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The study of the hydrodynamic limit of the Boltzmann equation with physical boundary is a challenging problem due to the appearance of the viscous and Knudsen boundary layers. In this paper, the hydrodynamic limit from the Boltzmann equation with the specular reflection boundary condition to the incompressible Euler equations in a channel is investigated. Based on the multi-scaled Hilbert expansion, the equations with boundary conditions and compatibility conditions for interior solutions, and viscous, and Knudsen boundary layers are derived under different scaling, respectively. Then, some uniform estimates for the interior solutions, and viscous, and Knudsen boundary layers are established. With the help of the L2–L∞ framework and the uniform estimates obtained above, the solutions to the Boltzmann equation are constructed by the truncated Hilbert expansion with multiscales, and hence the hydrodynamic limit in the incompressible Euler level is justified. [ABSTRACT FROM AUTHOR]

We show the lifespan of a strong solution to the incompressible 3D axially symmetric Euler system in R 3 can be arbitrarily large if the initial swirl is small enough. A precise lower bound of the lifespan, depending on the size of ‖ ∇ × (v 0 , θ e θ) ‖ L ∞ , is given. This extends the conclusion of Danchin (Proc Am Math Soc 141:1979–1993, 2012), in which the domain must be distant from the axis of symmetry. This may help us to have a better understanding of the nonlinear framework and the blowup mechanism inherent in the 3D axisymmetric Euler equations. [ABSTRACT FROM AUTHOR]

We consider the free boundary problem for a layer of incompressible fluid lying below the atmosphere and above a rigid bottom in the horizontally infinite setting. The fluid dynamics is governed by the incompressible Euler equations with damping and gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the global well-posedness of the problem with the small initial data in both 2D and 3D. One of key ideas here is to make use of the time-weighted dissipation estimates to close the nonlinear energy estimates; in particular, this implies that the Lipschitz norm of the velocity is integrable-in-time, which is significantly different from that of viscous surface waves (Guo and Tice in Anal PDE 6(6):1429–1533, 2013; Wang in Adv Math 374:107330, 2020). [ABSTRACT FROM AUTHOR]

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\geq 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\nu|^{-\alpha/2} $ with $ \alpha > 0 $.

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]

We prove that the dissipative measure-valued solutions of the 3D compressible Euler–Poisson equations converge to the solution of the incompressible Euler equations in an asymptotic regime, where the Mach number and Debye length tend to zero simultaneously. Furthermore, the convergence rates are also obtained. [ABSTRACT FROM AUTHOR]

Many numerical methods have been developed since 1961, but unresolved issues remain. This study developed a numerical method to address these issues and determine the coefficients and properties of rotational waves with a shear current in a finite water depth. The number of unknown constants was reduced significantly by introducing a wavelength-independent coordinate system. The reference depth was calculated independently using the shooting method. Therefore, there was no need for partial derivatives with respect to the wavelength and the reference depth, which simplified the numerical formulation. This method had less than half of the unknown constants of the other method because Newton's method only determines the coefficients. The breaking limit was calculated for verification, and the result agreed with the Miche formula. The water particle velocities were calculated, and the results were consistent with the experimental data. Dispersion relations were calculated, and the results are consistent with other numerical findings. The convergence of this method was examined. Although the required series order was reduced significantly, the total error was smaller, with a faster convergence speed.

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]

An evidence of temporal discontinuity of the solution in F 1 , ∞ s (R d) is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in related spaces are also discussed. [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]

As a framework for handling low Mach number limits we consider a notion of measurevalued solution of the incompressible Euler system which explicitly formulates the usually suppressed dependence on the pressure variable. We clarify in which sense such a measure-valued solution is generated as a low Mach limit and state sufficient conditions for this convergence. For the special case of pressure-free solutions we are able to give such sufficient conditions only on the incompressible solution itself. As a necessary condition for low Mach limits we obtain a Jensen-type inequality. The same necessary condition actually holds true for limits of weak solutions or vanishing viscosity limits. We illustrate that this Jensen inequality is not trivially fulfilled. Since measure-valued solutions in the classical sense are always generated by weak solutions, this shows that our solution concept contains more information and that low Mach limits lead to a partial selection criterion for incompressible solutions. [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]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the inviscid and zero Froude number limits of the viscous shallow water system. We prove that the limit system is represented by the incompressible Euler equations on the whole space. Furthermore, the rate of convergence is also obtained.

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]

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived. [ABSTRACT FROM AUTHOR]

Abstract 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.

In this paper, we study the desingularization of multiscale rotating vortex patches with fixed angular velocity to the 2D Euler equations in a disk. We prove the existence of two-parameter multiscale V-states concentrating near two points. These two points are determined by the Robin function to the Green’s function, the Green’s function and angular velocity, and not symmetric about the origin, which is new to the former results. The existence of multiscale V-states with N-folds for N ≥ 2 is also proved. [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]

The three-dimensional incompressible Euler equations with a passive scalar θ are considered in a smooth domain with no-normal-flow boundary conditions = 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: = {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. © 2013 Springer Science+Business Media New York.

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]

In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for multi-dimensional linear transport equations without operator splitting developed in Cai et al. (J Sci Comput 73(2–3):514–542, 2017) to the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation and the guiding center Vlasov model. We adopt a local DG method for Poisson's equation of these models. For tracing the characteristics, we adopt a high order characteristics tracing mechanism based on a prediction-correction technique. The SLDG with large time-stepping size might be subject to extreme distortion of upstream cells. To avoid this problem, we propose a novel adaptive time-stepping strategy by controlling the relative deviation of areas of upstream cells. [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]

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.

A theory for the evolution of a metric g driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash–Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving C 1 manifold. The theory is applied to the incompressible Euler and Navier–Stokes equations. One practical outcome of the theory is a computation of critical profile initial data for what may be interpreted as the onset of turbulence for the classical incompressible Navier–Stokes equations. [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 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 generalize an h-adaptive Runge-Kutta discontinuous Galerkin scheme developed earlier in Zhu et al. (J Sci Comput 69:1346-1365, 2016) for the 1D Vlasov-Poisson system to the guiding center Vlasov model and the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation. The main difficulty of this generalization lies in solving the 2D Poisson equation due to the irregular adaptive mesh with hanging nodes. We adopt a local discontinuous Galerkin method to solve the Poisson equation. The full adaptive algorithm and the related numerical implementation details are included. Extensive numerical tests have been performed to showcase the effectiveness of the adaptive scheme and its advantage over the fixed-mesh scheme in saving computational cost and improving solution quality. [ABSTRACT FROM AUTHOR]