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We consider the Kuramoto-Sivashinsky equation uₜ+uuₓ+uₓₓ+uₓₓₓₓ=0 with L-periodic boundary conditions and zero spatial mean. We prove that every smooth solution satisfies \|u(t)\|L²≤ CL5/6+CL3/2/t for $t>0$, where C is a universal constant. In particular, \|u(t)\|L²≤ CL5/6 for all t≥ L2/3, independently of the initial norm.
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Ingimarson et al. (2026) studied this question.
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