New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations. (English) Zbl 1452.35168
Summary: We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author [J. Funct. Anal. 257, No. 7, 2188–2245 (2009; Zbl 1194.35082)], at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by F. Golse and B. Perthame [Rev. Mat. Iberoam. 29, No. 4, 1477–1504 (2013; Zbl 1288.35343)].
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B35 | Stability in context of PDEs |
| 35K55 | Nonlinear parabolic equations |
| 35B44 | Blow-up in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
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