On global solutions and blow-up for Kuramoto-Sivashinsky-type models, and well-posed Burnett equations

TL;DR

This paper investigates the existence, blow-up, and bounds of solutions for Kuramoto-Sivashinsky and Burnett equations using advanced mathematical techniques, providing new insights into their global behavior and well-posedness.

Contribution

It offers new proofs of global existence and boundedness for higher-order PDEs, and extends classical regularity results to Burnett-type equations.

Findings

Global solutions exist under certain boundary conditions.

Blow-up phenomena are characterized using scaling arguments.

Generalized regularity results for Burnett equations are established.

Abstract

The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global existence and blow-up as well as uniform bounds are reviewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry's version of weighted Gronwall's inequalities, and (vi) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirichlet and "Navier" boundary conditions. For some related higher-order PDEs in N dimensions uniform boundedness of global solutions of the Cauchy problem are established. As another related application, the well-posed Burnett-type equations, which are a higher-order extension of the classic Navier-Stokes equations, are studied. As a simple illustration, a generalization of the famous Leray-Prodi-Serrin-Ladyzhenskaya regularity results is obtained.