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Spinodal decomposition for the stochastic Cahn-Hilliard equation
| Content Provider | Semantic Scholar |
|---|---|
| Author | Blömker, Dirk Maier-Paape, Stanislaus Wanner, Thomas |
| Copyright Year | 2000 |
| Abstract | We address spinodal decomposition for the stochastic Cahn-Hilliard equation. Solutions starting at the homogeneous equilibrium u(0) 0 will leave a neighborhood of 0 along a strongly unstable subspace X + " with high probability. This produces solutions of a characteristic wavelength, as discussed in 5]. All estimates are established for the linearized stochastic equation. We explain the characteristic pattern formation in the stochastic Cahn-Hilliard equation by looking at a strongly unstable space X + " as in 5, 6]. The dominance of X + " is established in two steps. Firstly, we have to bound the exit time of a solution from a ball in a suitable Hilbert space from below by looking at fast growing modes in the space X + ". Secondly, the solution component orthogonal to X + " is shown to be small up to the exit time. In contrast to the deterministic case, where initial conditions close to but diierent from the homogeneous state are considered (cf. 5, 6]), we always consider u(0) 0. The instability that leads to phase separation is now due to a random additive force, and randomness is canonically described by the underlying probability space. Note also that we can therefore naturally consider the dynamics on the whole space, and not only on an inertial manifold as in 6]. The stochastic Cahn-Hilliard equation, rst introduced in 2], is given by u t = ?(" 2 u + f(u)) + in G ; @u @ = @u @ = 0 on @G; (1) with initial condition u(0) 0 and domain G IR d (d = 1; 2; 3) with suuciently smooth boundary (e.g. G = 0; 1] d). The function ?f is the derivative of a double-well potential, the standard choice being f(u) = u ? u 3. More generally it is also possible to consider u(0) m, as long as m lies in the spinodal region (i.e., f 0 (m) > 0). The stochastic force is given by the noise strength > 0 and the generalized time derivative = @ t W of a Q-Wiener process fW(t)g t0 given by the series expansion W(t) = X k2IN k k (t)f k in L 2 (G) (cf. 4]). We assume that the f k g k2IN are normalized to supf k : k 2 INg = 1, and that they decay/grow like 2 k ?s k as k ! 1 for some … |
| Starting Page | 1265 |
| Ending Page | 1267 |
| Page Count | 3 |
| File Format | PDF HTM / HTML |
| DOI | 10.1142/9789812792617_0235 |
| Alternate Webpage(s) | http://www.math.umbc.edu/~wanner/papers/stosdequa.ps.gz |
| Alternate Webpage(s) | https://doi.org/10.1142/9789812792617_0235 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
