Dispersive Perturbations of Burgers and Hyperbolic Equations I: Local Theory

Abstract

The aim of this paper is to show how a weakly dispersive perturbation of the inviscid Burgers equation improve (enlarge) the space of resolution of the local Cauchy problem. More generally we will review several problems arising from weak dispersive perturbations of nonlinear hyperbolic equations or systems.1 for any β ∈ R. It is established in [11] (see also [29] for the case β = 1 2 ) that for 0 ≤ β < 1, there exist initial data u 0 ∈ L 2 (R) ∩ C 1+δ (R), 0 < δ < 1, and T (u 0 ) such that the corresponding solution u of (1.5) satisfiesThis rules out the case −1 < α < 0 in our notation. As observed in [49], the proof in [11] extends easily to non pure power dispersions, such as (1.3) and thus to the Whitham equation (1.1). Note however that it is not clear whether or not the blow-up displayed in the aforementioned papers is shocklike. The solution is proven (by contradiction) to blow-up in a C 1+δ norm and the sup norm of the solution and of its derivative might blow-up at the same time.The case 0 < α < 1 is much more delicate (see the discussion in the final Section).Remark 1.1. Similar issues have been addressed in [43] for the Burgers equation with fractionary dissipation.

A numerical approach to blow-up issues for dispersive perturbations of Burgers’ equation

“…Taking into account the dispersion, it has been established in [42] that the Cauchy problem for (8) is locally well posed for initial data in H s (R), for s > s α = 3 2 − 3α 2 > α 2 , which does not allow to globalize the solution using the conservation laws.…”

“…One does not expect solitary waves to exist when α < 1 3 since then the Hamiltonian does not make sense (see a formal argument in [37] and a rigorous proof in [42] which also proves nonexistence of solitary waves when α < 0).…”

“…The following local well-posedness result is established in [42]. One aim of our numerical simulations is to investigate what happens when 1 2 < α < 1.…”

“…As in [42] the motivation is to study the influence of dispersion on the dynamics of solutions to the Cauchy problem for "weak" dispersive perturbations of hyperbolic quasilinear equations or systems, as for instance the Boussinesq systems for surface water waves.…”

On Whitham and Related Equations

“…Note also that the Whitham equation with surface tension looks, for high frequencies, like the following fractional KdV (fKdV) equation with We refer to , , and for some properties of the fKdV equations viewed as toy models to study the influence of a “weak” dispersive perturbation on the dynamics of the Burgers equation.…”