Search

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 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]

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]

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]

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]

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]

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 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]

In this paper, we are concerned with the rigorous proof of the convergence of the quantum Navier–Stokes-Poisson system to the incompressible Euler equations via the combined quasi-neutral, vanishing damping coefficient and inviscid limits in the three-dimensional torus for general initial data. Furthermore, the convergence rates are 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]

Highlights: [•] We study the evolution of shoaling internal solitary waves. [•] We used hyperbolic-tangent stratifications with thin pycnocline. [•] We did high resolution numerical simulations of full equations. [•] We compared results with predictions of weakly-nonlinear models. [•] The cubic RLW equation and a version of the Gardner equation gave best results. [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]

Abstract: This work is the continuation of the recent paper (Danchin, 2010) devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Hölder spaces and of the endpoint Besov space . For such data and under the non-vacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda (1984) . In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations. [Copyright &y& Elsevier]

This paper deals with the Kato Ponce-type commutator estimates in the Besov space Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and the Triebel-Lizorkin space Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. related to the Euler equations describing the motion of perfect incompressible fluids. We investigate the relation between the optimal bound of the commutator estimates and the solvability of the Euler equations. In particular, we show that these commutator estimates fail in Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. with the critical differential order s = n/p + 1 and various exponents p and q. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the convergence of the time-dependent and non-isentropic Euler–Maxwell equations to incompressible Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and energy method. [Copyright &y& Elsevier]

Abstract: In this paper, we study the quasi-neutral limit of compressible Euler–Poisson equations in plasma physics in the torus . For well prepared initial data the convergence of solutions of compressible Euler–Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl–div decomposition of the gradient. [Copyright &y& Elsevier]

Abstract: The limit of the vanishing ratio of the electron mass to the ion mass in the isentropic transient Euler–Poisson equations with periodic boundary conditions is proved. The equations consist of the balance laws for the electron density and current density for a given ion density, coupled to the Poisson equation for the electrostatic potential. The limit is related to the low-Mach-number limit of Klainerman and Majda. In particular, the limit velocity satisfies the incompressible Euler equations with damping. The difference to the zero-Mach-number limit comes from the electrostatic potential which needs to be controlled. This is done by a reformulation of the equations in terms of the enthalpy, higher-order energy estimates and a careful use of the Poisson equation. [Copyright &y& Elsevier]

Abstract: The quasineutral limit of compressible Navier–Stokes–Poisson system with heat conductivity and general (ill-prepared) initial data is rigorously proved in this paper. It is proved that, as the Debye length tends to zero, the solution of the compressible Navier–Stokes–Poisson system converges strongly to the strong solution of the incompressible Navier–Stokes equations plus a term of fast singular oscillating gradient vector fields. Moreover, if the Debye length, the viscosity coefficients and the heat conductivity coefficient independently go to zero, we obtain the incompressible Euler equations. In both cases the convergence rates are obtained. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to the study of the incompressible Euler limit of the Boltzmann equation via the relative entropy method. It extends the convergence result for well-prepared initial data obtained by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47–80]. It explains especially how to take into account the acoustic waves and relaxation layer, and thus to obtain convergence results under weak assumptions on the initial data. The study presented here requires in return some nonuniform control on the large tails of the distribution, which is satisfied for instance by the classical solutions close to a Maxwellian built by Guo [Y. Guo, The Vlasov–Poisson–Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002) 1104–1135]. [Copyright &y& Elsevier]

Abstract: The incompressible Euler equations are obtained as a weak asymptotics of the Boltzmann equation in the fast relaxation limit (the Knudsen number goes to zero), when both the Mach number (defined as the ratio between the bulk velocity and the speed of sound) and the inverse Reynolds number (which measures the viscosity of the fluid) go to zero. The entropy method used here consists in deriving some stability inequality which allows us to compare the sequence of solutions of the scaled Boltzmann equation to its expected limit (provided that it is sufficiently smooth). It thus leads to some strong convergence result. One of the main points to be understood is how to deal with the corrections to the weak limit, i.e. the contributions converging weakly but not strongly to 0 such as the initial layer or the acoustic waves. [Copyright &y& Elsevier]

In this paper we study the combined quasineutral and non-relativistic limit of compressible Euler-Maxwell equations. For well prepared initial data the convergences of solutions of compressible Euler-Maxwell equations to the solutions of incompressible Euler equations are justified rigorously by an analysis of asymptotic expansions and a careful use of ε-weighted Liapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to ε is to use the curl-div decomposition of the gradient. [ABSTRACT FROM AUTHOR]

The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus ?? d , d = 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data. [ABSTRACT FROM AUTHOR]

This paper studies the pressureless Euler–Poisson system and its fully nonlinear counterpart, the Euler–Monge–Ampère system, where the fully nonlinear Monge–Ampère equation substitutes for the linear Poisson equation. While the first is a model of plasma physics, the second is derived as a geometric approximation to the Euler incompressible equations. Using energy estimates, convergence of both systems to the Euler incompressible equations is proved. [ABSTRACT FROM AUTHOR]

This work is concerned with a steady state Euler–Poisson system for potential flows arising in mathematical modeling for plasmas and semiconductors. We study the zero electron mass limit and zero relaxation time limit of the system by using the method of asymptotic expansions. These two limits are expressed by the Maxwell–Boltzmann relation and the classical drift-diffusion model, respectively. For each limit, we show the existence and uniqueness of profiles and justify the asymptotic expansions up to any order. These results also give new approaches for the convergence of the Euler–Poisson system to incompressible Euler equations, which has already been obtained via the quasi-neutral limit. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence of time‐dependent Euler–Poisson equations to incompressible type Euler equations via the quasi‐neutral limit. The local existence of smooth solutions to the limit equations is proved by an iterative scheme. The method of asymptotic expansion and the symmetric hyperbolic property of the systems are used to justify the convergence of the limit. [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]

Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating time-dependent incompressible inviscid flow which combines a projection method with a \Cartesian grid" approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volume-of-fluid representation. A redistribution procedure is used to eliminate time-step restrictions due to small cells where the boundary intersects the mesh. The projection step uses an approximate projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and for the time-dependent algorithm in two dimensions. The method is also demonstrated on flow past a half-cylinder with vortex shedding. [ABSTRACT FROM AUTHOR]

In this paper, we study the incompressible inviscid limit of the viscous two-fluid model in the whole space R 3 with general initial data in the framework of weak solutions. By applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of the densities and the velocity, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Euler equations in the time interval provided that the latter exists. Moreover, thanks to the Strichartz's estimates of linear wave equations, we also obtain the convergence rates. The main ingredient of this paper is that our wave equations include the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolution of the solutions. [ABSTRACT FROM AUTHOR]