Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
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Bibliographic record
Abstract
Abstract We study the following parabolic nonlocal 4-th order degenerate equation: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mrow> <m:mo maxsize="160%" minsize="160%">[</m:mo> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>π</m:mi> <m:mo></m:mo> <m:mi>H</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>x</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>ln</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>a</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mfrac> <m:mn>3</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>a</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> <m:mo maxsize="160%" minsize="160%">]</m:mo> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> u_{t}=-\Bigl{[}2\pi H(u_{x})+\ln(u_{xx}+a)+\frac{3}{2}(u_{xx}+a)^{2}\Bigr{]}_{% xx}, arising from the epitaxial growth on crystalline materials. Here H denotes the Hilbert transform, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {a>0} is a given parameter. By relying on the theory of gradient flows, we first prove the global existence of a variational inequality solution with a general initial datum. Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>a</m:mi> </m:mrow> </m:math> {u_{xx}+a} approaches zero. Thus we show that, if the initial datum <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> {u_{0}} is such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo>
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it