Gradient flow approach to an exponential thin film equation: global existence and latent singularity

Abstract

In this work, we study a fourth order exponential equation, $u_t=\Delta e^{-\Delta u},$ derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data.

latent singularity, curve of maximal slope. arXiv admin note: text overlap with arXiv:1711.07405 by other authors

References in corpus (3)

• Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime
• Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface
• Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces

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• Analysis of a fourth order exponential PDE arising from a crystal surface jump process with Metropolis-type transition rates
• Global existence and decay to equilibrium for some crystal surface models
• Partial regularity for an exponential PDE in crystal surface models
• Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects
• Global Strong Solution With BV Derivatives to Singular Solid-on-Solid model With Exponential Nonlinearity