Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

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 Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν1/2.     

Euler method vs. GESS method for dynamical systems

In this paper we introduce GESS method and show that dynamics of the systemy′ =A(s,t,y)y is more faithfully approximated by GESS method than by Euler method. Numerical experiments are given for the comparison of GESS method with Euler method.     

On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.     

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases

The Riemann solutions for the Euler system of conservation laws of energy and momentum in special relativity for polytropic gases are considered. It is rigorously proved that, as pressure vanishes, they tend to the two kinds of Riemann solutions to the corresponding pressureless relativistic Euler equations: the one includes a delta shock, which is formed by a weighted δ-measure, and the other involves vacuum state.     

The truncated Euler–Maruyama method for stochastic differential delay equations

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (Appl. Math. Comput. 217, 5512–5524 2011), and the theory there showed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L ^p ) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao (J. Comput. Appl. Math. 290, 370–384 2015) to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.     

Exact periodic traveling water waves with vorticityOndes d'eau avec tourbillons

For the classical inviscid water wave problem under the influence of gravity, described by the Euler equation with a free surface over a flat bottom, we construct periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use global bifurcation theory to construct a connected set of such solutions. This set contains flat waves as well as waves that approach flows with stagnation points. To cite this article: A. Constantin, W. Strauss, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 797–800.     

Velocity averaging in L1 for the transport equationMoyennisation en vitesse dans L1 pour l'équation de transport

A new result of L¹-compactness for velocity averages of solutions to the transport equation is stated and proved in this Note. This result, proved by a new interpolation argument, extends to the case of any space dimension Lemma 8 of Golse–Lions–Perthame–Sentis [J. Funct. Anal. 76 (1988) 110–125], proved there in space dimension 1 only. This is a key argument in the proof of the hydrodynamic limits of the Boltzmann or BGK equations to the incompressible Euler or Navier–Stokes equations. To cite this article: F. Golse, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 557–562.     

Some paranormed Euler sequence spaces of difference sequences of order m

The main purpose of this work is to extend the sequence spaces which are defined in [KARAKAYA, V.—POLAT, H.: Some new paranormed sequence spaces defined by Euler and difference operators, Acta Sci. Math. (Szeged) 76 (2010), 87–100] and [POLAT, H.—BASAR, F.: Some Euler spaces of difference sequences of order m, Acta Math. Sci. Ser. B Engl. Ed. 27 (2007), 254–266] by using difference operator of order m, and to give their alpha, beta and gamma duals. Furthermore, we characterize some classes of the related matrix transformations.     

An asymptotically stable discretization for the Euler–Poisson system in the quasi-neutral limit

We are interested in the modeling of a plasma in the quasi-neutral limit using the Euler–Poisson system. When this system is discretized with a standard numerical scheme, it is subject to a severe numerical constraint related to the quasi-neutrality of the plasma. We propose an asymptotically stable discretization of this system in the quasi-neutral limit. We present numerical simulations for two different one-dimensional test cases that confirm the expected stability of the scheme in the quasi-neutral limit. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Sur quelques limites de la physique des particules chargées vers la (magnéto)hydrodynamiqueOn some limits in charged particle physics towards (magneto)hydrodynamic equations

We discuss the connection between different scalings limits of the quantum-relativistic Dirac–Maxwell system. In particular we give rigorous results for the quasi-neutral/non-relativistic limit of the Vlasov–Maxwell system: we obtain a magneto-hydro-dynamic system when we consider the magnetic field as a non-relativistic effect and we obtain the Euler equation when we see it as a relativistic effect. A mathematical key is the modulated energy method. To cite this article: Y. Brenier et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 239–244.     

Note for “Some Exact Blowup Solutions to the pressureless Euler equations in R” [Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2993-2998]

In this supplementary note, we can generalize the exact solutions for the pressureless Euler equations in [Yuen MW. Some exact blowup solutions to the pressureless Euler equations in R^N, Commun. Nonlinear Sci. Numer. Simulat. 2011;16:2993-8]. Here, the solutions are constructed in implicit or explicit forms.     

Blowup phenomena of the compressible Euler equations

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.     

Existence of topologically cylindrical shocks

In this Note the multidimensional stability of cylindrical shock profiles and the existence of a nearby perturbed structure is presented for the full Euler equations. This provides an example of a nonplanar structure for which the uniform Kreiss–Lopatinski–Majda stability condition can be explicitly verified. To cite this article: N. Costanzino, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Résolution de problèmes hyperélastiques de recalage d'imagesResolution of some hyperelastic image-matching problems

We focus on some image-matching problems that are based on hyperelastic strain energies. We design an algorithm that solves numerically the Euler–Lagrange equations associated to the problem. This algorithm is formulated in terms of an ODE (Ordinary Differential Equation). We give a theorem which states that the ODE has a unique solution and converges to a solution of the Euler–Lagrange equations. To cite this article: F.J.P. Richard, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 295–299.     

Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations

We are concerned in this paper with the non-relativistic global limits of the entropy solutions to the Cauchy problem of 3 × 3 system of relativistic Euler equations modeling the conservation of baryon numbers, momentum, and energy respectively. Based on the detailed geometric properties of nonlinear wave curves in the phase space and the Glimm’s method, we obtain, for the isothermal flow, the convergence of the entropy solutions to the solutions of the corresponding classical non-relativistic Euler equations as the speed of light c → +∞.     

The Euler characteristic and valuations on MV-algebras

Every finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each a ∈ A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero.     

Limite non visqueuse pour le système de Navier–Stokes dans un espace critique

In a recent paper, Vishik proved the global well-posedness of the two-dimensional Euler equation in the critical Besov space B_2,1². In the present paper we prove that Navier–Stokes system is globally well-posed in B_2,1², with uniform estimates on the viscosity. We prove also a global result of inviscid limit. The convergence rate in L² is of order ν. To cite this article: T. Hmidi, S. Keraani, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On Euler products and multi-variate Gaussians

In this Note, we extend a recent result of A. Selberg concerning the asymptotic value distribution of Euler products to a multi-dimensional setting. Under certain conditions, an asymptotic development of Edgeworth type is found. To cite this article: D.A. Hejhal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Cocycles and Mañe sequences with an application to ideal fluids

Exponential dichotomy of a strongly continuous cocycle Φ is proved to be equivalent to existence of a Mañe sequence either for Φ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension.