arXiv subjects

Livio Pizzocchero

Publications and source records attributed to Livio Pizzocchero.

36 records · Page 2Linked to original sources

arXiv2013

On the Reynolds number expansion for the Navier-Stokes equations

In a previous paper of ours [Nonlinear Anal. 2012] we have considered the incompressible Navier-Stokes (NS) equations on a d-dimensional torus T^d, in the functional setting of the Sobolev spaces H^n(T^d) of divergence free, zero mean vector fields (n > d/2+1). In the cited work we have presented a general setting for the a posteriori analysis of approximate solutions of the NS Cauchy problem; given any approximate solution u_a, this allows to infer a lower bound T_c on the time of existence of the exact solution u and to construct a function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for t in [0,T_c). In certain cases it is T_c = + infinity, so global existence is granted for u. In the present paper the framework of [Nonlinear Anal., 2012] is applied using as an approximate solution an expansion u^N(t) = Sum_{j=0}^N R^j u_j(t), where R is the Reynolds number. This allows, amongst else, to derive the global existence of u when R is below some critical value R_{*} (increasing with N in the examples that we analyze). After a general discussion about the Reynolds expansion and its a posteriori analysis, we consider the expansions of orders N=1,2,5 in dimension d=3, with the initial datum of Behr, Necas and Wu [M2AN, 2001]. Computations of order N=5 yield a quantitative improvement of the results previously obtained for this initial datum in [Nonlinear Anal. 2012], where a Galerkin approximate solution was employed in place of the Reynolds expansion.

math.AP↗

arXiv2012

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.

math.AP↗

arXiv2012

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 τ_3 in the H^3 Sobolev space, with 0.32 < τ_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 τ_3, our results agree with the original computations of Behr, Necas and Wu (yielding in fact to conjecture that 0.32 < τ_3 < 0.33). Moreover, our analysis supports the following conclusions: (a) The finiteness of τ_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 τ_3. In fact, the solution is likely to exist, at least, up to a time θ_3 > 0.47. (c) Pade' analysis gives a rather weak indication that the solution might blow up at a later time.

math.AP↗

arXiv2011

Local zeta regularization and the Casimir effect

In this paper, whose aims are mainly pedagogical, we illustrate how to use the local zeta regularization to compute the stress-energy tensor of the Casimir effect. Our attention is devoted to the case of a neutral, massless scalar field in flat space-time, on a space domain with suitable (e.g., Dirichlet) boundary conditions. After a simple outline of the local zeta method, we exemplify it in the typical case of a field between two parallel plates, or outside them. The results are shown to agree with the ones obtained by more popular methods, such as point splitting regularization. In comparison with these alternative methods, local zeta regularization has the advantage to give directly finite results via analitic continuation, with no need to remove or subtract divergent quantities.

math-ph↗

arXiv2011

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.

math.AP↗

arXiv2010

On the constants in a Kato inequality for the Euler and Navier-Stokes equations

We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v . D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + 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. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.

math.AP↗

arXiv2009

An H^1 setting for the Navier-Stokes equations: quantitative estimates

We consider the incompressible Navier-Stokes (NS) equations on a torus, in the setting of the spaces L^2 and H^1; our approach is based on a general framework for semi- or quasi-linear parabolic equations proposed in the previous work [9]. We present some estimates on the linear semigroup generated by the Laplacian and on the quadratic NS nonlinearity; these are fully quantitative, i.e., all the constants appearing therein are given explicitly. As an application we show that, on a three dimensional torus T^3, the (mild) solution of the NS Cauchy problem is global for each H^1 initial datum u_0 with zero mean, such that || curl u_0 ||_{L^2} <= 0.407; this improves the bound for global existence || curl u_0 ||_{L^2} <= 0.00724, derived recently by Robinson and Sadowski [10]. We announce some future applications, based again on the H^1 framework and on the general scheme of [9].

math.AP↗

arXiv2009

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.

math-ph↗

arXiv2009

New results on multiplication in Sobolev spaces

We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \leqs K_{ell m n d} || f ||_{m} || g ||_n (f in H^m(R^d, C), g in H^n(R^d, C); 0 <= ell <= m <= n, m + n - ell > d/2); we derive upper and lower bounds K^{+}_{ell m n d}, K^{-}_{ell m n d} for this constant. As examples, we give a table of these bounds for d=1, d=3 and many values of (ell, m, n); here the ratio K^{-}_{ell m n d}/K^{+}_{ell m n d} ranges between 0.75 and 1 (being often near 0.90, or larger), a fact indicating that the bounds are close to the sharp constant. Finally, we discuss the asymptotic behavior of the upper and lower bounds for K_{ell, b ell, c ell, d} when 1 <= b <= c and ell -> + Infinity. As an example, from this analysis we obtain the ell -> + Infinity limiting behavior of the sharp constant K_{ell, 2 ell, 2 ell, d}; a second example concerns the ell -> + Infinity limit for K_{ell, 2 ell, 3 ell, d}. The present work generalizes our previous paper [16], entirely devoted to the constant K_{ell m n d} in the special case ell = m = n; many results given therein can be recovered here for this special case.

