1,721,015 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
A posteriori estimates from approximate solutions of the Euler or Navier-Stokes equations
No full textThis communication deals with the Cauchy problem for the incompressible Euler or Navier-Stokes (NS) equations
on a d-dimensional torus T^d, in a setting based on the Sobolev spaces H^n(T^d) (n > d/2 + 1; typically, d = 3).
Following [Morosi and Pizzocchero, Nonlinear Analysis, 2012], an approach will be presented to obtain fully quantitative information on the exact solution u of the Euler or NS Cauchy problem from a posteriori analysis of any approximate solution u_a.
This approach allows to derive estimates on the interval of existence [0, T) of the exact solution u, and
on the Sobolev distance between the exact and the approximate solution. The latter estimate has the form
||u(t) − u_a(t)||n ≤ R_n(t) where R_n(t) is a real, nonnegative function of time t, obtained solving a differential
“control inequality”. In particular, the exact solution u of the Cauchy problem is granted to be global in time
if the control inequality has a global solution R_n : [0,+∞) → [0,+∞).
The quantitative implementation of the above setting requires accurate estimates on the constants in a
number of inequalities, in the Sobolev setting for the Euler/NS equations. For example, it is necessary to use
estimates [Morosi and Pizzocchero, CPAA, 2012] on the constants in the celebrated Kato inequality for _n (with v, w two velocity fields).
The above scheme will be compared with the setting proposed by [Chernyshenko et al, J. Math. Phys., 2007] for the approximate solutions of the Euler or NS equations (and with other works on this subject by Morosi and Pizzocchero [Rev. Math. Phys. 2008; Nonlinear Analysis, 2011]).
Finally, as an application, some results will be presented on the Euler or NS equations on T^3 with the
Behr-Neˇcas-Wu initial datum [ESAIM: M2AN, 2001]; such a datum was proposed by the cited authors as a candidate for finite-time blow-up of the Euler equations
Rigorous existence results for the Euler or Navier-Stokes equations from a posteriori analysis of approximate solutions
No full textThis communication presents results from joint work with Carlo Morosi (Politecnico di Milano) and Mario Pernici (Istituto Nazionale di Fisica Nucleare, Sezione di Milano). The Cauchy problem is considered for the incompressible Euler or Navier-Stokes (NS) equations on a d dimensional torus T^d, in the framework of the Sobolev spaces H^n(T^d)(n > d/2 + 1).
In papers [Morosi and Pizzocchero: Nonlinear Analysis, 2012, Commun. Pure Appl. Anal. 2012, Appl. Math. Lett. 2013] (partly inspired by, or related to [Chernyshenko, Constantin, Robinson and Titi, J.Math. Phys. 2007],[Morosi and Pizzocchero: Rev. Math. Phys. 2008, Nonlinear Analysis 2011]) an approach has been developed to obtain rigorous and fully quantitative results on the exact solution u of the Euler or NS Cauchy problem from the a posteriori analysis of any approximate solution u_a. This approach allows to derive estimates on the interval of existence
[0,T) of the exact solution u, and on the Sobolev distance between the exact and the approximate solution. The latter
estimate has the form || u(t)-u_a(t) ||_n <= R_n(t) where R_n(t) is a real, nonnegative function of time t, obtained solving a differential ``control inequality''. In particular, the exact solution u of the Cauchy problem is granted to be global
in time if the control inequality has a global solution R_n :
[0,+\infinity) -> [0,+\infinity).
In the present communication, the above general setting is exemplified working in dimension d=3 with simple initial data, such as the Behr-Necas-Wu vortex [ESAIM:M2AN 2001].
The approximate solutions in these examples are of the following types:
(i) Galerkin solutions for Euler or NS equations (corresponding to suitable sets of Fourier modes) [Morosi and Pizzocchero, Nonlinear Analysis 2012]; (ii) Large order Taylor expansions in time for the Euler equations [Morosi, Pernici and Pizzocchero, ESAIM:M2AN 2013][Morosi, Pernici and Pizzocchero, submitted];
(iii) Large order expansions in the Reynolds number for the NS equations [Morosi and Pizzocchero, arXiv:1304.2972][Morosi, Pernici and Pizzocchero, in preparation].
Under specific conditions (on the datum and/or the viscosity), the approach based on (i) or (iii) allows to infer the global existence of the exact solution u for the NS Cauchy problem.
In cases (ii)(iii), the construction of the approximate solution and
its a posteriori analysis is performed using tools for automatic symbolic
computation, finally yielding ``computer assisted proofs'' of
existence and regularity for the Euler or NS equations
Soluzioni approssimate di equazioni evolutive semi- o quasi- lineari, con applicazioni alle equazioni di Navier-Stokes
No full textIn questa comunicazione si presentera' un metodo generale per ricavare stime a posteriori dalle soluzioni approssimate di una equazione di evoluzione in uno spazio di Banach, con una parte lineare generante un semigruppo ed una parte non lineare sufficientemente regolare. Data una soluzione approssimata del problema di Cauchy, il metodo fornisce stime sull'intervallo di esistenza della soluzione esatta, e sulla distanza tra le soluzioni esatta e approssimata. Si illustreranno alcune applicazioni del metodo alle equazioni alle derivate parziali semi- o quasi- lineari di tipo evolutivo, ambientate in spazi di Sobolev; tra le soluzioni approssimate si esamineranno, ad esempio, quelle di Galerkin. In particolare, saranno considerate le equazioni di Navier-Stokes per un fluido incomprimibile, in dimensione spaziale tre; facendo riferimento a tali equazioni si derivera' un limite superiore (completamente quantitativo) per la vorticita' del dato iniziale, al di sotto del quale e' possibile garantire l'esistenza globale della soluzione
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 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 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
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 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 INTEGRABILITY OF QUANTUM-MECHANICS AS AN INFINITE DIMENSIONAL HAMILTONIAN SYSTEM
No full textBy a suitable definition of integrability for infinite-dimensional Hamiltonian systems, we give a rigorous meaning to the integrability of Schrodinger quantum mechanics
- …
