857 research outputs found
Large order Reynolds expansions for the Navier-Stokes equations
We consider the Cauchy problem for the incompressible homogeneous
Navier-Stokes (NS) equations on a d-dimensional torus, in the C^infinity
formulation described, e.g., in [25]. In [22][25] it was shown how to obtain
quantitative estimates on the exact solution of the NS Cauchy problem via the
"a posteriori" analysis of an approximate solution; such estimates concern the
interval of existence of the exact solution and its distance from the
approximate solution. In the present paper we consider an approximate solutions
of the NS Cauchy problem having the form u^N(t) = sum_{j=0}^N R^j u_j(t), where
R is the "mathematical" Reynolds number (the reciprocal of the kinematic
viscosity) and the coefficients u_j(t) are determined stipulating that the NS
equations be satisfied up to an error O(R^{N+1}). This subject was already
treated in [24], where, as an application, the Reynolds expansion of order N=5
in dimension d=3 was considered for the initial datum of Behr-Necas-Wu (BNW).
In the present paper, these results are enriched regarding both the theoretical
analysis and the applications. Concerning the theoretical aspect, we refine the
approach of [24] following [25] and use the symmetries of the initial datum in
building up the expansion. Concerning the applicative aspect we consider two
more (d=3) initial data, namely, the vortices of Taylor-Green (TG) and
Kida-Murakami (KM); the Reynolds expansions for the BNW, TG and KM data are
performed via a Python program, attaining orders between N=12 and N=20. Our a
posteriori analysis proves, amongst else, that the solution of the NS equations
with anyone of the above three data is global if R is below an explicitly
computed critical value. Our critical Reynolds numbers are below the ones
characterizing the turbulent regime; however these bounds have a sound
theoretical support, are fully quantitative and improve previous results of
global existence.Comment: Some overlaps with our works arXiv:1405.3421, arXiv:1310.5642,
arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832, arXiv:1009.2051,
arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670. These overlaps aim to make
the paper self-cointained and do not involve the main result
Localization of supergravity on the brane
A supersymmetric Randall-Sundrum brane-world demands that not merely the
graviton but the entire supergravity multiplet be trapped on the brane. To
demonstrate this, we present a complete ansatz for the reduction of (D=5,N=4)
gauged supergravity to (D=4,N=2) ungauged supergravity in the Randall-Sundrum
geometry. We verify that it is consistent to lowest order in fermion terms. In
particular, we show how the graviphotons avoid the `no photons on the brane'
result because they do not originate from Maxwell's equations in D=5 but rather
from odd-dimensional self-duality equations. In the case of the fivebrane, the
Randall-Sundrum mechanism also provides a new Kaluza-Klein way of obtaining
chiral supergravity starting from non-chiral.Comment: 12 pages, Latex, minor improvements, references adde
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