20 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
On power series solutions for the Euler equation, and the Behr-Necas-Wu initial datum
We consider the Euler equation for an incompressible fluid on a three
dimensional torus, and the construction of its solution as a power series in
time. We point out some general facts on this subject, from convergence issues
for the power series to the role of symmetries of the initial datum. We then
turn the attention to a paper by Behr, Necas and Wu in ESAIM: M2AN 35 (2001)
229-238; here, the authors chose a very simple Fourier polynomial as an initial
datum for the Euler equation and analyzed the power series in time for the
solution, determining the first 35 terms by computer algebra. Their
calculations suggested for the series a finite convergence radius \tau_3 in the
H^3 Sobolev space, with 0.32 < \tau_3 < 0.35; they regarded this as an
indication that the solution of the Euler equation blows up. We have repeated
the calculations of Behr, Necas and Wu, using again computer algebra; the order
has been increased from 35 to 52, using the symmetries of the initial datum to
speed up computations. As for \tau_3, our results agree with the original
computations of Behr, Necas and Wu (yielding in fact to conjecture that 0.32 <
\tau_3 < 0.33). Moreover, our analysis supports the following conclusions: (a)
The finiteness of \tau_3 is not at all an indication of a possible blow-up. (b)
There is a strong indication that the solution of the Euler equation does not
blow up at a time close to \tau_3. In fact, the solution is likely to exist, at
least, up to a time \theta_3 > 0.47. (c) Pade' analysis gives a rather weak
indication that the solution might blow up at a later time.Comment: 34 pages, 8 figure
A posteriori estimates for Euler and Navier-Stokes equations
The first two sections of this work review the framework of [6] for
approximate solutions of the incompressible Euler or Navier-Stokes (NS)
equations on a torus T^d, in a Sobolev setting. This approach starts from an
approximate solution u_a of the Euler/NS Cauchy problem and, analyzing it a
posteriori, produces estimates on the interval of existence of the exact
solution u and on the distance between u and u_a. The next two sections present
an application to the Euler Cauchy problem, where u_a is a Taylor polynomial in
the time variable t; a special attention is devoted to the case d=3, with an
initial datum for which Behr, Necas and Wu have conjectured a finite time
blowup [1]. These sections combine the general approach of [6] with the
computer algebra methods developed in [9]; choosing the Behr-Necas-Wu datum,
and using for u_a a Taylor polynomial of order 52, a rigorous lower bound is
derived on the interval of existence of the exact solution u, and an estimate
is obtained for the H^3 Sobolev distance between u(t) and u_a(t).Comment: AUTHORS' NOTE. In Sect.s 1 and 2, some overlap with our previous
works on the Euler/NS equations (arXiv:1203.6865, arXiv:0709.1670,
arXiv:0909.3707, arXiv:1009.2051, arXiv:1104.3832, arXiv:1007.4412,
arXiv:1304.2972). These overlaps aim to make the present paper
self-contained, and do not involve the main results of Sect.s 3, 4. To appear
in the Proceedings of Hyp 201
The one-loop effective action of noncommutative {\cal N}=4 super Yang-Mills is gauge invariant
We study the gauge transformation of the recently computed one-loop
four-point function of {\cal N}=4 supersymmetric Yang-Mills theory with gauge
group U(N). The contributions from nonplanar diagrams are not gauge invariant.
We compute their gauge variation and show that it is cancelled by the variation
from corresponding terms of the one-loop five-point function. This mechanism is
general: it insures the gauge invariance of the noncommutative one-loop
effective action.Comment: LaTex, 13 pages, 4 figure