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Optimal Bounds on the Kuramoto-Sivashinsky Equation Felix Otto ...

**on** **the** **Kuramoto**-**Sivashinsky** Equati**on** **Felix** **Otto** no. 386 Diese Arbeit ist mit Unterstützung des v**on** der Deutschen Forschungsgemeinschaft getragenen S**on**derforschungsbereichs 611 an der Universität B**on**n entstanden und als Manuskript vervielfältigt worden. B**on**n, März 2008

- Page 2 and 3: Optimal bounds on
- Page 4 and 5: the rate at which a wave of length
- Page 6 and 7: for all 0 ≤ α ≤ 2. Nicolaenko,
- Page 8 and 9: 1.3 Bounds by the
- Page 10 and 11: 1.4 Main result of this paper We st
- Page 12 and 13: for all (α, p) with ii) and for al
- Page 14 and 15: any L ≥ 2 and any smooth function
- Page 16 and 17: Definition 4. For u(x) L-periodic a
- Page 18 and 19: Indeed, we notice that because of (
- Page 20 and 21: and ∑ 2 −(1/2)k (2 4k 〈〈|u
- Page 22 and 23: These estimates in turn follow from
- Page 24 and 25: and noticing that 1 = 3(2 − β) 5
- Page 26 and 27: M = L 8 , with (42) and Step 1 to
- Page 28 and 29: Indeed, let A(z) be a smooth approx
- Page 30 and 31: for any smooth L-periodic function
- Page 32 and 33: Thus the triangle inequality and (1
- Page 34 and 35: For (119) we argue as follows: (∫
- Page 36 and 37: ≤ C ∑ k≤l We further remark t
- Page 38 and 39: From this representation we obtain
- Page 40 and 41: We thus obtain by Hölder’s inequ
- Page 42 and 43: the Littlewood-Paley decomposition
- Page 44 and 45: Notice that by definition of { ˜φ
- Page 46 and 47: Inequalities (165) & (166) follow f
- Page 48 and 49: On the other hand, we have ∫ |φ
- Page 50 and 51: Remark 3. i) Notice that the linear
- Page 52 and 53:
The role of the r. h. s. of (187) w

- Page 54 and 55:
which yields (191). Step 3. Conclus

- Page 56 and 57:
We note that because of A ′′ (z

- Page 58 and 59:
we have by definition (196) of A:

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References [1] J. Bergh, J. Löfstr

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