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 I upsilon(t) + upsilon(xxxx) + upsilon(xx) = 1/2(upsilon(2))(x) 2 and other related 2mth-order semilinear parabolic partial differential equations in one dimension and in R-N are considered. Global existence and blow-up as well as LOO-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, (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 R-N x R+, uniform boundedness of global solutions of the Cauchy problem are established. As another related application, the well-posed Burnett-type equations v(t) + (v . del)v = -del p - (-Delta)(m)v, div v = 0 in R-N x 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 L-P(R-N)-bound on locally sufficiently smooth v(x, t) for p > N/2m-1 implies a uniform L-infinity(R-N)-bound, hence the solutions do not blow-up. For rn = 1 and N = 3, this gives p > 3, which reflects the famous Leray-Prodi-Serrin-Ladyzhenskaya regularity results (L-p,L-q criteria), and re-derives Kato's class of unique mild solutions in R-N. Truly bounded classic L-2-solutions are shown to exist in dimensions N < 2 (2m - 1). Crown Copyright (c) 2008 Published by Elsevier Ltd. All rights reserved.