Algebraic spiral solutions of the 2d incompressible Euler equations

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals.     

Blow up of incompressible Euler solutions

We present analytical and computational evidence of blowup of initially smooth solutions of the incompressible Euler equations into non-smooth turbulent solutions. We detect blowup by observing increasing L ₂-residuals of computed solutions under decreasing mesh size. AMS subject classification (2000)  35Q30, 65M60     

Existence of weak solutions for the incompressible Euler equations

Using a recent result of C. De Lellis and L. Székelyhidi Jr. (2010) [2] we show that, in the case of periodic boundary conditions and for arbitrary space dimension d?2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data v₀, where v₀ may be any solenoidal L²-vectorfield. In addition, the energy of these solutions is bounded in time.     

Exact solutions of the Euler equations for some two-dimensional incompressible flows

Two-dimensional (plane and axisymmetric) steady flows of an ideal incompressible fluid are considered in a potential field of external forces. An elliptic partial differential equation is obtained such that each of its solutions is a stream function of a flow described by a certain solution of the Euler equations. Examples of such new exact solutions are given. These solutions can be used, in particular, for testing numerical algorithms and computer programs.     

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data

Using Strichartz estimates, it is possible to pass to the limit in the weakly compressible 2-D Euler system, when the Mach number ε tends to zero, even if the initial data are not uniformly smooth. This leads to results of convergence to solutions of the incompressible Euler system whose regularity is critical, such as vortex patches or Yudovich solutions. To cite this article: A. Dutrifoy, T. Hmidi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Stabilization of the 2D incompressible Euler system in an infinite strip

The paper is devoted to the study of a stabilization problem for the 2D incompressible Euler system in an infinite strip with boundary controls. We show that for any stationary solution (c,0)$(c,0)$ of the Euler system there is a control which is supported in a given bounded part of the boundary of the strip and stabilizes the system to (c,0)$(c,0)$.     

A class of large solutions to the 3D incompressible MHD and Euler equations with damping

In this paper, we derive the global existence of smooth solutions of the 3 D incompressible Euler equations with damping for a class of laxge initial data, whose Sobolev norms H~s can be arbitrarily large for any s ≥ 0. The approach is through studying the quantity representing the difference between the vorticity and velocity. And also, we construct a family of large solutions for MHD equations with damping.     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

The Euler Characteristic of a Random Algebraic Hypersurface

We calculate the mean value of the Euler characteristic of the hypersurface in (d odd) defined by a random polynomial of given degree under the condition that the polynomial has normal O(d+1)-invariant distribution with zero mean. Bibliography: 3 titles.     

Helicity and Kirillov forms in incompressible Euler flow

The Hamiltonian structure induced by the Kirillov symplectic form on coadjoint orbits of incompressible inviscid flow is studied. Helicity is shown to be associated to a degeneracy of the corresponding Kirillov-Poisson bracket.     

Convergence of compressible Euler-Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler-Poisson equations in plasma physics in the torus T^d. 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.     

Remarks on the nonlinear instability of incompressible Euler equations

In this paper, we consider the nonlinear instability of incompressible Euler equations. If a steady density is non-monotonic, then the smooth steady state is a nonlinear instability. First, we use variational method to find a dominant eigenvalue which is important in the construction of approximate solutions, then by energy technique and analytic method, we obtain the dynamical instability result.     

The incompressible limit of solutions of the two‐dimensional compressible Euler system with degenerating initial data

Using Strichartz estimates, it is possible to pass to the limit in the weakly compressible 2‐D Euler system, when the Mach number ? tends to zero, even if the initial data are not uniformly smooth. More precisely, their norms in Sobolev spaces embedded in C¹ can be allowed to grow as small powers of ?^?1. This leads to results of convergence to solutions of the incompressible Euler system whose regularity is critical, such as vortex patches or Yudovich solutions. © 2000 Wiley Periodicals, Inc.     

Weak solution of incompressible Euler equations with decreasing energy

Weak solution of incompressible Euler equations are L²-vector fields, satisfying integral relations, which express the mass and momentum balance. They are believed to describe the turbulent fluid motion at high Reynolds numbers. We justify this conjecture by constructing a weak solution with decreasing kinetic energy. The construction is based on Generalized Flows, introduced by Y. Brenier.     

A note on incompressible limit for compressible Euler equations

This paper presents a simple justification of the classical low Mach number limit in critical Besov spaces for compressible Euler equations with prepared initial data. As the first step of this justification, we formulate a continuation principle for general hyperbolic singular limit problems in the framework of critical Besov spaces. With this principle, it is also shown that, for the Mach number sufficiently small, the smooth compressible flows exist on the (finite) time interval where the incompressible Euler equations have smooth solutions, and the definite convergence orders are obtained. Copyright © 2011 John Wiley & Sons, Ltd.