaneously. Furthermore, the rate of convergence is also obtained. [ABSTRACT FROM AUTHOR]

The flow of chemically reacting gaseous mixture is associated with a variety of phenomena and processes. We study the combined quasineutral and inviscid limit from the flow of chemically reacting gaseous mixture governed by Poisson equation to incompressible Euler equations with the ill-prepared initial data in the unbounded domain $$\mathbb {R}^2\times {\mathbb {T}}$$ . Furthermore, the convergence rates are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the weak solutions of the three-dimensional incompressible Euler equations with helical symmetry in the whole space when the helical swirl vanishes. Specifically, we establish the global existence of weak solutions when the initial vorticity lies in L 1 ∩ L p with p > 1 . Our result extends the previous work [2] , where the initial vorticity is compactly supported and belongs to L p with p > 4 / 3 . The key ingredient in this paper involves the explicit analysis of Biot–Savart law with helical symmetry in domain R 2 × [ − π , π ] via the theories of singular integral operators and second order elliptic equations. [ABSTRACT FROM AUTHOR]

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 L t ∞ B 3 , ∞ 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 C t , x 1 / 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] . [ABSTRACT FROM AUTHOR]

For a nonlinear operator T satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) T is surjective, ii) T is open at zero, and iii) T has a bounded right inverse. The theorem applies to numerous scale-invariant PDEs in regularity regimes where the equations are stable under weak⁎ convergence. Two particular examples we explore are the Jacobian equation and the equations of incompressible fluid flow. For the Jacobian, it is a long standing open problem to decide whether it is onto between the critical Sobolev space and the Hardy space. Towards a negative answer, we show that, if the Jacobian is onto, then it suffices to rule out the existence of surprisingly well-behaved solutions. For the incompressible Euler equations, we show that, for any p < ∞ , the set of initial data for which there are dissipative weak solutions in L t p L x 2 is meagre in the space of solenoidal L 2 fields. Similar results hold for other equations of incompressible fluid dynamics. [ABSTRACT FROM AUTHOR]

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained. [ABSTRACT FROM AUTHOR]

Energy stable and conservative nonlinear weakly imposed interface conditions for the incompressible Euler equations are derived in the continuous setting. By discretely mimicking the continuous analysis using summation-by-parts operators, we prove that the numerical scheme is stable and conservative. The theoretical findings are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found. [Copyright &y& Elsevier]

The three-dimensional incompressible Euler equations with a passive scalar θ are considered in a smooth domain $\varOmega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\boldsymbol{u}\cdot\hat{\boldsymbol{n}}|_{\partial\varOmega} = 0$ . It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B =∇ q×∇ θ, provided B has no null points initially: $\boldsymbol{\omega} = \operatorname{curl}\boldsymbol {u}$ is the vorticity and q= ω ⋅∇ θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (Phys. Fluids 12:744–746, 2000 ) on the non-existence of Clebsch potentials in the neighbourhood of null points. [ABSTRACT FROM AUTHOR]

Standard hyperbolic solvers for the compressible Euler equations cause increasing approximation errors and have severe stability requirement in the low Mach number regime. It is desired to design numerical schemes that are suitable for all Mach numbers. A second order in both space and time all speed method is developed in this paper, which is an improvement of the semi-implicit framework proposed in [5]. The second order time discretization is based on second order Runge-Kutta method combined with Crank-Nicolson with some implicit terms. This semi-discrete framework is crucial to obtain second order convergence, as well as maintain the asymptotic preserving (AP) property. The AP property indicates that the right limit can be captured in the low Mach number regime. For the space discretization, the pressure term in the momentum equation is divided into two parts. Two subsystems are formed correspondingly, each using different space discretizations. One is discretized by Kurganov-Tadmor central scheme (KT), while the other one is reformulated into an elliptic equation. The proper subsystem division varies with time and the scheme becomes explicit when the time step is small enough. Compared with previous semi-implicit method, this framework is simpler and natural, with only two linear elliptic equations needed to be solved for each time step. It maintains the AP property of the first order method in [5], improves accuracy and reduces the diffusivity significantly. [ABSTRACT FROM AUTHOR]

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 R 3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R 3 with bounded and piecewise smooth vorticities. [ABSTRACT FROM AUTHOR]

The quasineutral limit of the isothermal Navier–Stokes–Poisson system is rigorously proved when the combined quasineutral and vanishing viscosity limit is considered in a domain with boundary. The convergence of the global weak solution for Navier–Stokes–Poisson system to the strong solution for incompressible Euler equations is obtained. [ABSTRACT FROM AUTHOR]