Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.