We study an Hamiltonian system of N particles in ?3 interacting by a short-range repulsive and a long-range attractive potential. It is shown that the empirical measures associated to the positions and velocity of the system converge to the solutions of Euler equations for a self-gravitating fluid, in the limit as the particle number tends to infinity, for a suitable scaling of the interactions. 相似文献
In this paper we prove nonexistence of stationary weak solutions to the Euler–Poisson equations and the Navier–Stokes–Poisson equations in ?N, N ≥ 2, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler–Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier–Stokes–Poisson equations in ?N this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case. 相似文献
Global solutions of the nonlinear magnetohydrodynamic (MHD) equations with general
large initial data are investigated. First the existence and uniqueness of global solutions are
established with large initial data in
H1.
It is shown that neither shock waves nor vacuum and
concentration are developed in a finite time, although there is a complex interaction between the
hydrodynamic and magnetodynamic effects. Then the continuous dependence of solutions upon
the initial data is proved. The equivalence between the well-posedness problems of the system
in Euler and Lagrangian coordinates is also showed. 相似文献
Under the hypothesis that the initial perturbation has small BV norm, we prove that in any bounded domain the L1 norm of the difference between solutions to the isentropic Euler system of steady supersonic flow and the system of steady
irrotational supersonic flow with the same initial data can be bounded by the cube of the total variation of the initial perturbation. 相似文献
We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity and initial data in the space of Radon measures M (R2). The solutions of this regularized PDE, also known as the isotropic Lagrangian averaged Euler or Euler-α equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in M (R2) such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes as are common in practical vortex computations. 相似文献
Under the hypothesis that the initial perturbation has small BV norm, we prove that in any bounded domain the L1 norm of the difference between solutions to the isentropic Euler system of steady supersonic flow and the system of steady
irrotational supersonic flow with the same initial data can be bounded by the cube of the total variation of the initial perturbation. 相似文献
Eventhough existence of global smooth solutions for one dimensional quasilinear hyperbolic systems has been well established, much less is known about the corresponding results for higher dimensional cases. In this paper, we study the existence of global smoothe solutions for the initial-boundary value problem ofo Euler equtions satisfying γ law with damping and exisymmetry, or spherical symmetry. When the damping is strong enough, we give some sufficient conditions for existence of global smooth solutions as 1<γ< 5 3 and 5 3 <γ<3 . The proof is based on technical estimation of the C1 norm of the solutions. 相似文献
In this paper, a compensated compactness framework is established for sonicsubsonic approximate solutions to the n-dimensional (n ≥ 2) Euler equations for steady irrotational flow that may contain stagnation points. This compactness framework holds provided that the approximate solutions are uniformly bounded and satisfy H 1 loc (Ω) compactness conditions. As illustration, we show the existence of sonic-subsonic weak solution to n-dimensional (n ≥ 2) Euler equations for steady irrotational flow past obstacles or through an infinitely long nozzle. This is the first result concerning the sonic-subsonic limit for n-dimension (n ≥ 3). 相似文献
We consider the elliptic equation ? Δu = f(u) in the whole ?2m, where f is of bistable type. It is known that there exists a saddle-shaped solution in ?2m. This is a solution which changes sign in ?2m and vanishes only on the Simons cone 𝒞 = {(x1, x2) ∈ ?m × ?m: |x1| = |x2|}. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established. 相似文献
We prove consistency, stability, and convergence of a point vortex approximation to the 3-D incompressible Euler equations with smooth solutions. The 3-D algorithm we consider here is similar to the corresponding 3-D vortex blob algorithm introduced by Beale and Majda; see [3]. We first show that the discretization error is second-order accurate. Then we show that the method is stable in lp norm for the particle trajectories and in w?1.p norm for discrete vorticity. Consequently, the method converges up to any time for which the Euler equations have a smooth solution. One immediate application of our convergence result is that the vortex filament method without smoothing also converges. 相似文献
Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable
less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady
version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The
knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations
provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler
equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler
equations are determined, and used to build unsteady solutions. 相似文献
We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3, For macroscopic times τ = t/?N, ? « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ?N?1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t0, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t0 which are close, to O(?2) in a suitable norm, to the local Maxwellian [p/(2πT)d/2]exp{?[v ? ?u(x,t)]2/2T } with constant density p and temperature T . This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density. 相似文献
We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i?tu = Δu + ??2u(1 ? |u|2) on ?2 with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ?. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [19
Schochet , S. ( 1996 ). The point vortex method for periodic weak solutions of the 2D Euler equations . Comm. Pure Appl. Math. 49 : 911 – 965 .[Crossref], [Web of Science ®], [Google Scholar]]. 相似文献
We consider the periodic problem for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor. By choosing a suitable symmetrizers and using an induction argument on the order of the time‐space derivatives of solutions in energy estimates, we obtain the global stability of solutions with exponential decay in time near the nonconstant steady‐states for 2‐fluid nonisentropic Euler‐Poisson equations. This improves the results obtained for models with temperature diffusion terms by using the pressure functions pν in place of the unknown variables densities nν. 相似文献