Linear stability of selfsimilar solutions of unstable thin-film equations
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Bibliographic record
Abstract
We study the linear stability of selfsimilar solutions of long-wave unstable thin-film equations with power-law nonlinearities u_t = −(u^n u_{xxx} + u^m u_x)_x\qquad \text{for }0 < n < 3, \ n≤ m. Steady states, which exist for all values of m and n above, are shown to be stable if m ≤ n + 2 when 0 < n ≤ 2 , marginally stable if m ≤ n + 2 when 2 < n < 3 , and unstable otherwise. Dynamical selfsimilar solutions are known to exist for a range of values of n when m = n + 2 . We carry out the analysis of the stability of these solutions when n= 1 and m = 3 . Spreading selfsimilar solutions are proven to be stable. Selfsimilar blowup solutions with a single local maximum are proven to be stable, while selfsimilar blowup solutions with more than one local maximum are shown to be unstable. The equations above are gradient flows of a nonconvex energy on formal infinite-dimensional manifolds. In the special case n = 1 the equations are gradient flows with respect to the Wasserstein metric. The geometric structure of the equations plays an important role in the analysis and provides a natural way to approach a family of linear stability problems.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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