math.FA↗

arXiv2008

On the averaging principle for one-frequency systems. Seminorm estimates for the error

We extend some previous results of our work [1] on the error of the averaging method, in the one-frequency case. The new error estimates apply to any separating family of seminorms on the space of the actions; they generalize our previous estimates in terms of the Euclidean norm. For example, one can use the new approach to get separate error estimates for each action coordinate. An application to rigid body under damping is presented. In a companion paper [2], the same method will be applied to the motion of a satellite around an oblate planet.

math-ph↗

arXiv2008

On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations

In our previous paper [12] (Rev. Math. Phys. 16, 383-420 (2004)), a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work the abstract framework of \cite{uno} is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper this extended framework is applied to theincompressible Navier-Stokes equations, on a torus T^d of any dimension. In this way a number of results are obtained in the setting of the Sobolev spaces H^n(T^d), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e., giving the values of all the necessary constants; this makes a difference with most of the previous literature). Nextly, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their H^n distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

math.AP↗

arXiv2006

On the average principle for one-frequency systems

We consider a perturbed integrable system with one frequency, and the approximate dynamics for the actions given by averaging over the angle. The classical theory grants that, for a perturbation of order epsilon, the error of this approximation is O(epsilon) on a time scale O(1/epsilon), for epsilon -> 0. We replace this qualitative statement with a fully quantitative estimate; in certain cases, our approach also gives a reliable error estimate on time scales larger than 1/epsilon. A number of examples are presented; in many cases our estimator practically coincides with the envelope of the rapidly oscillating distance between the actions of the perturbed and of the averaged systems. Fairly good results are also obtained in some "resonant" cases, where the angular frequency is small along the trajectory of the system. Even though our estimates are proved theoretically, their computation in specific applications typically requires the numerical solution of a system of differential equations. However, the time scale for this system is smaller by a factor epsilon than the time scale for the perturbed system. For this reason, computation of our estimator is faster than the direct numerical solution of the perturbed system; the estimator is rapidly found also in cases when the time scale makes impossible (within reasonable CPU times) or unreliable the direct solution of the perturbed system.

math-ph↗

arXiv2005

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.

math.FA↗

arXiv2004

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 $G$, we give a "tame" bound on the $H^n$ norm of this composite function in terms of a linear function of the $H^n$ norm of $f$, with a coefficient depending on $G$ and on the $H^a$ norm of $f$, for all integers $n, a, d$ with $a > d/2$. In comparison with previous results on this subject, our bound is fully explicit, allowing to estimate quantitatively the $H^n$ norm of the function $x \mapsto G(f(x),x)$. When applied to the case $G(f(x), x) = f^2(x)$, this bound agrees with a previous result of ours on the pointwise product of functions in Sobolev spaces.

math.FA↗

arXiv2004

On the Treves theorem for the AKNS equation

According to a theorem of Treves, the conserved functionals of the AKNS equation vanish on all pairs of formal Laurent series of a specified form, both of them with a pole of the first order. We propose a new and very simple proof for this statement, based on the theory of Bäcklund transformations; using the same method, we prove that the AKNS conserved functionals vanish on other pairs of Laurent series. The spirit is the same of our previous paper on the Treves theorem for the KdV, with some non trivial technical differences.

nlin.SI↗

arXiv2004

On a theorem by Treves

According to a theorem of Treves, the conserved functionals of the KdV equation vanish on each formal Laurent series 1/x^2 + u0 + u2 x^2 + u3 x^3 + >... . We propose a new, very simple geometrical proof for this statement.

nlin.SI↗

arXiv2004

On approximate solutions of semilinear evolution equations

A general framework is presented to discuss the approximate solutions of an evolution equation in a Banach space, with a linear part generating a semigroup and a sufficiently smooth nonlinear part. A theorem is presented, allowing to infer from an approximate solution the existence of an exact solution. According to this theorem, the interval of existence of the exact solution and the distance of the latter from the approximate solution can be evaluated solving a one-dimensional "control" integral equation, where the unknown gives a bound on the previous distance as a function of time. For example, the control equation can be applied to the approximation methods based on the reduction of the evolution equation to finite-dimensional manifolds: among them, the Galerkin method is discussed in detail. To illustrate this framework, the nonlinear heat equation is considered. In this case the control equation is used to evaluate the error of the Galerkin approximation; depending on the initial datum, this approach either grants global existence of the solution or gives fairly accurate bounds on the blow up time.

math-ph↗

arXiv2003

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.

math-ph↗