158 research outputs found
On the constants in a basic inequality for the Euler and Navier-Stokes equations
We consider the incompressible Euler or Navier-Stokes (NS) equations on a
d-dimensional torus T^d; the quadratic term in these equations arises from the
bilinear map sending two velocity fields v, w : T^d -> R^d into v . D w, and
also involves the Leray projection L onto the space of divergence free vector
fields. We derive upper and lower bounds for the constants in some inequalities
related to the above quadratic term; these bounds hold, in particular, for the
sharp constants K_{n d} = K_n in the basic inequality || L(v . D w)||_n <= K_n
|| v ||_n || w ||_{n+1}, where n in (d/2, + infinity) and v, w are in the
Sobolev spaces H^n, H^{n+1} of zero mean, divergence free vector fields of
orders n and n+1, respectively. As examples, the numerical values of our upper
and lower bounds are reported for d=3 and some values of n. Some practical
motivations are indicated for an accurate analysis of the constants K_n.Comment: LaTeX, 36 pages. The numerical values of the upper bounds K^{+}_{5}
and K^{+}_{10} for d=3 have been corrected. Some references have been
updated. arXiv admin note: text overlap with arXiv:1009.2051 by the same
authors, not concerning the main result
On the constants for multiplication in Sobolev spaces
For n > d/2, the Sobolev (Bessel potential) space H^n(R^d, C) is known to be
a Banach algebra with its standard norm || ||_n and the pointwise product; so,
there is a best constant K_{n d} such that || f g ||_{n} <= K_{n d} || f ||_{n}
|| g ||_{n} for all f, g in this space. In this paper we derive upper and lower
bounds for these constants, for any dimension d and any (possibly noninteger) n
> d/2. Our analysis also includes the limit cases n -> (d/2) and n -> +
Infinity, for which asymptotic formulas are presented. Both in these limit
cases and for intermediate values of n, the lower bounds are fairly close to
the upper bounds. Numerical tables are given for d=1,2,3,4, where the lower
bounds are always between 75% and 88% of the upper bounds.Comment: LaTeX, 45 page
Quantitative functional calculus in Sobolev spaces
In the framework of Sobolev (Bessel potential) spaces H^n(\reali^d, \reali
{or} \complessi), we consider the nonlinear Nemytskij operator sending a
function x \in \reali^d \mapsto f(x) into a composite function x \in
\reali^d \mapsto G(f(x), x). Assuming sufficient smoothness for , we give a
"tame" bound on the norm of this composite function in terms of a linear
function of the norm of , with a coefficient depending on and on
the norm of , for all integers with . In comparison
with previous results on this subject, our bound is fully explicit, allowing to
estimate quantitatively the norm of the function .
When applied to the case , this bound agrees with a
previous result of ours on the pointwise product of functions in Sobolev
spaces.Comment: LaTex, 37 pages. Final version, differing only by minor typographical
changes from the versions of May 23, 2003 and March 8, 200
Smooth solutions of the Euler and Navier-Stokes equations from the a posteriori analysis of approximate solutions
The main result of [C. Morosi and L. Pizzocchero, Nonlinear Analysis, 2012]
is presented in a variant, based on a C^infinity formulation of the Cauchy
problem; in this approach, the a posteriori analysis of an approximate solution
gives a bound on the Sobolev distance of any order between the exact and the
approximate solution.Comment: Author's note. Some overlaps with our previous works arXiv:1402.0487,
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-contained and do not involve the main
results. Final version to appear in Nonlinear Analysi
On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities
We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d,
formulated in terms of the Laplacian Delta and of the fractional powers D^n :=
(-Delta)^(n/2) with real n >= 0; we review known facts and present novel
results in this area. After illustrating the equivalence between these two
inequalities and the relations between the corresponding sharp constants and
maximizers, we focus the attention on the L^2 case where, for all sufficiently
regular f : R^d -> C, the norm || D^j f||_{L^r} is bounded in terms of || f
||_{L^2} and || D^n f ||_{L^2} for 1/r = 1/2 - (theta n - j)/d, and suitable
values of j,n,theta (with j,n possibly noninteger). In the special cases theta
= 1 and theta = j/n + d/2 n (i.e., r = + infinity), related to previous results
of Lieb and Ilyin, the sharp constants and the maximizers can be found
explicitly; we point out that the maximizers can be expressed in terms of
hypergeometric, Fox and Meijer functions. For the general L^2 case, we present
two kinds of upper bounds on the sharp constants: the first kind is suggested
by the literature, the second one is an alternative proposal of ours, often
more precise than the first one. We also derive two kinds of lower bounds.
Combining all the available upper and lower bounds, the Gagliardo-Nirenberg and
Sobolev sharp constants are confined to quite narrow intervals. Several
examples are given.Comment: LaTex, 63 pages, 3 tables. In comparison with version v2, just a few
corrections to eliminate typo
On the expansion of the Kummer function in terms of incomplete Gamma functions
The expansion of Kummer's hypergeometric function as a series of incomplete
Gamma functions is discussed, for real values of the parameters and of the
variable. The error performed approximating the Kummer function with a finite
sum of Gammas is evaluated analytically. Bounds for it are derived, both
pointwisely and uniformly in the variable; these characterize the convergence
rate of the series, both pointwisely and in appropriate sup norms. The same
analysis shows that finite sums of very few Gammas are sufficiently close to
the Kummer function. The combination of these results with the known
approximation methods for the incomplete Gammas allows to construct upper and
lower approximants for the Kummer function using only exponentials, real powers
and rational functions. Illustrative examples are provided.Comment: 21 pages, 6 figures. To appear in "Archives of Inequalities and
Applications
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
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
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
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