A model system for strong interaction between internal solitary waves

Abstract

A mathematical theory is mounted for a complex system of equations derived by Gear and Grimshaw that models the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratified fluid. For the model in question, the Cauchy problem is of interest, and is shown to be globally well-posed in suitably strong function spaces. Our results make use of Kato's theory for abstract evolution equations together with somewhat delicate estimates obtained using techniques from harmonic analysis. In weak function classes, a local existence theory is developed. The system is shown to be susceptible to the dispersive blow-up phenomenon investigated recently by Bona and Saut for Korteweg-de Vries-type equations.

Gain of Regularity for an Nonlinear Dispersive Equation Korteweg - De Vries - Burgers Type

“…We show that solution u in (1) also satisfies a persistence property. Indeed, we prove that if the initial data ϕ lies in a certain weighted Sobolev space, then the unique solution u of the nonlinear equation (1) lies in the same Sobolev space. At the conclusion of sections, we give a formal proof of our gain in regularity theorem for nonlinear equation (1).…”

“…Indeed, we prove that if the initial data ϕ lies in a certain weighted Sobolev space, then the unique solution u of the nonlinear equation (1) lies in the same Sobolev space. At the conclusion of sections, we give a formal proof of our gain in regularity theorem for nonlinear equation (1). In section six we state our main results on the gain of regularity for the nonlinear equation (1) and prove the a priori estimate used in the main Theorem 7.2.…”

“…At the conclusion of sections, we give a formal proof of our gain in regularity theorem for nonlinear equation (1). In section six we state our main results on the gain of regularity for the nonlinear equation (1) and prove the a priori estimate used in the main Theorem 7.2. In the section seven, we state and prove our main results concerning the gain of regularity for solutions to the nonlinear equation (1), including the main estimates for the remainder terms.…”

“…Where b (1) = b (1) (∧w) and 1 , ∧w) are smooth bounded coefficients with w ∈ H ∞ (IR). The solution is defined in any time interval in which the coefficients are defined.…”

“…In section six we state our main results on the gain of regularity for the nonlinear equation (1) and prove the a priori estimate used in the main Theorem 7.2. In the section seven, we state and prove our main results concerning the gain of regularity for solutions to the nonlinear equation (1), including the main estimates for the remainder terms. Specifically, we prove the following principal theorem.…”