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 vt+vxxxx+vxx=12(v2)x and other related 2m2mth-order semilinear parabolic partial differential equations in one dimension and in RNRN are considered. Global existence and blow-up as well as L∞L∞-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 2m2mth-order PDEs in RN×R+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−(−Δ)mv,divv=0in RN×R+,m≥1, are considered. For m=1m=1 these are the classic Navier–Stokes equations. As a simple illustration, it is shown that a uniform Lp(RN)Lp(RN)-bound on locally sufficiently smooth v(x,t) for p>N2m−1 implies a uniform L∞(RN)L∞(RN)-bound, hence the solutions do not blow-up. For m=1m=1 and N=3N=3, this gives p>3p>3, which reflects the famous Leray–Prodi–Serrin–Ladyzhenskaya regularity results (Lp,qLp,q criteria), and re-derives Kato’s class of unique mild solutions in RNRN. Truly bounded classic L2L2-solutions are shown to exist in dimensions N<2(2m−1).