Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

Abstract

In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton-Kantorovich type argument (the radii polynomial approach), to obtain rigorous proofs of existence of the periodic orbits in a weighted 1 Banach space of space-time Fourier coefficients with geometric decay. We present several computer-assisted proofs of existence of periodic orbits at different parameter values.

Numerical validation of blow-up solutions of ordinary differential equations

“…The second problem is the direct calculation of trajectories that blow up using validated computations. Several methodologies exist for computing rigorous trajectories of PDEs, such as in [3,6,10,25,32], all of which focus on validations of bounded trajectories. All of these methodologies employ either power estimates of eigenvalues or the properties of analytic semigroups.…”

“…Interval arithmetic enables us to compute enclosures in which mathematically correct objects are contained. In dynamical systems theory, there are many applications for interval arithmetic, including validations of global trajectories, determining stability of invariant sets, and determining parameter ranges where dynamical bifurcations occur (see, e.g., [3,6,10,21,22,23,25,30,32]). In our case, affine arithmetic [17,18], an enhanced version of interval arithmetic, is applied to validate explicit enclosures of blow-up solutions and their blow-up times.…”

Numerical Computations and Computer Assisted Proofs of Periodic Orbits of the Kuramoto--Sivashinsky Equation

“…Therefore, we impose local uniqueness for Equation (5) by requiring that the correction δ should be perpendicular to the approximate parameterization u 0 . That is,…”

“…This methodology could be used for the proof of the existence of periodic orbits in other type of PDEs See [19] for a systematic study. Remarkably, in [5] the methodology has been extended to validate numerical periodic solutions of…”

A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations

“…However, it is worth remarking that, since we are only aiming at obtaining some particular solutions, we do not need that the evolution is defined nor well-posed. Indeed, methods similar to the ones in this paper have been applied to obtain existence of some specific solutions in some ill-posed equations such as the Boussinesq equation and the Boussinesq system [6,12,23,27] (numerical computations for some of these problems is work in progress). Similar ideas work for delay differential equations, neutral delay equations (with advanced and retarded delays) and for fractional evolution PDEs.…”