986 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 approximate solutions of the incompressible Euler and Navier-Stokes equations
We consider the incompressible Euler or Navier-Stokes (NS) equations on a
torus T^d in the functional setting of the Sobolev spaces H^n(T^d) of
divergence free, zero mean vector fields on T^d, for n > d/2+1. We present a
general theory of approximate solutions for the Euler/NS Cauchy problem; this
allows to infer a lower bound T_c on the time of existence of the exact
solution u analyzing a posteriori any approximate solution u_a, and also to
construct a function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for all t in
[0,T_c). Both T_c and R_n are determined solving suitable "control
inequalities", depending on the error of u_a; the fully quantitative
implementation of this scheme depends on some previous estimates of ours on the
Euler/NS quadratic nonlinearity [15][16]. To keep in touch with the existing
literature on the subject, our results are compared with a setting for
approximate Euler/NS solutions proposed in [3]. As a first application of the
present framework, we consider the Galerkin approximate solutions of the
Euler/NS Cauchy problem, with a specific initial datum considered in [2]: in
this case our methods allow, amongst else, to prove global existence for the NS
Cauchy problem when the viscosity is above an explicitly given bound.Comment: LaTex, 44 pages, 18 figure
The relation between the Toda hierarchy and the KdV hierarchy
Under three relations connecting the field variables of Toda flows and that
of KdV flows, we present three new sequences of combination of the equations in
the Toda hierarchy which have the KdV hierarchy as a continuous limit. The
relation between the Poisson structures of the KdV hierarchy and the Toda
hierarchy in continuous limit is also studied.Comment: 11 pages, Tex, no figures, to be published in Physics Letters
On the averaging principle for one-frequency systems. An application to satellite motions
This paper is related to our previous works [1][2] on the error estimate of
the averaging technique, for systems with one fast angular variable. In the
cited references, a general method (of mixed analytical and numerical type) has
been introduced to obtain precise, fully quantitative estimates on the
averaging error. Here, this procedure is applied to the motion of a satellite
in a polar orbit around an oblate planet, retaining only the J_2 term in the
multipole expansion of the gravitational potential. To exemplify the method,
the averaging errors are estimated for the data corresponding to two Earth
satellites; for a very large number of orbits, computation of our estimators is
much less expensive than the direct numerical solution of the equations of
motion.Comment: LaTeX, 35 pages, 12 figures. The final version published in Nonlinear
Dynamic
Separation of variables in multi-Hamiltonian systems: an application to the Lagrange top
Starting from the tri-Hamiltonian formulation of the Lagrange top in a
six-dimensional phase space, we discuss the reduction of the vector field and
of the Poisson tensors. We show explicitly that, after the reduction on each
one of the symplectic leaves, the vector field of the Lagrange top is separable
in the sense of Hamilton-Jacobi.Comment: report to XVI NEEDS (Cadiz 2002): 15 pages, no figures, LaTeX. To
appear in Theor. Math. Phy
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
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