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1.
We prove the existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler–Poisson
(Euler–Poisson) equations in three spatial dimensions with prescribed angular momentum and total mass. This problem can be
formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having
less symmetry than those functions considered in the classical Auchmuty–Beals paper. We prove the non-linear dynamical stability
of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove
finite time stability of solutions where the perturbations are entropy-weak solutions of the Euler–Poisson equations. Finally, we give a uniform (in
time) a priori estimate for entropy-weak solutions of the Euler–Poisson equations. 相似文献
2.
A. Babin A. Mahalov B. Nicolaenko 《Theoretical and Computational Fluid Dynamics》1998,11(3-4):215-235
Three-dimensional nonhydrostatic Euler–Boussinesq equations are studied for Bu=O(1) flows as well as in the asymptotic regime of strong stratification and weak rotation. Reduced prognostic equations for
ageostrophic components (divergent velocity potential and geostrophic departure/thermal wind imbalance) are analyzed. We describe
classes of nonlinear anisotropic ageostrophic baroclinic waves which are generated by the strong nonlinear interactions between
the quasi-geostrophic modes and inertio-gravity waves. In the asymptotic regime of strong stratification and weak rotation
we show how switching on weak rotation triggers frontogenesis. The mechanism of the front formation is contraction in the
horizontal dimension balanced by vertical shearing through coupling of large horizontal and small vertical scales by weak
rotation. Vertical slanting of these fronts is proportional to μ−1/2 where μ is the ratio of the Coriolis and Brunt–V?is?l? parameters. These fronts select slow baroclinic waves through nonlinear
adjustment of the horizontal scale to the vertical scale by weak rotation, and are the envelope of inertio-gravity waves.
Mathematically, this is generated by asymptotic hyperbolic systems describing the strong nonlinear interactions between waves
and potential vorticity dynamics. This frontogenesis yields vertical “gluing” of pancake dynamics, in contrast to the independent
dynamics of horizontal layers in strongly stratified turbulence without rotation.
Received 8 April 1997 and accepted 29 March 1998 相似文献
3.
This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity. 相似文献
4.
Sébastien Novo 《Journal of Mathematical Fluid Mechanics》2005,7(4):485-514
In the paper [7], author gives a definition of weak solution to the nonsteady Navier–Stokes system of equations which describes
compressible and isentropic flows in some bounded region Ω with influx of fluid through a part of the boundary ∂Ω. Here, we
present a way for proving existence of such solutions in the same situation as in [7] under the sole hypothesis γ > 3/2 for
the adiabatic constant. 相似文献
5.
Thomas Alazard 《Archive for Rational Mechanics and Analysis》2006,180(1):1-73
The low Mach number limit for classical solutions of the full Navier-Stokes equations is here studied. The combined effects
of large temperature variations and thermal conduction are taken into account. In particular, we consider general initial
data. The equations lead to a singular problem, depending on a small scaling parameter, whose linearized system is not uniformly
well-posed. Yet, it is proved that solutions exist and they are uniformly bounded for a time interval which is independent
of the Mach number Ma ∈ (0,1], the Reynolds number Re ∈ [1,+∞] and the Péclet number Pe ∈ [1,+∞]. Based on uniform estimates
in Sobolev spaces, and using a theorem of G. Métivier & S. Schochet [30], we next prove that the penalized terms converge
strongly to zero. This allows us to rigorously justify, at least in the whole space case, the well-known computations given
in the introduction of P.-L. Lions' book [26]. 相似文献
6.
Fractional conservation laws in optimal control theory 总被引:1,自引:0,他引:1
Using the recent formulation of Noether’s theorem for the problems of the calculus of variations with fractional derivatives,
the Lagrange multiplier technique, and the fractional Euler–Lagrange equations, we prove a Noether-like theorem to the more
general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian
does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian
a new term which depends on the fractional-order of differentiation, the generalized momentum and the fractional derivative
of the state variable.
Partially presented at FDA ’06—2nd IFAC Workshop on Fractional Differentiation and its Applications, 19–21 July 2006, Porto, Portugal. 相似文献
7.
