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

The initial-boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto–Sivashinsky equation v_t + v_xxxx + v_xx = 1/2 (v^2 )_x and other related 2mth-order semilinear parabolic partial differential equations in one dimension and in R^N are considered. iewed 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, (iv) 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 2mth-order PDEs in RN × R+ , uniform boundedness of global solutions of the Cauchy problem are established. As another related application, the well-posed Burnett-type equations vt +(v·∇)v=−∇p−(−∆)mv, divv=0 inRN ×R+, m≥1, are considered. For m = 1 these are the classic Navier–Stokes equations. As a simple illustration, it is shown that a uniform Lp(RN)-bound on locally sufficiently smooth v(x, t ) for p > N implies a uniform L∞ (RN )-bound, hence the solutions do not 2m−1 Crown Copyright © 2008 Published by Elsevier Ltd. All rights reserved. blow-up. For m = 1 and N = 3, this gives p > 3, which reflects the famous Leray–Prodi–Serrin–Ladyzhenskaya regularity results (Lp,q criteria), and re-derives Kato’s class of unique mild solutions in RN . Truly bounded classic L2 -solutions are shown to exist in dimensions N < 2 (2m − 1).