On the existence of chaos for the viscous van Wijngaarden–Eringen equation

Abstract

Abstract. We study the viscous van WijngaardenEringen equation:∂ 4 u ∂t 2 ∂x 2 which corresponds to the linearized version of the equation that models the acoustic planar propagation in bubbly liquids. We show the existence of an explicit range, solely in terms of the constants a 0 and Re d , in which we can ensure that this equation admits a uniformly continuous, Devaney chaotic and topologically mixing semigroup on Herzog's type Banach spaces.

$L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

“…When the Knudsen number a 0 is less than 1, it was proved in [7] that model (1.1) can exhibit chaotic behavior. However, the analysis of mathematical behavior of the model for the case a 0 > 1 was left open.…”