Using Hamilton’s principle the coupled nonlinear partial differential motion equations of a flying 3D Euler–Bernoulli beam
are derived. Stress is treated three dimensionally regardless of in-plane and out-of-plane warpings of cross-section. Tension,
compression, twisting, and spatial deflections are nonlinearly coupled to each other. The flying support of the beam has three
translational and three rotational degrees of freedom. The beam is made of a linearly elastic isotropic material and is dynamically
modeled much more accurately than a nonlinear 3D Euler–Bernoulli beam. The accuracy is caused by two new elastic terms that
are lost in the conventional nonlinear 3D Euler–Bernoulli beam theory by differentiation from the approximated strain field
regarding negligible elastic orientation of cross-sectional frame. In this paper, the exact strain field concerning considerable
elastic orientation of cross-sectional frame is used as a source in differentiations although the orientation of cross-section
is negligible. 相似文献
8.
Near-resonance highly nonlinear ideal perfect gas oscillations in tubes are studied numerically for boundary conditions of
various types. The oscillations are initiated by weak periodic perturbations at one end of the tube. As distinct from earlier
studies [1–10], the oscillation amplitudes were not assumed to be small and the entropy increase at the shock waves formed
was taken into account. Periodic flow regimes result as a limit of the solution of a Cauchy problem for one-dimensional time-dependent
gasdynamic equations. The frequency responses of the oscillations under consideration are determined for boundary conditions
of various types. It is shown that in specific cases the attainment of a periodic regime is accompanied by the appearance
of long-wave modulations. The “repeated resonance” effect is revealed. This is due to the change in the tube's natural acoustic
frequency, which takes place during the heating of the gas in the tube by the shock waves traveling in it.
Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–157, July–August, 1994. 相似文献
9.
David C. P. Ellis François Gay-Balmaz Darryl D. Holm Vakhtang Putkaradze Tudor S. Ratiu 《Archive for Rational Mechanics and Analysis》2010,197(3):811-902
The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible
continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are
nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal
dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that
illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods.
In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion
equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry
group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids.
On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic
structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant
formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation
of the nonlocal equations of molecular strand motion is transformed into quaternionic form. 相似文献
10.
Yu. A. Chirkunov 《Mechanics of Solids》2009,44(3):372-379
We perform the group foliation of the system of Lamé equations of the classical dynamical theory of elasticity for an infinite subgroup contained in a normal divisor of the main group. The resolving system of this foliation includes the following two classical systems of mathematical physics: the system of equations of vortex-free acoustics and the system of Maxwell equations, which allows one to use wider groups to obtain exact solutions of the Lamé equations. We obtain a first-order conformal-invariant system, which describes shear waves in a three-dimensional elastic medium. We also give examples of partially invariant solutions. 相似文献
11.
E. V. Rozhkova 《Mechanics of Solids》2009,44(4):526-536
The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial
equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses.
In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists
in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three
equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical
solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing
the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of
three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator
method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their
systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary
analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive
relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined
directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also
be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary,
special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear
combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation
method. 相似文献
12.
We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations
that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a
family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the
initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first
mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical
nonlinear hyperbolic waves. 相似文献
13.
In the regime of weakly transverse long waves, given long-wave initial data, we prove that a non-dimensionalized water wave
system in an infinite strip under the influence of gravity and surface tension on the upper free interface has a unique solution
on [0, T/ e{0, T/ \varepsilon} ] for some e{\varepsilon} independent of constant T. We shall prove in the subsequent paper (Ming et al., The long wave approximation to the three-dimensional capillary gravity
waves, 2011) that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev–Petviashvili
(KP) equations. 相似文献
14.
Holm Altenbach Victor A. Eremeyev Leonid P. Lebedev Leonardo A. Rendón 《Archive of Applied Mechanics (Ingenieur Archiv)》2010,80(3):217-227
Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility
relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression
for the acoustic tensor are derived. The condition for acceleration wave’s propagation is formulated as an algebraic spectral
problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. As an example, a quadratic
form for the specific free energy is considered and the solutions of the corresponding spectral problem are presented. 相似文献
15.
