1,721,096 research outputs found
Smooth solutions of the Euler and Navier-Stokes equations from the a posteriori analysis of approximate solutions
No full textThe 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
On the Euler equation: bi-Hamiltonian structure and integrals in involution
No full textWe propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the 'physical' phase space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion
On the Reynolds number expansion for the Navier-Stokes equations
No full textIn a previous paper of ours [Morosi and Pizzocchero, Nonlinear Analysis 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 all 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 [Morosi and Pizzocchero, Nonlinear Analysis 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
[ESAIM:M2AN, 2001]. Computations of order N=5 yield a quantitative improvement of the results previously obtained for this initial datum
in [Morosi and Pizzocchero, Nonlinear Analysis 2012], where a Galerkin approximate solution was employed in place of the Reynolds expansion
On the continuous limit of integrable lattices III. Kupershmidt systems and sl(N+1) KdV theories
No full textWe discuss the connection between the zero-spacing limit of the N- fields Kupershmidt lattice and the KdV-type theory corresponding to the Lie algebra sl(N + 1). The case N = 2 is worked out in detail, recovering from the Limit process the Boussinesq theory with its infinitely many commuting vector fields, their Lax pairs and Hamiltonian formulations. The 'recombination method' proposed here to derive the Boussinesq hierarchy from the limit of the N = 2 Kupershmidt system works, in principle, for arbitrary N
On the constants in a basic inequality for the Euler and Navier-Stokes equations
No full textWe 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 . Dw, 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 . Dw)||_n <= K_n || v ||_n || w ||_{n+1}, where n ∈ (d/2,+∞) 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
On the continuous limit of integrable lattices .1. The Kac-Moerbeke system and KdV theory
No full textKdV theory is constructed systematically through the continuous limit of the Kac-Moerbeke system. The infinitely many commuting vector fields, the conserved functionals, the Lax pairs and the biHamiltonian structure are recovered as the limits of suitably defined linear combinations of homologous objects for the Kac-Moerbeke system. The combinatorial aspects of this recombination method are treated in detail
Large order Reynolds expansions for the Navier–Stokes equations
No full textWe consider the incompressible homogeneous Navier-Stokes (NS) equations on a torus (typically, in dimension 3); we improve previous results of Morosi and Pizzocchero (2014) on the approximation of the solution via an expansion in powers of the Reynolds number. More precisely, we propose this approximation technique in the C-infinity setting of Morosi and Pizzocchero (2015) and present new applications, based on a Python program for the symbolic computation of the expansion. The a posteriori analysis of the approximants constructed in this way indicates, amongst else, global existence of the exact NS solution when the Reynolds number is below an explicitly computable critical value, depending on the initial datum; some examples are given
On the equivalence of two super Korteweg–deVries theories: A bi‐Hamiltonian viewpoint
No full textFrom a bi-Hamiltonian viewpoint the equivalence of two supersymmetric Korteweg-deVries theories, introduced by Manin-Radul and Laberge-Mathieu, is discussed herein. It is shown that the transformation connecting the two theories (proposed recently in the literature) preserves the bi-Hamiltonian structures; moreover, another derivation of this transformation, stemming from bi-Hamiltonian reduction theory and strongly emphasizing the geometrical meaning of the above equivalence, is presented
On approximate solutions for semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations
No full textIn a previous paper of ours (Rev.Math.Phys.34 (2006) 319-363), 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 the cited paper 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 the incompressible Navier–Stokes equations, on a torus of any dimension d. In this way, a number of results are obtained in the setting of the Sobolev spaces H^n on the torus, 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 or 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 existing literature on this topic). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for heir 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)
On the constants in a Kato inequality for the Euler and Navier-Stokes equations
Full text linkWe continue an analysis, started in a previous paper of ours, of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map B(v, w), where v and w 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 Gn in the Kato inequality _n d/2 + 1, and v, w are in the Sobolev spaces 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 our previous paper 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
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