New bounds for the Kuramoto-Sivashinsky equation. (English) Zbl 1062.35113
Summary: We show that every \(L\)-periodic mean-zero solution \(u\) of the Kuramoto-Sivashinsky equation is on average \(o(L)\) for \(L \gg 1\), in the sense that for any \(T > 0\) the space average of \(|u(t)|\) is bounded by \(\frac{L}{T}\) for any \(t > T\) and any \(L\) sufficiently large. For this we argue that on large spatial scales, the solution behaves like an entropy solution of the inviscid Burgers equation. The analysis of this non-standard perturbation of the Burgers equation is based on a so-called div-curl argument.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B35 | Stability in context of PDEs |
| 35B10 | Periodic solutions to PDEs |
References:
| [1] | ; Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equation. Preprint. · Zbl 1057.65069 |
| [2] | Berestycki, Interfaces Free Bound 3 pp 361– (2001) |
| [3] | Chow, Phys D 84 pp 494– (1995) |
| [4] | Collet, Comm Math Phys 152 pp 203– (1993) |
| [5] | Cross, Rev Mod Phys 65 pp 851– (1993) |
| [6] | De Lellis, Quart Appl Math |
| [7] | DiPerna, Arch Rational Mech Anal 82 pp 27– (1983) |
| [8] | Golovin, Phys Rev Lett 86 pp 1550– (2001) |
| [9] | Goodman, Comm Pure Appl Math 47 pp 293– (1994) |
| [10] | Il?yashenko, J Dynam Differential Equations 4 pp 585– (1992) |
| [11] | Kuramoto, Prog Theor Phys 55 pp 356– (1976) |
| [12] | Liapounov exponents for the Kuramoto-Sivashinsky model. Macroscopic modelling of turbulent flows (Nice, 1984), 319-326. Lecture Notes in Physics, 230. Springer, Berlin, 1985. · doi:10.1007/3-540-15644-5_26 |
| [13] | Michelson, Phys D 19 pp 89– (1986) |
| [14] | Murat, Ann Scuola Norm Sup Pisa Cl Sci (4) 8 pp 69– (1981) |
| [15] | Nicolaenko, Phys D 16 pp 155– (1985) |
| [16] | Robinson, Phys Lett A 184 pp 190– (1994) |
| [17] | Sivashinsky, SIAM J Appl Math 39 pp 67– (1980) |
| [18] | Sivashinsky, Prog Theor Phys 63 pp 2112– (1980) |
| [19] | Smyrlis, Proc Nat Acad Sci U S A 88 pp 11129– (1991) |
| [20] | The compensated compactness method applied to systems of conservation laws. Systems of nonlinear partial differential equations (Oxford, 1982), 263-285. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 111. Reidel, Dordrecht, 1983. · doi:10.1007/978-94-009-7189-9_13 |
| [21] | Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, 68. Springer, New York, 1988. · doi:10.1007/978-1-4684-0313-8 |
| [22] | Wittenberg, Phys Lett A 300 pp 407– (2002) |
| [23] | Zaleski, Phys D 34 pp 427– (1989) |
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