M. Chandrasekar 《Heat and Mass Transfer》2003,40(1-2):157-165
The Governing Principle of Dissipative Processes (GPDP) formulated by Gyarmati into non-equilibrium thermodynamics is employed
to study the effects of heat transfer, two dimensional, laminar and constant property fluid flow in the boundary layer with
suction and injection. The flow and temperature fields inside the boundary layer are approximated by simple third degree polynomial
functions and the variational principle is formulated over the region of the boundary layer. The Euler–Lagrange equations
of the principle are obtained as polynomial equations in terms of momentum and thermal layer thicknesses. These equations
are solvable for any given values of Prandtl number Pr, wedge angle parameter m and suction/injection parameter H. The obtained
analytical solutions are compared with known numerical solutions and the comparison shows the fact that the accuracy is remarkable. 相似文献
16.
At present, the problems of unsteady waves initiated by surface perturbations in an elastic half-space have been studied sufficiently
well (see, e.g., [1–5]; a detailed bibliography on this problem can be found in [6]). At the same time, the analytical solutions
of the corresponding unsteady problems of bulk perturbations are practically absent. It is these questions as applied to the
plane problem that are considered in this paper. 相似文献
17.
Laurent Baratchart Monique Chyba Jean-Baptiste Pomet 《Journal of Dynamics and Differential Equations》2007,19(1):75-107
We consider the problem of locally linearizing a control system via topological transformations. According to [2,3], there is no naive generalization of the classical Grobman–Hartman theorem for ODEs to control systems: a generic control system, when viewed as a set of under-determined differential equations parametrized by the control, cannot be linearized using pointwise transformations on the state and the control values. However, if we allow the transformations to depend on the control at a functional level (open loop transformations), we are able to prove a version of the Grobman–Hartman theorem for control systems. 相似文献
18.
The investigation of codimension-2 bifurcations, in particular in systems with cylindric symmetry, enables us to deduce new
types of secondary regimes branching-off from the symmetric regimes. This investigation also allows us the unique possibility
of a rigorous treatment of chaotic solutions to Navier–Stokes and other nonlinear PDE’s. The central manifold approach combined
with the reduction to the normal form lead to the so-called amplitude systems. These ODE systems describe the nonlinear interaction
between the neutral modes, and always include several nonlinear terms due to so-called intrinsic resonances. However, sometimes
additional resonances appear. In this paper we present the complete list of all possible resonances in dynamic systems with
cylindric symmetry and the corresponding forms of the amplitude equations. Further, we present the results of extensive numerical
investigation of the resonant codimension-2 bifurcations in the Couette–Taylor problem, thus creating an intriguing subject
for further investigation. 相似文献
19.
M. Boileau S. Pascaud E. Riber B. Cuenot L. Y. M. Gicquel T. J. Poinsot M. Cazalens 《Flow, Turbulence and Combustion》2008,80(3):291-321
An extension of the large eddy simulation (LES) technique to two-phase reacting flows, required to capture and predict the
behavior of industrial burners, is presented. While most efforts reported in the literature to construct LES solvers for two-phase
flow focus on Euler–Lagrange formulation, the present work explores a different solution (‘two-fluid’ approach) where an Eulerian
formulation is used for the liquid phase and coupled with the LES solver of the gas phase. The equations used for each phase
and the coupling terms are presented before describing validation in two simple cases which gather some of the specificities
of real combustion chamber: (1) a one-dimensional laminar JP10/air flame and (2) a non-reacting swirled flow where solid particles
disperse (Sommerfeld and Qiu, Int. J. Multiphase Flow 19(6):1093–1127, 1993). After these validations, the LES tool is applied to a realistic aircraft combustion chamber to study both a steady flame
regime and an ignition sequence by a spark. Results bring new insights into the physics of these complex flames and demonstrate
the capabilities of two-fluid LES. 相似文献
20.
K. V. Avramov 《International Applied Mechanics》2006,42(2):169-175
Forced vibrations of cylindrical shells described by a system of three ordinary differential equations are studied. There
are two internal resonances. Standing and traveling waves in the shells are described by a system of six modulation equations
derived using the multiple-scales method. These waves are analyzed for stability
__________
Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 51–58, February 2006. 相似文献