A coupled system of Korteweg de Vries equations as singular limit of the Kuramoto-Sivashinsky equations

Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equations as a singular limit of the Kuramoto-Sivashinsky system

“…The purpose of this paper is to show how a coupled system of Kortewegde Vries (KdV) equations known to describe strong interactions of two long internal gravity waves in a stratified fluid may be obtained as a singular limit of a system of Kuramoto-Sivashinsky equations so that the decay rate of the energy as t → +∞ remains uniform as the singular parameter tends to zero. Our analysis improves earlier work by the authors [16] and extends the results in [13] where the same issue was addressed for the corresponding scalar models.…”

“…We introduced the parameter ν in the above system with the purpose of justifying in mathematical terms the "proximity" of the KS system to a nonlinear system of KdV equations derived in [5]. A rigorous proof for this fact was given in [16] where we deduced that, as ν → 0, the solution {u, v} of the above model converges, in the weak sense, to the solution of a nonlinear coupled system of KdV equations, namely…”

“…Consequently, in view of (1.8) and the available literature on the asymptotic behavior of (1.4)-(1.6) (including our previous work [16]), the following questions make sense: (i) Does E ν (t) decay exponentially to zero, as t → ∞, for some value L? (ii) If yes, can one obtain the damped system of KdV equations as a singular limit of the KS system so that the decay rates remain uniform as the singular parameter tends to zero?…”

“…In order to obtain the results we proceed as in [16]. Classical a priori estimates (energy estimates) of (1.1)-(1.3) provide uniform bounds, with respect to ν, on the solutions which allow us to obtain differential inequalities leading to the exponential decay of E ν (t) and E(t).…”

“…Clearly, in order to avoid ambiguities in the limiting process, it is necessary to guarantee the existence and uniqueness of solutions for both models. In this regard we mention the works [3] and [16].…”