We present two different existence and uniqueness algorithms for constructing global mild solutions in C([0, T);L3(ℝ3)) to the Cauchy problem for the Navier-Stokes equations with an external force.
We present two different existence and uniqueness algorithms for constructing global mild solutions in C([0, T);L3(ℝ3)) to the Cauchy problem for the Navier-Stokes equations with an external force.
We consider a few cases of homogeneous and isotropic turbulence differing by the mechanisms of turbulence generation. The advective terms in the Navier-Stokes and Burgers equations are similar. It is proposed that the...
We consider a few cases of homogeneous and isotropic turbulence differing by the mechanisms of turbulence generation. The advective terms in the Navier-Stokes and Burgers equations are similar. It is proposed that the longitudinal structure functions Sn(r) in homogeneous and isotropic three-dimensional turbulence are governed by a one-dimensional (1D) equation of motion, resembling the 1D Burgers equation, with the strongly nonlocal pressure contributions accounted for by Galilean invariance-breaking terms. The resulting equations, not involving parameters taken from experimental data, give both scaling exponents and amplitudes of the structure functions in an excellent agreement with experimental data. The derived probability density function P(Δu,r)≠P(−Δu,r), but P(Δu,r)=P(−Δu,−r), in accord with the symmetry properties of the Navier-Stokes equations. With decrease of the displacement r, the probability density, which cannot be represented in a scale-invariant form, shows smooth variation from the Gaussian at the large scales to close-to-exponential function, thus demonstrating onset of small-scale intermittency. It is shown that accounting for the subdominant contributions to the structure functions Sn(r)∝rξn is crucial for a derivation of the amplitudes of the moments of the velocity difference.
In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonhar...
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In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see (3]). In the case of an orthonormal basis, our estimate reduces to Kadec' optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.
The detection and unfolding of degenerate local bifurcations provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension. Using the Brusselator...
The detection and unfolding of degenerate local bifurcations provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension. Using the Brusselator partial differential equations (PDEs) (Prigogine and Lefever, 1968) as motivation and main example, we critically review this method. We extend and correct previous calculations, presenting explicit formulae from which normal forms accurate to third order may be computed, and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold, and with Galerkin and finite difference approximations of the original PDE. While judicious use of symbolic manipulations makes feasible such high-order center manifold and normal form calculations, we show that the conclusions drawn from them are of limited use in understanding spatio-temporal complexity and chaos. As Guckenheimer (1981) argued, the method permits proof of existence of quasi-periodic motions and, under mild genericity assumptions, Sil'nikov chaos (sub-shifts of finite type), but the parameter and phase space ranges in which these results may be applied are extremely small.
B. D. Coller, P. Holmes, John Lumley; Erratum: ‘‘Interaction of adjacent bursts in the wall region’’ [Phys. Fluids 6, 954 (1994)], Physics of Fluids, Volume 9,
B. D. Coller, P. Holmes, John Lumley; Erratum: ‘‘Interaction of adjacent bursts in the wall region’’ [Phys. Fluids 6, 954 (1994)], Physics of Fluids, Volume 9,
A method is developed for the optimal estimation of the parameters in a fully nonlinear model of flow in a channel. The data assimilated consist of values of the water surface elevation during a given interval. The me...
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A method is developed for the optimal estimation of the parameters in a fully nonlinear model of flow in a channel. The data assimilated consist of values of the water surface elevation during a given interval. The method is based on the adjoint method of optimal control. It is shown that accurate values of the parameters can be estimated, and the estimates are stable with respect to random perturbations of the data provided that data from a sufficient number of locations are available for assimilation.
We show that the statistical properties of the large scales of the Kuramoto-Sivashinsky equation in the extended system limit can be understood in terms of the dynamical behavior of the same equation in a small finite...
The purpose of this paper is to study the motion of a spinless axisymmetric rigid body in a Newtonian field when we suppose the motion of the center of mass of the rigid body is on a Keplerian orbit. In this case the ...
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The purpose of this paper is to study the motion of a spinless axisymmetric rigid body in a Newtonian field when we suppose the motion of the center of mass of the rigid body is on a Keplerian orbit. In this case the system can be reduced to a Hamiltonian system with configuration space of a two-dimensional sphere. We prove that the restricted planar motion is analytical nonintegrable and we find horseshoes due to the eccentricity of the orbit. In the case I-3/I-1 > 4/3, we prove that the system on the sphere is also analytical nonintegrable.
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